TOPICS IN GENERAL TOPOLOGY
North-Holland Mathematical Library Board of Advisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, F.P. Peterson, I.M. Singer and A.C. Zaanen
VOLUME 41
NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO
Topics in General Topology
Kiiti MORITA and Jun-iti NAGATA Osaka Kyoiku University Japan
1989 NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, lo00 AE Amsterdam, The Netherlands Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010. U.S.A.
Library of Congress Catalogingin-Publication Data Morita, Kiiti, 1915Topics in general topology / Kiiti Morita and Jun-iti Nagata. p. cm. -- (North-Holland mathematical library: v. 41) Includes bibliographies and index. ISBN 0-444-70455-8 I. Topology. I. Nagata, Jun-iti, 1925- 11. Title. 111. Series. QA611.M675 1989 5 14--dc20 89-9374 CIP
ISBN: 0 444 70455 8
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic. mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. (North Holland), P.O. Box 103,1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.
Printed in The Netherlands
PREFACE
The primary purpose of this book is to provide an advanced account of some aspects of general topology for students as well as working mathematicians in the field relevant to the subject. Although some of the topics discussed here are quite new, all of them are not necessarily the newest, because what we want to present here is not a . collection of newest research papers that could become out of date rapidly, but rather a collection of results that represent recent developments of general topology and could be a foundation for coming developments. This book does not necessarily cover all aspects of modem general topology, because what we intend here is not to give an encyclopedic exposition of the subject but to give a wider scope of various trends in recent developments of general topology while avoiding overlap with other books of the same type such as the Handbook of Set Theoretic Topology. We assume the reader to have a rudimentary knowledge of set theory, algebra and analysis while a little more is presupposed from general topology which may be obtained in the undergraduate course (e.g. J. Nagata’s book, Modern General Topology). Thus graduate students (and some undergraduate students as well) with sufficient knowledge of basic general topology would feel little difficulty in reading the book. We would be especially happy if our book could help them in writing their theses. This book consists of fifteen chapters, and each chapter is written independently from the others (with a few exceptions). Thus the reader could begin with any chapter, though he is advised to begin with Part I for topics divided into two chapters. This book was planned and edited under support of the “Symposium of General Topology”. Thus we conclude this preface with our thanks to the staff of the “Symposium”; especially to Prof. Y. Yasui, Mr. K. Yamada, Mr. H.Chimura, Miss K. Iwata, Mr. T. Nagura and Mr. S. Okada, who assisted us in editing the manuscripts written by different authors. Kiiti MORITA Jun-iti NAGATA Editors
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CONTENTS PREFACE
V
CONTENTS
vii
OF MAPPINGS 1 CHAPTER 1 EXTENSIONS by Kiiti Morita Introduction 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem References
1 2 10 25 35 38
CHAPTER 2 EXTENSIONS OF MAPPINGS 11 by Takao Hoshina Introduction 1. Preliminaries 2. C*-, C-, P"- and P-embeddings 3. Unions of C*-embedded subsets 4. C*-embedding in product spaces 5. Homotopy extension property References
41 42 51 62 69 75
78
CHAPTER 3 NORMALITY OF PRODUCT SPACES 1 by Masahiko Atsuji Introduction 1. Fundamental results 2. The first of Morita's three conjectures 3. The second and third of Morita's three conjectures References
81 82 92 109 116
OF PRODUCT SPACES 11 CHAPTER 4 NORMALITY by Takao Hoshina 1. Products of normal spaces with metric spaces 2. Products of normal spaces with generalized metric spaces References
121 140 158
viii
Contents
CHAPTER 5 GENERALIZED PARACOMPACTNESS by Yoshikazu Yasui Introduction 1. Preliminaries 2. Characterizations of paracompactness 3. Characterizations of submetacompactness 4. Characterizations of metacompactness 5. Characterizations of subparacompactness 6. Shrinking properties 7. Examples References
161 162 164 175 178 184 186 191 198
CHAPTER 6 THE TYCHONOFF FUNCTOR AND RELATED TOPICS by Tadashi Ishii Introduction 1. The Tychonoff functor 2. Product spaces and the Tychonoff functor 3. w-Compact spaces 4. A space X such that r ( X x Y ) = z ( X ) x r ( Y ) for any space Y 5 . A space X such that r ( X x Y ) = r ( X ) x r ( Y ) for any k-space Y 6. Products of w-compact spaces 7. w-Paracompact spaces and the Tychonoff functor 8. A generalization of Tamano’s theorem 9. Rectangular products and w-paracompact spaces References
203 204 207 209 215 218 22 1 224 230 234 24 1
7 METRIZATION I CHAPTER by Jun-iti Nagata Introduction 1 . Metrizability in terms of g-functions 2. Metrizability in terms of base of point-finite rank 3. Metrizability in terms of hereditary property 4. Some other aspects References
245 245 251 259 265 27 1
8 METRIZATION I1 CHAPTER by Yoshio Tanaka Iritroduction 1. Spaces which contain a copy of S, or S, 2. Spaces dominated by metric subsets
27 5 276 284
Contents
3. 4. 5. 6.
Spaces with a-hereditarily closure-preserving k-networks Spaces with certain point-countable covers Quotient s-images of locally separable metric spaces Ic-Metrizable spaces and b-metrizable spaces References
CHAPTER 9 GENERALIZED METRIC SPACES I by Jun-iti Nagata Introduction 1. Lagnev space and K-space 2. Developable space 3. M-space and related topics 4. Universal spaces References CHAPTER 10 GENERALIZED METRIC SPACES I1 by Ken-ichi Tamano Introduction 1. Review of basic results 2. Closure-preserving collectionsand definitionsof various stratifiable spaces 3. Expandability, extension property, and sup-characterization of stratifiable spaces 4. Classes of M,-spaces 5. Embeddings 6. Closed maps and perfect maps 7. Related topics 8. Problems References CHAPTER 11 FUNCTION SPACES by Akihiro Okuyama and Toshiji Terada 1. Introduction and notation 2. Some properties of C,*(X) 3. Some properties of C , ( X ) 4. Some properties of C,(X) 5. Topological properties and linear topological properties 6. Topological properties and /-equivalence References
ix
287 29 1 30 1 304 311
315 315 326 337 351 365
367 368 371 382 388 396 400 405 406 407
41 1 416 417 427 436 447 457
X
Contents
CHAPTER 12 N-COMPACTNESS AND ITS APPLICATIONS by Katsuya Eda, Takemitsu Kiyosawa and Haruta Ohta Introduction 1. Conventions and notation 2. N-compactness and N-compactifications 3. N-compactness vs. realcompactness 4. N-compactness of k,X 5. Rings and lattices of continuous functions 6. Applications to abelian groups 7. Applications to non-Archimedean Banach spaces 8. Problems References
459 460 46 1 467 474 479 483 497 516 518
13 TOPOLOGICAL GAMES AND APPLICATIONS CHAPTER by Yukinobu Yajima Introduction 1. The games and winning strategies 2. K-scattered spaces 3. Closure-preserving covers by compact sets 4. Covering properties of product spaces 5. The games in product spaces 6. Applications to dimension theory References
523 524 529 536 543 549 553 560
14 CATEGORICAL TOPOLOGY CHAPTER by Ryosuke Nakagawa Introduction 1. Categories and functors 2. Monomorphisms and epimorphisms 3. Diagrams and limits 4. Complete categories 5. Factorizations of morphisms 6. Reflective subcategories 7. Characterization theorem 8. (E, M)-categories 9. Epireflective vs. bireflective in Top 10. Separation axioms and connectedness 11. Simplicity of epireflective subcategories 12. Topological functors 13. Topological categories References
563 565 569 575 579 582 588 592 598 602 606 61 1 614 618 622
Contents
xi
CHAPTER 15 TOPOLOGICAL DYNAMICS by Nobuo Aoki Introduction 1. Orbit structures 2. Expansive behaviours 3. Expansivity and dimension 4. Pseudo-orbit-tracing property 5. Coordinate systems and topological stability 6. Representations of maps with hyperbolic coordinates 7. Chain components and decompositions 8. Markov partitions and subshifts 9. Topological entropy References
625 627 635 658 667 680 687 695 707 727 736
SUBJECT INDEX
741
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 1
EXTENSIONS OF MAPPINGS I
Kiiti MORITA Professor emeritus, University of Tsukuba, Ibaraki, 305, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces. 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem . References.. . . . . . . . . . . . . . . . . . .
............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............
1 2 10 25 35 38
Introduction Let X and Y be topological spaces, A a subspace of X and f:A + Y a continuous map. In case there exists a continuous map g : X + Y such that g I A = f (that is, g ( x ) = f ( x ) for all x in A), g is called an extension (or a continuous extension) off over X ;in this casefis said to be extendable (or continuously extendable) over X,or more precisely, to be extendable to a continuous map g from X to Y. Here it is to be noted that the image of g should lie in the range space of the original mapf; if the range of g is admitted to be enlarged beyond Y, thenf: A + Y is always extended to a continuous map from X to the adjunction space X u, Y (e.g. see Morita [198l]). In genera1,fis not extendable over X. For example, if Xis the closed unit interval Z = [0, 11, A = Y = (0, 1) and f : A + Y is the identity map A, then f is not extendable over X. 1, :A To discuss conditions under which a given continuous map f:A + Y is extendable over X , is one of the most important problems in topology. The purpose of this chapter and the next is to investigate this problem. We distinguish two cases, according to the subspace A being dense or not. This chapter is devoted to the study of the case of A being dense.
K. Morita
2
A basic result in this case is a theorem asserting that if A is a dense subspace of a metric space X , then every uniformly continuous map from A to a complete metric space Y is extendable to a uniformly continuous map from X to Y. As is well known, this theorem was generalized to the case of uniform spaces. However, those spaces which are obtained as uniform spaces are restricted to the class of completely regular spaces. The most satisfactory results in the extension theory concerned here are obtained in the realm of regular spaces. Therefore, we need a generalization of uniform spaces so that it defines any regular space. Such a generalization is provided by semi-uniform spaces introduced by Morita [1951]. In this chapter we shall first discuss semi-uniform spaces in the framework of generalized uniform spaces with applications to strict T,-extensions of spaces (Sections 1 and 2), and then establish the most general extension theorem for continuous maps from dense subspaces (Section 3). In the Appendix, it will be shown that semi-uniform spaces are available for obtaining a generalization of the Ascoli Theorem.
1. Generalized uniform spaces and semi-uniform spaces 1.0. Notation and terminology. Let X be a set, let x E X, A c X,B c X , and let d and A? be collections (or families) of subsets of X.
ud
nd
= ~ { A I A E ~ } , = n{AIAEd}, d v A? = {A u BI A E d , B E A?}, d A W = {AnBIAEd,BEa}, d A B = {AnBIAEd}, d < W o for each A E d there exists B E A? such that A c B o d is a refinement of A? ( d refines A?), St(B, at) = U { A E d l A n B # S}, St(x, d ) = St({,}, d ) = U { A E d l x E A } , St"+'(B, d ) = St(St"(B, d),d ) , n = 1, 2 , . . . , d is a cover of X o Uat = X.
Let X be a topological space and Y a subspace of X. C1 A = ClxA = the closure of A in the space X, C1,B = the closure of B c Y in the subspace Y, Int A = Int,A = the interior of A in the space X, Int,B = the interior of B c Y in the subspace Y, c1 d = {CI A I A E d } , d is an open (resp. a closed) cover of X o U d = X and each A E d is open (resp. closed) in X.
Extensions of Mappings I
Let f:X + Y be a map and d (resp. (resp. Y).
fW)=
{f(A)IA
E
a) a
3
collection of subsets of X
4, f - W
=
{f-'(B)lBE a}
Let us begin with recalling the definition of uniform spaces. There are several ways to approach the theory of uniform spaces. Here we adopt the approach of J. W.Tukey [1940].
1.1. Definition. A uniformity (or uniform structure) on a set Xis a nonempty family 0 of covers of X satisfying the following conditions: (I)
If diE 0 for i = I , 2, then there exists W 3 < dl A d 2 .
(2)
If d
(U)
For each d d.
E
0 and 8 is a cover of X such that d < E
E
0 such that
W,then a E 0.
@ there exists W E 0 which is a star-refinement of
Here a cover W of X is called a star-refinement of a cover d if {St(B, a)I B E W } < d. A set X together with a uniformity 0 on X is called a uniform space and will be denoted by (X, 0). By weakening the axiom (U) gradually we have the concepts of semiuniform spaces and generalized uniform spaces.
1.2. Definition. A semi-uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) from Definition 1.1 and the following: (SU)
For each d E 0 there exists W ment of d in a.
E
0 which is a local star-refine-
Here a cover W of X is called a local star-refinement of another cover d of X in a family 0 of covers of X, if for each B E W there exist dBE 0 and A E d such that St(B, dB) c A. A set X together with a semi-uniformity 0 on X is called a semi-uniform space and will be denoted by ( X , 0).
1.3. Definition. A generalized uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) of Definition 1.1 and the following:
K. Morita
4
(GU)
For each d E @ the collection { I n t A I A which is refined by some W E @.
Here we define for B by
(3)
t
E
d } is a cover of X
X the set Int, B, the interior of B with respect to 0,
I n t B = {x E XI there exists d
E
@ such that St(x, d ) c B}
A set X together with a generalized uniformity '-D on X is called a generalized uniform space and will be denoted by (X, 0).
1.4. Proposition. Every uniform space is a semi-uniform space and every semi-uniform space is a generalized uniform space.
Proof. The first statement is obvious since a star-refinement is a local star-refinement. To prove the second let @ be a semi-uniformity. Let 9 E @ be a local star-refinement of d in 0.Then for each B E 9 there exist 9 E @, A ~d such that St(B, 9)c A. This shows that B c I n t A . Hence 9 refines {IngA I A E d } . 0 1.5. Definition. A generalized uniformity (a semi-uniformity or a uniformity) @ on a set Xis called T I ,resp. Hausdorf, when condition (TI), resp. (H), below is satisfied:
(TI)
For any distinct x, y E X there exists d y $ S t k d).
E
@ such that
(H)
For any distinct x, y E X there exists d St(x, d )n St( y , d ) = 8.
E
0 such that
Clearly (H) implies (TI). Conversely, if @ is a semi-uniformity, then (TI) implies (H). Such a naming will be justified by Theorem 1.10 (c). In the discussion about generalized uniformities it is sometimes useful to deal with generalized uniformity bases.
1.6. Definition. (a) A subfamily CDo of a generalized uniformity @ is called a base for CD if for each d E @ there exists W E 0,such that W < d ,and a subbase for @ if the family { d lA . . A d n l d CD, i ~i = 1, . . . , n; n = 1, 2, . . .} is a base for @. (b) A nonempty family CD of covers of a set X is called a generalized uniformity base (resp. a semi-uniformity base or a uniformity base) if @ satisfies (1) and (GU) (resp. (SU) or (U)).
-
5
Extensions of Mappings I
1.7. Propasition. (a) A basefor a generalized uniformity (resp.semi-uniformity or uniformity) is a generalized uniformity base (resp. semi-uniformity base or uniformity base). (b) Let 0 be a generalized uniformity base (resp. semi-uniformity base or uniformity base). Then the family 0'of those covers of X which are refined by some members of 0 is a generalized uniformity (resp. semi-uniformity or uniformity) and 0 is a base for W ; W is said to be generated by 0.
The proof is straightforward and is left to the reader. 1.8. Examples. (a) Let (X,e) be a metric space. For any positive number > 0, let us define an open cover @(E)of X by
E
@(E)
= {U(x; E ) ( XE X} where U(x;
E)
=
{ y E Xle(x, y ) c E}.
Then @(+E)is a star-refinement of @(E),because V (y ; $ 8 ) n U(x; + E ) # 8 implies U( y ; ) E ) c U(x; E ) ; that is, St(U(x; +E), @ ( ; E ) ) c U(x; E ) E a(&). Therefore, @,,(p) = {@(E)I E > 0} is a base for a uniformity on X , which is called the metric uniformity induced by e and will be denoted by 0(4). (b) Let R be the set of all real numbers and e the Euclidean metric on R (that is, p(x, y ) = I x - y I). With the notation from (a), let us put 4Vn = {U(n
+ k; +(1
- 2P))lk
= 1, 2,
. . .} u @(l/2") f o r n = 1,2,
... .
Then we have
St(U(n
+ k; +(I
c V((n - 1)
St(U(x; 2-"), @,)
- P)), @),
+ (k +
1); +(l - 2-@+'))) for m > k
c U(x; 2 4 " - ' ) ) for rn
> max(x
+
+2 1, n).
Hence @, is a local star-refinement of en-, in 0,where 0 = {@,,In = 1, 2, . . .}. Therefore 0 is a base for a semi-uniformity0' on R. However, 0' is not a uniformity. 1.9. Proposition. Let 0 be a generalized uniformity base on a set X . Then the
of ) all the subsets G of X such that I n t G = G is a topology on collection ~ ( 0 X which is called the topology underlying 0. If0 is a base for a generalized uniformity W , then 7 ( 0 ) = T(W).Moreover, I n k is the interior operation in ( X t(@)).
K. Morita
6
Proof. It suffices to verify the conditions: (i) (ii) (iii) (iv)
I n t A c A and IntJ = X, A i c A2 * I n t A , c IntA,, Int,(B, n B2) = In@, n Int&, Int,(Int,B) = Int,B, where A, A l , A * , B, B , , B2 are subsets of X.
(i) and (ii) are obvious. If x E X and St(x, di) c Bi, i = 1, 2 for d l , dzE 0, then St(x, d ) c BI n B2 for d E 0 with d < di A d2.This proves (iii) by (ii). Next, for each d E 0, let us put Intd
=
{IngA IA E d}.
(1.1)
Then I n b d E 0 by (2) and (GU). To prove St(x, Int,d) c Int,[St(x, d ) ] ,
(1.2)
let y E St(x, I n t d ) . Then there exists A E d such that x, y E I n t A . Since y E Int,A, there exists W E 0 such that St( y , %) c A. Since x E Int,A c A, we have St(y, W) c A c St(x, d).This shows that y E Int,[St(x, d ) ] . Thus, (1.2) holds. Let x E Int,B. Then there exists d E 0 such that St(x, d ) c B. Hence we have by (1.2), St(x, Int,d) c Int,[St(x, d ) ]c Int,B. This shows that I n t B c Int,(Int,B), which proves (iv) by (i).
0
1.10. Theorem. Let ( X , a) be a generalized uniform space. Then the following hold in the topological space ( X , ~(0)).
(a)
{St(x, d )I d each x E X .
(b)
If B c ,'A
(c)
The space (A', ~(0)) is Ti or Hausdorffaccording as 0 is Ti or Hausdorff.
E
0}is a base for n b h ( = neighborhoods) of x for
the closure C1 B of B equals n{St(B, d )I d E a}.
Proof. (a) Suppose that G c X is open and that x E G. Since I n t G = G, there exists d E 0 such that St(x, d ) c G. On the other hand, I n t d is an open cover of X by Proposition 1.9. Thus {St(x, &)Id E 0}is a base for nbds of x.
Extensions of Mappings I
7
(b) By (a) we have Cl B = {x E XI St(x, d)n B #
0 for all d E 0}.
Since St(x, d)n B # 0 iff x E St(B, d), the assertion (b) is proved. (c) follows readily from (a).
0
1.11. Proposition. If 0 is a semi-uniformity on a set X , then {St2(x,d)I d E 0}is a base for nbds of each point x E ( X , ~(0)). Proof. Let d E 0.Let d,and d2be local star-refinements of d and dlin 0 respectively. For each x E X there exists A, E d, with x E A 2 . Let us first find W,E 0,A , E d, such that &(A,, a,)c A, and then $3, E 0,A E d such that St(A,, W,) c A. Thus, for W E 0 with W < W,A W,we have St2(x,93) c St(St(A,, a,),9,) c St(A,, W,) c A c St(x, d). Thiscompletes the proof by Theorem 1.10 (a). 1.12. Proposition. Every generalized uniformity 0 on a set X has a base Q0 which consists of open covers of ( X , ~(0)). Proof. = { I n g dI d E 0}(see (1.1)) is a desired base for 0,since Int,d is an open cover of the space ( X , ~(0)) by Proposition 1.9. 0
1.13. Definition. Let (X,t)be a topological space. A generalized uniformity (resp. uniformity base) 0 on the set X is said to be compatible with the =)T. Sometimes, by a generalized uniform space ( X , 0) topology of Xif ~ ( 0 we refer to a topological space X together with a generalized uniformity compatible with the topology of X. In applications the following theorem is fundamental. Indeed, in view of Proposition 1.12, it provides a method of introducing a generalized uniformity on a given topological space. 1.14. Theorem. Let ( X , t )be a topological space and a non-empty family of open covers of X which satisfies condition (1) in Definition 1.1. (a) is a generalized uniformity base compatible with the topology ifl {St(x, 4V) 1% E O 0 }is a base for nbds of x for each x E X . (b) #o is a semi-uniformity (resp. uniformity) base compatible with the topology if {St(x, 4V) I 9 E Q0] is a base f o r nbrls of x for each x E X and satisfies (SU) (resp. (U)).
8
K. Morita
Proof. The “only if” part of (a) follows directly from Theorem 1.10. Conversely, suppose that {St(x, @) 1% E Q0} is a base for nbds of x for each x E X. Then for B c X,the interior Int B of B in the space ( X , T) coincides with I n h B defined by (3). Hence CPo satisfies (GU) and ~ ( 0= ~r. )This proves the “if” part of (a). The assertion (b) follows readily from (a). 0 To discuss the problem: “when does a topological space admit a generalized uniformity compatible with the topology?”, we give the following definition. 1.15. Definition. A topological space Xis called weakly regular if for x and for any open set G with x E G we have Cl{x} c G.
E
X
Clearly every T,-space as well as every regular space is weakly regular. One of the important properties of semi-uniform spaces is shown in the following theorem. 1.16. Theorem. Let X be a topological space. (a) If X admits a generalized uniformity (resp. semi-uniformity) compatible with the topology, then X is weakly regular (resp. regular). (b) If X is weakly regular (resp. regular), then the family of all covers of X which are rejined by open covers of X is a generalized uniformity (resp. semi-uniformity) on X compatible with the topology.
Proof. Let 0 be a compatible generalized uniformity on the space X . Let x E X and G be an open set of X with x E G. Then there exists an d E @ such that St(x, d ) c G. By Theorem 1.10 we have Cl{x} c St(x, d ) ,that is, CI { x} c G. Thus X is weakly regular. Suppose further that 0 is a semi-uniformity. Then by Proposition I . 11 {St2(x,d )I d E 0 }is a base for nbds of x for each x E X . Since by Theorem 1.10 we have C1 St(x, d ) c St2(x, d),it follows that Xis regular. Thus, (a) is proved. Next, let CP, be the family of all the open covers of X. Then CPo satisfies condition ( I ) . Let x E G where G c Xis open. Let X be weakly regular. Then Cl{x} c G and hence @o = {X - Cl{x}, G} is an open cover of X . Hence % ’ , E Q0 and St(x, @), c G. This shows that {St(., @) I @ E Q0} is a base for nbds of x for each x E X. Hence by Theorem 1.14, 0, is a generalized uniforrhity base compatible with the topology. Therefore, 0 which is generated by O0, is a generalized uniformity on X compatible with the topology.
Extensions of Mappings I
9
Finally, suppose that X is regular. Then the family defined in the preceding paragraph satisfies condition (SU).To see this, let 4 be any open cover of X. Then for each x E X there exists U(x) E such that x E U(x). Since Xis regular, there is an open nbd V ( x )of x such that CI V(x) c U(x). Let us put V = { V ( x )Ix E X} and Wx = {X - CI V(x), U(x)} for x E X. Then we have St( V ( x ) ,Wx)c U(x). This shows that Y is a local star-refinement of 4 in (Po. Hence, by Theorem 1.14, O0 is a semi-uniformity base compatible with the topology of X. 0 1.17. Remark. A topological space X admits a uniformity compatible with the topology iff X is completely regular (the proof is found in Tukey [ 19401 and in textbooks on general topology). The family @ described in (b), however, is not always a uniformity; indeed @ is a uniformity iff X is paracompact regular. This is an essential difference between uniformities and semi-uniformities. 1.18. Definition. Let ( X , 0)and (Y, Y) be generalized uniform spaces. A map f from X to Y is called a uniformly continuous map (or a uniformitypreserving map) from ( X , @) to (Y, Y) if f-’(W) E @ for each W E $. A uniformly continuous mapf: (X,@) --* (Y, Y) is called a uniform isomorphism or unimorphism iffis a bijection from X to Y and the inversef-’ off is also a uniformly continuous map from (Y, Y) to (X,a). If O0(resp. Yo)is a base for @ (resp. Y)J: X + Y is a uniformly continuous map from ( X , @) to ( Y , Y) if for each W E Yothere exists d E such that cc9 c f-’(W). The uniformly continuous maps are often used in this form. The naming “continuous” is justified by the following proposition. 1.19. Proposition. A uniformly continuous map f:( X , @) + ( Y , Y) induces a continuous map f:(X,~(0)) --* ( Y , ~(‘4’)). Proof. For each W E Y and each x E X we have f(St(x,f-’(W)) c St(f(x), 9 ) . This proves the continuity off by Theorem 1.10. 0 1.20. Definition. Let X c Y. Then a generalized uniform space ( X , @) is called ’a subspace of a generalized uniform space (Y, Y) if Y 1 X = {W A XI W E Y } is a base for 0.A subspace (X, @) of (Y, Y) will be denoted by (X,YI X ) .
10
K. Morita
Products of generalized uniform spaces can be defined analogously as in the case of uniform spaces but the definition is omitted here because we have no oDportunity of using them in this chapter. 1.21. Remarks. (a) The concepts of generalized uniformities and semiuniformities were introduced by Morita [ 195I] for topological spaces, under the name of T-uniformities and regular T-uniformities agreeing with the topology. Indeed, in that paper we started our study by taking Theorem 1.14 as the definition of these uniformities and established Theorem 1.16. (b) A. K. Steiner and E. F. Steiner [1973] gave a topology-free axiomatization to regular T-uniformities, which they renamed semi-uniformities;the axiom (SU) is due to Morita [1951]. A topology-free axiomatization was given by Herrlich [ 1974bl to T-uniformities agreeing with the topology, which we call, in the text, generalized uniformities. (c) Herrlich [I9741 introduced the concept of nearness spaces. As was pointed out by Herrlich [1974b], the category of nearness spaces and nearness-preserving maps is equivalent to the category of generalized uniform spaces and uniformly continuous maps. (d) T-uniformities in the sense of Morita with a certain condition are discussed by Rinow [1967], [I9751and Poppe [1974]. In Poppe [1974], spaces with such T-uniformities are called generalized uniform spaces, and semiuniform (resp. generalized uniform) spaces are called regular (resp. weakly regular) generalized uniform spaces. For various generalizations of uniform spaces, see Herrlich [ 1974bl.
2. Completeness, completions and extensions of spaces Among various generalizations of uniform spaces the most important is the class of semi-uniform spaces, which will play a fundamental role in the subsequent sections of this chapter. Although our main concern lies in semi-uniform spaces, we shall discuss completeness and completions for generalized uniform spaces. Throughout this section let us assume that ( X , 0)is a generalized uniform space and let CP,, = {%!a I a E a} be a base for CP which consists of open covers of x. Filters, filter bases and their convergence will be used in what follows (for the definitions see any textbook on general topology). The collection of all (not necessarily open) nbds of a point x is called the neighborhoodfilter of x and will be denoted by %!(x).
11
Extensions of Mappings I
2.1. Definition. A filter base d is called Cauchy, resp. strictly Cauchy, with respect to Oowhen (I), resp. (2), below holds: (1)
For each a E R there exist A E d and U E 9YUsuch that A c U .
(2)
For each a E R there exist A E d , U E St(A, %& c u.
aUand /3 E R such that
The words “with respect to (Do’’ are omitted unless there is a fear of ambiguity. 2.2. Examples. (a) Let (X,e) be a metric space and let U ( x ; E ) and (Do(@) be the same as in Examples 1.8 (a). Let {a,,}be a sequence of points of X and A, = {aiI i 2 n } , n = 1,2, . . . . {a,,}is called Cauchy if for any E > 0 there exists no such that e ( x i , xi) < E for i, j 2 no. Hence the filter base {A,} is Cauchy with respect to (Do(@) iff the sequence {a,} is Cauchy. (b) The nbd filter is a Cauchy filter. (c) Let d ( x ) be a collection which consists of a single-point set {x}, where x E X. Then d ( x ) is a strictly Cauchy filter base; because for each a E R there exists U E 4?la with x E U and for such U there exists j? E R with St(x, aP) c U since {St(x, as) I /3 E R} is a base for nbds of x. The filter % ( x ) is not always a strictly Cauchy filter unless X is a regular space.
2.3. Proposition. Let (Do be a semi-uniformity base. Then afilter base d is strictly Cauchy if it is Cauchy.
Proof. Since A c St(A, GVP), every strictly Cauchy filter base is Cauchy. Conversely, let d be a Cauchy filter base. For a E R there exist A E d and VE such that A c V , where ”pcs is a local star-refinement of au. For V there exist y E R and U E 9ZU such that St(V, aY) c U . Hence we have St(A, aY) c St(V, ‘By)c U . This shows that d is strictly Cauchy. 0
-
2.4. Definition. A strictly Cauchy filter base d is said to be equivalent to A? in notation, if for each a E R another strictly Cauchy filter base A?, d and A E d there exist /3 E R and B E W such that
--
2.5. Umma. (a) Ifd (b) Ifd
--
Let d,B and V be strictly Cauchyfilter bases. W,then A? d. W and A? V, then d 5%.
-
K. Morila
12
Proof. (a) Let a E R and B E A?. Since d is strictly Cauchy, there exist E R, A, E d , U, E %a such that St(A,, t?Za0) c U,. Since d A?, there c St(A,, %%).Since B n B, # 0, exist /3, E R, B, E A? such that St(B,, and Bo c St(A,, 42%) c V,, we have U, c St(B, %a). This shows that d. St(Ao, 42%) c U, c St(B, @a), that is, A? (b) This assertion follows directly from Definition 2.4 0
-
a,
-
-
Since d d for any strictly Cauchy filter base d , the equivalence defined above is an equivalence relation. 2.6. Proposition. I f 0,is a semi-uniformity base and d , 93 are strictly Cauchy Jilter bases, then the following are equivalent.
-
(i) d A?. (ii) For each a E R and A E d there exists B E A? such that B c St(A, ea). (iii) For each a E R there exist A E d , B E A? and U E %a such that A v B c U (that is, d v A? is a CauchyJilter base). Proof. (i) * (ii) is a direct consequence of Definition 2.4. (ii)-(iii). Assume (ii) and let a E R. Since d is strictly Cauchy, there exist A E d , U E %a, /3 E R such that St(A, %)) c U. Then by (ii) there exist B E A? such that B c St(A, %,). Hence we have A u B c St(A, '4!lp)c U . This shows that (iii) holds. (iii)*(i). Assume (iii) and let A E d , a E R. Since d v A? is Cauchy by (iii), d v A? is strictly Cauchy by Proposition 2.3. Hence for a there exist A, E d , B, E 93, U E @a and /3 E R such that St(A, u B,, 4Yp) c U. Since A, c U and A, n A # 0,we have A n U # 8 and hence U c St(A, %a), and consequently
St(&,
@p)
This shows that d
-
c St(A0 u Bo, %p)
c U c St(A, %a).
A?.
-
2.7. Lemma. (a) Ifd and A? are strictly Cauchy Jilter bases and d A?, then nCl d = nCl 93. (See Section 1.0.). (b) Ifd is afilter base and converges to x E X , then x E n C l d . (c) Ifd is a strictly Cauchy Jilter base and x E n C l d , then d converges to x.
(d) If X is a T,-space and d is a strictly Cauchy Jilter base, then n C l d is either empty or composed of a single point.
Extensions of Mappings I
- a,
13
= n{St(A, S ) I A E d,u E R}. d and u E Q, we have
Proof. (a)ByTheorem 1.lOwehavenCl d Since d
for each A
nci
w
E
= n{st(B, % p ) ~B E 8, p
E
n} c
st(A, %J
and therefore nC1 W c n{St(A, %u) I A E d , u E Q} = nCl d , that is, nCl c nCl d . Since W d by Lemma 2.5 we have also nCl d c nC1 W,and therefore nC1 d = nCl $3. (b) Is well known and easy to prove. (c) Let a E R. Since d is strictly Cauchy, there exist A E d ,p E R, U E eU such that St(A, %& c U.Since x E nCl d = n{St(A, a a ) l A E d ,a E Q}, we have x E St(A, %p) c U . Since x E U , we have U c St(x, %u) and hence A c St(A, %p) c St(x, %u). Since {St(x, 9 )I u E R} is a base for nbds of x, d converges to x. (d) Suppose that nCl d # 8 and that x E nCl d .Let y E X with y # X. Then there exists a E R such that y 4 St(x, aU). As was shown in the proof of (c), there exist A E d , /? E R such that St(A, %p) c St(x, qU). Hence we have y # St(A, %$). Since nCl d = n{St(A, % u ) l A E d,u E Q}, this shows that y 4 ncl d. 0
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2.8. Definition. For a strictly Cauchy filter base d the collection {St(A, 9 )I A E d , a E R} is a filter base and the filter generated by it is called the star-filter of d with respect to and will be denoted by S t ( d ; moo). Every star-filter is Cauchy. The nbd filter %(x) of x is St(d(x); O0) (see Examples 2.2 (c)).
2.9. Proposition. Let d and W be strictly CauchyJilter bases. Then d tfSt(d; 0 0 ) = St(W; 00). Proof. This follows directly from Definition 2.4 and Lemma 2.5.
-W 0
It follows from Proposition 2.9 that there is a bijection between the set of all the equivalence classes of strictly Cauchy filter bases and the set of all the star-filters. To express an important property of star-filters we need two more definitions.
2.10. Definition. A Cauchy filter % is called a weak star-Jilter if for each F E 9 there exists u E R such that U E and U E % imply U c F, that is, n %u) c F.
u(%
K. Morira
14
2.11. Definition. A nonempty collection W of subsets of X is called a stack if B E W and B c C imply C E W. A stack is called Cauchy if it contains at least an element of 42, for each u. A Cauchy filter (resp. Cauchy stack) d is called a minimal CauchyJilter (resp. minimal Cauchy stack) if it contains no proper subfamily which is a Cauchy filter (resp. stack). 2.12. Proposition. Every star-Jilter is a weak star-filter. Proof. Let d be a strictly Cauchy filter base and let 9 = S t ( d ; Q0). Let F E 9. Then there exist A E d , u E R, such that %(A, 42,) c F. Let U E 9 n 42,. Since d v 9 has the finite intersection property, we have U n A # 0 and hence U c St(A, 42,) c F. 0 2.13. Proposition. A Cauchy stack is a minimal Cauchy stack i f i t is a weak star-Jilter. Proof. Assume that W is a weak star-filter and that W' c W is a Cauchy stack. Then for each u E R there exists U, E @' n 42,. Let B be any element of W.Since g is a weak star-filter, there exists u E R such that u(g n 42,) c B. Since U, E a' n 42, and W' c W, we have U, c U(W n 42,) c B. Since 9.3' is a stack and U, E W', we have B E W'.Hence W c W'. Thus W is a minimal Cauchy stack. Conversely, assume that 43 is a minimal Cauchy stack. Let us put
G, = lJ(W n 42,)
for u E R.
Since W is Cauchy, we have W n 42, # 0 and hence G, # 0 for each a E R. Let B be any element of W.Suppose that B $ G, for all u E R, that is, (X - B) n G, # 0 for all u E R. Then there exists, for each u, an element n& such I that (X - B) n U, # 0. Let us put U, E #
=
{A c XI A contains U, for some u E R}.
Then A?' is a Cauchy stack and W' c W.Since W is a minimal Cauchy stack, we have = W. On the other hand, it follows from the definition of g'that &?I
(X - B) n A #
0
for all A E W = W.
However, this is a contradiction since B is itself an element of 9.Thus we have B, 3 G, for some u E R. If 42y < A 42, (a, 8, y E R), then G, c G, n G,. Therefore W is a filter. Thus W is a weak star-filter. 0
Extensions of Mappings I
15
2.14. Corollary. A weak star-filter is a minimal Cauchyfilter. In general, a minimal Cauchy filter is not necessarily a weak star-filter and a weak star-filter is not necessarily a star-filter. These filters, however, are coincident for some cases which we are interested in.
2.15. Theorem. Let Oobe a semi-uniformity base. Then every Cauchyjilter contains a weak star-filter, and the three types of Cauchy filters - star-filters, weak star-filters and minimal Cauchyfilters - are all coincident.
Proof. A Cauchy filter 9 is strictly Cauchy by Proposition 2.3 and hence it contains the star-filter S t ( 9 ; (Do). In particular, if 9is a minimal Cauchy filter, then 9 = S t ( 9 ; O,,). From Proposition 2.12 and Corollary 2.14 it follows that the three types of Cauchy filters mentioned in the theorem are all coincident. There are two more cases in which the minimal Cauchy filters are precisely the weak star-filters; one is given in the next theorem and the other in Theorem 2.36 at the end of this section.
2.16. Theorem. Suppose that every open cover of X is refined by some cover in (Do. Then every Cauchyjilter converges. Therefore the three types of Cauchy filters mentioned in Theorem 2.15 are all coincident.
Proof. Let 9be a Cauchy filter. Suppose, on the contrary, that % does not converge to any point of X.Then for each x E X there is an open nbd U(x) of x such that V(x) Ffor all F E %. Then %o = { U ( x )I x E X} is an open cover of X and hence a0> @a for some a E R. On the other hand, since 9 is a Cauchy filter, 9contains some element Uof %a, and U c U(x) for some x E X. But this is in contradiction with the property of V(x). Therefore, 9 converges to a point x of X,that is, 9 3 @(x). If % is a minimal Cauchy filter, then 9 = @(x). On the other hand, we have % ( x ) = St(d(x); Oo) (see Examples 2.2 (c)). 0 2.17. Definition. X i s said to be complete with respect to (Do (or simply, Oo is complete) if every weak star-filter with respect to Ooconverges to a point in X . 2.18. Lemma. Let 0, be another base for O which consists of open covers of X . Then X is complete with respect to OoijTX is complete with respect to (D,. Hence in case X i s complete with respect to Oo,we shall say that ( X , O ) or (D is complete.
K. Morita
16
Proof. Since every cover in CDI is refined by some cover in CD, and vice versa, a filter is a weak star-filter with respect to 0,iff it is a weak star-filter with respect to CDl. From this the lemma follows immediately. 0 Theorem 2.16 immediately yields the following theorem. 2.19. Theorem. Zf(X, CD) is a generalized uniform space such that every open cover of X is contained in CD, then ( X , 0)is complete. The following theorem, together with Theorem 2.19, is important in Section 3. 2.20. Theorem. Let ( X , 0)be a semi-uniform space. Then the following conditions are equivalent. (a)
( X , 0)is complete.
(b)
Every Cauchy filter base converges.
(c)
For every Cauchyfilter base d , we have nCl d #
0.
{a
Proof. We shall assume that CDo = I a E R} is a base for CD as before. It suffices to prove the theorem for CD,. (a)*(b). Let d be a Cauchy filter base. Then d is strictly Cauchy (see Proposition 2.3) and S t ( d ; CD,) is a weak star-filter (see Proposition 2.12). By (a) S t ( d ; CDo) converges to some point x in X , that is, S t ( d ; CDo) = %!(x). This shows that each nbd of x contains %(A, eU) for some A E d , a E R. Hence d converges to x. (b)o(c) is a direct consequence of Lemma 2.7. (b)*(a) is obvious. 0 2.21. Example. Let ( X , e) be a metric space and @(e)its metric uniformity (see Examples 1.8 (a)). Then a metric space ( X , e) is complete iff the uniform space ( X , CD(e)) is complete. Thus the concept of completeness above is a generalization of the completeness for metric spaces. The following construction of completions of generalized uniform spaces is a highly elaborated application of Hausdorff’s method for completing metric spaces. Let us denote by E the set of all the weak star-filters with respect to CDo which do not converge to any point in X . Let us put
x*
=
XUE
Extensions of Mappings I
17
and for any open set G of X let us define
G* = G u { ~ E E I G E ~ } . Then we have the following lemma.
2.22. Lemma. Let G and H he open sets of X . (a)
G c H implies G * c H * ; G* n X = G .
(b)
(G n H ) *
=-
G* n H*.
The proof is straightforward and is left to the reader. From Lemma 2.22 it follows that the collection { G * I G open sets of X} is a base for a topology of X * and we shall consider X * as a topological space with this topology.
2.23. Lemma.
=
{ U * I U E an} is an open cover of X * .
Proof. Let 9 E E. Since 9 is a Cauchy filter, there exists U E U E 9. This shows that 9 E U*.
with
0
2.24. Lemma. Let y E X * and y E G * for an open set G of X . Then there exists a E R such that St( y , a:) c G * .
Proof. Since the lemma is obvious for y E X , assume that y E X * - X and Since 9 is a weak star-filter, that y = 9 E E n G*. Then we have G E 9. there exists a E R such that U ( 9 n an)c G. Suppose that 9, X E U * with U E an. Then U E 9 n and hence U c G. On the other hand, U E X . Therefore G E X , that is, X E G*. If x E X n U* = U , then x E U c G . Thus, we have St( y , a:) c G * . 2.25. Lemma. St(G*, a:)
c
[St(G, an)]* for any open set G of X .
Proof. If G* n U* # 0 with U E ea,then G n U # 0 and hence U c St(G, an). Therefore, U* c [St(G, aa)]*. This proves the lemma. 0 Let us now put @$ =
{a: I a E R}.
2.26. Lemma. (a) @$ is a generalized uniformity base on X * which is compatible with the topology.
18
K. Morita
(b) Let @* be a generalized uniformity generated by @$. Then ( X * , @*) does not depend on the choice of a base (Do, and is called the completion of ( X , 0). (c) If@ is a semi-uniformity, then @* is a semi-uniformity. The same holds for uniformities.
Proof. (a) If %, < +!$ A C?Zy, then % : < %$ A 4.3‘: Hence (a) follows from Lemma 2.24, by virtue of Theorem 1.14. = {V,1 fl E W} be another base for 0 which consists of open (b) Let covers of X.As was shown in the proof of Lemma 2.18, a filter in Xis a weak star-filter with respect to iff it is a weak star-filter with respect to (Do. Hence, even if we use @I instead of O0, we obtain the same set X * and the same topology on X*. If 43, < Vpand Vp< %, then we have % : < V,* and V,* < % .: This shows that 0: = {4$‘I a E R} and @? = {V,*1 fl E W} generate the same generalized uniformity which is @*. is a semi-uniformity base. Hence (c) Let @ be a semi-uniformity.Then for each a E R there is a local star-refinement%, of%, in (Do.For each V E 42, there exist 6 E R, U E %, such that St(V, %a) c U . Hence we have, by ): c [St(V, %a)]* c U*. This shows that %$ is a Lemma 2.25, St(V*, % local star-refinement of % : in 0;. Therefore 0; is a semi-uniformity base. 0 2.27. Lemma. (a) I f A is a weak star-filter with respect to 0$, then A‘ = { A c XI there exists U* E A n 43: with a E R such that U c A } is a weak star-filter with respect to (Do. (b) I f 9 is a weak star-filter with respect to a,, then 9*= { B c X * I there exists U E 9 n %, with a E R such that U* c B} is a weak star-filter with respect to @$. (c) [A’]*= A, [9*]’ = 9 for A in (a) and 9 in (b).
e}.
Proof. (a) First we observe that A‘ n %, = { U c XI U* E A n To see this, let U E A‘ n 42,. Then there exist /IE R, V E %, such that V c U , V* E A n 42;. Since V* c U*, we have U* E A n .% : Thus A ’ n %a c { U I U* E A n q}. Since the converse inclusion holds by the definition of A’,we have A’ n %, = { U (U* E A n 4%:). Let A E A‘. Then there exists U E A’ n 92, such that U c A. Since U* E A and A is a weak star-filter, there exists fl E R such that U ( A n 92;) c U*. Hence we have U ( A ’ n 42,) c U c A. This shows that A‘ is weak star. Let a, B E $2. Then there exists y E R such that %, < A %., If we define G, = U ( A ’ n 4 ‘ 2,) and define G,, G, similarly, then G, c G, n G,. Since any element of A’ contains some GayA‘ is a filter.
Extensions of Mappings I
19
Since A' n %a # 0 for all a E R, A' is Cauchy, and hence a weak star-filter with respect to Qo. (b) Similarly as in the proof of (a) we can prove that 9*n 922 = { U* I U E 9 n %a} and also that 9* is a weak star-filter with respect to Q:. (c) follows readily from (a) and (b). 0 2.28. Lemma. Let A and A' be the same as in (a) of Lemma 2.27.
(a)
I f A' converges to x in X , then A converges to x in X*.
(b)
I f A' does not converge to any point in X , then A' E E and defines a point y in X * - X and A converges to y in X*.
Proof. (a) Let G be any open set of X with x E G*. Then x E G and, since A' converges to x in X , there exists M' E A' with M' c G. Since M' E A', there exists U E A' n %a for some a E R such that U c M'. Hence we have U c G, and consequently U* c G * with U* E A n .% : This shows that A converges to x in X*. (b) Let y E G* where G is an open set of X. Then there exists a E R such that St( y , 922)c G*. Let y E U,+ with U, E Then Uo E A'. Since A' n = { U I U * E A n a:}, we have U,*E A. Since U,* c G*, this shows that G * E A. Thus, A converges to y in X*. 0
As further properties of X* we have the following lemmas.
2.29. Lemma. For any open set G of X we have G* = X * - Cl(X - G )
where CI means the closure operation in the space X*.
Proof. Since X * - G * is closed in X * and (X* - G*) n X = X - G, it is obvious that CI(X - G ) c X * - G*. Conversely, let y E X * - G* and let y E H * for an open set H of X. Then H * n (X - G ) # 0, because, otherwise we would have H n (X - G) = 0 and hence H c G which implies H* c G*. Therefore y E Cl(X - G). 0 2.30. Lemma. For x
E
X * the following holds:
n{St(x, %*)la
E
R}
=
i
{x}
ij-XEX* -
Cl,{x}
if x E
In particular, each point of X * - X is closed.
x.
x,
20
K. Morira
Proof. Let x E X and y E X* - X. Suppose that y is defined by a weak star-filter 9 in X which converges to no point in X. Then there exists a E R such that St(x, an) 4 9.If x E U E an,then U 4 9, that is, y 4 U*. Hence y 4 St(x, 4%';). Therefore n{St(x, a:) I a E R} c X, and n{St(x, 43:) I a E R} = a E n} = ci,{x>. n{st(x, an)[ On the other hand, the relation y 4 St(x, 99:) implies that x 4 St( y, 43:). Therefore we have n{St( y , )I %: a E Q} c X * - X. If z E X* - X and z # y, then the weak star-filter Y which defines z is different from 9. Hence there exists a E R such that U ( 9 n an) 4 3, which shows that z 4 St( y , %.): This proves the lemma. 0 2.31. Definition. A topological space Y is called an extension of a topological space X if Y contains X as a dense subspace. An extension Y of X is called a T,-extension of X if { y } is a closed set of Y for each y E Y - X, and called strict if { E , ( G ) I G open sets of X} is a base for the open sets of Y , where
E y ( G ) = Y - CI,(X - G ) . The results which have been obtained hitherto show that the completion ( X * , a*) satisfies the conditions in Theorem 2.32 below. 2.32. Theorem. Let ( X , #) be a generalized uniform space. Then the completion ( X * , #*) of (X,#) is characterized as a generalized uniform space ( Y , Y) satisfying conditions (i) to (iii):
(i) (Y, t(Y)) is a strict T,-extension o f ( X , ~(0)). (ii) For each a E R the collection E Y ( a n )= { E y ( U ) I U E '%a} is an open cover of Y and 'Po = {EY(42n) I a E R} is a base for Y. (iii) ( Y , Y) is complete. Here (Do = {an I a E R} is a base for Q, as before. Moreover, i f ( X , #) is a semi-uniform space (resp. a uniform space), then ( X * , #*) is a semi-uniform space (resp. a uniform space). Proof. Let (i)-(iii) hold for (Y, Y). Let us first observe the following lemma. 2.22'. Lemma. Let G and H be open sets of X .
(a)
G c H implies E,(G) c E , ( H ) ; E y ( G ) n X = G ;
(b)
E,(G n H ) = E , ( G ) n E , ( H ) .
The proof of Lemma 2.27, as a careful examination will show, is based on Lemma 2.22 only. Hence, analogously we have the next lemma on the basis of Lemma 2.22'.
21
Extensions of Mappings I
2.27'. Lemma. (a) Zf A is a weak star-jilter with respect to Y o , then R,(A) = { A c XI there exists E y ( U )E A n E Y ( q a )with a E R such that U c A } is a weak star-jilter with respect to (Do. (b) Zf 9 is a weak star-jilter with respect to O0, then E y ( 9 ) = { B c Y I there exists U E 9 n %a with a E R such that E y ( U ) c B} is a weak star-jilter with respect to Yo. ( 4 EY[RX(&I = M&49-)l = 8. A
9
Let y E Y - X and %( y ) the nbd filter of y . Since { y } is closed in Y and hence { y } = Cl{ y } = (){St( y , Ey(%J) I a E R}, %( y ) does not converge to any point of X . Similarly as in the proof of Lemma 2.28, we can prove that R,[%(y)] does not converge to any point in X . Hence R X [ % ( y ) E] 9 and defines a point z in X* - X . Let us put cp(y) = z. For any x E X we put cp(x) = x . Thus cp defines a map from Y to X*. Let z E X * - Xbe defined by a weak star-filter 9 with respect to (Dowhich does not converge in X.Then E y ( 9 )is a weak star-filter with respect to Y o . Since (Y, Y ) is complete, E y ( 9 ) converges to a point y E Y - X , that is, E y ( 9 ) = % ( y ) . Then by Lemma 2.27' we have R,[%(y)] = 8. Hence cp( y ) = z. This shows that cp is surjective. If y , , y, E Y - X and y , # y,, then %( y , ) # %( y,) and hence It,[%( y , )] # Rx[%( y,)], and consequently we have cp( y , ) # cp( y2). Thus cp is a bijection. Let G be any open set of X , and let y E Y - X . Then we have Y E EY(G) 0 EY(G) E W Y ) * G E R,[Q(y)l*
cp(Y) E G * .
This shows that cp(E,(G)) = G*. Therefore cp:(Y,Y) + (X*, a*) and cp-': ( X * , @*) + (Y, Y ) are both uniformly continuous, and cp is a uniform isomorphism which leaves each point of X fixed. This completes the proof of the theorem. The following is an application of Theorem 2.32.
2.33. Theorem. Let X and Y be weakly regular spaces and let Y be a strict TI-extension of X . Then there exists a generalized uniformity @ on X compatible with the topology, such that Y is homeomorphic to the underlying space of the completion of ( X , 0). Proof. Let Yobe the collection of all the open covers of Y which consist of the sets of the form E y ( G )with G open in X . Since { E y ( G )I G open in X } is a base for the open sets of Y, every open cover of Y is refined by some member of Yoand hence Yois a base for Y , where Y is a generalized uniformity on Y such that every member of Y is refined by some member of Yo; of course,
K. Moriia
22
Y is compatible with the topology of Y since Y is weakly regular (see Theorem 1.16). By Theorem 2.19, (Y,Y) is complete. is a generalized uniformity base on Let Q = {*lr A XI *lr E Y o }Then . X compatible with the topology. Let CD be the family of covers of X which are refined by some covers in B0.Then Q, is a generalized uniformity on X compatible with the topology, and (Do is a base for Q,. Therefore by Theorem 2.32 there is a uniform isomorphism between (Y, Y) and ( X * , @*). 0
Since a regular space is a strict extension of each of its dense subspaces, we have the following theorem from Theorem 2.33. 2.34. Theorem. Let X be a regular TI-space. Then every regular TI-space which contains X as a dense subspace can be obtained as the completion of a semi-uniform space ( X , a) with CD as a semi-uniformity compatible with the topology of x. The following theorem will be used in the next chapter.
2.35. Theorem. Let ( X , (D) be a generalized uniform space and (X*, a*) its 1 a E R} be a base for Q, consisting of open covers of completion. Let (Do = {aa X , and {G, I ilE A} a collection of open sets of X . Let us put Wu = { G A I i l ~ A }
u ( U I U n ( X - U { G A l l ~ A }#) ~ , U E @ ~U} E, R . (a) If for each u E R the cover Wuis refined by some 4fpwith p E R, then = tU(G, 13, E AH*. U{G? I A E (b) In case each of covers aU, a E R, is afinite cover of X , then the converse of (a) holds.
Proof. (a) Since we have clearly U{G: 11 E A} c [U{G, I 1 E A}]*, it suffices to prove the converse inclusion. From the assumption in (a) we have U{G: 11 E A} u St(X - UG,, @!,*) = X*,a E R and consequently X * - U{Gf 11E A} c St(X - UG,, 4)2:'
for a E R.
(2.2)
Therefore it follows from Theorem 1.10 that X* - U(G:IAEA}
c Cl(X - U { G , I L E A } ) .
(2.3)
Consequently, by Lemma 2.29 we have the desired inclusion [U{Gn I 1 E All*
= U { G f I 1 E A}.
(2.4)
23
Extensions of Mappings I
(b) Suppose that U{Gt 11 E A} = [U{G, 11 E A}]*. Since the implications (2.4)*(2.3)*(2.2) hold in the proof of (a), and for U E 42,, ( X - UG,) n U # 0 iff (X - UG,) n U* # 0, we have U { W * l W E K } = X*. Now Theorem 2.36 below shows that W, is refined by some 42p, /3 E R. 0
2.36. Theorem. Let ( X , O ) and Oo = {42, I u E R} be the same as in Theorem 2.35. Assume, in addition, that each 42, with u E R is afinite cover of X . (a)
Every Cauchy stack with respect to m0(see Definition 2.1 1) contains a minimal Cauchy stack (= a weak star-flter by Proposition 2.13).
(b)
Every open cover of X * is refined by
(c)
X* is compact.
422
with some u E R.
Proof. (a) Let W be a Cauchy stack. Let r be the family of all the Cauchy stacks which are contained in W.Then there exists a minimal element in r, ordered by incluson. To prove this, let (W(1) I 1 E A} be a subfamily of such that A is a linearly ordered set of indices and that W(1) =I W ( p ) if 1 < p in A. Let us put cp,(l) = { U l U
E
W(1) n %,},
u
E
R.
Since 42, is a finite set and cp,(1) # 0 for all 1 E A, there exists 1, E A such that cp,(1) = cp,(1,) for 1 > 1,. Now, let us put W o = n(W(1) 11 E A}. Then W ois clearly a stack, and for u E R we have W on 42, = cp,(L,). Because, if U E cp,(A,), then U E 42, n W(1)for 1 > 1,and hence for all 1 E A and consequently cp, (1,) c gon 42,. Thus W ois a Cauchy stack and Wo t W ( 1 ) for all 1 E A. Therefore by Zorn's lemma there exists a minimal element W'in r. W'is a minimal Cauchy stack. This proves (a). (b) Let X be any open cover of X*. Then there exists u E R such that 42; < 2.To see this, assume, on the contrary, that 42: 4 2 for all a E R. Then for each u E R there exists U, E 42, such that U$ 4: H for all H E 2, that is, U,* n ( X * - H ) #
0
for all H E X .
Let us put W = {B c X * I B n ( X * - H) # 0 for all H E X } .Then W is a stack in X * and W n 422 # 0 for all u E R since U$ E W.By applying (a) to the generalized uniform space ( X * , O*), one sees that W contains a weak star-filter W'. Since ( X * , 0*)is complete, there exists y E X * such that 93' coincides with the nbd filter of y in X*. Hence each nbd of y meets every
K. Morita
24
element X * - H with H E &'. This shows that y E CI(X* - H). Since H i s open in X*, we have y E X* - Hfor all H E &',which, however, contradicts the assumption that M is an open cover of X*. Finally, (c) is a direct consequence of (b). 0 In concluding this section, we shall show that Shanin's compactification, which is a generalization of the Wallman compactification, is obtained as the completion of a certain generalized uniform space. Let X be a weakly regular space and Y a base for the open sets of X satisfying conditions below: (i) X E Y, (ii) if G, H E Y, then G n H E 9, (iii) if x E G for G E Y, then there exist Gi E Y, i = I , . . . , k such that x 4 Gi for 1 < i < k and G u (U{GiI i = 1, . . . , k } ) = X . Let be the collection of all the finite open covers of Xconsisting of open sets from 9.Then Qois a generalized uniformity base on X compatible with the topology by virtue of Theorem 1.14. The collection @ of all the covers of X which are refined by members of Qois a generalized uniformity and O0 is a base for @. Let ( X * , @*) be the completion of (X, 0). Then by Theorem 2.36 the space X * is compact. By Theorem 2.35 we have [U{Gili = I , .
. . ,n}]*
= U{Gi*li = 1 , .
..,n}
f o r G i E Y , i = 1, . . . , n. 2.37. Theorem (Shanin [1943]). Let X be a weakly regular space and 9 a base for the open sets of X satisfying conditions (i), (ii) and (iii) above. Then there exists a compact space Y with the properties below: (a)
Y is a TI-extensionof X ,
(b)
{ E,(G) I G E Y} is a base for the open sets of Y,
(c)
&(GI u . . . u G,) = E Y ( G , )u . . . u E,(G,) holds for any finite number of elements G I ,. . . , G, E Y.
Moreover, such a space Y is essentially unique.
Proof. Let 9 = {X - GI G E Y}. Then 9 is a base for the closed sets of X. Shahin's theorem is stated in terms of 9.The existence of Y with the properties mentioned in the theorem is provided by X * constructed above. Conversely, suppose that Y has the properties mentioned in the theorem.
Extensions of Mappings I
25
Let y E Y - X and y E E , ( G ) for some G E S. Since C1,B is a base for the closed sets of Y and { y } is a closed set in Y , there exists a collection { F , I I E A } such that F i e B for all A E A and { y } = n{CI,F,IAEA}. Since n{ClyFi1AE A} c E,(G) and Y is compact, there is a finite set {A,, . . . , A,} such that n { C l , F , , J i= 1, . . . , n} c E,(G). Then by taking intersections with X we have n{F,,li = 1, . . . , n} c G. Let us put q 0 ={ X - F J i = 1, . . . , n } ~ { G } . T h e n ~ ~ i s a n o p e n c o v e r o f X a n d q0E (Do. Therefore we have St(y, E y ( q 0 ) )c E y ( G ) . On the other hand, if x E X and St(x, 42) c G for 4 E (Do and G E 9,then St(x, Ey(42)) c E,(G). Therefore, Yo = { E Y ( 4 2 ) ) qE (Do} is a generalized uniformity base on Y compatible with the topology by Theorem 1.14. Since { E , ( G ) I G E S}is a base for the open sets of Y , every open cover of Y is refined by some cover Ey(42)with 42 E (Do, Yois complete by Theorem 2.19. Thus Theorem 2.32 applies to the present case and we obtain Theorem 2.37 0 2.38. Remarks. (a) The completeness and completions for spaces with T-uniformities were discussed in Morita [1951J by means of star-filters, and it was proved there that completions of uniform spaces and Shanin’s compactification are obtained as such completions. (b) The concept of weak star-filters was introduced by Rinow [1967]; it was defined also by Harris [1971] as round Cauchy filters, and an equivalent concept of minimal Cauchy stacks was given by Herrlich [1974b]. Rinow [ 19671 announced that replacement of star-filters by weak star-filters yields important results such as Theorems 2.32 and 2.33 (which he stated for T, -spaces), although the completions of semi-uniform spaces are the same as those in Morita [1951]. Our description in this section follows the line of thoughts in Morita [1951] in the main. (c) For corresponding results on nearness spaces, see Herrlich [1974]. (d) Some of the results in this section hold for spaces with T-uniformities. In Morita [1951], Theorem 2.37 (Shanin’s Theorem) was obtained by means of completions for the case that X is a topological space which is not necessarily weakly regular.
3. Extensions of continuous maps from dense subspaces Throughout this section let X be a topological space and A a dense subspace of X unless otherwise specified. As in Section 2 for an open subset G of the subspace A we define an open subset E,(G) of X by E,(G)
=
X - Cl,(A - G).
(3.1)
26
K. Morita
Then the following hold for open subsets G, H of A:
E,(G) n A = G; E,(G) = U { M c XI M open in X and M n A
=
G},
if G c H, then E,(G) c E,(H); E,(G n H ) = E,(G) n E,(H).
(3.2) (3.3)
For a collection Y of open subsets of A, we define
The following is a fundamental theorem in this chapter, which enables us to deduce all the extension theorems for continuous maps as well as for uniformly continuous maps.
3.1. Theorem (Morita [1951]). Let f : A + Y be a continuous map where Y is a regular Hausdorf space. Let Y be a complete semi-uniformity on Y compatible with the topology and Yo = {Y,,la E R} a base for Y which consists of open covers of Y. Let us put H ( Y ~ )=
n { u w f - w )i a
(3.5)
E QI.
Then there exists uniquely a continuous map g :H ( Y o ) Y which is an extension o f f ; moreover, if Ypis a local star-refinement of Y, in Y o ,then Ex(f-'(V8)) A H ( Y o ) is a refinement ofg-I(Y,).
Proof. Let x E H ( Y o ) .Since x E UE,( f - ' ( f for aeach ) ) a E R, there exists V , E % such that x E Ex(f - I ( V,)). Then the collection { V,I a E Q} has the finite intersection property. To prove this, let { a I , . . . , a,} be any finite subset of R. Since x E (){Ex(f - I ( & , ) ) I i = 1 , . . . , n} and A is dense, we have n{E,(f-'(V,,))Ii = 1 , . . . , n} n A # 8 and by (3.2) n { f - ' ( V J l i = 1, . . . , n } # 8. Thus n{V,, 1 i = 1, . . . ,n } # 8. Therefore the collection d
=
{ V , , n . . - n V , n l a i ~ R , i =, .l . . , n ; n = 1,2 , . . .}
of subsets of Y constitutes a Cauchy filter base with respect to Y o .Since (Y, Y) is complete, d converges to a point of Y (see Theorem 2.20 and Lemma 2.7). This point will be denoted by g(x). Then we have g ( x ) = nC1 d If { W, I a
E
=
n{Cl (&, n . . . n V.,)lai E R, i = 1 , . . . , n ; n = 1,2 , . . .}.
Q} is another collection of subsets W, E YEwith a
E
R such that
Extensions of Mappings I
xE
E , ( f - ' ( K ) )then , the collection of subsets of Y W = { W . , n * * - nW a n I a i ~ R , i I=, . . . , n ; n =
27
1,2, . . . }
is also a Cauchy filter base with respect to Y o . For any ai E R, i = 1, . . . , n and E R,we have x
E
n{Ex(f-'(K,)li = 1, . . . , n} n E,(f-'(q))
and hence, by an argument which has been used above, one sees that n { K , l i = 1 , . . . , n } n W, # 8, that is, W, c St(K, n . . . n Kn,V,). This shows by Proposition 2.6 that d is equivalent to W. Hence by Lemma 2.7 we have nCl d = nCl W.Therefore the value g(x) does not depend on the choice of the collection { K} and is determined uniquely by x. If x E A, thenf(x) E V, for all a E R and since {St(y, Va) I a E R} is a base for nbds of y for each y E Y,we have g(x) = f(x) for x
E
A
.
Now, let Vpbe a local star-refinement of Vain Y o .Let W E V,.Then there exist 6 E R and V E Vasuch that St( W, V6)c V. Then we obtain from the definition of g g(E,(f-'(W))
n H(Yo))c CI,W c st(w, "y) c V .
This shows that the open cover E,(f-'(Vp)) A H ( Y o )refines g-'(V#). This implies the continuity of g. Because for x E H ( Y o ) we have
g-wm c St(g(x), m.
g(St(x9 E*(f-'(%)) A WYO)))= g(St(x9
Since {St(g(x), Va) I a E R} is a base for nbds of g(x), g is continuous at x. Since Y is Hausdorff, such an extension map g is unique. Here we make a supplement to the above theorem.
3.2. Lemma. Let X,A , Y, Y , Yoand f:A + Y be the same as in Theorem 3.1. Let Yl = {W,ly E r} be a subbase for Y and Y o = {W,, A . . * WJyi E r, i = 1, . . . , n; n = 1, 2, . . .>.Then
H(\Y,) =
nwwf-l(-Wj))I Y E ri.
Proof. By a repeated application of (3.3) we have
A
28
K. Morita
Therefore we have the desired equality.
0
The following is the first application of Theorem 3.1.
3.3. Theorem. Let ( X , #) be a generalized uniform space, ( A , # I A ) a dense subspace of ( X , #), and ( Y , @) a complete semi-uniform Hausdorffspace. Then every uniformly continuous map f from ( A , # I A ) to,( Y , Y) can be extended to a uniformly continuous map from ( X , #) to ( Y , Y).
Proof. Let Yo = { VaI a E R} be a base for Y which consists of open covers of Y. Since f is uniformly continuous, for each a E R there exists an open cover %a E # such that %a A A < f -'(Va). Let U E %fa. Then there is V E Va such that U n A c f - ' ( V ) . Hence by (3.2) and (3.3) we have U c E,(U n A) c E , ( f - ' ( V ) ) . Therefore %a < Ex(f -'(Va))and hence X = Uaac U E x ( f - ' ( V m ) )that , is, U E x ( f - ' ( V a ) )= X for each a E 0. With the notation in Theorem 3.1 we have H ( Y o ) = X and hence Theorem 3.1 shows that there exists a map g : X -, Y such that if Vbis a local < g-'(Va).Since%@< Ex(f -'(V@)), star-refinementof Va,then Ex(f -'(V@)) we have < g--'(Va).This shows that g : (X, (0) + (Y, Y) is a uniformly continuous map, completing the proof. 0
3.4. Remark. As was mentioned in Section 1, the category of nearness spaces and nearness-preserving maps is equivalent to that of generalized uniform spaces and uniformly continuous maps. In the former category Herrlich [ 1974al proved a theorem corresponding to Theorem 3.3. The following are concerned with semi-uniformities (see Section 2).
3.5. Corollary. Let ( X , #) be a semi-uniform Hausdorffspace. Then any two complete semi-uniform Hausdorff spaces each of which contains ( X , 9)as its dense subspace are uniformly isomorphic by a uniform isomorphism which leaves invariant each point of X .
Extensions of Mappings I
29
Proof. Let ( q ,'Pi), i = 1, 2, be complete semi-uniform Hausdorff spaces each of which contains (X, #) as a dense subspace. By Theorem 3.3 the inclusion map from ( X , #) into (Y,, Y , ) is extended to a uniformly continuous map f : ( Y ,, Y I ) --* ( K ,Y 2 ) ,and similarly we have a uniformly continuous map g : ( Y , , Y,) + ( Y , , Y l ) such that g ( x ) = x for X E X . Then g f : Yl + Y , is a continuous map which coincides with the identity map on the dense subspace X and hence g f = 1 y, since Y, is Hausdorff. Similarly f o g = 1 ,. This proves Corollary 3.5. 0 0
0
3.6. Corollary (=Theorem 2.34). Let X be a dense subspace of a regular Hausdorfl space Y. Then Y is obtained as the completion ( Y , Y ) of a semiuniform space ( X , #), where Y is a complete semi-uniformity on Y which is compatible with the topology of Y and # = Y I X . (See Theorems 1.16 and 2.19.)
Proof. Let ( X * , #*) be the completion of the semi-uniform space (X, #). Then X * is a regular TI-spaceby Theorem 2.32. Applying Corollary 3.5 to the semi-uniform space (Y, Y ) and ( X * , #*), we obtain Corollary 3.6 immediately. 0 Returning to the general case, the necessity condition for extendability of continuous maps will be discussed.
3.7. Lemma. Let f : A + Y be a continuous map. I f f is extendable over B with A c B c X , then the following hold: (a)
If #' is an open cover of
(b)
I f 9 is a family of closed subsets of Y with n c i , f - l ( s ) = 0.
Y , then UE,( f
-I(#'))
3
B.
09 = 8,
then
Proof. Let g : B -, Y be a continuous extension off. Let H c Y be open and F c Y closed. Then f - ' ( H ) = g - ' ( H ) n A , f - ' ( F ) = g - ' ( F ) n A . Since g - ' ( H ) is open in B and g - ' ( F ) is closed in B, we have, by applying (3.2) to EB, g - ' ( H ) c E B ( f - I ( H ) ) = B n E , ( f - ' ( H ) ) c E x ( f - ' ( H ) ) , and C1, f - ' ( F ) c g-'(F). Thus (a) and (b) follow from these relations. 0 Now Theorem 3.1, together with Lemmas 3.2 and 3.7, leads us to the following, which is the main theorem in this section. 3.8. Theorem. Let f : A + Y be a continuous map, where Y is a regular Hausdorfspace. Let Y be a complete semi-uniformity on Y compatible with
K. Morifa
30
the topology, and Y o= { Va1 a E R} a subbase for Y which consists of open covers of Y , and let us put H ( Y ~ ) = n { u E X ( f - m )I a E q. Then the following hold: (a)
f is extended to a continuous map g : H ( Y o ) + Y
(b)
H(Yo) is the largest subspace of X which contains A and over which f is extendable.
(c)
H ( Y o ) = { x E XI f ( @ ( x ) A A ) converges}, where @(x) is the nbdfilter of x. From now on, H ( Y o ) will be denoted by H( f ).
Proof. (a) follows from Theorem 3.1 and Lemma 3.2. (b) Suppose that g : B + Y is an extension off, where A c B c X. By Lemma 3.7, for each a E R we have B c U E x ( f -I(%)) and hence B c H(Yo). (c) Let x E X - A. Suppose that f ( @ ( x ) A A) converges to a pointy of Y. For each a E R, there exists V E Vawith y E V. Sincef ( @ ( x ) A A) converges to y, there is a nbd U of x such that f ( U n A) c V. Hence we have x
E
u = E A U n A ) = E x ( f - I ( V ) E E x ( f -I(%)),
that is, x E UEx(f -'(Va)). Therefore x E H ( Y o ) . Conversely, if x E H ( Y o ) ,then f ( @ ( x ) A A) converges to g ( x ) by (a).
0 As corollaries to Theorem 3.8 we obtain the following theorems.
3.9. Theorem. Let ( Y , Y ) be a complete semi-uniform Hausdorff space, f : A + Y a continuous map, and {VaI a E R} a subbase for Y which consists of open covers of Y. Then f is extendable over X iy Ex(f -'(Va)) is a cover of X for each a E R. 3.10. Theorem (see Dugundji [ 1966, p. 2 161). Let f : A + Y be a continuous map, where Y is a regular Hausdorff space. Then f is extendable over X iff the filter base f ( @ ( x ) A A ) converges for each x E X . The following is a dual form of Theorem 3.9. 3.11. Theorem. Let X , A , Y , Y and {Va1 a E R} be the same as in Theorem 3.9. Then a continuous map f : A + Y is extendable over X i y f o r each a E R we have n { c i , f - l ( Y - V ) IV E
va}= 0.
(3.6)
Extensions of Mappings I
31
If we require the validity of condition (3.6) for all covers Y E Y . Theorem 3.1 1 becomes a theorem proved by Wooten [1973], which he derived from the following theorem.
3.12. Theorem (Wooten [1973]). Let (Y, Y) be a complete semi-uniform Hausdorff space. Then a continuous map f : A + Y is extendable over X iff U{Int,Cl,f-'(V)l V E Y } = X f o r each Y EY. Proof. For any open set V of Y we have E,(f-'(V))
=
x - Cl,(A
- f-'(V))
c
cl,f-'(v),
that is, Ex(f - I ( V ) ) c Int,Cl,f-'(V). Hence the condition in Theorem 3.9 implies the condition in Theorem 3.12 for each open cover Y E Y, and hence for all Y E Y , since any cover in Y is refined by an open cover in Y. To prove the converse of this implication, let Wand V be open sets of Y such that C1, W c V , and put H = Int,Cl, f -'(Cl, W). Then we have A n H c A n Cl,f -'(Cl, W) = C1, f -'(Cl, W) =
f -I(Cl,w) c f -I(v),
and hence H c E,(H n A ) c Ex(f - ' ( V ) ) . This shows that if Ypis a local star-refinement of Yain Y then {Int,Cl, f - I ( W) 1 W E V p }< Ex(f -'(YE)). Thus, the condition in Theorem 3.12 implies the condition in Theorem 3.9.
0 As was proved by Theorems 1.16 and 2.19 the collection of all the open covers of a regular TI-space Y generates a complete semi-uniformity Yl on
Y compatible with the topology. If is a base for the open sets of Y,then the collection of all the open covers of Y consisting of members of 4?is clearly a base for Yl. Hence we have the following theorem which is free from the terminology concerning semi-uniform spaces. 3.13. Theorem. Let be a basefor the open sets of a regular Hausdorflspace Y. Then a continuous map f : A + Y is extendable over X i#lJE,( f -'(Y))= X for any 3 ' c 9d with UY = Y .
The dual form of Theorem 3.13 reads as follows.
3.14. Theorem (Herrlich [1967]). Let f :A + Y be a continuous map where Y is a regular Hausdorfspace. Let d be a base for the closed sets of Y . Then f is extendable over X z y n{Clxf I B E W } = 0 for any 9d c d with
na
=
0.
32
K. Morita
3.15. Remark. (a) The assumption of regularity for the image space Y in Theorem 3.13 cannot be weakened to Hausdorffness. To see this, let us consider the set Roof all non-negative real numbers and define two topologies t and t' : t is the usual Euclidean topology and t' = {G u (H - D) I G, H E t}, where D = { l / n l n = 1, 2, . . .}. Let us put X = (!&, t), A = &,- D , Y = (R,,t')anddefinef:A+ Y b y f ( x ) = x f o r x E A . T h e n A is dense in X , Y is a Hausdorff space which is not regular, andf is continuous. But f is not extendable over X . To prove this, suppose that there exists a continuous map g : X + Y with g I A = f. Then we have g ( x ) = x for x E D also and hence g is the identity map. Therefore, we have g([O, r)) [0, s) - D for any r, s > 0. This shows that g is not continuous at x = 0, contrary to the continuity of g . On the other hand, UE,( f -I(&')) = X for any open cover &' of Y. To see this, let us consider the base A9 = { [0, a ) - D I a > 0 } u {(a, b)10 < a < 6 ) for t'. Since E,(f-'([0, a ) - D)) = [0, a) and Ex(f - I @ , b)) = (a, b), we have UE,( f -I(&')) = X for any open cover 8 ' of Y consisting of elements in A9, and hence for any open cover 8 'of Y. Thus, the condition in Theorem 3.13 is satisfied. (b) Theorem 3.3 is not true if we assume (Y, Y ) to be a complete generalized uniform space. Indeed, let 0 and Y be the collections of all covers refined by open covers of X and Y in (a) respectively. Then f :(A, 0 I A) + (Y, 4')' is a uniformly continuous map which is not extendable to a uniformly continuous map on (X, 0). (c) It is to be noted that f is extendable over A u F for any finite subset F of D, in particular over A u {x} for each x E X - A, although f is not extendable over X . This fact is interesting in view of Proposition 3.16 below.
+
3.16. Proposition (Bourbaki and DieudonnC [1939]). Let Y be a regular T,-space. Then a continuous map f : A + Y is extendable over X i f f f is extendable over A u {x} for each x E X - A.
Proof. This proposition follows readily from Theorem 3.8 (c).
0
In case the range space Y is compact Hausdorff, the family of all the finite open covers of Y generates a complete semi-uniformity(in fact, a uniformity) on Y compatible with the topology of Y. Hence the following is an immediate consequence of Theorem 3.9. 3.17. Proposition (Eilenberg and Steenrod [1952]). Let Y be a compact Hausdorffspace. Then a continuous map f :A + Y is extendable over X ifffor anyfinite open cover Y of Y there exists afinite open cover Q of X such that 9 A A < f-'(Y).
33
Extensions of Mappings I
The following is more convenient in applications.
3.18. Theorem (Taimanov [ 19521). Let Y be a compact Hausdorf space. Then a continuous map f:A + Y is extendable over X i f CI, f -'(C)n CI, f -l(D) =
0
holds for any closed subsets C and D of Y with C n D =
0.
Proof. Let Y be the collection of all covers of Y which are refined by finite open covers. Then Y is a complete uniformity on Y compatible with the topology. Since every finite open cover {GI, . . . , G,,} of Y has an open star-refinement X , the cover A {&;I i = I , . . . , n } where Xi = {G;,St(X - G;,X ) } ,refines {G,, . . . , G,,}. Thus the collection of all the binary open covers of Y is a subbase for Y. Hence Theorem 3.18 follows 0 immediately from Theorem 3.9. 3.19. Remark. Any binary open cover {HI,H,} is refined by a binary cozero-set cover of Y. Hence in Theorem 3.18 it suffices to assume the condition merely for any two disjoint zero-sets C and D. 3.20. Corollary (Engelking [1964]). Let Y be a realcompact space. Then a continuous map f:A + Y is extendable over X if we have nCl f -'(9) = 0 for any countable collection 9 of zero-sets of Y with 09 = 0. Proof. For a completely regular Hausdorff space Y the collection of all countable normal covers of Y is a uniformity on Y compatible with the topology, and Y is called realcompact if this uniformity is complete. Since every countable normal cover is refined by a countable cozero-set cover (see Section 1 in Chapter 2), as an immediate consequence of Theorem 3.9 we have Corollary 3.20. 0 In concluding this section we shall prove a generalization of a theorem of Lavrentieff.
3.21. Theorem. Let ( X , 0 )and ( Y , Y ) be complete semi-uniform Hausdorf spaces, and let A and B be dense subspaces of X and Y respectively. Let f:A + B be a homeomorphism. Then there exist the largest subspaces A, of X and B, of Y such that A c Ao, B c Bo and f is extendable to a homeomorphism fo : A, 4 B,. In case each of 0 and Y has a subbase of cardinality < m, each of A. and Bo is the intersection of m open sets, where rn is an infinite cardinal number.
K. Morira
34
Proof. Let g :B + A be the inverse off, Let us considerf and g as continuous mapsf : A + Y and g: B + X, and apply Theorem 3.8 to these maps. Then, with the notation in Theorem 3.8 there exist continuous mapsf, :H( f ) + Y and g, : H(g) + X. Let us put A, = H ( f ) n f i - v w ) ,
Bo = H(g) n g ; W f ) ) .
Then g, ofi :A, + Xcoincides with the inclusion map iA:A + Xon A. To see this,letx E A.Thenfi(x) = f(x) E Bandg,(fi(x)) = g(f(x)) = x.SinceX is Hausdorff and A is dense. in A,, we have (8, ofi)(x) = x for x E A,. Similarly we have (fi og,)(y) = y for y E Bo. Since g,(fi(A,)) = A,, we have f i ( A , ) c g;'(A,) c g;l(H(f)). On the other hand, since A, c A-'(H(g)), it follows that f i ( A , ) c H(g). Therefore f i ( A , ) c H(g) n g;l(H(f)) = B,, that is,fi(A,) c B,. Similarly g,(B,) c A,. Now, let us Put f, = fiIA,:A, B,, go = g,IB,:B, + A,. +
As has been proved above g,(f,(x)) = x for x E A , and henceg,(f,(x)) = x for x E A,. Similarly we havefo(g,( y)) = y for y E B,. Thereforef, : A, + Bo is a homeomorphism onto. Suppose that A c C, B c D and f is extended to a homeomorphism cp : C + D . Let $ :D + C be the inverse of cp. Let us consider cp and $ as continuous maps cp : C + Y and $ : D -+ X respectively. Then cp : C + Y is an extension off: A + Y and hence by Theorem 3.8 we have C c H ( f ) and cp = fi I C. Similarly we have D c H(g) and $ = g, ID. Therefore, we have C = cp-'(D) c fi-'(H(g)), and consequently, C c H c f ) nf,-'(H(g)) = A,. Similarly D c B,. Thus, the first statement of the theorem is proved. Suppose that 0, = {el I a E R} (resp. Yo = {
H(f1 =
n{uJ%(f-l<<)) I Y E r>,
H(g) = n { u w - i w i a E and hence H( f ) (resp. H(g)) is the intersection of at most m open sets of X (resp. Y). This proves the second statement. 0
3.22. Corollary (Lavrentieff Theorem, Lavrentieff [1924]). Let X and Y be complete metric spaces, let A c X and B c Y, and let f :A + B be a homeomorphikm. Then f is extendable to a homeomorphismf,:A , + B, such that A c A, c CI,A, B c B, c CI,B and A, (resp. B,) is a G,-set o f X (resp.
Y).
Extensions of Mappings I
35
Proof. Since CI,A and C1,B are complete metric spaces, we can apply Theorem 3.21 to the case that A c CI,A and B c CI,B, and we see that there exist G,-sets A , and B, of C1,A and CI,B and that f is extendable to a homeomorphismf,: A , -,B,. Since C1,A and C1,B are G,-sets of X and Y respectively, A, and B, are also Gd-setsof X and Y respectively. 0
Appendix. A generalization of the Ascoli Theorem For topological spaces X and Y , let C ( X , Y ) be the space which consists of all continuous maps from X to Y and has the compact-open topology. In Kelley [I9551 the two types of generalizations of the Ascoli Theorem are given: Theorem A. Let X be a regular locally compact space, ( Y , Y) a Hausdorf ungorm space. Then a closed subset F of C(X, Y ) ' i scompact iy (a)
F is equicontinuous,
(b)
C1 F[x] is compact for each x
E
X , where F [ x ] = { f(x) If
E
F}.
Theorem B. Let X be a regular locally compact space and Y a regular Hausdorff space. Then a closed subset F of C ( X , Y ) is compact i f f (a)
F is evenly continuous,
(b)
CI F[x] is compact for each x
E
X.
Here a family F of maps from X to Y is called evenly continuous if for each X , each y E Y and each nbd G of y there exist a nbd U of x and a nbd H o f y s u c h t h a t f ~ F a n d f ( x ) ~ H i m p l y f ( U )c G. On the other hand, the definition of equicontinuity presupposes a compatible uniformity Y on Y :F c C(X, Y ) is called equicontinuous if for each X E XandeachV E Y thereexistsanbd Uofxsuchthatf(U) c St(f(x), V ) for all f E F. Since Y has such a uniformity iff Y is completely regular, there is a discrepancy between Theorems A and B. It was perceived first by Poppe [I9671 that a good remedy for this situation is provided by semi-uniformities. In this appendix we shall define the concept of equicontinuity for a semiuniforrh space ( Y , Y) in the same way as above and prove the following theorem, which was given by Poppe [I 9671 with a stronger definition of equicontinuity than ours.
x
E
36
K. Morita
A.l Theorem. Let X be a topological space and ( Y, Y) a semi-uniform space. Let F c C(X, Y ) .
If F is equicontinuous, then F is evenly continuous. If F is evenly continuous and CI F [ x ] is compact for each x E X ,
(a) (b)
then F is equicontinuous.
Proof. (a) Let x E X , y E Y and let G be any nbd of y. Then there exists Y E Y such that St*(y, Y ) c G (see Proposition 1.1 1). Since F is equicontinuous, there is a nbd U of x such thatf(U) c S t ( f ( x ) , Y ) .Let us put H = St(y, Y ) .Then, iff ( x ) E H, thenf ( U ) c StCf(x), Y ) c St2(y,Y ) c G. (b) Let x E X , and let Y be any open cover of Y. For each y E C1 F[x] there is G( y ) E Y with y E G( y). Since F is evenly continuous, there exist a nbd U, of x and a nbd H( y) ofy such that f (x) E H( y ) impliesf(U,) c G( y). Since C1 F [ x ]is compact, there exist y j E CI F[x],i = 1, . . . ,n ( n finite) such that CI F[x] c U { H ( y , ) l i= 1 , . . . , n } . Let US put U = n { U , l i = 1 , . . . , n } . Then U is a nbd of x . If f ( x ) E H ( y , ) , then f ( U ) c f(U,,) c G ( y J c
0
S t ( f ( x ) , Y).
A.2. Corollary. Theorem A holh if (Y, Y ) is a Hausdor- semi-uniform space. A direct proof of Corollary A.2 without appealing to Theorem B was given by Poppe [ 19671 for the special case of X being compact Hausdorff and by Rolicz [ 19751 generally. As for the assumption on X Bagley and Yang [I9661 proved that Theorems A and B hold if X is a Hausdorff k-space. Here a topological space Xis called a k-space if G c X is open whenever G n K is open in K for each compact subset K of X . Any locally compact space is a k-space, and for such a space Hausdorffness implies regularity but not conversely. Therefore, Theorems A and B are not contained in the above results of Bagley and Yang. Our next theorem extends these theorems and results simultaneously. A.3. Theorem. Let X be a k-space and ( Y , Y) a semi-uniform space. Let F c C ( X , Y ) . Then C1 F is compact iff
F is equicontinuous (resp. evenly continuous),
(a) (b)
’
CI F[x] is compact for each x
E
X.
Our proof rests upon the following lemma.
Extensions of Mappings I
31
A.4. Lemma. Let X be a k-space, Y a regular space and F a locally compact subset of C(X, Y ) . Then the evaluation map o :F x X + Y, which is dejined by o ( f ,x ) = f ( x ) , is continuous. Proof. (i) Let K be a compact subset of X . Then o I F x K is continuous. To see this, let x E K, f E F, y = f ( x ) . Let G be any nbd of y . Since Y is regular, there is a nbd H of y with C1 H c G. Let U be a nbd of x in K such that f ( U ) c H. Then f(C1,U) c Clf(U) c C1 H c G. Thus, if we put C = C1,U and M(C, G ) = { g E F l g ( C ) c G } , then C is compact, M(C, G ) is a nbd off in F and w(M(C, G ) x U ) c G . . (ii) Since Y is regular, so is C ( X , Y) (see Kelley [1955, Theorem 7.41) and hence Fis regular. The proof of Lemma 3 in Morita [1953], in which it suffices to assume local compactness and regularity for the space Y there, shows that G c F x Xis open whenever G n (F x K ) is open in F x K for each compact subset K of X (and hence F x X is a k-space). Hence it follows from (i) that o is continuous. 0
Proof of Theorem A.3. Let us first observe that if F in Lemma A.4 is compact, then F is equicontinuous. To see this, let x E X and let Y be any open cover of Y. Then for each f E F there are a nbd W ( f )off and a nbd 4)for some 5 E Y .Since F is compact, U’of x such that W ( f ) x U, c o-’( . . . , W ( f , ) } of F. Let us put there is a finite open cover {W(f,), U = n{U,li = 1 , . . . , n } . I f f € F, thenfe W ( f ; )for somef; and hence f ( U ) c f(v,) c v/, c S t ( f ( x ) , Y ) .Now, assume C1 F is compact. Then Cl F, and hence F, is equicontinuous and o(C1 F x { x } ) = (Cl F ) [ x ] is compact for each x E X by Lemma A.4. Since the closure of a compact subset of a regular space is compact, C1 F [ x ] is compact. This proves the necessity of (a) and (b). Conversely, assume (a) and (b). Let us consider the product space Y x = Il{Y, I x E X } where Y, = Y for each x E X; Y x consists of all maps f :X + Y with f ( x ) = p , ( f ) where p , : Y x + Y, is the projection. Let P be the closure of F in Y x . It suffices to prove: (1) E is compact, (2) P is evenly continuous and hence P c C(X, Y), (3) the subspace topology B of P c Y x coincides with the compact-open topology % on P, (4) P = C1 F as topological spaces. Assettion (1) follows from the relation F c n{Cl F [ x ]I x E X } by virtue of (b). By Theorem A.l, F is evenly continuous. Hence for x E X,y E Y and a closed nbd G of y there exist a nbd U of x and an open nbd H of y such
38
K. Morita
that if f~ F then f ( x ) E H implies f(U) c G, that is, F c Q where Q = p ; ' ( Y - H ) u n { p ; ' ( G ) l x ' E U}.SinceQisclosed,wehaveP c Q which implies the even continuity of P since Y is regular. Thus (2) is proved. If we define a map w : P x X + Y by w ( g , x ) = g ( x ) , then E c Q implies w(p;'(H) x V ) c G. Hence w is continuous. Therefore, for K c X compact and V c Y open, B(K, V) = { g E El { g } x K c w - ' ( V ) } is open in F. Since B(K, V) = { g E El g ( K ) c V } and {B(K, V) I K c X compact, V c Y open} is a subbase for V ,we have V c 8.Since 8 c V we have (3), and hence (4). 0
For various geiieralizations of the Ascoli theorem see Poppe [ 19741. References Bagley, R. and J. Yang [I9661 On k-spaces and function spaces, Proc. AMS 17, 703-705. Bourbaki, N. and J. Dieudonnt [I9391 Note de teratopologie 11, Revue Scientifque 77, 180-181. Dugundji, J. [ 19661 Topology, Boston. Eilenberg, S. and N. Steenrod [ 19521 Foundations of Algebraic Topology, Princeton.
Engelking, R. [I9641 Remarks on real-compact spaces, Fund. Math. 55, 303-308. [I9771 General Topology, Warsaw. Harris, D. [I9711 Structures in topology, Mem. AMS 115. Herrlich, H. [I9671 Fortsetzbarkeit statiger Abbildungen und Kompaktheitsgrad topologischer Raume, Math. Z . 96, 64-72. [I9741 A concept of nearness, General Topology Appl. 4, 191-212. [1974a] On the extendability of continuous functions, General Topology Appl. 4, 213-215. [1974b] Topological structures, Math. Centre Tracts 52, 59-122. Isbell, J. R. [I9641 CJni/brmSpaces, Providence. Kelley, J. L. [I9551 General Topology, New York. Morita, K. [I9511 On the simple extension of a space with respect to a uniformity, I-IV, Proc. Japan Acad. 27, 65-72, 130-137, 166-171, 632636. [I9531 On spaces having the weak topology with respect to closed coverings, Proc. Japan Acad. 29, 537-543. [I9811 Theory of Topological Spaces (Japanese), Tokyo.
Extensions of Mappings I
39
Nagata, J. [19851 Modern General Topology, Amsterdam. Poppe, H. [I9671 Ein Kompaktheitskriterium fur Abbildungsraume mit einer verallgemeinerten uniformen Struktur, Proceedings of the Second Prague Topology Symposium 1966. Praha 1967, pp. 284-289. [ 19741 Compactness in General Function Spaces, Berlin.
Know, W. [1967] uber die verallgemeinerten uniformen Strukturen von Morita und ihre Vervollstandigung, Proceedings of the Second Prague Topology Symposium 1966, Praha 1967, pp. 297-305. [1975] Lehrbuch der Topologie, Berlin. Rolicz, P. [1975] Generalizations of Ascoli’s theorems, Colloq. Math. 33, 213-218. Shanin, N. A. [I9431 On the theory of bicompact extensions of topological spaces, Doklady URSS 38, 154-156. Steiner, A. K. and E. F. Steiner [I9731 On semi-uniformities, Fund. Math. 83, 47-58. Taimanov, A. D. [I9521 On the extension of continuous mappings of topological spaces (in Russian), Mat. Sb. 31, 459-462. Tukey, J. W. [1940] Convergence and Uniformity in Topology, Princeton. Wooten, D. F. [I9731 On the extension of continuous functions, Fund. Math. 83, 59-65.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 2
EXTENSIONS OF MAPPINGS I1
Takao HOSHINA Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. C*-, C-, P”- and Pembeddings. . . . . . . . . . . . . . . . . . . . . . . 3. Unions of C*-embedded subsets . . . . . . . . . . . . . . . . . . . . . . 4. C*-embedding in product spaces . . . . . . . . . . . . . . . . . . . . . . 5. Homotopy extension property . . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 42 51 62 69 75 78
Introduction
In this chapter we shall consider cases, such as C*-, C- or P-embedding, of extending continuous maps defined on a (not necessarily dense) subspace to the whole space. For a space Xand its subspace A we shall say that A is C*-embedded (resp. C-embedded) in X if every bounded real-valued (resp. real-valued) continuous function on A can be extended to a continuous map on X. These notions, which apparently come from the well-known Tietze Extension Theorem for normal spaces, were investigated with nice expositions by Gillman and Jerison [1960] and are now familiar to us. Another useful notion we mention is that of the P-embedding. The P-embedding was introduced by Shapiro [ 19661in connection with the extension of continuous pseudometrics defined on a subspace to the whole space. Afterwards it has been investigated in various studies, in consideration of several equivalent but distinct forms, some of which will be our main concern in this chapter. One of such forms, which will be adopted here as our
T. Hoshina
42
definition of the P-embedding, is described in terms of extending normal covers of a subspace to normal ones on the whole space. However, the P-embedding will also answer the purpose of this chapter because it will be characterized in terms of extending continuous maps from a subspace into Cech-complete AR spaces to the whole space. On the other hand, C*- and C-embeddings will also be characterized by using finite and countable normal open covers respectively in a form analogous to our .definition of the P-embedding. Thus, C*-, C- and P-embeddings can be approached by each of two methods: extension of continuous maps or extension of normal covers. The latter will be a good tool to develop our study. Section 1 is devoted to some preliminary notions and results. In Section 2 we shall prove some basic properties of C *-, C-, P" - and P-embeddings and their characterizations as stated above, by introducing the notion of P" -embedding which is an intermediate one between C-embedding and P-embedding. In Section 3, C*-embeddability of finite, or certain infinite, unions of C*-embedded subsets will be discussed. It will be found that, among other results, C*-embeddability of unions of merely two C*embedded subsets is the most fundamental. In Section 4, C*-embedding in product spaces will be discussed. We shall characterize C-, P"- or P-embeddings in terms of C*-embedding on product spaces. Results obtained in these sections are interesting in themselves, yet the more interesting lie in applications to the homotopy extension theorems. In the final section we shall completely describe and generalize the homotopy extension theorems which were originally studied by K. Borsuk and afterward by C. H. Dowker. Throughout this chapter, by a space we shall mean a topological space without any separation axiom, unless otherwise specified.
1. Preliminaries
We use the same notation and terminology as in Section 1 .O of Chapter I . N and I denote the set of all positive integers and the closed unit interval respectively. For a cover 9 of a space X we put
9* = {St(U, %)I
u E 9}, a"
= {St(.,
a)lx
E
X}.
Hence, 9*c Y for another cover Y of Xmeans that 9is a star-refinement of Y . If 9"< Y , 91 is said to be a A-refinement of Y . By a simple calculation we always have 9 c 42" c a* c (a!")".
Extensions of Mappings II
43
First let us recall a normal cover of a space. A sequence {%,,In E N} of open covers of a space Xis said to be normal if < a,, for each n E N. An open cover 9 of X is said to be normaI if there exists a normal sequence {Q,,} such that Q1refines Q. Let Q be a normal cover of a space X , and @ = {a,,I n E N} be a normal For any subset A of X we define sequence of open covers of X with Q, < 9. Int(A; @) = {x E XI St(x, Q,,) c A for some n } . Then Int(A; @) is an open subset of X. For, let x E Int(A; 0).Then St2(x,a,,+') c A for some n since < Q,. Hence we have c Int(A; a), which shows Int(A; @) is open. St(x, For each cover Q, of 0 let us put V,,= {Int(U; @)I U E Q,,}. Since < Q, , one can prove that Y,,is an open cover of X and for each n E N (1) Vn+l is a star-refinement of Y,,, (2) % + I < "IT, < QnThat is, {V,,,(nE N} is a normal sequence, and it'is denoted by @*. Let us denote by (X,@) the topological space obtained from X by taking {St(x, Q,,) In E N} as a nbd base at each point x of X. Let X / @ denote the quotient space obtained from (X, 0)by defining those two points x and y equivalent for which y E St(x, 9,,) for each n E N. Let cp be the composite of the identity map from X onto (X, @) and the quotient map cjj from ( X , @) onto X / @ . Then we have @-'(cjj(Int(A;@))) = Int(A; @) for any A c X. Therefore, cp is continuous. Taking also views of (1) and (2), {cp(V,,) I n E N} is a normal sequence of open covers of X / @ , which generates a uniformity of X / @ compatible with the topology. Thus, X/@is metrizable. These arguments and results can be seen in Morita [1970]. Finally, we note that cp-'(cp(%)) = % < Ql. 1.1. Theorem (Morita [1970]). An open cover % ' o f a space X is normal if there exist a metric space T, a continuous map cp :X + T and an open cover Y of T such that cp-'(V)reJnes 9.
Proof. The "only if" part immediately follows from the arguments above. Conversely, suppose that T, cp : X + T and V of the theorem exist. Then V is a normal cover of T since every open cover of a metric space is normal. Consequently, cp-'(*Y) is also a normal cover of X and so is Q. 0 A subset Z of a space X is called a zero-set if 2 can be written as Z = {x E Xlf(x) = 0} with some continuous mapf:X + I. A cozero-set is the complement of a zero-set. A cozero-set (resp. a zero-set) cover is a cover consisting of cozero-sets (resp. zero-sets).
T. Hoshina
44
As another characterization of normal covers we have the following theorem. 1.2. Theorem (Morita [1962, 19641). For an open cover 4 of a space X the following are equivalent. (a) 9 is normal. (b) 42 is refined by a local&Jinite cozero-set cover. (c) 9 is refined by a o-locallyfinite cozero-set cover. (d) 4 is refined by a o-discrete cozero-set cover. (e) 9 has apartition of unity subordinated to it, that is, a family (cpl I A E A} of continuous maps cpl : X -+ Zsuch that Zcpl(x) = 1for eachpoint x of Xand {{x E XI cpl(x) > 0} 1 rl E A} refines %. Here, Lp1(x) means sup{X:,,, cpl(x) 1 y is afinite subset of A}.
Proof. (a)*(b). Suppose 9 is normal. By Theorem 1.1 there exist a metric space T, a continuous map cp :X -+ T and an open cover V of T such that cp-'(V)refines 4. Since T is paracompact, we can take a locally finite open refinement W of V ;each member of W is a cozero-set since T is metric. Hence, cp-'(W)is also a locally finite cozero-set cover of X,and refines 9. (a)*(d). By replacing "locally finite" by "a-discrete" the same argument as in (a)*(b) can be applied to this case. (b)*(c), (d)*(c). These are obvious. (c)*(e). Suppose that there exists a cozero-set cover Y = UncN Vnsuch that V refines 4 and each V, = { V,, Ia E Q,} is locally finite. Let cpna :X + I be a continuous map such that V,, = {x E XI rp,,(x) > O}. Since Vnis locally finite, cp,(x) = cpna(x),x E X defines a continuous function cp, over X.If we put
then rp is also continuous over X.Since V covers X,for every point x E X we have ~ ( x > ) 0. Let us put +na(x) = Pna(x)/2"
E
X-
Then co
1 JInU(x) =
1 for X E X and
n = l asn,
Ka
Hence
= {x~XlICIna(x)> 01-
I a E Q,, n E N} is a partition of unity subordinated to 9.
Extensions of Mappings 11
45
(e)*(a). Let {cpl I 1 E A} be a partition of unity subordinated to 9. Let us consider a metric space M which consists of { xl I 1E A} with xl E Z such that C xl = 1 and has a metric d(x, y ) = C IxI - y l l . Define cp:X -+ M by cp(x) = {cpl(x)11 E A} for x E X. We show cp is continuous. Let xo E X and E > 0. Since C cpl(xo) = 1, for a finite subset y of A and a nbd U of xowe have
Then we have for x E U,
Let 9 be a collection of cozero-sets of a space X. Then it is well known that U 9 = U{U 1 U E 9}is also a cozero-set in case either Card 9 ( = cardinality of 9)< Noor 9 is locally finite. Hence, further in case 9 is a-locally finite, U 9 is also a cozero-set; this can be assured by the similar construction of cp such as in the proof of (c)=.(e) of Theorem 1.2. For collections of zero-sets the corresponding results need not be true (see Example 3.14 below). The following lemma is useful in discussing unions of zero-sets. 1.3. Lemma (Morita and Hoshina [1976]). Let {FaI a E R} be a collection of zero-sets of X . Assume that there exists a locally Jinite collection { G, I a E n} of cozero-sets of X with Fa c G, for each a E R. Then the union U{F,I a E R} is a zero-set.
Proof. Since Fais a zero-set and G, a cozero-set of X with Fa c G,, there exists a continuous map f, : X -,Z such that I
Fa
=
{XE
X l f , ( x ) = l},
G, =
{ X E
Xlf,(x) > O}.
T. Hoshina
46
Define g : X
-, Z by
g ( x ) = s u p { ~ ( x ) I uE R}
for x
Then g is continuous, and we have U{F,I u is, F, I a E f2} is a zero-set of X.
u{
E
E
f2}
X. = { x E Xlg(x) =
l}, that
0
In calculating normal covers the following theorem will be useful. 1.4. Theorem. For any normal cover 42 = { U, I u E R} of a space X , there exist a locally finite cozero-set cover { V ,1 u E R} and a zero-set cover {F, I u E R} of X such that F, c V , c U, for each a E f2. Proof. By Theorem 1.2 there exists a locally finite cozero-set cover Y of X refining 42. For each Y E Y choose uv E R so that V c UUv, and define a map s: Y -, R by s ( V ) = u,,. Let us put for each u E f2 V, = U{ V E V Is(V) = u } . Then V, is a cozero-set with V, c U, and { V ,I u E R} is locally finite and covers X. Hence, { V ,I u E R} is constructed. Next, by Theorem 1.2 there exists a locally finite cozero-set cover W such that W* < { V , l u ~ R } . I f w e p u t F , = X - S t ( X - V , , W ) f o r e a c h u ~ R , then one can prove that F, is a desired zero-set. 1.5. Lemma. Let {G, I u E R} be a locallyfinite collection of cozero-sets and
{ F, I u E R} be a collection of zero-sets of a space X such that F, c G, for u E R, then the collection
where r is the set of all finite subsets of R, is a locally finite cozero-set cover of X such that St(F,, W ) c G, for each u E R.
nuEy
Proof. For y E r let W(y) = G, n (X - UBdY F8). Then W(y) is a F, is a zero-set of X by Lemma 1.3. Let x be any point cozero-set since UBCy of X. Let y I and y2 be finite subsets of R such that x E G, iff a E y I (yl may be empty) and for a suitable nbd V of x, B 4 y2 implies V n G, = 8. Then x E W(y,),and y Q y z implies V n W(y) = 0. Hence W is a locally finite cozero-set cover. Finally, if F, n W(y) # 8, then u E y, that is, W(y) c G,. Hence, St(F,, W ) c G,. 0 Remark. In the lemma above we note that Card W < KO or Card W = Card Y according as Card Y < No or KO < Card Y, and that in case 9 is further a cover of X then so is Y and we have W A< Y.
Extensions of Mappings 11
47
1.6. Theorem. Any normal cover Q of a space X admits a normal sequence {Q,,} such that Ql < Q and either Card Q, < KOfor each n E N or Card Qn = Card Q for each n E N according as Card Q < KOor KO < Card Q. Proof. By Lemma 1.5 and its remark we can inductively construct a sequence {W,,}of open covers such that W: < W n pwhere l , Wo= 42, and either Card Wn< KOfor each n E N or Card W,, = Card Q for each n E N according as Card 42 < KOor KO < Card 9. Let Qn = W2,,for each n E N. Then {Q,,} is the desired normal sequence since we have = W2fn+I) < 0 (Ktn+I)IA < %:+I < Kn = Qn (n > 1).
Theorem 1.6 strengthens Theorem 1.1 as follows. 1.7. Theorem. For any normal cover 42 of a space X there exist a metric space T which is either compact or tech-complete with weight < Card Q according as Card 42 < KOor KO < Card Q, a continuous map cp : X + T and an open cover Y of T such that q-'(V)< 42. Proof. Let @ = {Q,,} be a normal sequence obtained in Theorem 1.6. Applying the same arguments as above to @, the metric space X/@, a normal sequence {Y,,} and a continuous map cp : X + X/@ are constructed so that refines Ql, Card = Card a,,, cp-'cp(Vn) = V,,,and Y = {cp(Vfl)} is a normal sequence of open covers of X / @which generates a uniformity of X / @ compatible with the topology. Let T be the completion of X/@with respect to Y. Regard cp as a map from X into T , and finally put Y = { T - ClT(X/@ - V')I V' E rp(Y1)).
Then T , cp : X + T and V satisfy all the required properties.
0
By a linear topological space we mean a real vector space L with the topology such that a : L x L + L and m : R x L + L are both continuous, where a(x, y ) = x + y , m(a, x ) = ax. A linear topological space L is locally convex if each point x of L has a nbd base consisting of convex nbds of x. It is well known that every Banach space is a locally convex linear topological space.
1.8. Theorem (Dugundji [ 195 11). Let L be a locally convex linear topological space. Let X be a metric space and A its closed subspace. Then every continuous map f : A + L is extended to a continuous map g : X + L so that g ( X ) c the convex hull of f ( A ) in L.
T. Hoshina
48
Proof. Let d be a metric on X. Let B ( x ; E ) = { y E XI d ( x , y ) < E } . Let x E X - A and let E, = d ( x , A ) . Then B ( x ; 38,) c X - A and W = { B ( x ; +e,)lx E X - A} covers X - A. Since X - A is paracompact, W is refined by a locally finite open cover 4 of X - A. First we shall prove: if U E 4 and U n B(a; E ) # 8 for some a E A, then 6 ( U ) (=diameter of U ) < 2 ~ For, . choose z E U n B(a; E ) . Let B ( x ; + e x ) E W containing U . Then 6(U) <
Ex
=
d ( x , A ) 6 d ( x , 2)
+ d(z, a) < $ E x + E.
Hence, we have 6 ( U ) < 2.5. Let U be any member of 9 with U # 0. Take a point xu E U . Then d(x,,, A) > 0, and so a point a, E A can be chosen so that d(x,,, a,,) < 2d(x,,, a). Then the following assertion holds: (*)
For each a E A and a nbd W of a in X , there exists a nbd V of a with V c W such that whenever U n V # 0 and U E 9 then U c Wanda,,€ W.
Suppose that Indeed, take E > 0 so that B(a; E ) c W. Put V = B(a; h~). U n V # 8 and U E 4. Then by the fact proved above 6 ( U ) < + E . Hence, U c B(a; * E ) . Consequently, U c Wand we have
d(a,,
4 < d(a,,, xu)
< 3d(x,,
+ d(x,,, a) < 2d(x,,, A ) + d(x,, 4
a) <
ZE,
that is, a,, E B(a; E ) c W. Hence (*) is proved. Since X is a metric space, each U E 9 can be written as U = { x E XI g,,(x) > 0 } for some continuous map g , :X + Z. Define K,, : X + Z by K,,(x) = g,,(x)/C,,.,,g,.(x) for x E X - A . Now let us define h :X + L by
x
E
A,
Clearly, h I A = f. We prove the continuity of h. Since X - A is open in X and 4 is locally finite, h is continuous at each point of X - A. We show h is continuous at each a E A. Let 0 be an arbitrary nbd of h(a) = f ( a ) . Since f is continuous and L is locally convex, there exist a nbd W of a in X and a convex nbd C off@) in L such thatf( W n A) c C c 0. For W find a nbd V of a satisfying (*). Then we have h( V) c C. To see this, it suffices to prove h(x) E 'C for each x E X ,- A. Then for this point x we have x E Q for at most finitely many U , , . . . , U, E 4. Then n V # 0, which implies a,, E W by (*). Therefore,f(a,,) E C, i = 1, . . . , n. Hence, h ( x ) E C since
Extensions of Mappings II
49
C is convex. Thus, we have h ( V ) c C c 0, which shows that h is continuous at a E A. By construction h ( X ) c the convex hull of f ( A ) . Thus, h is the desired extension off. 0 1.9. Theorem (Kuratowski-Wojdyslowski, see Hu [1965, p. 851). Let Y be a metric space of weight < m (m 2 No).Then there exist a Banach space L and an isometrical embedding cp : Y + L such that cp( Y ) is a closed subset of its convex hull Z in L and weight of Z < m.
Proof. Let C * ( Y ) denote the set of all bounded real-valued continuous maps defined on Y. Then it is known that C*( Y) is a Banach space endowed with the usual linear structure and a norm )If 1) by taking )If )I = supy, If ( y) I. Let d be a metric on Y. Choose p E Y, and p will be fixed throughout the proof. For a point a E Y definef, by f , ( Y ) = d(Y, 4 - d(Y, P ) f o r y E y* Then f, is continuous, and If,( y) I < d(a, p). Hence, f, E C*( Y). Define cp:Y+ C * ( Y ) by c p ( 4 =A,, Y . Since If,(yj - h ( y ) I = I d ( y , a ) d( y, b) I < d(a, b ) for each y E Y and If,(b) - fb(b)1 = d(a, b), we have
II c p ( 4 -
cpw I1
= SUP If,(y) - h(Y ) I =
4 2 ,
b).
YEY
Hence, cp is an isometrical embedding. We shall show that p ( Y ) is closed in its convex hull Z in C * ( Y )and weight of Z < m. To see cp( Y) is closed in Z, let g be any point of Z - cp( Y). Since Z is the convex hull of cp( Y), there exist al, . . . , a, E Y such that g =
AIL,+
*
n
- + A,f,
where li > 0 with
1 li = 1.
i= I
Since g # q ( Y ) implies 11 g - f,,11 > 0 for i = 1, . . . , n, 6 > 0 can be chosen so that 6 < 11 g - f,,11 for every i = 1, . . . , n. Let us put V = { f E 21llg - f I1 < a}. Then Vis an open nbd ofgin Z. We will show V n cp(Y) = 8. Suppose not and choose a point a E Y with cp(a) = f, E V. Then for every i = 1, . . . , n d(a, ail =
IILi - f a l l 2 llg
- L,ll - Ilg
- LII > 26
-
Consequently, we must have
Aid(a, ai) >
= i= I
(il i )6
a contradiction. Hence, V n cp(Y) =
= 6,
i= I
8,and cp(Y) is closed in Z.
6 = 6.
T. Hoshina
50
Next we show weight of Z < m. Since Z is metrizable, it suffices to prove that Z has a dense subset of cardinality < m. Since q( Y) is homeomorphic to Y, q( Y) has also weight < m, and has a dense subset D of Card D < m. Let H ( D ) be the convex hull of D in C*(Y). Then H ( D ) c Z . Let r be the set of all finite subsets of D.Denote by H(y) the convex hull of y in C*(Y) for a y = { g l , . . . ,g,} in r. Then H(y) is a finite union of closed simplexes with vertices g , , . . . , g, and hence it is separable. Therefore, H ( D ) has a dense subset of cardinality < m since Card < m and H ( D ) = U { H ( y )I y E r}.Hence, it remains to show that H ( D ) is dense in Z . To see this, let f E Z and E > 0. Then there exist fi, . . . ,fn E q( Y) such that
f
=
A,fi + . . . + A,f, where A, > 0 with
n
Izi = 1. i= I
Since D is dense in q(Y), there exist g , , . . . , g, in D such that llx - g,II < $ E for i = 1, . . . , n. Let g = Algl + * * * + Angn. Then g E H(y), where y = { g , , . . . , g , } , hence g E H ( D ) . Moreover, we have
Thus, H ( D ) is dense in Z . The theorem is proved.
0
Let X be a space, which is a subspace of another space Y. Then X is a retract (resp. neighborhood retract) of Y if there is a retraction from Y (resp. a neighborhood of X in Y = an open subset of Y containing X ) onto X . Let V be a class of spaces. Then a space Xis said to be an absolute retract or shortly AR (resp. absolute neighborhood retract or shortly ANR) for V if X belongs to V and whenever X is a closed subspace of a space Y in V X is a retract (resp. neighborhood retract) of Y. A space Yis said to be an absolute extensor or shortly AE (resp. absolute neighborhood extensor or shortly ANE) for V if Y belongs to 9 and for every continuous mapf: A --* Y from every closed subspace A of any space X in V into Y there exists a continuous extension off from X (resp. a neighborhood of A in X ) into Y. It is obvious that for any class V every AE (resp. ANE) for %? is an AR (resp. ANR) for V , and it is known that the converse also holds for many classes %? (see Hu [1965]); in particular the class A of all metric spaces is such one asis shown by the following. 1.10. Theorem. A space X i s an A R (resp. A N R ) f o r A iflit is an A E (resp. A N E ) f o r A.
Extensions of Mappings II
51
Proof. We shall prove an ANR for I is an ANE for A. The proof for the case of an AE space for 4 is similar and simpler. Let Y be a space which is an ANR for A. Let X be a metric space and A its closed subspace. Let f:A + Y be a continuous map. Then by Theorem 1.9 there exists a Banach space L such that Y is closed in its convex hull Z in L. Since Y is an ANR for .M and 2 metrizable, there exists a nbd V of Y in Z together with a retraction r : V + Y. On the other hand, by Theorem 1.8f: A + Y is extended to a continuous map g : X -+ L such that g ( X ) t the convex hull off(A). Hence g ( X ) c Z. Let U = g - ' ( V ) . Then U is a nbd of A in X , and we see the composite r o ( g 1 U ) :U -+ Y is clearly an extension off. Thus, Y is an ANE for A. 0 In the sequel of this chapter an AR (resp. ANR) for A will be called simply an AR (resp. ANR). By Theorems 1.8 and 1.10 every locally convex linear topological space is an AR if it is metrizable. 2. C*-, C-, P"- and P-embeddings
Let us begin with notions of C*- and C-embeddings and their basic properties. 2.1. Definition. Let X be a space and A its subspace. Then A is said to be C*-embedded (resp. C-embedded) in X if every bounded real-valued (resp. real-valued) continuous mapfon A is extended to a continuous map g over X. We note in the definition above that iffis bounded, that is, there exist a, b E R such that a < f(x) d b for any x E A , then the extension g can be taken so that a < g(x) 6 b for any x E X. For, let g' be a continuous extension off. Then define g by g(x) = min{b, max{g'(x), a } } for x E X . Let E and F be a pair of disjoint subsets of a space X . Then E and F a r e said to be completely separated in X (or E is said to be completely separated from F ) if there exists a continuous mapf: X -+ I that takes values 0 on E and 1 on F, that is,f(x) = 0 for x E E andf(x) = 1 for x E F. 2.2. Lemma. E and F are completely separated in X i f l there exist disjoint zero-sets Z , and Z , such that E c Z , and F c Z , .
Proof. ' If there exists a continuous mapf: X
+
I such thatf
=
0 on E and = l } are
f = 1 on F, then Z, = {x E Xlf(x) = 0} and Z , = {x E Xlf(x)
disjoint zero-sets and we have E c Z , and F c Z 2 .
52
T. Hoshina
Suppose conversely, then there exist continuous mapsf, g : X + I such that E c Z, = { X E X l f ( x ) = 0},F c Z, = { X E X l g ( x ) = 0 } andZ, n 2, = 8. Define h : X + I by h ( x ) = f ( x ) / (f ( x ) g ( x ) ) ,x E X . Then h is continuous and is equal to 0 on E and to 1 on F. 0
+
The following is fundamental (cf. Gillman and Jerison [1960]).
2.3. Lemma. Let A be a subspace of a space X . Then we have the following: (a) A is C*-embedded in X iffevery pair of completely separate subsets of A are completely separated also in X . (b) A is C-embedded in X i f A is C*-embedded in X and is completely separated from any zero-set of X disjoint from A . Proof. (a) Assume A is C*-embedded in X. Let E and F be a pair of subsets of A and f : A + I be continuous with f = 0 on E and f = 1 on F. If g : X -, I is an extension off, then clearly g equals 0 on E and 1 on F. Hence E and F are completely separated in X , which proves the “only if” part. To prove the “if” part, assume that the condition described in (a) is satisfied. Let f be a bounded real-valued continuous map on A . Let If I < c for some c > 0, where If I = sup{/f ( x ) 1 1 x E A } . Define r,, = $c($)” for each n E N. We shall define inductively for each n E N a real-valued continuous map f,, on A so that If,[ < 3rn. Define fi = f. Then clearly Ifi I = If I < c = 3r,. Assumef, is given. Let us put En
=
{ X E
Alf,(x)
<
-r,,},
Fn = { x E A I f , ( x )
> r,,}.
Then it is easy to see that Enand F,, are completely separated in A and so by assumption they are completely separated in X . Hence, there exists a continuous real-valued map gn on X such that g, = - r,, on E,, and r, on F,, and with I g,, I < r,, . Then we define
L+.l(X)
= f,(4
- gnw, x
E
A.
Thenf,,, is continuous on A and satisfies If,+,I < 2rn = 3r,,+,.Thereforef, can be defined inductively. Let us now put g ( x ) = X.“=,gn(x)for x E X . Since I gnI < r,, X,”=,g,,(x)converges uniformly on X . Hence g defines a continuous map on X . Finally, observe that X b , g i I A = (fi - fi) + (fi - h ) + . . . + (f,- A+,) = fi f,,,. Hence we have gl A = f, because for x E A ,-
(b) Assume A is C-embedded in X . Clearly A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Let g : X + I be continuous with
Extensions of Mappings I1
53
Z = { x E Xlg(x) = O}. Then f = l/g is defined on A and continuous. By assumption take h which is a continuous extension off over X . Then clearly gh = 0 on 2 and 1 on A. Hence, A and Z are completely separated in X. This proves the “only if” part. Suppose the converse, and let f be any real-valued continuous map on A. Define g : A + (-in, in) by g(x) = tan-’(f(x)), x E A. Regard g as g : A + [-in, i n ] and by assumption take a continuous extension h : X + [-+n,+n]ofg.SinceZ = { x ~ X I l h ( x ) = I +n}isazero-setofX and 2 n A = 8, by assumption there exists a continuous map cp : X + I such that cp = 1 on A and 0 on Z. Let us now define k on X by k = tan (cph). Then clearly k is a continuous extension off over X . 0 2.4. Corollary. Every C*-embedded zero-set is C-embedded.
Here we shall give some comments about when a subspace A can be C*-embedded in a space X. Familiar and easy facts are that every compact subspace in a Tychonoff (=completely regular and T,) space is C-embedded, and that every C*-embedded pseudo-compact subspace of a space is C-embedded. By Tieze’s Extension Theorem we have the following.
2.5. Theorem. For a space X the following are equivalent. (a) X is normal. (b) Every closed subspace is C*-embedded in X . (c) Every closed subspace is C-embedded in X . Let us denote flX the Stone-Cech compactification of a Tychonoff space X . flX is shown as the compactification in which Xis C*-embedded. A fact related to Chapter 1 is that flX is the completion of a Tychonoff space X with respect to the uniformity generated by all the finite normal covers of X . Also, many basic and important facts on flX have been studied until now (see for details Gillman and Jerison [1960] and Walker [1974]). In this text we shall use, without proof, the following fact: a subspace A is C*-embedded in a Tychonoff space X iff flA = CI,,A. Lemma 2.3 can be applied to another description of C*- and C-embeddings as follows: (a) was proved in Morita and Hoshina [I9751 and (b) was proved by Gantner [ 19681.
2.6. Theorem. For a subspace A of a space X the following hold.
T.Hoshina
54
(a) A is C*-embedded in X iyfor everyfinite normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 . (b) A is C-embedded in X i f f o r every countable normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 .
Proof. (a) To prove the “if” part, assume that the condition in (a) is satisfied. Let E and F be completely separated subsets of A. By Lemma 2.2 there exist disjoint zero-sets Z , and Z , of A such that E c Z, and F c Z,. Let 4 = {X - Z,, X - Z,}. Then 4 is a binary cozero-set cover, and hence a normal cover of A by Theorem 1.2. By assumption there exists a normal cover Y of X such that Y A A < 4. Since Y is normal, there exists a locally finite cozero-set cover W of X such that W * < Y . Let us now put Z;= X - U { W E W I W ~ Z ~ = ~ }1 , 2~. T=h e n e a c h o f Z ; , i = 1,2, is a zero-set of X , and we have Z, c Z,‘ and Z , c Z;. Since W * < Y and Y A A < 4, we have Z,’ n Z; = 0. Hence by Lemma 2.2 E and F are completely separated in X . Thus, A is C*-embedded in X by Lemma 2.3 (a). Conversely, assuming that A is C*-embedded in X , let 4 = { U,,. . . , U,,} be any finite normal cover of A. Then by Theorem 1.4 there exist a cozero-set K and a zero-set 4 of A for i = 1, . . . , n such that F, c K c 17and , A = Ui&. Since A - r/; is a zero-set of A and t;; n (A - K) = 0, by assumption and Lemma 2.3 (a), there exist disjoint zero-sets Zi, and Zi, of X for i = 1, . . . , n such that t;; c Zi, and A - I( c Zi,. Let us put
w,
=
x - z,,v z,, v
. . . v Znl,
K = X - Z i 2 , i = l , . . . , n. y ,j = 0, 1, . . . , n is a cozero-set of X
Then each and W = { ylj = 0, 1, . . . , n } covers X . Hence W is a normal cover of X . Since W, n A = 0 and K n A c K c U, for i = 1, . . . ,n, we have W A A < 4, which proves the “only if” part. (b) To prove the “if” part, assume that the condition in (b) is satisfied. Then by (a) A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Letf: X + I b e a continuous map such that Z = { x E XI f ( x ) = O } . For each n E N let us put U,= {x E XI f (x) > I / n } . Then U,, is a cozero-zet of X , and since Z n A = 0, 4 = {V,, n A In E N} covers A. Hence, by assumption, there exists a normal cover -‘9 of X such that Y A A < 42. By Theorem 1.4 there exist a locally finite cozero-set cover W = { W,I a E R} and a zero-set cover 9 = {F,I a E fZ} of X such that W refines Y and Fa c Wafor a E R. Let a E R. Since 9 A A < 4,there exists an n(a) E N such that F, n A c U,(,)n A. Let us put
E, = F, n { x E XI f ( x ) 2 l/n(a)}.
Extensions of Mappings 11
55
Since { x E X l f ( x ) 2 l/n(a)} is a zero-set of X , E, is also a zero-set of X. Moreover we have
E, c W, for each a E R, E, n 2 = 8 for each c1 E R and u{E,la E R}
3
(2.1) A. (2.2)
Since W is locally finite, by (2.1) and Lemma 1.3 Z’ = U { E , I a E a} is a zero-set of A’, and by (2.2) we have A c Z’ and Z’ n Z = 0.Hence, A and 2 are completely separated in X.Thus, A is C-embedded in X by Lemma 2.3 (b). This proves the “if” part. To prove the “only if” part, suppose that A is C-embedded in X.Let 4 be a countable normal cover of A. Let *Y = { V ,I n E N} be a countable cozeroset cover of A that refines 9. Let V , = { x E A Ifn(x) > 0} with some continuous mapf, : A + I. Since A is C*-embedded in X , there is a continuous extension g, off, over X . Let us put W, = {x E X ( g , ( x ) > O}. Then W, is a cozero-set of X with W, n A = V,. Note that A = U V , c W,. Therefore, if we put 2 = X W, I n E N}, then Z is a zero-set of X and we have 2 n A = 0. By assumption and Lemma 2.3 (b) we can take a zero-set Z’ of X such that A c Z’ and Z’ n Z = 8. Let us put W = {X - Z ’ , W,ln E N}. Then W is a countable cozero-set cover and, therefore, a normal cover of X,and we have W A A c 9,which proves the “only if” part. 0
u
u{
Theorem 2.6 leads us naturally to the following definitions of P“- and P-embeddings. Let m be an infinite cardinal number. 2.7. Definition. A subspace A of a space Xis said to be P”-embedded in X if for every normal cover 9 of A with Card 9 < m there exists a normal cover Y of X such that Y A A c 4. A is said to be P-embedded in X if A is PI”-embedded in X for any m.
In view of Theorem 2.6 Puo-embedding coincides with C-embedding. Therefore we have the following implications. P-embedding P”-embedding Puo-embeddingo C-embedding
C*-embedding
For the converses it will be shown later that there exist spaces, one of which contains a closed C*-embedded but not C-embedded subspace and the other one contains a closed C-embedded but not P-embedded subspace. Moreover, concerning extensions of mappings, the following theorem makes it clear how these embeddings relate to each other; (c) is due to Morita [1975a].
T. Hoshina
56
2.8. Theorem. Let A be a subspace of a space X . Then each of the following holds. (a) A is C*-embedded in X i f f any continuous map from A into any compact AR is continuously extended over X . (b) A is C-embedded in X iff any continuous map from A into any techcomplete separable AR is continuously extended over X . (c) A is P“-embedded in X for any continuous map from A into any techcomplete AR with weight < m is continuously extended over X . (d) A is P-embedded in X @any continuous map from A into any techcomplete AR is continuously extended over X .
Proof. (c) To prove the “if” part, assume that the condition in (c) is satisfied. Let 9 be any normal cover of A of Card 9 < m. By Theorem 1.7 there exist a complete metric space T of weight < m , a continuous map f : A + T and an open cover W of T such that f - ’ ( W ) refines 9. By Theorem 1.9 there exists a Banach space L such that T is isometrically embedded in L and closed in its convex hull Z, in L and the weight of Z, < m. Let Z = C1 Z,. Then Z is also convex with weight < m and it is complete as a closed subset of L. Therefore if we consider f as the map f:A + Z, by assumption and Theorem 1.9 there exists a continuous map g : X -P 2 such that g I A = f.Since T is a complete metric subspace of Z, T is closed in Z. Hence, there exists an open cover Y of Z such that Y A T = W . Since 2 is metrizable, Y is a normal cover of Z. Hence g - ’ ( Y ) is a normal cover of X , and Y A T = W and f - ‘ ( W ) < 9 imply that g - ’ ( Y ) A A refines 9. Hence, A is P”-embedded in X. This proves the “if” part. Conversely, suppose that A is P”-embedded in X . Let Y be a cechcomplete AR of weight < m. Then there exists a normal sequence { *w; I i E N} of open covers of Y such that {St( y , K )I i E N} is an nbd base at each point y of Y , the cardinality of K < m for each i E N, and such that Y is complete with respect to { K } . Let f : A + Y be a continuous map. T h e n f - ’ ( K ) is a normal cover of A with cardinality <m. Since A is P”-embedded in X , we can inductively construct a sequence {qI}of open covers of X such that for each i E N,
9, I is a star-refinement of 9, ,
(2.3)
911 A A refines f
(2.4)
-I(%).
Therefore 0 = {qII i E N} is a normal sequence of open covers of X . Let us now apply the arguments given in Section 1 to 0. Then we can construct a
Extensions of Mappings II
57
metric space S, a continuous map cp:X + S and a normal sequence {.Y; I i E N} of open covers of S such that
cp-’(c)refines for each i E N, n St(x, cp-l(c))= St(x, a,) for each x E X , m
I=
(2.5)
XI
I
{St(t,
I=
c)I i
I
(2.6)
N} is a nbd base at each point t of S.
(2.7) Let us define a map g : cp(A) + Y by g( y ) = f ( x ) , where y = cp(x), x E A. Then, by (2.4), (2.5) and (2.6) g is single-valued, and, since for each i E N we have that A cp(A) refines g-’(“w;), g is uniformly continuous when we regard cp(A) as a uniform subspace of S with the uniformity {.Y; 1 i E N} and Y as a uniform space with the uniformity {“w; I i E N). Since Y is complete with respect to {W,},g is extended to a continuous map g : Cl,cp(A) + Y. Since Y is an AR, there exists a continuous map h : S + Y such that h (Cl,cp(A) = g. Now consider the composite h0cp:X + Y. Then we see that h cp is an extension off. This proves the “only if” part of (c). (a) Since any closed interval [a, b] in (w is compact AR, the “if” part is obvious. Using Theorem 2.6 (a), the “only if” part can be proved similarly as (c); in the present case we further take “w; so that Card “w; < KO. (b) By putting m = KO,(b) follows from (c) and Theorem 2.6 (b). (d) This is a restatement of (c) for arbitrary cardinal m. 0 E
0
As was mentioned in the introduction, Shapiro [I9661 defined the notion of P-embedding, where P”-embedding was also defined, using terms of continuous pseudo-metrics. A pseudo-metric d on a space X is said to be continuous if as a function d : X x X + R, where X x X is a product space, it is continuous. d is said to be m-separable if the pseudo-metric space ( X , d ) has a dense set with cardinality < m. The following is easy to prove. 2.9. Lemma. If d is a (resp. an m-separable) continuous pseudo-metric on a space X , then there exist a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T such that d ( x , y ) = e(cp(x),cp(x)).Conversely, f there exists a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T, then the pseudo-metric d on Xdefined by d ( x , y ) = e(cp(x), cp( y)), x, y E X is continuous (resp. and m-separable).
Now Shapiro’s definition of P”- and P-embeddings is as follows: a subspace A of a space Xis P”-embedded in Xiff for every rn-separable continuous pseudo-metric d on A there exists a continuous pseudo-metric d on X
T. Hoshino
58
such that dl ( A x A ) = d. P-embedding is defined to be P“-embedding for every m. We shall show that the definitions of Shapiro’s P”-embedding and ours are identical. 2.10. Theorem. Let A be a subspace of a space X . Then A is P“-embedded in X in the sense of Definition 2.7 i f l A is P”-embedded in X in the sense of Shapiro.
Proof. To prove the “only if” part, assume that A is P”-embedded in X in the sense of Defnition 2.7. Let d be an m-separable continuous pseudo-metric on A . Then by Lemma 2.9 there exist a metric space (T, e) with weight < m and a continuous map cp: A + T such that d ( x , y) = e(cp(x), cp( y)). By Theorem 1.9 ( T , e) is isometrically embedded in a Banach space B and the convex hull Z of ( T , e) in B has weight < m. Let Z’ be the closure of Z. By assumption and Theorem 2.8, as is shown in the proof of Theorem 2.8, cp is extended to a continuous map $: X -+ Z’, where we regard cp as the map cp : A + 2’. Now define a pseudo-metric d on X by 4 x 9 Y ) = G($(x), $ ( Y h
x9 Y
E
x,
where 6 is a metric on B. Then clearly d is continuous. Moreover, since + I A = cp, we have for x , y E A &x,
r)
$(A) = G(cp(4, d Y ) ) e(cp(4,cp!~)) = d(x, v),
= G(+(x), =
that is, dl ( A x A ) = d as desired. This proves the “only if” part. Conversely, suppose that A is P”-embedded in X in the sense of Shapiro. Let 42 be a normal cover of A with Card 9 < m. By Theorem 1.6 there exists a locally finite cozero-set cover Y of A such that Y A= {St(x, Y ) I x E A } refines 9. Let Y = { & 1 A E A}. For each A E A take a continuous map q i : A + Z such that 6 = { x ~ A I c p ~ (>x )0). Define + A : A+ I by + l ( x ) = cp,(x)/C,,,cp,(x). Then $i is continuous, X )(IA(x)= 1 for x E A, and 5 = { x E A I t,hi(x) > O } . Let M be the set which consists of { x AI A E A} with xi E Zsuch that Z xi = 1 and x , = 0 for all but a finite number of A’s, and define a metric e ( x , y) = C I x , - y , 1 on M. Define a map : A + M by $ ( x ) = {+,(x)l A E A} for x E A . Then, similarly as in the proof of (d)*(a) of Theorem 1.2, the map $ is continuous. Moreover, it should be noted that the metric space (M, e) has weight < m. Therefore, if we put for x , y E A , d(x, y ) = e ( $ ( x ) , +( y)), then by Lemma 2.9 d is an m-separable
+
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continuous pseudo-metric on A. By assumption there exists a continuous pseudo-metric d o n X such that d ( ( A x A) = d. Let us put 9Y = { B ( x ;4)lx E X}, where B(x; 4) = { y E X l d ( x , y ) < 3). In view of Lemma 2.9 and Theorem 1.1 it is easy to see that B is a normal cover of X. Now suppose that B ( x ; f) n A # 8. Choose a point xoE B ( x ; 4) n A . Let y be any point of B ( x ; f) n A. Then we have d(xo, y ) =
d(x0,
y)
< d(x0, X)
+ d(x, JJ) < 4 + +
= 1.
Hence e($(xo), $ ( Y ) ) = 4 x 0 , Y ) < 1.
(2.8)
Let yo = {A E Alx,, E q} and y = {A E h l y E K}. Then by (2.8) we must have yo n y # 8. Therefore, there is a 1 E A such that xo,y E K . That is, y E St(xo, V ) .Thus, we have
B(x; 4) n A c St(xo, Y ) . Since Y * < 4, we have B A A < 4. Hence A is P”-embedded in X in the sense of Definition 2.7, which proves the “if” part. 0 As for when a subspace of a space is P”-or P-embedded, one sees that every compact subspace in a Tychonoff space is P-embedded and that every C*-embedded pseudo-compact subspace of a space is P-embedded, since every locally finite cozero-set cover of a pseudo-compact subspace is nothing else a finite cozero-set cover. Analogous to Theorem 2.5, we shall obtain spaces in which every closed subspace is P”-embedded. Recall that a space X is said to be m-collectionwise normal if for every discrete collection {FaI a E R} of closed sets of X with Card R < m there exists a discrete collection (G, I a E R} of open sets of X such that Fa c G, for each a E R. Clearly, X is collectionwise normal iff X is m-collectionwise normal for every m.
2.11. Theorem (Dowker [1952]). A space X is m-collectionwise normal z f f every closed subspace is P”-embedded in X . Proof. The “if” part. Let 9 = {FaI a E R} be a discrete collection of closed subsets of X with Card R < m. Since each Fa is closed and open in the closed subspace A = U{FaI a E R}, 9can be regarded as a locally finite cozero-set cover and so a normal cover of A. By assumption there exists a normal cover V of X such that Y A A < 9. Yet take a locally finite open cover W such
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60
that W * < -Y-. Let G, = St(F,, W ) for a E R. Then we have F, c G,, and easily see that {G, I a E R} is discrete. Hence X is m-collectionwise normal. This proves the “if” part. For the proof of the “only if” part, we need a lemma; this lemma will be used also in Section 4. 2.12. Lemma (Morita and Hoshina [1976]). A subspace A of a space X is P” -embedded in X if A is C-embedded in X and for every discrete collection {G,l a E R} of cozero-sets of A with Card R Q m and every collection {F, I a E R} of zero-sets of A with F, c G, for each a E R, there exists a locally Jinite collection {HaI a E R} of cozero-sets of X such that Fa c Ha n A t G,
for a
E
R.
Proof. Suppose A is P”-embedded in X . Then A is C-embedded in X . Let {G, 1 a E R} and {F,l a E R} be collections described in the lemma. Let us put
Since UaGnF, is a zero-set of A by Lemma 1.3, 9 is a locally finite cozero-set cover of A with Card 9 Q m. Moreover, note that St(F,, 9)c G , for a E R. By assumption there exists a locally finite cozero-set cover -Y- of X such that -Y- A A refines 42. We then have for V E -Ya #
p, a,
E
R
* either
Y n Fa =
8 or
Y n FB =
8.
Therefore, if we put Ha = St(F,, “f) then H, is a cozero-set of X , and the above and the locally finiteness of -Y- imply that { H ,I a E R} is locally finite. Since F, c H, n A c St(F,, 9)c G, for a E R, the “only if” part is proved. Conversely, suppose that the condition in the lemma is satisfied. Let 9 be a normal cover of A with Card 9 Q m. Then by Theorem 1.2, 9 is refined by a a-discrete cozero-set cover -Y- = UlsN of A , where = { I a E a,} is discrete with Card R, Q m. By Theorem 1.2, Y is normal. Hence by and Theorem 1.4 there exists a zero-set F,, of A such that F,, c {F,, I a E Q,, i E N} covers A . By assumption there exists a locally finite collection { H I ,I a E R,} of cozero-sets of X such that F,, c HI, n A c for each a E R,. Let us put D = u{H,,Ia E R,, i E N}. Since {H,,la E R,, i E N} is a a-locally finite collection of cozero-sets of X , D is a cozero-set of X . Note that D 3 A . Since A is C-embedded in X , by Lemma 2.3 (b) there exists a cozero-set G of X such that G n A = 0,G v D = X . Hence if we put 2 = { G } u { H I ,I a E R,, i E N}, then the above arguments show that
<
<
v,
v,
vu
Extensions of Mappings II
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.% is a a-locally finite cozero-set cover and so a normal cover of X such that .% A A < 42. Thus, A is P”-embedded in X . 0
Proof of Theorem 2.11 (continued). The “only if” part. Assume that X is m-collectionwise normal. Let A be a closed subspace of X. By Theorem 2.5, A is C-embedded in X . Let (G, I a E R} and {F,I a E R} be a discrete collection of cozero-sets and a collection of zero-sets of A with Card R < m respectively such that F, c G,, a E R. Since A is closed, {F, 1 a E R} is discrete also in X . By assumption there exists a discrete collection {K,I a E Q} of cozero-sets of X such that F, c K, for a E R. On the other hand, since A is C*-embedded in X , there exists a cozero-set L, of X such that L, n A = G,. Finally, let H, = K, n L, for a E R. Then we see that ( H , 1 a E R} is a discrete collection of cozero-sets of X , and F, c H , n A c G, for a E R. Hence, by Lemma 2.12 A is P“-embedded in X . 0 2.13. Corollary. A space X is collectionwise normal iff every closed subset of X is P-embedded in X .
Remark. Bing’s Example G (Bing [ 19513) is a normal space but not collectionwise normal. By Theorem 2.5 and Corollary 2.13 such a space contains a closed C-embedded but not P-embedded subspace.
For the case Y in Theorem 2.8 (c) ranges over all (not necessarily Cechcomplete) AR spaces with weight < m, we have the following theorem. 2.14. Theorem (Morita [1975]). Assume that A is a P”-embeddedzero-set of a space X . Then every continuous map from A into any A R with weight < m is extended continuously over X .
Proof. Let Y be an AR with weight < m and f:A + Y a continuous map. Then the proof of the “only if” part of Theorem 2.8 (c) shows that there exist a metric space T of weight < m and continuous maps g : X + T and cp : g ( A ) + Y such that f = cp ( g I A ) . On the other hand, since A is a zeroset, there is a continuous map k :X + I such that A = { x E XI k ( x ) = O } . Consider the subspace S = ( ( g ( x ) ,k ( x ) )1 x E X ) of X x Z and define a map h : X + S by h ( x ) = ( g ( x ) , k ( x ) ) for x E X. Then h is continuous. Let $ : S + T be the projection map. Then it should be noted that h ( A ) is closed in S arid g = h holds. Since Y is an AR, the composite cp 0 (I) I h ( A ) ): h(A) + Y is extended to a continuous map q:S + Y. Then we see that h o q : X + Y is the desired extension off. 0 0
$ 0
T. Hoshina
62
Remark. Theorem 2.14 suggests us to characterize a subspace A of a space X such that every continuous map from A into any AR space with weight < m is continuously extended over X. Such a subspace A is said to be M" -embedded in X after Sennott [19781; M-embedding means M" -embedding for every m. Theorem 2.14 reads that every P"-embedded zero-set is M" -embedded. Characterizations of M" -embedding were given in Sennott [1978] (see also Hoshina [1977'J)and results analogous to Morita and Hoshina [1975] were also obtained there. As for spaces in which every closed subspace is M"-(or M-) embedded, it was shown in Hoshina [1977] that, as compared with Theorem 2.12, not even paracompact spaces are such one; indeed the Michael line (see Michael [1963])contains Q ( = rational numbers) as a closed subspace which is not MKo-embedded. A remarkable result due to Sennott [1978a] is that if X is a collectionwise normal P-space in the sense of Morita [I9641( = a collectionwisenormal space whose product with any metric space is normal), then every closed subspace is M-embedded in X (for an extended result, see WaSko [1984]). We note that the converse of this result is not true because the Rudin's Dowker space (Rudin [19711) is collectionwise normal but not countably paracompact, and in which every closed subspace is shown to be M-embedded.
3. Unions of C*embedded subsets In discussing C*-embeddabilityof a given subspace, we often meet whether any union of C*-embedded subspaces is again C*-embedded or not. But, it need not be C*-embedded even for two P-embedded subspaces: 3.1. Example. For any ordinal number a let a itself mean the space of all ordnals < a with the usual order topology. Let
X =
(0,
+ 1) x
A = w1 x (wl},
(0,
+ 1) -
B =
((01,
( 0 , )x
0,>},
wI,
where wIis the first uncountable ordinal. Each of A and B is a closed, C*-embedded countably compact subspace of X. Hence, they are Pembedded in X. But the union A u B is not C*-embedded in X. In this section we shall first discuss when the union A u B of two C*-embedded subspaces A, B is C*-embedded, and secondly give results related to certain unions of infinitely many C*-embedded ones. The following two theorems explain how A u B can be C*-embedded.
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3.2. Theorem (Morita and Hoshina [1975]). Let A and B be C*-embedded in a space X . Zf for anyfinite normal cover 9 of X the cover { X - A, X - B, U I U n A n B # 8, U E 9}is refined by a normal cover of X , then A u B is C*-embedded in X . Proof. Let Y be a finite normal cover of C = A v B. Take a finite normal cover X of C such that X * < Y. Since each of A and B is C*-embedded in X , by using Theorem 2.6 twice, we can take a finite normal cover 9 of X such that 9 A A and Q A B refine X . By assumption there exists a finite normal cover Y of X such that Y < {X - A, X - B, U ( U n A n B # 8, U E a}.Let W = 9 A V .Then W is also a finite normal cover of X . We show W A C < Y. Let U E 4,V E Y with U n V n C # 8. Incaseeither V c X - A or V c X - B, then our purpose is clear. Otherwise, there exists a U' E 9 such that V c U' with U' n A n B # 8.Then there exist HI,H2 E X such that U' n A c H I and U' n B c H 2 . Hence H I n H, # 8 and consequently we have U n V n C c U' n C c H , v H, c St(H,, X ) c G for some G E Y. Hence in any case we see W 2.6, C = A v B is C*-embedded in X .
A
C < Y. Thus, by Theorem 0
3.3. Theorem (Morita and Hoshina [1975]). Let A, B be subsets of a space X which are closed in A u B. Zf A v B is C*-embedded in X , then for any normalcover@ofXthecover{X - A,X - B , U I U n A n B # ~ , U E % } is refned by a normal cover of X . Proof. Let Q be a normal cover of X . Using Theorem 1.4, we can take a locally finite cozero-set cover V = { K I 1E A}, a cozero-set cover { H AI 1E A} such that Y < 9and for some continuous maph : X -P I for 1E A we have 0 if x h(x) =
1
E
HA,
ifxEX-K.
Now, suppose that HA - St(A n B, V ) # 8 for some 1E A. Then we have 5 n A n B = 8. Define a map g A :A u B + I by g,l A = & and g, I B = 1; then g, is single valued and continuous on A u B sincef, = 1 on A n B'and each of A and B is closed in A u B. By assumption there is a continuous extension h,: X + I of g,. Then h, is 0 on HA n A and 1 on HAn B. Let us define cozero-sets HA,and HA2by HA, = { x E H, I h,(x) < $},
T. Hoshina
64
Hi, = { x E Hj.I h,(x) >
i}.Finally, let us put
W = { H i i )i = I, 2; H , - St(A n B, V ) # u {V,lV, n A n B # & P E A } .
8,
E
A}
It is easy to see that W is a locally finite cozero-set (and so a normal) cover of X that refines {X - A, X - B, UI U n A n B # 8, U E 4 } ,completing the proof. 0 Remark. In Theorem 3.3, the assumption that A and B be closed in A u B is essential, even when both A and B are C*-embedded in X . Indeed, let X = wI + 1, A = wl, B = {wl}. Then A, B and A u B = X are C*embedded in X. But {X - A, X - B) cannot be refined by a normal cover; if otherwise B would be a zero-set of X , a contradiction. 3.4. Corollary. Let A and B be C*-embedded in X . Then A u B is C*embedded in X i f f A u B is C*-embedded in Cl(A v B) and for any finite normal cover 4 of X the open cover { X - C1 A, X - C1 B, UI U n C1 A n C1 B # 8, U E 4 } is normal.
Remark. In case X is Tychonoff, our condition that for any finite normal cover 4 of X {X - A, X - B, UI U n A n B # 8, U E 4 } is refined by a normal cover is equivalent to Cl,,(A n B) = Cl,,A n Cl,,B; this can be proved similarly as the proof of Theorem 2.35 of Chapter 1. The/following two theorems assert that for A u B to be C- or P"embedded the essential is C*-embeddability of A v B. 3.5. Theorem (Morita and Hoshina [1975]). Suppose A and B be Cembedded in X . Then A u B is C-embedded in X i f f A u B is C*-embedded in X.
Proof. We only have to prove the "if" part. Assume A u B is C*-embedded in X. By Theorem 2.3 (b) it suffices to prove that if Z is a zero-set of X with (A u B) n Z = 8 then A u B and Z are completely separated in X. Take such a zero-set Z. By assumption and Theorem 2.3 (b) there exist zero-sets Z, and'Z, of X such that Z, 3 A, Z, n Z = 8 and Z , B, Z , n Z = 8. Then Z' = Z, u Z , is a zero-set and we have A u B c Z', Z' u Z = 8. Hence A u B and Z are completely separated in X . 0 =)
Extensions of Mappings II
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3.6. Theorem (Morita and Hoshina [1975]). Suppose A and B be Pmembedded in X. Then the following are equivalent. (a)
A u B is P"-embedded in X .
(b)
A u B is C-embedded in X .
(c)
A u B is C*-embedded in X .
Proof. We shall prove (b)*(a) since (a)*(b) is obvious and (b)*(c) follows from Theorem 3.5. Assume A u B is C-embedded in X . Let 9 be a normal cover of A u B of Card 9 < m. Let Y be a locally finite cozero-set cover of A u B with Card Y < rn that refines @. By assumption, for every V E Y there is a cozero-set 9of X such that v n (A u B) = V.On the other hand, since each of A and B is P"-embedded in X , there exists a locally finite cozero-set cover .W of X such that .W A A < Y and .W A B < Y . For any H E .W with H n A # 0,choose V,, E Y so that H n A c V,, and put W,, = H n Let W-, = { W,l H E S,H n A # 0).Then W-,is a locally finite collection of cozero-setsof X and we have UW-, 3 A and W-, A ( A u B) refines Y .Similarly let us construct WBfor B. Let K = UW-,u UWB.Then K is a cozero-set which contains A u B. By assumption and Theorem 2.3 (b) there exists a cozero-set L of X such that L n ( A u B) = 8 and K u L = X . Now let us put W = {L} u WAu WB.Then the above shows that W is a locally finite cozero-set (and so a normal) cover of X such that W A ( A u B ) < Y < 9.Hence A u B is P"-embedded in X . 0
v,,.
3.7. Corollary. Suppose A and B be P-embedded in X . Then thefollowing are equivalent. (a)
A u B is P-embedded in X .
(b)
A u B is C-embedded in X.
(c)
A u B is C*-embedded in X
Remark. (1) By the same method one sees that Theorems 3.5, 3.6 and Corollary 3.7 are also valid for a finite union A, u . * . u A,. (2) A subspace A is z-embedded in a space X if any zero-set Z of A can be written as Z = Z' n A with some zero-set Z' of X (see Alo and Shapiro [1974]). C*-embedding implies z-embedding. It is not hard to see that A is C-embedded in X iff A is z-embedded in X and is completely separated from any zefo-set of X disjoint from A. In view of this, as R. L. Blair noted (cf. Morita and Hoshina [1975]), in Theorem 3.5 and 3.6 and Corollary 3.7 C*-embedding of A u B can be weakened to z-embedding.
T. Hoshina
66
We shall proceed to prove applications of results above, the first of which will be used also in Section 5. 3.8. Theorem (Morita and Hoshina [19751). Let A and B be C-embedded in X. If one of A and B is a zero-set of X,then A u B is C-embedded in X.
Proof. Let 9 be a finite normal cover of X. Let Y be a finite cozero-set cover of X that refines 9. Assume A is a zero-set. Then, if we put G = (X - A ) u U { V l V n A n B # 8, V E Y},Gisacozero-setcontaining B. Since B is C-embedded, there is a cozero-set H such that H n B = 8 and H u G = X.Thisimplies that {X - A, H, VI V n A n B # 8, V E Y }is a finite cozero-set (and so a normal) cover of X, which refines {X - A, X-B, V J V n A n B f 8 , V~Y}.HencebyTheorem3.2AuBis C*-embedded in X,and hence by Theorem 3.5 it is C-embedded in X. 0 3.9. Corollary. Let A and B be P"-embedded in'X. If one of A and B is a zero-set of X,then A u B is P"-embedded in X. Since a C*-embedded zero-set is C-embedded (2.4), we have the following corollary. 3.10. Corollary. Let A and B be C*-embedded zero-sets of X. Then A u B is C-embedded in X. For the case of C*-embedding, a theorem analogous to Theorem 3.8 does not hold.
+
3.11. Example. Let X = (aI 1) x BN - {a1} x (BN - N), A = a1x (BN - N ) and B = {a1} x N. Observe that BX = (aI 1) x BN, PA = (a1+ 1) x (BN - N ) = Cl,,A, BB = {a1}x /IN = Cl,,B. Hence A and B are C*-embedded in X. A is a countably compact zero-set of X,and hence A is a P-embedded zero-set of X. On the other hand, it is easy to see that A u B is not C*-embedded in X.
+
For the case of finite unions we have the following theorem. 3.12. Theorem. Let A l , . . . , A,, be C*-embedded (resp. C-embedded or P"-embedded) subsets of a space X. If any of A, u Aj (1 < i, j d n) is C*-embeddedinX,then A l u . . . u A,, is C*-embedded(resp. C-embeddedor P"-embedded) in X.
Extensions of Mappings 11
61
Proof. Let B, and B, be completely separated subsets of A = A , u . * u A,.
For each pair (i,j), 1 < i, j < n, by assumption and Lemma 2.3 (a) there exist disjoint zero-sets Z,(i,j ) and Z,(i, j ) of X such that Bk n ( A i u A j ) c Zk(i,j ) , k = 1, 2. Here, we may assume Z,(i, j ) = Z k ( j ,i ) . Let us put for 1 < i < n , k = 1,2,
Z k ( i ) = Z,(i, 1) n . . * n Z,(i, n). Then Z,(i) is a zero-set of X, and we have
Bk n A ,
c Z,(i),
Z , ( i )n Z,(j) =
k = 1, 2,
0,
1
< i, j < n.
-
Hence, if we put Z, = Z k ( l )u . u Zk(n),then Z, is a zero-set of X and we have that B, c Z,, k = 1,2, and 2, n Z, = 0.Thus, by Lemma 2.2, B, and B, are completely separated in X. Hence A is C*-embedded in X. The theorem has been proved for the case of C*-embedding. From this case together with (1) of the remark following Corollary 3.7 the other cases follow 0 easily. For infinite unions any corresponding results to those obtained above is not true, even for a countable union UieNAi such that each Ai is a one-point (and so P-embedded) set and {Ail i E N} is discrete.
3.13. Example. Let X and B be as in Example 3.11. Then B is a union of {(q, n)} and {{(q, n)} In E N} is discrete. But B is not C-embedded in X. 3.14. Example. Let X be the Niemytzki Space (= the space R, in Nagata [1984]), that is, X is the subset {(x, y) E R2I y > 0} of the Euclidean plane Rz, with the topology: nbds of ( x , y) with y > 0 are those usual ones in R2, and basic nbds of points z = (x, 0) are of the form {z} u { ( x ’ , y ’ ) E XI ( x - x’)* + ( E - y’)’ < E ’ } , E > 0. Then each point of X is a zero-set. Let A = { ( x , O ) I X E Q}, B = { ( x , O ) I X E P}, where Q = the set of rationals, P = the set of irrationals. Then B is a zero-set of X , while A is not. Consider disjoint subsets Q,and Q, of Q, each of which is dense in the subspace Q in R. Then Q,x (0) and Q, x {0} are obviously completely separated in A, but not in X. Hence A is not C*embedded in X. Similarly B is not C*-embedded in X.
T. Hoshina
68
3.15. Example. Let X be the space A given in Gillman and Jerison [1960, p. 971, that is, A = fiR - (/?N - N). Then each point of N is a zero-set of A and { { n } 1 n E N} is discrete. N is C*-embedded in A since fiA = flR and Cl,, N = Cl,, N = /?N.On the other hand, since A is pseudo-compact, N can not be C-embedded. We shall give a convenient notion for the union of a locally finite collection of C*-embedded subsets to be C*-embedded.
3.16. Definition (Morita [1980], Ohta [1977]). A collection d of subsets of a space X is said to be uniformly locallyfinite if there exists a normal cover Q of X such that each member of Q intersects at most finitely many members of d. 3.17. Theorem (Morita [1980], Ohta [1977]). A collection d = { A, 11 E A} of subsets of a space X i s uniformly locally finite fi there exist a cozero-set GI and a zero-set Z , of X for A E A such that A, c Z, c G, and { G II 1 E A} is locally finite. Proof. Suppose d is uniformly locally finite. Let 9 be a normal cover such as given in Definition 3.16. Take a locally finite cozero-set cover Y = {V.I a E R} and a zero-set cover 8 = {EuIa E R} of X such that Y * c Q and Eu c V,, a E R. Let us put GI = St(A,, Y ) ,
Z , = St(A,, 8).
Then G, is a cozero-set and by Lemma 1.3, Z, is a zero-set and we have A, c Z, c G,. Let I/ E Y .Then St(V, Y ) c U for some U E Q. Let y be a finite subset of A such that U n A, # 8 implies A E y. Then we have V n G, # 8 1 E y. Thus, {G,} is (uniformly) locally finite. Conversely, assume that G2 and Z,, 1 E A described in the theorem exist. For each finite y c A let W, = n l s y G , n ( X - U { Z , I A $ y } ) , and put W = { F l y is a finite subset of A}. Then by Lemma 1.5 W is a locally finite cozero-set (and so a normal) cover of 1.Since A , n 6 # 8 implies 1 E y , d is uniformly 0 locally finite. With the aid of uniformly local finiteness Theorem 3.12 is extended to the following theorem.
3.18. Theorem (Morita [1980]). Let GI = {A, I 1 E A} be a uniformly locally finite collection of C*-embedded (resp. C-embedded or P”-embedded) subsets of a space X . I f A n v A,, is C*-embedded in X for each A. p E A, then the union ud is C*-embedded (resp. C-embedded or P“-embedded) in X .
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69
Proof. Let 42 be a finite normal cover of U d . By assumption there exists a normal cover 9 = (G, I a E R} of X such that each a E R admits a finite subset y, of A with A, n G, # 8 = A E y,. We may assume B is a locally finite cozero-set cover. Let a E R. By assumption and Theorem 3.1 1 B, = U { A , l A E y,} is C*-embedded in X. Hence by Theorem 2.6 (a) there exists a locally finite cozero-set cover V, of X such that V; A B, refines 42. Let us put = ( V n Gal V E a E nt.
c,
Then V is a locally finite cozero-set (and so a normal) cover of X . Let a E R and V E Va.Then V n G, n ( U d ) = V n G, n B, c V n B,. Hence V A ( U d ) refines 42. Thus, U d is C*-embedded in X by Theorem 2.6 (a). Using Theorem 3.12, in view of Theorem 2.6 (b) and Definition 2.7, the other cases can be proved similarly. 0 3.19. Corollary. Let d be a ungormly locallyfinite collection of C*-embedded (resp. P"-embedded) zero-sets of X . Then U d ' i s a C*-embedded (resp. P'"-embedded)zero-set of X . Proof. By Corollary 3.10 Theorem 3.18 implies that U d is C*-embedded (resp. P"-embeeded) in X. On the other hand, by Lemma 1.4 and Theorem 3.17, U d is a zero-set. 0 4. C*-embedding in product spaces As we have learned in Section 2, C*-embedding (resp. C-embedding or P"-embedding) is precisely the notion of extending normal covers of cardinality < KO(resp. < KOor < m) of a subspace to normal ones of the whole space. On the other hand, we know that normal (resp. countably paracompact normal or m-paracompact normal) spaces are precisely those ones of which every open cover of cardinality < No (resp.
Theorem A (Dowker [1951]). equivalent.
For a Hausdorff space X the following are
(a)
X is countably paracompact normal.
(b)
X x Y is normal for any compact metric space Y
(c)
X x I is normal.
T. Hoshina
I0
Theorem B (Morita [1962]). For a Hausdorflspace X the following are equivalent. (a)
X is m-paracompact and normal.
(b)
X x Y is normalfor any compact Hausdor-space Y of weight
(c)
X x I" is normal.
< m.
Being motivated by these theorems, we shall consider in this section the problem of A x Y being C*-embedded in X x Y for a C*-embedded subspace A of a space X and a compact HausdorfT space Y. This problem was first studied by Alo and Sennott [1972] and afterward by Morita and Hoshina [ 19761 and Przymusinski [ 19781. First of all we note the following proposition. The proof is easy and has been omitted. 4.1. Proposition. For a subspace A of a space X and a space Y if A x Y is C*-embedded in X x Y , then A is C*-embedded in X . The following theorem, which was proved in Morita [1975], plays an essential role; for the proof see Chapter 3. 4.2. Theorem. Let X be a space and Y a compact Hausdorfspace. Let 9 be a normal cover of X x Y. Then there exist a normal cover Q = { U, I A E A} of X and a collection {V,I A E A} offinite open covers of Y such that the cover { V i x V J V E K , I E A } ~ ~Y rXe fXi n e s Y a n d C a r d Q <max{CardY, weight of Y } . Now the first theorem we shall prove is the following.
4.3. Theorem (Alo and Sennott [1972]). Let A be P"-embedded in a space X . Then A x Y is P"-embedded in X x Y for any compact Hausdorfspace Y with weight G m .
Proof. Let Y be a compact Hausdorff space with weight G m . Let Q be a
normal cover of A x Y of Card Y < m. By Theorem 4.2 there exist a normal cover 6 = { V, I A E A} of A with Card A < m and finite open covers VA, I E A of Y such that { V , x VI V E VA,I E A} refines Y. Here, we may assume that each set in VA is a cozero-set of Y since Y is normal. Now since
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A is P"-embedded in X , by using a technique as in the proof of Theorem 1.4 we can take a locally finite cozero-set cover W = { W, I 1 E A} of X such that W , n A c U,foreach1EI\.Let% = { W , x V I V E Y , , ~ E A } . T ~ ~ ~ ~ is a locally finite cozero-set cover of X x Y and we have % A ( A x Y ) refines 3. Hence A x Y is P"-embedded in X x Y. 0 In Section 2 we noted that every compact subspace in a Tychonoff space is P-embedded. But we obtain a stronger version with the following lemma. Using Theorem 4.2 the proof is straightforward and left to the reader.
4.4. Lemma. Let B be a compact subspace of a Tychonoff space Y. Thenfor any space X X x B is P-embedded in X x Y. The following lemma, which is another characterization of P"-embedding, is needed to prove Lemma 4.6 below. 4.5. Lemma (Morita and Hoshina [1976]). Let A be C*-embedded in a space X . Then A is P"-embedded in X iff for every normal cover Q of A with Card Q < m there exists a normal cover Y of X such that each set V n A , V E Y is contained in a union of ajinite number of sets in Q.
Proof. We only have to prove the "if" part. Let 3 = {G,Ia E R} be a normal cover of A with Card R < m. 3 may be assumed to be a locally finite cozero-set cover of A . Then there exist a cozero-set cover { H , I a E R} of A and a continuous map& : A -P Z for a E R such thatf, equals 0 on Ha and 1 on A - G,. Let g, :X + Z be an extension off, and put for a E R K,
= {XE
Xlg,(x)
=
0},
La
= {XE
X l g , ( x ) < l}.
Then K, is a zero-set and L, a cozero-set of X . On the other hand, for { H , I a E R}, there exists by assumption a locally finite cozero-set cover { U, I 1 E A} of Xwith the property in the lemma. For each 1 E A let us choose a finite subset y, of R so that U,n A c U { H ,I a E y l } . Then we have K, c La for a E R, H, c K, n A c La n A c G, for a E R, U, n A c U{K,Ia E y,}
(4.1) (4.2)
for 1 E A.
(4.3)
Let us put Y- = { u, n Lu I a E Y, 1 E A} u { u, n ( X 9
Uucyl
Ka) I 1 E A}.
Then by (4.1) Y is a locally finite cozero-set cover of X and by (4.2) and (4.3) Y A A refines 9.Hence A .is P"-embedded in X. 0
T. Hoshina
12
4.6. Lemma. Let A be a subspace of a space X and Y a non-discrete compact Hausdorff space. If A x Y is C*-embedded in X x Y, then A is C-embedded
in X .
Proof. Suppose A x Y is C*-embedded in X x Y. By Proposition 4.1 A is C*-embedded in X . Let Q = {G, I n E N} be a countable normal cover of A; assume each G, is a cozero-set of A. Since Y is nondiscrete Hausdorff, Y contains an infinite discrete subspace B = { y,ln E N}. Let us put
< n} I n E N}, (C1 B - { y i l i < n } ) I n E N}.
H , = U{G, x { yil i H, = U{G. x
Then each of HI and H , is a cozero-set of A x C1 B since each pointy. of B is isolated in C1 B, and it is easy to see that { H , , H , } covers A x c1 B. By Lemma 4.4, A x C1 B is C*-embedded in A x Y, and hence by assumption A x C1 B is C*-embedded in X x C1 B. Hence,. by Lemma 2.6 (a) and Theorem 4.2 there exist a locally finite cozero-set cover Q = { U, I I E A} of XandfiniteopencoversV,,I E AofC1 Bsuchthat{(U, n A) x V ( V E VA, I E A} < { H I ,H , } . Suppose that U, n A # 8. Since C1 B - B # 8, for some V E V, (C1 B - B) n V # 8. This implies (U, n A) x V c H , . Moreover, since Vcontains a y, of B, we have then U, n A c UiGnG,. Thus, by Lemma 4.5 A is C-embedded in X. 0 4.7. Corollary. A subspace A is C-embedded in a space X embedded in X x I.
iff
A x I is C*-
We shall now establish Theorem 4.9 below which is the main theorem in this section. Before proving this we state a lemma, where y denotes the initial ordinal with Card m. Let Y be a Tychonoff space of weight m. Thenfor each u < y there are subsets A , , B,, U, and V, of Y such that
4.8. Lemma.
(a)
A , and B, are zero-sets and A , c U, and B, c V,,
(b)
U, and V , are cozero-sets and disjoint,
(c)
Vg <
a, either A , Q U, or B, Q
V,.
In case it is only required that each of A, and B, be closed and each of U, and V, be open, Lemma 4.8 was proved by Starbird (cf. Rudin [1975]) (see, Chapter 4) and his proof can be modified easily so as to yield the lemma.
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Extensions of Mappings I1
4.9. Theorem (Morita and Hoshina [1976], Przymusinski [1978]). Let A be a subspace of a space X and Y a compact Hausdorfspace of weight m. Then A is P”-embedded in X i f f A x Y is C*-embedded in X x Y.
Proof. We shall prove the “if” part since the “only if” part follows readily from Theorem 4.3. Suppose that A x Y is C*-embedded in X x Y. We shall prove the theorem by applying Lemma 2.12. First let us note that A is C-embedded in X by Lemma 4.6. Let {G, I u < y} be a discrete collection of cozero-sets of A with Card y = m and {F,I u < y } a collection of zero-sets of A such that F, c G, for each u < y. Let A,, B,, U, and V , be subsets of Y with the properties described in Lemma 4.8. Let us put
Since F, x A, is a zero-set and G, x U, is a cozero-set of A x Y, F, x A , c G, x U, for tl < y and { G, x U , 1 u <’ y } is discrete, by Lemma 1.3, Z, is a zero-set of A x Y. This goes similar for Z , , and we see Z, and Z , are disjoint. Also, we have the disjoint zero-sets Z;
=
u { F , x B,[u< y } ,
< y}.
Z; = u { F , x ( Y -
By assumption and Theorems 2.6 (a) and 4.2 there exist a locally finite cozero-set cover A = { M ,I 1 E A} of X and finite open covers Mj,,I E A of Y such that { ( M , n A) x N ( N E Ni, 1E A} refines each of {A x Y - Z , , A x Y - Z , } and {A x Y - Z ; , A x Y - Z ; } . Then for a set Mj,of A we have (4.4) Mi n F, # 8 and Mi. n Fp # 8 =. u = /?. To see this, suppose /? < u. Then by Lemma 4.8 either A, - U p # 8 or B, - 5 # 8. For example, let A, - Up # 8. Then for some N in Mj, (A, - U p )n N # 8. We have then
((Mi n A ) x N ) n Z ,
= ((Mj.n A ) x
((Mi. n A ) x N ) n Z ,
3
N ) n (F, x A , ) #
8,
((M;.n A ) x N ) n (Fp x ( Y - U s ) #
0.
This contradicts that { ( M i n A ) x N ( N E Mi, I E A} refines {A x Y - Z , , A x Y - 2,). Thus, (4.4) is proved. Let us put H , = St(K,, A) for u < y. Then by (4.4) we see that { H , 1 u < y } is locally finite, and each Ha is a cozero-set of X containing F, . Since A is C*-embedded, there is a cozero-set c, of X such that G, n A = G, . Then { H , n G, I u < y } is a locally finite collection of cozerosets of X , and we have F, c H , n G, n A c G, for u < y. Therefore by Lemma 2.12 A is P” -embedded in X. 0
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T. Hoshina
4.10. Corollary. A subspace A is P"-embedded in a space X if A x I" is C*-embedded in X x I". Remark. It should be noted that Theorem 4.9 has been proved for a given compact Hausdorff space Y of weight m . As for normality of product spaces the corresponding result is not true. For, let A ( w , )be the one-point compactification of the discrete space of Card w , . Then weight of A ( w , ) is w , . It is known that wl x A ( w , ) is normal, but w , is not w,-paracompact. 4.11. Corollary (Starbird [1974]). Let Y be a compact Hausdorfspace with weight m. Then a space X i s m-collectionwise normal ifffor every closed set A of X A x Y is C*-embedded in X x Y . 4.12. Corollary (Rudin [1975]). If X x Y of a space X with a compact Hausdorflspace Y of weight m is normal, then X is m-collectionwise normal. In the results above we have been concerned with products with a compact factor. As for products with a metric factor Przymusinski [I9831 obtained several results. On such a product the following example is interesting, which asserts that the corresponding result to Theorem 4.3 does not hold.
4.13. Example (Morita [1977]). There exist a paracompact Hamdorff space X , a closed subspace A of X (hence it is P-embedded in X) and a metric space Y such that A x Y is not C*-embedded in X x Y. Indeed, let M be the Michael line given in Michael [1963]. Then as is proved there, Q is a closed set of M and M x P is not normal because Q x P and A(P) = {(x, x) I x E P} cannot be separated by open sets. Note that A(P) is a zero-set of M x P. Hence by Theorem 2.3(b) Q x P is not C-embedded in M x P. Let K be a non-discrete compact subset of P. Then by Theorem 4.9, Q x P x K is not C*-embedded in M x P x K, and by Lemma 4.4, Q x P x K is C*-embedded in Q x P x P. Hence Q x P x P is not C*-embedded in M x P x P. Since P x P = P, Q x P is not C*-embedded in M x P. Concerning products with a metric factor the following problem discussed in Przymusinski [1983] is still open.
4.14. Problem. Let Y be a non-discretemetric space. I f A x Y is C*-embedded in X x Y , must A x Y then be C-embedded in X x Y ? Finally, we shall discuss spaces X such that for any closed set A of X A x Y is C*-embedded in X x Y for any space Y.
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4.15. Theorem (Morita [1977]). Let X be a locally compact paracompact Hausdorfspace and A its closed subspace. Then A x Y is C*-embedded in X x Y for any space Y . Proof. By assumption on X there exist collections Y = {Galu E R} of cozero-sets, 9 = {F,I u E R} of zero-sets and V = {C, I u E f2} of compact subsets of X such that C, c F, c G,, u E R, V covers X and Q is locally finite. Let B, = (A n C,) x Y . Then by Lemma 4.4 for any u, B E R B, u B, = ( A n (C,u C,))x Y is C*-embedded in X x Y. Since B, c F, x Y c G, x Y for u E i2, by Theorem 3.17 {B,} is uniformly locally finite. Thus by Theorem 3.18 U { B , I u E f2} = A x Y is C*-embedded in X x Y . In case X being a metric space Theorem 4.15 is also valid; the proof is omitted because of its length.
4.16. Theorem (Michael, see Starbird [1974]). Let X be a metric space and A its closed subspace. Then A x Y is C*-embedded in X x Yfor any space Y . Remark. (1) As a common generalization of locally compact paracompact spaces and metric spaces, we know paracompact M-spaces (Morita [19641). WaSko [ 19841 showed by an example that in case Xis a paracompact M-space the same result as above need not hold. (2) Theorem 4.16 has been generalized by Fujii [1984] to the case X is an M 3 (= a stratifiable) space. 5. Homotopy extension property Let (X, A) be a pair of a space X and its subspace A, and let Y be another space. Then ( X , A) is said to have the homotopy extension property (abbreviated HEP) with respect to Y if every continuous map f:Xx{O}uAxZ+Y
can be continuously extended over X x Z. It is well known that the original Borsuk's homotopy extension theorem (cf. Hu [1965]) was generalized by Dowker [1956]:
Theorein A. Let Y be a tech-complete separable ANR space. Thenfor every closed subspace A of a countably paracompact normal space X ( X , A ) has the HEP with respect to Y.
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T. Hoshina
Theorem B. Let Y be an ANR. Then for every closed subspace A of a countably paracompact, collectionwise normal space X ( X , A ) has the HEP with respect Y , if either A is a Gs-set of X or Y is tech-complete. In this section, as an application of results obtained in preceding sections, we will show that Dowker's theorems above can be generalized to the case A being C-embedded or P-embedded in X , and conversely that the HEP of ( X , A ) with respect to an arbitrary Y as in theorems above implies C- or P-embeddability of A , under discussions based on Morita [1975a] and Morita and Hoshina [1975]. The following theorem is fundamental and gives a characterization for ( X , A) has the HEP with respect to an arbitrary (5ech-complete ANR of weight < m; (a)=>(d) was previously proved by Morita [1975a]. 5.1. Theorem (Morita and Hoshina [1975]). For a subspace A of a space X the following are equivalent. (a)
A is P"-embedded in X .
(b)
X x B u A x Y is €'"-embedded in X x Y for every compact Hausdorff space Y of weight < m and its closed subset B.
(c)
X x {0} u A x I is P"-embedded in X x I.
(d)
( X , A ) has the HEP with respect to any tech-complete ANR of weight < m .
Proof. (a)*(b). Assume (a). Then by Theorem 4.3 A x Y is P"-embedded in X x Y. By Lemma 4.4 X x B is P"-embedded in X x Y. Let % be a finite normal cover of X x Y. Let V be a finite cozero-set cover of X x Y that refines %. Let us put G = St(A x B, V ) .Then G is a cozero-set of X x Y. Letf : X x Y + I be a continuous map such that G = ((x, y ) If (x, y ) > O}. Definef ': X + I byf"(x) = inf{f(x, y ) I y E B } for x E X . Since the projection : X x B + Xis open and closed, one can prove thatf' is continuous. If we put H = { x E X l f ' ( x ) > 0}, then H is a cozero-set of X such that A x B c H x B c G. Since A is C-embedded in X , there is a cozero-set W of X such that A n W = 0, W u H = X . Since X x B is C-embedded in X x Y , there is a cozero-set L of X x Y such that L n ( X x B ) = 0, L u ((W x Y ) u G ) = X x Y. Now we have the cover { L , W x Y , V 1 V n (A x B ) # 0, V E Y } , which is a finite cozero-set cover of X x Y refining { ( X - A) x Y , X x ( Y - B), UI U n (A x B ) # 0,UE%}.Thus, byTheorem3.2X x B u A x Y
Extensions of Mappings 11
77
is C*-embedded in X x Y, and hence by Theorem 3.6 it is P"-embedded in x x Y. (b)*(c). This is obvious. (c)*(a). Assume (c). Let %! = { U ,12. E A) be a locally finite cozero-set cover of A with Card A < m. Let us put V = {U; x
(5,
l]lA E A} u { X x ( 0 ) u A x [0, 5)).
Then it is easy to see that V is a locally finite cozero-set cover of X x (0) u A x I . By (c) there exists a normal cover W of X x I such that W A ( X x ( 0 ) u A X I )< V . L e t u s s e t K = W A ( X x { I ) ) . T h e n K anormalcoverofX x {l}such that% A ( A x {I}) < V A ( A x (1)) = { U, x { 1) 11 E A}. This implies that A is P"-embedded in X . (d)*(c). This follows readily from Theorem 2.8 (c). (c)*(d). Assume (c). Let Y be a Cech-complete ANR of weight < m, and let f:( X x ( 0 ) ) u A x I + Y is any continuous map. By Theorem 1.9 there is a Banach space L, in which Y is embedded as a closed subset and the closed convex hull Z of Y in L has weight < m. Since Y is an ANR, there is an open subset U of Z with a retraction r : U + Y. By assumption and Theorem 2.8 (c) there exists an extension g : X x I + Z off. Since U is a cozero-set of Z , g - ' ( U ) is also a cozero-set of X x I containing A x I . Then as above there is a cozero-set H of X such that A x I c H x I t g - ' ( U ) . Since A is C-embedded in X by (c)-(a), there exists a continuous map cp : X x I such thatrp= 1 o n A a n d O o n X - H . D e f i n e h : X x I - , Z b y h(x, t ) = g ( x , cp(x)t), ( x , 1) E X x I .
Then h is continuous, and it is easy to see that h I ( X x { 0 } u A x I ) = f and h(X x I ) c U . Therefore, the composite ro h : X x I + Y is now a desired 0 extension off. 5.2. Corollary (Morita and Hoshina [1975], Morita [1975a]). For a subspace A of a space X the following are equivalent.
(a)
A is C-embedded in X .
(b)
X x B u A x Z is C-embedded in X x Z for every compact metric space Z and its closed subset B.
(c)
X x ( 0 ) u A x I is C-embedded in X x I .
(d)
( X , A ) has the HEP with respect to every separable tech-complete ANR.
78
T. Hoshina
Remark. (1) (a)*(b) of Corollary 5.2 is also implied by Theorem 3.8 since in this case X x B is a zero-set of X x Z. (2) (a)*(b) has been proved by Starbird [1975] in case Y is [0, 11 or (0, 1). 5.3. Corollary (Miednikov [1975]). Let A,, . . . , A, be closest subsets of a normal space X , and let B,, . . . , B,, be closed in I. Then A, x B, v ‘ . . v A, x B, is C-embedded in X x I. Proof. Since each Ai x Bi is C-embedded in X x I, by Theorem 3.12 the proof is sufficient for the case n = 2. Also, we may assume that X = A , u A , a n d I = B, v B,. Let C = A , x B, v A, x B,,andletfbea real-valued continuous map on C. Since C , = A, x B, v (A, n A,) x I is C-embedded in A , x I by Corollary 5.2,fJ C , is extended to g , over A , x I. Similarly f l C2 is extended to g , over A, x I, where C, = A2 x B2 u (A, n A2) x I. Then clearly the map g on X x Idefined by g = g, on A , x I and g = g, on A, x I is continuous and an extension off.
If, in addition, A is a zero-set of X , by using further the same method as in the proof of Theorem 2.14, the following theorem, generalizing Theorem B, can be proved similarly to Theorem 5 . I . 5.4. Theorem (Morita [1975a]). Let A be a P”-embedded zero-set of a space X . Then ( X , A) has the HEP with respect to every ANR of weight <m. Remark. Applying the methods in Morita [1975a] and Morita and Hoshina [1975] to the notion of M”-embedding, Sennott [1978] obtained that ( X , A) has the HEP with respect to every ANR of weight < m iff A is M”-embedded in X .
References AlO, R. A. and L. I. Sennott
[I9721 Collectionwise normality and the extensions of functions on product spaces, Fund. Maih.76, 231-243. Aid, R. A. and H.L. Shapiro [I9741 Normal Topological Spaces, Cambridge.
Bing, R.,H. [1951] Metrization of topological spaces, Canad. J . Math. 3, 175-186. Dowker, C. H. [I9511 On countably paracompact spaces, Canad. J . Math. 3, 219-224.
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[I9521 On a theorem of Hanner, A r k . f . Mar. 2, 307-313. [I9561 Homotopy extension theorems, Proc. London Math. SOC.6, 100-1 16. Dugundji, J. [1951] An extension of Tietze’s theorem, PaciJic J. Math . 1, 353-367. Fujii, S. [I9841 Ir-embedding and Dugundji extension theorem, MSc thesis, University of Tsukuba. Gantner, T. E. [I9681 Extensions of uniformly continuous pseudometrics, Trans. A M S 132, 147-1 57. Gillman, L. and M. Jerison [1960] Rings of Continuous Functions, New York. Hoshina, T. [I9771 Remarks on Sennott’s M-embedding, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 13, 284-289. Hu, S. T. [I9651 Theory of Retracts, Detroit. Michael, E. [I9631 The product of a normal space and a metric space neid not be normal, Bull. A M S 69, 375-376. Miednikov, L. E. [I9751 Embeddings of bicompacta into the Tychonoff cube and extension of mappings from subsets of products (in Russian), Dokl. Akad. Nauk SSSR 222, 1287-1290. Morita, K. [19621 Paracompactness and product spaces, Fund. Math. 50, 223-236. [1964] Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. I19701 Topological completions and M-spaces, kci. Rep. Tokyo Kyoiku Daigaku, Sect. A 10, 27 1-288. [I9751 Cech cohomology and covering dimension for topological spaces, Fund. Math. 87, 3 1-52. [1975a] On generalizations of Borsuk’s homotopy extension theorem, Fund. Math. 88, 1-6. [I9771 On the dimension of the product of topological spaces, Tsukuba J. Math. 1, 1-6. [1980] Dimension of general topological spaces, in: G. M. Reed, Ed., Surveys in General Topology, 297-336, New York. Morita, K. and T. Hoshina [1975] C-embedding and the homotopy extension property, General Topology Appl. 5,69-81. [1975a] Review of Morita and Hoshina [1975], Zentralbl. Math. 2%, #54014. [I9761 P-embedding and product spaces, Fund. Math. 93, 71-86. Nagata, J. [19851 Modern General Topology, Amsterdam. Ohta, H. [I9771 Topologically complete spaces and perfect maps, Tsukuba J. Math. 1, 77-89. Przymusinski, T. C. [19781 Collectionwise normality and extensions of continuous functions, Fund. Math. 98, 75-8 1. [I9831 Notes on extendability of continuous functions from products with a metric factor, Manuscript.
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Rudin, M. E. [I9711 A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. [I9751 The normality of products with one compact factor, General Topology Appl. 5,45-59. Sennott, L. 1. [I9781 On extending continuous functions into a metrizable AE, General Topology Appl. 8, 219-228. [1978a] Some remarks on M-embedding, Topology Proc. 3, 507-520. Shapiro, H.L. [I9661 Extensions of pseudometrics, Canad. J . Math. 18, 981-998. Starbird, M. (19741 The normality of products with a compact or a metric factor, PhD Thesis, University of Wisconsin. [I9751 The Borsuk homotopy extension theorem without the binormality condition, Fund. Math. 87,207-2 I 1. Walker, R. C . [I9741 The Stone-tech CompactiJication, Berlin. Waiko, A. [I9841 Extensions of functions defined on product spaces, Fund. Math. 124, 27-39.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 3
NORMALITY OF PRODUCT SPACES I
Masahiko ATSUJI Josai University, Sakado, Saitama, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental results. . . . . . . . . . . . . . . . . . . . . . 2. The first of Morita’s three conjectures . . . . . . . . . . . . . . 3. The second and third of Morita’s three conjectures . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
81 82 92 109 116
Introduction
Unless explicitly stated to the contrary, the product of spaces is the topological product space of two spaces. As a separation axiom normality is not a special one, and it is the natural one next to regularity or complete regularity. However, the situation is quite different in the theory of product spaces. A product space satisfies a separation axiom T,, 0 < i < 3+, if and only if each factor space of it does. However, surprisingly, even a product of the unit closed interval I on the real line with a normal space is not necessarily normal (Rudin [1971]), so normality is found to be a very interesting property in the theory of product spaces. On the other hand, Dowker [1951] characterized a structure of a space X , countable paracompactness, by normality of X x I, and Tamano [1960] did the same: a characterization of paracompactness of a completely regular space Xby normality of the product of Xwith its Cech-Stone compactification. The works of Dowker and Tamano have stimulated a study of general topology and contributed to a development of the study, and we have had many beautiful and important results on product spaces since then. Thus we
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recognize a special importance of normality of product spaces in general topology, and we expect further possibilities for finding a characterization of various spaces in this direction. Among other topologists, especially K. Morita has done much in this area, and various interesting problems derived from or connected with his works have been researched and a further study is expected. So, in this chapter, we will concern ourselves mainly with Morita’s work, particularly with his three conjectures and related topics which are about the characterization of spaces by means of normality of product spaces. Section 1 deals with some, not all, fundamental results on the normality of product spaces with a compact factor. In the proofs of many propositions we use a pseudometric in order to make them simpler. The main contents of Sections 2 and 3 are clear by their titles, which also include various topics derived from Morita’s conjectures. Throughout this article, a space is a topological Hausdorff space unless otherwise specified, whose cover is open. Let A be a subset of a product space X x Y and y a point of Y. A[ y ] = { x E XI (x, y ) E A} is called a slice of A at y- 1 A 1 is the cardinality of a set A, w ( X ) is the weight of a space X.w and w,are the first countably infinite and the first uncountable ordinal numbers respectively; rc is a cardinal number or an initial ordinal of the cardinal, which often stands for the set of all ordinals smaller than K, and also for a topological space equipped with the order topology or some topology indicated. A space X is said to be rc-paracompact if any cover of X with power < rc admits a locally finite refinement. If X is rc-paracompact for any infinite IC, then Xis said to be paracompact; so if w ( X ) < rc, then rc-paracompactness is equivalent to paracompactness for X . o-paracompactness is nothing but a countable paracompactness. For undefined notions and symbols, for facts used without mention and also for related subjects the reader is referred to Engelking [1977] or Nagata [1985]. 1. Fundamental results 1.1. Theorem (Tukey [ 19401). A cover 4 of a space X is normal if and only ifthere exists a pseudometric don X such that { U ( x ;E ) 1 x E X } is a refinement of 4 for some E > 0, where U ( x ; E ) = { y E XI d(x, y ) < E } . 1.2. Definition. The pseudometric d stated in this theorem is called the pseudometric associated with the cover 4 , and the topology induced by d is denoted by Fd. The following theorem is a generalization of the well-known Stone’s theorem (Corollary 1.6 below).
Normality of Product Spaces I
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1.3. Theorem (Morita [1981, Theorem 29.101). Let .T and Y be topologies and @, n E o,are on a nonempty set X . Suppose that Y is a cover of (X,9) covers of (X,9’) with the property thatfor any x E G E 3 ’ there exists a number n with StS(x,@,,) c G. Then there exists a locallyfinite, with respect to 9, cover of (X,9) which refines 9.
Proof. Suppose the index set R of 9 = {G, I a E R} is well-ordered. For each point x in X we take the smallest a with x E G, which is denoted by a,. We let Bna = Ana - U{GpIB < a}, An, = {xlStS(x,@ n ) c Ga}, Bn = U{&ala
E Q},
HI, = St2(Bla,@I),
H,,, = St2(B,,,, @,) - U{St(Bj, % ) l j < n} for n > 1, where the bar
H
-
means the closure by the topology 9,
= { H , , , ~ ~ EaW ER , }.
Then we have StS(Bn,, q n ) c StS(An,, an) c G,, and p < a implies St5(Bn,,@,,) n B,, = 8, hence, for any point x E X and a member U of @, including x , we have U n St2(Bn,,@,,) = 8 or U n St2(Bn,,@,,) = 8. Therefore, for every n E o,{St2(Bn,,@,J I a E R} is locally finite with respect to 9. For each x E X there exists an n E o with StS(x,@,) c Gax,and x is in BnaX, hence
X
= U{BnInE a}.
For any x E X there exists a smallest number n E o such that x E St(B,, @,,). Since St(B,,, @,,) = U{St(Bna,@,,) 1 a E R} and {St(B,,, @,,) 1 a E R} is locally finite, there exists an a with
x
E
St(Bna, @ n ) c st2(Bna,a n ) ;
hence, if n = 1, then x x
E
E
HI,, and if n > 1, then
St2(B,,,, @ n ) - U{St(Bj, @j)Ij < n}
=
Hna.
Consequently, .W is a cover of (X,Y ) ,which refines Y. For x E X,there exists an n with x E B,,, and St(x, @,,) c St(B,,, @,,), so rn > n implies H,, n St(x, a,,) = 8. Since { q . a l j= 1, . . . ,n, a E R} is locally’finite, H is locally finite with respect to 9’.
Remark. If IR( 2 w , then tH I = IRI. 2t‘ is o-discrete.
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M.4tsuji
1.4. Corollary. Every normal cover of a space has a locallyfinite refinement. The converse of this corollary is valid in a normal space, namely, a locally jinite cover of a normal space is normal. We can omit the normality of the space if we replace “cover” with “cover consisting of cozero sets”, and then the condition “locally finite” can be weakened to “a-locally finite” (Morita [1964, Theorem 1.21;see Nagata [1985, VS.D]), where G is called a cozero set of a space X if there exists a real-valued continuous map f on X such that G = {x E XI f ( x ) # 0). Since a space is normal if and only if every finite cover is normal, we obtain the following assertion by these facts: a space is normal and K-paracompact if and only if any cover of the space with power at most K is normal. We can restate Corollary 1.4 as follows.
1.5. Corollary. Let Y be a normal cover of a space and let d be a pseudometric associated with 9. Then B has a locallyfinite refinement consisting of Fd-open sets. Since every paracompact space is normal, we have the following corollary. 1.6. Corollary (Stone [1948]). A space X i s paracompact ifand only ifevery cover of X is normal.
We also mention the following proposition (cf. Nagata [1985, Theorem V. I]): If X is a regular space, then X is paracompact i f and only if every cover of X has a a-locallyfinite refinement (Michael [1953]). 1.7. Corollary (Tukey [1940], Stone [1948]). compact.
A pseudometric space is para-
1.8. Lemma (Morita [1975]). Let (X,Y) be a space, and let (C, 9)be a compact space. Then for any normal cover Y = { G a l aE R} of ( X x C, F x 9) there exist a pseudometric d on X and a normal cover &‘ = {Ha 1 a E R} of ( X x C, F d x 9’) such that Ha c G, for all a E R.
Proof. Let e be a pseudometric on X x C associated with Y. For any points x and x’ in X we define d(x, x’) = sup{e((x, Y ) , (x’, Y ) ) I Y
E
C};
then d is a pseudometric on X and the topology Y d is coarser than Y. It is Let also eaSy to see that the topology Y d x 9’ on X x C is finer than YQ. &‘ = {Ha I a E R} be a cover of (X x C,Yo)with Ha c G,, a E R; then it is normal and a cover of (X x C, Y d x 9’). 0
Normality of Product Spaces I
85
Let us here present two useful lemmas the first of which is found in Nagata [1985, Section V.5.E]. 1.9. Lemma (Morita [1962]). Suppose that U{Vi:,lIE A} is a cover of a space X and { 11 E A} is a normal cover of X , and that { Wi n V I V E "v;} is a normal cover of the subspace Wi for each I E A; then { K, n V I V E 9;. , Iz E A} is a normal cover of X , and also is U-V;.
1.10. Lemma (Morita [1975]). Let ( X , 9 )be a space, let (C, 9)be a compact space with a base a,1931 < K, o < K, and let Y = {G,la E R}, I R I < K, be a cover of X x C. Then there exists a cover 4 = { U, 11 E A}, I A I < K, of X satisfying conditions (i) and (ii) below. (i) There exists a family {Vi11 E A} of finite covers of C satisfying the condition that Vi c for every I E A, { U, x V I V E Vi,I E A} is a cover of X x C and { U i x VI V E Vi,I E A } refines 9. (ii) Y is a normal cover of X x C if and only if@ is a normal cover of X .
Proof. (i): We can take a family { V, c 9 l I 6 E A}, I A I < K, of finite covers of C such that any finite cover of C is refined by some V,, 6 E A. Let Vd= {hiI i = 1, . . . , r,}, and let r, be the set of all maps f of { 1, . . . , r,} to R. Then I r, I < K. For 6 E A and f E r, we denote U(6,f ) = { x E XI { x } x 6;c G,,;,for each i < r 6 } . Since is compact, U(6,f ) is open in X. For any point x of X there is a finite subset y of R such that {x} x C c U{G,la E y } . Hence there is a 6 E A such that { { x }x V E V,} refines { G,1 a E y } . Therefore, 4 = p(6,f)ifE
r,, 6 E A}
is a cover of X , (U(6,f ) x VI V E V,,f E r,, 6 E A} covers X x C , and { U ( 6 ,f ) x VI V E 6 , f E r,, 6 E A} refines 9.Denote A = ((6, f ) I f E r,, 6 E A} and 9'&) = 6 ; then 1 A I < K. Thus condition (i) is satisfied. (ii): Suppose that Y is normal. By Lemma 1.8 there exist a pseudometric don X and a normal cover & = { H , I a E R} of (X x C, Ydx 9) such that H , c G,, a E R. We let M(6,f ) = { x E XI { x } x
6, c
H,(i) for each i
< r,}.
As above, ( M ( 6 ,f ) } is a cover of ( X , Yd), and 4 = ( U ( 6 ,f ) } is normal because of M(6, f ) c U(6,f ) . Conversely, suppose that 4% = { U ,I I E A} is normal. Then { U , x CI I E A} is a normal cover of X x C, and { U i x VI V E V i }is a normal cover of U , x C, and hence, { U , x V I V E Yj,,I E A} is a normal cover of X x C by Lemma 1.9, which refines Y by (i), and thus Y is normal.
M. Alsuji
86
1.11. Definition. Let p be a point of a space X. We denote by ~ ( pX) , the character of p , namely ~ ( pX, ) = min{ I Y I I Y is a local base for p } . Let p be an accumuiation point with ~ ( pX) , = K . If there exist a local base 9 = { K 1 1 < K ] for p and a set A with p E A - A satisfying the condition that, for any 1 < K, there exist a point x E A n V, and an index a, < K such that x does not belong to any V, with a, < a < K, then we call p a simple accumulation point for 9 and A (or briefly a simple accumulation point). If { A n K I 1 < K} is monotone decreasing, then p is a simple accumulation point for Y and A ; particularly, if p has a monotone decreasing local base 9, then p is a simple accumulation point for Y and X. Another example is this. Denote X = (0, 1) x (w l), p = (w,, w), V,, = (a, w , ] x (n, 01, A = {(a, w ) l a < w , ) , and B = {(a,, n)ln < w>. Order { a n l a c w , , n < w } lexicographically; then p is a simple accumulation point for Y = {&,,la < w , , n < w } and A, but not for Y and B.
+
+
9.12. Theorem. Suppose that X i s a space and C is a compact space including a simple accumuiation point p , with ~ ( p ,Y, ) = v for each v < K . I f X x C is normal, then X is normal and K-paracompact. Proof. We select a local base 9 = { V ,I a < K} for p , and a subset A, of C for which p Kis simple. We shall verify the theorem inductively. Xis obviously wrmal, and we assume that, for any infinite cardinal K’ < K, our theorem :s valid. Let 93, 193 I = p, be a base for C consisting of members of Y and and let 0 = { 0,I a < K ] be a cover spen sets contained in C - q , Y, E 9, of X. We denote
G, = u{O, x V,la <
K}
and G2
=
X X C - X X {p,};
then 9 = {GI, G,} is a cover of X x C , and by Lemma 1.10 there exist a normal cover Q = { U, I 1 E A}, I A I < p, of X and a family {V,I I E A} of finite covers of C with V, c & such I that { U, x V I V E VL, 1 E A} is a cover o f X x Cand{U, x P I V ~ V , , A ~ A } r e f i n e s Y . F o r a n y 1 ~ A t h e r e e x i s t s a V E Y,includingp,. Vis in 2,and we rewrite it V,, ,j, < K. We then have (*)
ui. x V,,
c GI,
and we can find an a, < (**)
U, c U(0,la
K
such that
< a,}.
87
Normality of Product Spaces I
In fact, if we cannot, then select a point y in VS; n A, and an index a,, < K such that y is not in Vy E Y for all y > a,, in K. There exists a point x in U, - u{O,Ia < a,,}; then the point (x, y ) of U, x F$, does not belong K } , which contrato both u{O, x V,la < a y ) and U{O, x KIa, < a dicts (*). Since X and $! are normal, $! is refined by a normal cover W = {W,16EA}ofXsuchthat % t U,forsomeAEA,and
-=
c U, c u{O,Ia
< aI}.
Since W,x C is normal and a1 < K, u(0.l a < a l } has a normal refinement on W, by the inductive assumption. Therefore, by Lemma 1.9,O has a normal refinement. 0 1.13. Theorem (Morita [1961, 19621). Let X be a space, and let K be an infinite cardinal number; then the following conditions are equivalent: (1) X is normal and K-paracompact. (2) The product space X x C is normal and K-paracompact for any compact space C with w ( C ) < K . (3) X x C is normalfor any compact space C with w ( C ) < K . (4) X x M" is normal for any compact metrizable space M . ( 5 ) X x I" is normal for I = [0, 11. (6) X x D" is normal for D = (0, l} c I. (7) X x (K 1) is normal'.
+
Proof. (1)=42): Let Y = {G, I a E R} be a cover of X x C with 1 R I < K ; then we construct a cover 92 = { U, I 1E A} of X stated in Lemma 1.10.9 is normal because of 1A1 < K, and Y is also normal by condition (ii) in the same lemma. (2)*(3): Trivial. (3)*(4): Obvious because M" is a compact space with w ( M " ) < K. (4)*(5): Obvious. (5)*(6): Obvious because D" is a closed subset of I". (6)*(7): Obvious because K + 1 is closedly embedded in D". (7)*(1): Obvious by Theorem 1.12. 0 Probing Theorem 1.13, Y. Katuta obtains a result which we introduce here without proof. Let K and 1 be infinite cardinal numbers. A space X is ' S o w mathematicians, e.g., Przymusinski [1984], say that the equivalence of (1) and (7) is Kunen's theorem. Some others, e.g., Rudin [1985], call it Morita's theorem, and she shows a proof of the theorem which is largely the same as the one of Lemma 2.5 in Morita [ 19621. Though the lemma does not explicitly assert the equivalence, it states an essential part of the proof.
88
M.A fsuji
K-collectionwise normal if, for every discrete family {F,I a < K} of closed subsets of X,there exists a family {C,I a < K } of disjoint open subsets of X with F, c G, for all a < K. A family .W of subsets of a space Y is said to separate points of Y if, for any two distinct points of Y, there exists a member of .W including only one of them. We denote by w( Y) the smallest cardinal A such that there exists a family 3Ep = U{.W’( fl < A} separating points of Y, where JE., is a family consisting of disjoint cozero sets of Y for each fl < A. For the one-point compactification Y of a discrete space, we have v ( Y) = w and w( Y) = I Y I; in general, it is known that for a compact space Y with I Y I 2 w it holds that t( Y) < w( Y ) < w( Y ) , where t( Y) is the tightness of Y (= the smallest cardinal number a such that, for every point y in Y and A c Y, if y E A then there exists a subset E of A with x E B and I E I < a). We denote by I @ ) the subspace of the product I Aof A copies of I consisting of points p such that at most one coordinate of p is not 0. D(A) is similarly defined by using the discrete D = (0, l} instead of I. D(A) is nothing but the one-point compactification of a discrete space with A points. Theorem 1.13 is the case K = A of the following theorem.
1.14. Theorem (Katuta [ 1977al). Thefollowing statements are equivalenrfor a space X : (1) X is K-paracompact and A-collectionwise normal. (2) The product X x C is normal for any compact space C with v(C) < K and w ( C ) < A. (3) X x I(A)l is normal. (4) X x D(A)l is normal. We can derive the following corollary from this theorem (cf. Corollary 2.23): A space X is u-paracompact and collectionwise normal if and only if X x C is normal for every compact space C with w(C) < K . In particular, X is countably paracompact and collectionwise normal if and only if X x C is normal for any compact space C with w(C) = w.
Remark. Ohta [ I985/86] improves Katuta’s theorem above by replacing w(C) in the statement (2) with a smaller cardinal u(C), which is defined by replacing “disjoint cozero sets” in the definition of v(C) by “boundedly pointfinite cozero sets”, where a family of subsets of C is called boundedly pointfinite if there exists a natural number n such that every point of C is included in at host n members of the family. Trivially, u(C) < w(C). A compact space C with u(C) < w is precisely a uniform Eberlein compact space shown in Benyamini and Starbird [1976](cf. Negrepontis [1984, Section 6]), namely,
Normality of Product Spaces I
89
Cis homeomorphic to a subset of a Hilbert space in its weak topology. In the note cited above Ohta also discusses many topics together with problems on cardinal functions of compact spaces in connection with Katuta’s theorem. Let us present another form of result giving normality of the product with a compact factor. 1.15. Theorem (Morita [1962]). Let X be a subspace of a compact C . Then X is paracompact if and only if X x C is normal.
Proof. The “only if” part is obvious by Theorem 1.13. Now, suppose that X x C is normal and Y = { G, I a E R} is a cover of X, then for every a E R there exists an open set Ha of C with G, = X n H a . Denote H = U { H , I a E R} and A = {(x, x) I x E X}, then A is a closed subset of X x C contained in X x H. For the cover A? = {X x H , ( X x C) - A} of X x C, by Lemma 1.10, we obtain a normal cover 9 = { U, I I E A } of X and a family {V,11 E A} of finite covers of C such that { U, x VI V E f,, I E A} covers X x C and {U, x PI V E V,, I E A} refines A?. If U, n # 8, then U, x P meets A, and so U, x c X x H. Hence the compact set U { U, n P # 0, V E .U;} is contained in H and contains U,, and so there exists a finite subset y of R such that U, c U { H , I a E y } . Thus we have U, c U{G, I a E y } , and Y has a locally finite refinement. 0 This theorem can be generalized a bit as follows: Let X be a subspace of a space Z , let C be a compact space, and let f be a continuous map on C to Z with f ( C ) 3 X . Then X is paracompact if and only if X x C is normal. However, this is an immediate consequence of the theorem above because the map ix x f:X x C -, X x f(C) is perfect, where ix is the identity map on X. An investigation of a role of maps in the theory of product spaces is of great interest, which is one of main subjects of Hoshina’s Chapter 4. The following classical results are easily deduced from some of the propositions above. 1.16. Corollary (Dieudonnt [1944]). The product space of a compact space and a paracompact space is paracompact.
1.17. Corollary (Dowker [1951]). A normal space X is countably paracompact if and only if X x I is normal.
Proof. Since D” is a closed subset of I,X x D” is normal if X x I is normal. 0
M. Atsuji
90
1.18. Corollary (Tamano [1960]). A completely regular space X i s paracompact if the product of X and its cech-Stone compactijication flX is normal.
The next theorem is one of the natural generalizations of Corollary 1.16. 1.19. Theorem (Morita [1963a]). If X is a paracompact and a-locally compact space, then X x Y is paracompact for any paracompact space Y.
Proof. Let Y be a cover of X x Y. Since X is a-locally compact and paracompact, X is covered by U { d i li E a},where di = { A , I a E Q i } is a locally finite family consisting of compact subsets of X,and we can construct, for each i E w ,a locally finite open family { L i ,1 a E Q }with Aia c L, for all a E 0,.We apply Lemma 1.10 to the space Ai, x Y, and we have a locally finite cover qiaof Y and a family {“v;,, 11 E Aim}of finite covers of Aia such that every member of Kdis contained in Li,; moreover, each product of a is contained in some member of Y. Since member of qi,and a member of KaA {Ux V I U E ~ ~ V ~ ~, ~, ~ , ~ ~ A , , a ~ R , } i s l o c a l l y f i n i t e f o r e a c h i ~ w , B is refined by a a-locally finite cover, and the proof is completed by the proposition stated after Corollary 1.6. 0 After this theorem of Morita’s was published, a lot of topologists have searched for further generalizations. One of the results obtained is the following theorem (a space is C-scattered if every nonempty closed subspace S has a point with a compact neighborhood in S). 1.20. Theorem (Telgarsky [1971]). If X is paracompact and a-C-scattered, then X x Y is paracompact for any paracompact space Y.
Theorem 1.19 develops to Theorem 3.6 and to Morita’s Conjecture I11 stated later on the one hand, and on the other hand it is followed by Theorem 1.20 and others which are also considered partial solutions of Tamano’s problem (Tamano [1962]): Characterize a space X with which the product X x Y is normal for any paracompact Y. Applying Corollaries 1.16 and 1.18, Tamano himself verified the following theorem. 1.21. Theorem (Tamano [1962]).
Thefollowing conditions on a space X are
equivalknt: (1) X x Y is normal for any paracompact space Y . (2) X x Y is paracompact for any paracompact space Y .
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91
Though Katuta [1971] gives a solution of Tamano’s problem, a simpler characterization is also desired. Telgarsky [19751 defines an interesting notion: topologicalgumes, which can also be applied effectively to our subject. The reader will find some of the results concerned in Yajima’s Chapter 13 in this volume. Let us conclude this section by presenting two more theorems for which we first mention a well-known proposition without proof. Let I be an infinite cardinal. We recall that a well-ordered set { y , I a < A} of points of a space Y is said to be afree sequence of length I in Y if { y , I /3 < a } n { y , I /3 2 a } = 0 for every a in I.
1.22. Lemma (Arhangel’ski [1971]). For a nondiscrete compact space C, it h o l h that t ( C ) = sup{I I there exists a free sequence of length A in C } . ‘1.23. Lemma (Przymusinski [1984]). For a nondiscrete compact space C, it holh that t ( C ) = sup{A I there exist a closed subset F of C and a continuous m a p f : F + A + l o f F o n t o A + I}. Proof. We denote by p the cardinal number of right-hand side of the equality in our lemma, and by { y , I a < I } a free sequence of length 1in C. We let C, = { y,IS 6 a } for each B < I , C, = n{C,l/3 < A}. The map f:C, -+ 1 I defined by f ( y ) = sup{BI y E C,} is clearly surjective and continuous because the C, are closed and open in C,, and so we have t ( C ) 6 p. If t ( C ) < p, then there exists a cardinal 1‘ > t ( C ) in p + 1 satisfying the condition stated in the definition of p. However, this is impossible because a continuous map on a compact space cannot increase a tightness. 0
+
1.24. Theorem (Katuta [1977b]). r f the product X x C of a space X with a compact space C is normal, then X is I-paracompact for every A < t ( C ) . Furthermore, if t ( C ) is not weakly inaccessible, then X is t(C)-paracompact. Proof. This elegant proof is taken from Przymusinski [ 19841. Suppose that A < t(C);then, by Lemma 1.23, there exists a v > I in t ( C ) 1 such that a closed set F c C is continuously mapped onto v + 1. Since X x ( v + 1) is a perfect image of the normal space X x F, it is also normal, and, by Theorem 1.13, X is A-paracompact. When t ( C ) is not a limit cardinal, then the above proved fact immediately implies that Xis t(C)-paracompact; when t ( C ) is singular, then the fact that X is I-paracompact for all I < t ( C ) implies that X is t(C)-paracompact. 0
+
M.Atsuji
92
Giving an example, Katuta also notes in his paper cited above that t ( C ) being not weakly inaccessible is essential.
1.25. Theorem (Nogura [1976]). For an infinite cardinal K and a compact space C, the product K + x C is normal ifand only i f t ( C ) < K , where K + is the successor cardinal for K .
Proof. The “only-if” part is immediately obtained by Theorem 1.24 because -paracompact. = { G , , G,} be a binary cover of K + x C. For any y in C there exists an a < K + such that the interval (a, K + ) is contained in GI[ y ] or G,[ y]. Suppose (a, K + ) c GI[ y]; then there exists an open neighborhood V of y in C with (a, K + ) x P c G I .In fact, if not, then, for any ?‘ni&?, a local base for y in C, there exist a fly > a in K + and a z y in P such that the point (fly, zv)is not in GI. Since t ( C ) < K, there exists a set Z c {zyl V E U }with 121 < K and y E 2. We have s ~ p { f lZ} ~ = l ~y ~< ~K + , a < y and (a, K + ) $ G, [ y], a contradiction. Since the set a + 1 is compact and open in K + , it is a small step to the conclusion that Y is normal. 0 K+
is not
K+
“If” part: Let Y
Citing Kombarov” 19721, Chiba [I9761 remarks that the sufficiency of the theorem above is valid for a paracompact space C ;and, giving an example, she shows that the necessity does not hold for C. Afterward, K. Chiba [I9801 notes that in Nogura’s theorem we can replace “compact space” by “paracompact k-space” because it holds that t( Y) = sup{t ( H ) I Y 3 H: compact} for a k-space Y.
2. The first of Morita’s three conjectures Morita [1977] publicly posed the following three problems. (I) If X x Y is normal for any normal space Y, is X discrete? (11) If X x Y is normal for any normal P-space Y, is X metrizable? (111) If X x Y is normal for any countably paracompact space Y , is X metrizable and o-locally compact? Throughout this article, the term “P-space” is used in Morita’s sense; namely, it is defined in Morita [1964] and differently in Gillman-Jerison [1960], and it is characterized, together with its normality, by the property that its product with every metric space is normal (cf. Nagata [1985, Theorem VI.241). Further details will be introduced in Section 3 (see also Hoshina’s Chapter 4).
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In this section we discuss Problem I and related results. Conjecturing affirmatively, Morita raised Problem I mainly from the fact that we knew only few kinds of cases where products of normal spaces were normal. As we shall see in this section and also in the next one, conditions around one point of a factor space of a product space frequently plays an important role in deciding a structure of another factor space by normality of the product. We have already caught a glimpse of such circumstances in Theorem 1.12. Apparently, it seems special, but actually the way of approach from the local structure of a factor space to the characterization of another space through normality of their product has been shown to be effective in many important results, from the classical Dowker’s Theorem on countable paracompactness to the latest Chiba, Przymusinski and Rudin’s proof of Morita’s conjectures. 2.1. Definition. Let p be a point of a space Y. We denote by Y ( p ) the local base for p . Let { A , I y E Z } be a family of subsets of a space X with index set Z c Y . We call the following subset of X an upper intersection of the family at p :
n
u
Y E Y ( P )y s Y n Z
~ , . 2
The following lemma is an immediate consequence of this definition.
2.2. Lemma. Let X and Y be spaces, and let { A , I y E Z } be a family of subsets of X with index set Z c Y . Thefamily has the empty upper intersection at a point p of Y ifand only ifevery point x of X has a neighborhood which does not meet any A, with index y E V n Z for some neighborhood V of p . 2.3. Remarks. (1) The upper intersection at p E Y of a family d = {A, t X J Y E2 c Y } is the slice A [ p ] of A at p , where A = U { A , x { y } I y E Z } and A is the closure of A in X x Y, so the upper intersection is an invariant of local buses for p . The upper intersection at p of the family is empty if and only if A is disjoint from X x { p } . Consequently, A is a closed subset of X x Y if and only if for each y E Y the upper intersection at y is A, ( A , is empty for y 4 2).Trivially, the upper intersection at p of a family contains the intersection of all closures of sets of the family, not always the same. *Thissubset is denoted by limp sup A, in Atsuji [1977]. The notation was cited from Nobeling [1954, Section 8.21. The subset for a countable family is treated by some mathematicians with other notations, for instance, Isiwata [1964], and Alexandroff and Hopf [1935, p. 11 I) in which it is called “oberer topologischen Limes”. See also Comment 2.29.
M.Atsuji
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(2) If I Z - VI < y for every basic neighborhood V of p in the lemma above, then the family has the empty upper intersection at p if and only if the family is, as it were, locally less than y ; especially if y = w, it is locally finite. 2.4. Theorem. Let X and Y be topological spaces, and let d = { A , 1 y E Z , } and 1= {ByI y E Z z }be families of subsets of X with index sets in Y such that the upper intersections of d and 1at every point of Y are disjoint. If X x Y k normal, then there exkt open families 9 = {D, I y E 2,} and 6 = {E, I y E Z 2 } such that (i) 0,3 A, for y E 2,and Ey 3 Byfor y E Z,, and (ii) the upper intersections of 9 and 6 at every point of Y are dkjoint.
Proof. Denote A = U { A p { Y } I Y E Z I } and B = U{B,X { V } I Y E Z * } ; then the closures A and B of A and B in X x Yare disjoint, so there exist open sets D and E in X x Y containing A and 1 respectively and having disjoint closures. { D [y ] 1 y E Z , } and { E [y ] I y E Z , } are the desired open families. 0 In connection with this theorem it is interesting to find conditions under which the upper intersection is the same with the usual intersection. In order to present one of the conditions, we need some terms. Suppose that an order < is defined on a subset Z of a space Y whose relative topology on Z is not necessarily the same as the one defined by the order. A family {A,I y E Z c Y} of subsets of a space X is said to be increasing (decreasing) if A,, c A, whenever y , < y2 ( y , 2 yz).When Z is linearly ordered, the increasing (decreasing) family is also said to be monotone increasing (monotone decreasing). 2.5. Definition. Let an order be defined on a subset Z of a space Y. Z is said
to befit if Z satisfies the following two conditions: (i) for any point p in Z - Z there exists a subset Zpof Z with p E 2, - Z, such that for each point y in 2,there exists a neighborhood V of p satisfying y’ 2 y for all points y’ in V n Z; (ii) any point q in Z has a neighborhood U such that y’ 2 q for all y’ in U n Z. A family of subsets of a space X with a fit index set is also called to befit. The set Zpstated in condition (i) is called afit part of Z for p . If we require
Normality of Product Spaces I
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for Z only condition (i) for some point p E 2 - Z, then we say that Z (and the family of subsets with the index set Z ) is primarilyfit at p , Suppose that Z is a subset of Y and p is an accumulation point of Z. An accumulation degree of p is defined by @(PI = minil AllP E A - { P I } .
Let p be an accumulation point of a space Y with an accumulation degree K. Then there exists a subset D of Y such that ID I = K and p E b - D. Well-order D = { y,I ct < K}; then for any point y, of D there exists a neighborhood V of p with { ys 1 /3 < a } n V = 0,and thus D is primarily fit at p. The sets A and B defined in the example of the simple accumulation point, stated just before Theorem 1.12, are both primarily fit at the point p given there. The following lemma is interesting in the sense that many results on normality of product spaces include conditions which imply the equality of the upper intersection and the intersection of a family of subsets playing central roles in the proof of the results, and the equality comes from the fitness of the family assured by the conditions. This situation will be seen in many places in this chapter, too.
X and Y be spaces and let an order be deJined on a subset Z c Y, and let (A,I y E Z c Y } be a decreasingjit family of subsets of X . Then the upper intersection of the family at p E 2 - Z is the same as the intersection of {A,1 y E Z,,} for any jit part Z, of Z for p ; and the upper intersection of the family at q E Z is A,. 2.6. Lemma. Let
Proof. It will be enough to verify the first assertion, for which it suffices to show that the upper intersection of the family is contained in the intersection. Take an arbitrary pointy, in Z,; then we can find a neighborhood V, ofp such that y 2 y , for all y in V, n Z. H =
n u n A,.
Vc.%’(p) y c V n Z
H
=
A,
= yeynZ
= A,,,
Y‘ZP
2.7. Corollary. Let X and Y be spaces and { A , I y fit family of closed subsets of X . If
E
Z c Y } be a decreasing
M.Atsuji
96
for each point p in z - Z and for a fit part Z, of Z for p , then
UP,
x {YHY E
z>
is a closed subset of X x Y. The proof is obvious by Remarks 2.3(1) and the lemma above. 2.8. Definition. Let Z c Y and Z be ordered, and let 3 be a family of
subsets of Z. If, for any decreasing closed family f = {F, I y E Z } in a space X with the property that every family {F, E 9 1y E Z’}, Z’ E 3,has the empty intersection, there exists an open family 59 = {G,I y E Z } such that Fy c G, for all y E Z, and n{GyI y E Z ’ } = 8 for every Z’ E 3,then X is said to have a property Q(Z, a). Especially, in case %” = {Z},the property 9 ( Z , 3)is renamed a property 9(Z).
2.9. Corollary. Let Z c Y be ordered, and let %” be a point finite family of subsets Z, of Z, u E R, such that Z, is a primarily fit set at a point p , with fit part Z, for every u E R. Z f X x Y is normal, then X has the property Q(Z, 3).
Proof. Suppose that 9 = {FYIy E Z } is a decreasing closed family in X with the property that the intersection of fa= {F, E f I y E Z , } is empty for each u E a. Since Z, is a primarily fit set at pa with fit part Z , , the upper intersection of faat pa is empty by Lemma 2.6. Applying Theorem 2.4 to the families fa and { B y [y E { p a } } , Bpa= X , we have an open family 8, = {E,I y E Z,} such that Ey 3 F, for y in Z, and the intersection of {E, 1 y E Z,} is empty. Each point y in Y is included in only finitely many Z , , , . . . ,Zan,and wedefine G, = n{EZIEz3 F,, z E Zui,1 < i < n}. Thus we have the property 9(2,9’) of X . 0 The following corollary will be used in the next section.
2.10. Corollary. Let Z , and Z , be disjoint subsets of a space Y which are similarly ordered and fit, where the similarity is established by a map f : Z, + Z , , and let z.,i = 0, 1, be a family of subsets Z,, of Z , , u E R, such that (i) the similarity between Z,, and Z , , is established by f for each u E R, (ii) Z,, n z,, # 8for every u in R, and (iii)for any point y in 2, - Z , there exists an u in R such that Zi, is afit part of Zifor y , i = 0, 1. Zf X x Y is normal, then X has the property 9(Zo,3,).
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Proof. Let 9, = {F, I y E Z , } be a decreasing closed family in X such that the intersection of 9, = {F, E 9 , l y E Z,,} is empty for each a in R. We for every z in ZI and 9, = {F, I z E Zl}; then 2FI is a define F, = decreasing closed family. The upper intersections of Soand 9, at any point y of Yare disjoint. Indeed, if y E Zi - Z i ,then the upper intersection of at y is empty by assumption (iii) and ()Fa= 0 and by Lemma 2.6. If y E Z i , then y 4 Z,, j-= 1 - i, so the upper intersection of 5. at y is empty in both cases of y 4 Z, and y E Ziby Lemma 2.6. Therefore, by Theorem 2.4 there exist open families 9 = {D,ly E Z , } and I = {E, I z E Z,} such that G, = 0,n E’(,,, 2 F, for y in Z,, and the upper intersections of 9 and 6 at any point of Y are disjoint. Let a E R and x E X , and let p be in Z,,, n Z,,. x does not belong to the upper intersection of either 9or 6 atp. Suppose that
e.
X #
n
= H;
v a ~ ( p z) E v n Z l
then
n $,
x 4 H 3
ZEZlpl
and x is not in EZfor some z quently, we have (-{G,ly
E
Z,,}
=
= f( y),
0
y E Z,, , hence x is not in
G,. Conse-
for every a in R.
It is obvious that we can express the property 9 ( Z ) in other words as follows: any increasing open cover {G, I z E Z} of X can be shrunk (Rudin [1985]); namely, there exists a cover { H , I z E Z} of X with R2c G, for all z E 2 ; the latter cover is called a shrinking of the former. In general, a space is said to be (A-)shrinking,A an infinite cardinal, if every cover (of cardinality < A) of the space can be shrunk. Obviously, a A-shrinking space is normal and countably paracompact. 2.11. Definition. Suppose that I Z I is an infinite cardinal K and Z u { p} has an order similar to K + 1. In this special case 9 ( Z ) is called a property ~ ( I c ) . A space with the property 9 ( ~ for) any infinite K is said to have a property 9.’ If an open family {G, I y E Z}, whose existence is required corresponding to a monotone decreasing closed family in the definition of ~ ( I c ) is, also required to be monotone decreasing, then the property 9 ( ~ (or) 9)is called a property W ( K )(or W)(cf. Zenor [1970]). ’The author of this chapter temporarily called the property 9 ( ~ a property ) B*(K)in Atsuji [1977]. The property D was named a weak I-property by Yasui [1972]. Now, following Rudin [1985], the author would like to use the reasonable new term.
98
M. Alsuji
Trivially, the property 9 ( ~ implies ) countable paracompactness, and K shrinking =$ 9 ( ~ 6)1 ( ~ ) . Let Y be a space including an accumulation point p with accumulation degree K. Then, as stated before, Y contains a linearly ordered subset 2 which is similar to K and primarily fit at p , so, by Corollary 2.9, we have a following corollary. 2.12. Corollary (Atsuji [1977]). Suppose that a space Y includes an accumulation point p with an accumulation degree K. r f X x Y is normal, then X has the property 9 ( ~ ) .
A normal space without the property 9 ( ~ is called ) a K-Dowker space. An existence of K-Dowker space for an arbitrary infinite K is given by Rudin [1978], and it will be introduced in Definition 2.24 and Theorem 2.28. Thus Morita’s conjecture I is now answered affirmatively by the existence of ic-Dowker space along with the corollary above:
2.13. Theorem. Morita’s Conjecture I holds, namely, i f X x Y is normal for any normal space Y, then X is a discrete space. This theorem shows us that the property 9 ( ~ is a) rather interesting notion in the study of product spaces, and we have so many results on the property. Let us mention here some of the known basic properties of the property 9 ( ~ ) and the property B(K). 2.14. Facts. (1) The property 9(o)is equivalent to the countable paracompactness in topological spaces (cf. Ishikawa [19551). Yasui [ 19721 gives, among others, the following two examples: (2) o,is the normal space with the property 9 and without the property 1; (3) The Sorgenfrey line has the property 1,but its square does not. It is not hard to see that (4) i f X is K-paracompact, then X has the property B(K)(cf. a n o r [1970]). Using an example of Navy [ 19811, Rudin [1983](and [1985]) shows that the converse of this fact is not true (see also Fleissner [1984]), namely, ( 5 ) the property B, and also shrinkabiliiy, of a normal space does not imply paracompactness. She furthermore notices in the same note that (6) I-paracompactness for all I < cf(K) of a normal space does not always imply the property 9 ( ~where ) , Cf(K) is a confinality of K (cf. Theorem 2.28).
Normality of Product Spaces I
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Ohta [1987] obtains the following result on this issue (for details, see Yasui’s Chapter 5, Section 2; the definition of “FCP” is given at the end of this list): (7) Let X be a regular space. X is paracompact if and only if jor every increasing cover Y = (G, I a < u} of X there exists an FCP increasing cover (H,I a c u} of X with R, c G, for each a < K. K. Chiba [1984] shows that (8) for any regular different cardinals K and I the property 9 ( u ) does not imply the property 9 ( A ) ; and (9) the property 9 ( u ) is not productive for any regular cardinal u. (10) (Ohta [1983]) If X is a normal space that is a closed continuous image of a space with the property 9,then X also has the property 9. However, a closed continuous map does not preserve the property 9 in Tychonoff spaces. (1 1) (Rudin [19831). Closed continuous maps do not necessarily preserve the property W in normal spaces. (12) (Yasui [1984]). Every cover of a topological’spaceby open subsets with perfectly normal closures can be shrunk. Particularly, every perfectly normal space has the property 9. (K. Chiba [1984]). See also Yasui [1988a] and Chiba-Przymusinski-Rudin [19861.
A family 9 of subsets of a space X is said to be finitely closure preserving (FCP)in X if for any subfamily 9’ of 9 and for any point x in X there exist a neighborhood U of x and a finite subfamily 9” of 9‘ such that U n (Up’) t Cl(U9”). We have many interesting results connected with this subject (cf., e.g., Yasui [1986, 1988b] and Chiba [1987, 19881). For instance, results on equivalence of the property 9 and countable paracompactness on Z-product spaces are given in Chiba [1982, 19851. Moreover, including an answer to Chiba’s question, Yajima [1986] shows results on the property 9 and shrinkability of Z-products. As for the relation between shrinking spaces and the normality of products we additionally cite Hoshina [1984] and Rudin [1985]. For further information on this subject the reader is referred to Yasui’s Chapter 5 in this volume. The next corollary is an immediateconsequence of Corollary 2.12 and Facts 2.14 (1). Finding conditions on X under which the converse of Corollary 2.15 holds is an interesting problem. As we shall see in Section 3, one of the conditions is the property 9(Z, 9’) for some 2 and 9’ (Theorems 3.1 and 3.3). Taking another point of view in Chapter 4 of this volume, Hoshina shows another form of these conditions and develops it to the case of a Lainev space M :a closed image of a metric space.
M. Atsuji
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2.15. Corollary. Zfthe product X x M of a space X and a nondiscrete metrizable space M is normal, then X is normal and countably paracompact.
Let us now consider the converse of Corollary 2.9. When spaces X and Y are neither compact nor metrizable, we know quite few cases of normality of X x Y, so in order to search conditions for normality, we cannot help fairly restricting the structure of one of the factor spaces as we shall see in the following theorem (cf. Theorem 2.13). 2.16. Theorem! Let X be a normal space, and let Y = Z u { p } be a space in which every point of Z is isolated. The product space X x Y is normal if and only if any closed disjoint subsets K and X x ( p } in the product space can be
separated by open sets.
Proof. It suffices to prove the “if” part. Let A and B be nonempty disjoint closed subsets in the product. We denote
B = UP, x
(4Iz E Z } u (B, x
(PI).
For any y in Y, the sets A, and By are disjoint closed subsets of X,and there exist disjoint open subsets C, and D, of X containing A, and By respectively. Since the upper intersection of {A, I z E Z } at p is contained in A, by Remark 2.3 (I), the upper intersection of the family {(A, - C,)lz E Z } at p is empty, and by our assumption we have an open family { E , J zE Z } with A, - C, c E, c C,for every z E Z whose upper intersection at p is empty. Any point t of B, has an open neighborhood U, c D, satisfying Qn Iz E = 8 for some neighborhood of p .
(u{&
v})
F = u{V, x v l t ~ B,}
and
G
= ~{E,X(Z}~ZEZ}U(C,XY)
are the disjoint open sets containing Bp x { p } and A respectively. In the same way we have disjoint open sets F‘ and G’ containing A, x { p } and B. Thus A and B are separated by the open sets
G n (U{C, x { z } I z
E
Z}) u
F‘
4This theorem is explicitly stated in Starbird [1974], but it is essentially the same as a fact proved in Theorem 1 in Alas [1971], and the ideas of proofs in both papers are also identical. Our way of proving the theorem will follow Alas.
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and G’ n (U{Q x { z } I z E Z } ) u F.
0
2.17. Corollary. Let Z be a directed set, let Y = Z u { p } be a space in which every point except p is isolated and p has a local base Y consisting of all sets of the form V, = {z‘ E Z I z’ 2 z } u { p } for z in Z , and let Y be pointfinite at every point of Z . The product space X x Y of a space X with Y is normal if and only i f X is a normal space with the property 9(Z).
Proof. The “only-if‘’ part follows from Corollary 2.9 because Z is primarily fit at p . “If” part: By the theorem above it suffices to show that disjoint closed subsets A and X x { p } of the product space can be separated by open sets. Define B, = { x E X l ( x } x V, n A # S}; then, since {BzI z E Z} is the decreasing closed family with empty intersection, by the property 9 ( Z ) , there exists an open family { C, I z E Z} such that n{c,I z E Z} = 8 and C, 3 1,for all z E Z. Since (X - C,) x V , n A = 8 and only finitely many V , include a given point in Z,
(*I
U{(X - C,) x
V,lz E
z>
is a clos.:d subset of X x Y disjoint from A and containing X x { p } .
c]
The assumption of this corollary that Y is point-finite on Z is used only to say that the set (a) in the proof is closed, so we can replace the assumption by any one which assures the closedness of the set, for instance, by the condition “Z is linearly ordered and the family { C, I z E Z} is monotone decreasing”. The following example of Ohta is situated at an interesting place in Morita’s Conjecture I1 and suggests a new conjecture concerning Morita’s one on which we shall touch in Section 3. 2.18. Example (Ohta [1981]). Let r be the family of all finite subsets of wI, r = w;“; then r is an ordered set with respect to set inclusion, and we define wI > a for all a E r. We denote Y = r u {aI} and define a topology on Y in such a way that wIhas basic neighborhoods { fl E Y I fl 2 a } for each a in r and ,every element in is isolated. Then, by Corollary 2.17, X x Y is normal if and only if X is a normal space with the property 9(r),namely, every increasing open cover {G, I a E r}of X has a shrinking.
I02
M.Atsuji
Let X be a class of spaces. We denote by N ( X )the class of those spaces x Y is normal for every Y in X . Theorem 2.13 shows that, for the class X of all normal spaces, N ( X )is the class of discrete spaces. In many other cases of characterizing N(.X),all members of the class X are contained in a class of compact spaces (cf., e.g., Theorem 1.13, Corollaries 2.21 and 2.23). The following theorem of Yasui gives a characterization of N ( X )for a class X which is almost disjoint from the compact class. We equip YO&) = K + 1 with a topology in which K has neighborhoods in the usual order topology of K + 1 and each element in K is isolated.
X such that X
2.19. Corollary (Yasui [1983]). The product X x YO(x) of a space X with &(K) is normal ifand only i f X is a normal space with the property W ( K ) .Thus a space X is a normal space with the property W if and only if X x Y0(x) is normal for every infinite K.
Proof. Noticing the remark after the proof of Corollary 2.17, we see immediately that the "if" part of the first assertion in this corollary follows from Corollary 2.17. The "only-if" part of the first assertion follows from the relation, derived from fitness of YO&),
for a monotone decreasing closed family {FaI c1 < K}, and from the fact that { G, I o! < K} is monotone decreasing. 0
If K is a limit cardinal with cf(rc) = w , then K has a countable local base in YO@),and by Corollaries 2.17 and 2.19 we have the next corollary. 2.20. Corollary. Let K be a limit cardinal with cf (K) = w , then the following are equivalent 1 (1) X is normal and countably paracompact. (2) X is a normal space with the property W ( K ) . (3) X x Y,(K)is normal. Combining Corollary 1.17 with a special case of the corollary above, we have the following corollary.
2.21. Corollary. The following are equivalent for a space X : (1) X is normal and countably paracompact. (2) X x Z is normal.
Normality of Product Spaces I
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(3) X x (w + 1) is normal. (4) X is a normal space with the property W ( o ) . ( 5 ) X is a normal space with the property 9 ( w ) . One more interesting application of Theorem 2.16 is the following Alas’ result, for which we need the next well-known Dowker-KatEtov’s theorem. 2.22. Lemma (cf. Nagata [1985, V.3.D]). Let K be an infinite cardinal. A normal space X is countably paracompact and K-collectionwise normal if and only i f , for any locallyfinite family ( H aI a < K} of closed subsets of X,there exists a locallyfinitefamily {G,I a < K} of open subsets of X with Ha c G, for each a < K.
Let YI(rc)= K + 1 be the one-point compactification of the discrete space Remembering Remarks 2.3, we recognize immediately that the following three statements are equivalent. For a family d =’ {A, I a < K} of subsets of a space X and the subset A = U { A , x { a } I a < K} of X x Y,(K), (1) d is locally finite in X ; (2) A n (X x {K}) = 0; (3) the upper intersection of d at K is empty. Thus we obtain easily the following from Theorem 2.16 and the lemma above (cf. Theorem 1.14). K.
2.23. Corollary (Alas [19711). X x Y I @ )is normal if and only if X is countably paracompact and K-collectionwise normal. Following Starbird [1974], we call a space X a test space for a topological property P if the product X x Y is normal if and only if Y has the property P,namely, N ( { X } ) = P. Summing up, in connection with test spaces, the results mentioned above from Section 1 and some questions, we have the following Table 1. Now, let us introduce the well-known example of a K-Dowker space given by Rudin [1985], which plays important roles in the theory of product spaces: for K = w ,in virtue of the existence of an w-Dowker space, we can conclude that normality does not imply countable paracompactness, settling a long unsolved problem; for an arbitrary K, the existence of K-Dowker space is an essential half of Theorem 2.13; furthermore, the K-Dowker space X, constructed below has a fairly strong property so that it serves to clear relationship between various properties of spaces (cf., e.g., Facts 2.14(6); Rudin
M.Atsuji
104
TABLE I Test space I" D" K +
I
I
Property
Refer to
normal and K-paracompact
Theorem 1. I3
ltollectionwise normal and K-paracompact
Theorem 1.14
normal and countably paracompact
Corollary 1. I7
normal and property g ( K ) countably paracompact and K-collectionwise normal normal and property W K K-shrinking
Corollary 2.19 Corollary 2.23
)
[1983, 1984, 19851). Though its proof is not simple, we shall show the details of X, because, besides the importance observed above, a nicer example of a K-Dowker space is wanted; cardinal functions of X, seem larger than necessary as Rudin herself feels (cf. Rudin [1985]). Let K be an infinite regular cardinal and IC+ the successor cardinal for K. For a cardinal I we say that a Hausdorff space X is I-ultraparacompact if every open cover with the cardinality 1of X has a disjoint open refinement. Trivially, a I-ultraparacompact space has the property 9 ( I ) and the property A?@),and it is I-shrinking, I-paracompact and normal. 2.24. Definition. Let {A, I a < K}be a monotone increasing family of regular cardinals with I, = A: and I,, > K + . X, is the subspace of the box product (A, 1) consisting of mapsf for which there exists a /3 < K such that K < cf(f(a)) < I, for all a < K.
+
We denote F =
n (I, + 1)
U
and C =
{YEFly(.)
< I, for all a <
K}.
Forfand g in F we definef < g (orf < g) iff(a) c g(a) ( o r f ( a ) < g(a)) foralla < ~ . F o r g< f i n F w e l e t B , / = {hEX,lg < h < f } , a n d i t i s a
Normality of Product Spaces I
105
basic open neighborhood off in X , iff is in X,. It is easy to see that X , is Hausdorfland has the property that the intersection of < K open subsets of X , is open. For the proof that X , is a K-Dowker space we need three lemmas, the first ), of which is the one for the proof that X , does not have the property 9 ( ~and the other two are for A-ultraparacompactnessof X,. Note that in the first half of the proof of Theorem 2.28, actually, we verify that X , has a monotone increasing cover {G, 1 a < K} which cannot have any, not necessarily open, shrinking. 2.25. Lemma. Let a < K and G, = { f E X , If (B) < A, for all B 2 a in K}, and let H be a subset of X , with G, 3 R ( = the closure of H in X,). There exists an f E C with {g E C n HI f < g} = 8.
Proof. Index the set of maps P = {p:(a + 1) -,A,lp(B) < A, for all fl < a} as { p, I u < A,} in such a way that eachp isp, for A,-many u’s. If our lemma is false, then, for each u < A,, making an extension ofp, to K so that it is in C, we can inductively find a g, E C n H for each u < 1, such that P,(B) < g,(B) for B G a and sup{g,(B)It < 4 < g,(B) for B > a in K. Then we definef E X , byf(B) = A, for B < a andf(B) = sup{gU(/3)Iu < A,} for B > a in K. f is in X , and not in G,, so there exists a g < f i n F with Bd n H = 8.For each B, a < B < K, there exists a u, < 1, with g(B) < g,,(B). There exists a uo < 2, such that uo > sup{o,la < fl < K} and p,(B) = g(B) for all B < a because of g I (a + 1) E P. We already have gu0 in C n H, which is also in Bg,, a contradiction. 0 Let A < cf(K) and { 17, I p < A} be a cover of X,, and let V be an open subset of X,. We define the top of V: t V = t E F by t(a) =
sup{f(a)lf E V >
for all a < K, and we denote for y <
V,
= {g E VI cf(g(a)) <
K
1, for all a <
K}.
2.26. Lemma. ZfA < cf(K) and cf(t(a)) > K for all a < K, then for every y < uthereexistanf, < tinFandap, < Asuchthat{gE V , l f , < g} c U,,?.
Proof. Suppose that our lemma is false. Then, for some y < K and for any f E F w i t h f < t a n d f o r a n y p < 1 , w e c a n f i n d a g E V, - U,withf < g.
106
M.Atsuji
We denote A = {a < KIcf(t(a)) Q A,,}.
For any a E A we choose a T, c t(a) which is cofinal with t(a) and I T,I = cf(t(a)). Since I,, = I; and I < K < I,, we can index Q = nUEA T, as{p6,,lb < h a n d p < 1)insuchawaythatpE Qisp6,,forA,,many6’sfor each fixed p. Define the order /3v < 8 p lexicographically. By the induction on the index, we choose gap E V, - U,,for all 6 < A,, and p < I as follows. Definefa,, E FbYf,,(a) = P6p(a) for a E A andfa&(a)= sup{g,v(a) I Bv < dp} for a E K - A. Since gayis in V,, we have g,,(.)
Q r(a)
and cf(g,v(a)) < A,, < cf(t(a))
for a E K - A, and sofa,,(a) < t(a) for a E K - A because the number of /3v is less than A,,, and hence,fa,, < t . By our assumption we can select gJp> fa,, in V , - Up. Finally, define g in F by g(a) = t(a) for a E A and g(a) = sup{g,,(a) 16 < A,,, p < I}for a E K - A; then it is in X, and in some Urn, and hence there exists an h < gin Fwith Bhg c Urn.Since ~r, < I < K , there exists a 6, < A,, with g,#,(a) > h(a) for a in K - A; and by our choice o f p we can find 6 > sup(8,l a E K - A} withp6,(a) = h(a) for a in A. For a in A we have h(a) = p6p,3(a) =
< g6m(.)
t(a) = d a ) ,
and so h < gJrn< g; hence we have g6rnE Bhgc Urn,a contradiction. 0 2.27. Lemma. Suppose I < cf(u). Let t be the top of an open subset V of X, which is covered by {U,, I p < I}.Ifcf(t(a)) > K for all a < K , then there exist anf < t i n F a n d a p < Isatisfying{gEVlf < g } c Up.
Proof. For every y < K we have& < t and 4 < I stated in Lemma 2.26. Define f by f(a) = sup{&(a)ly < K } for all a < K; then f < t . Since I < cf(K), there exists a p < I such that p = c ~ yfor K many y’s less than K. Supposef < g E V . Since V = U,<, V , by the definition of V , , g is in some V,, v < K, and there exists a y > v in K w i t h 4 = p, and s o g is in V, 3 V,. Since& < f < g, g is in Uy.= U,,by Lemma 2.26. 0 2.28. Theorem. space.
If I < cf(K),
then X, is a I-ultraparacompact Ic-Dowker
Proof. , Let us first show that X,does not have the property 9(ic).We define an open set for every a < K C, = {fE X,If(fi) < I, for all /3 2 a in K}.
Normality of Product Spaces I
107
Suppose that the monotone increasing cover { GaI a < K } of X , has a shrinking {HaI a < .}. By Lemma 2.25 there exists anf. E C for every a < K such that
(*I
{ g E C n HUIL < g> =
8.
We may consider j#?) > sup{f,(B) I a < y} for every y < Define g in F by g(B) = sup{fa(b)Ia < }.
+ K+
for B <
K
and
B<
IC.
K;
then we have g(B) < I,, cf(g(B)) = K + and g E C n X,, and hence, g is in C n Hu for some a < K, which contradicts (*). Let us next show that X,is I-ultraparacompact. Let 4 = {Up1 p < I}be a cover of X,.We shall construct an open disjoint refinement W of 9. For this purpose we first define, by the induction on 0 < K + , a cover W, of X, consisting of disjoint open sets with the property that 7 < 0 < K + and W E Woimply the existence of V E WTsuch that . (1) w c V; (2) if W c t Up for any p < I, then tw # t,; (3) if V c U,, for some p < 2, then W = V. We let Wo= { X , } , and assume that W7has already been defined for all z < Q, where 0 < Q < u+. (i) When Q is a limit, then W, = W,l W , E WT for z < 0 and W, 3 W , for e < T < Q } is a desired cover satisfying the three conditions above because the intersection of < K open sets is open. (ii) When Q = T 1, then for each V E W7we define “w; as follows. If V c U,,for some p < I , then we define W, = { V}. Otherwise, dividing into two cases, we shall define W, below as an open cover of V by disjoint subsets of V which do not have t , as their top if they are not contained in any Up. If we obtain such W, for every V E *w,, then W, = U{WvI V E WT}is a desired cover of X , satisfying the three conditions. Now suppose that V Up for all p < I, and denote t = t,. Case 1: cf(r(a)) < K for some a < K. Let R = {adld < cf(t(a))} be a monotone increasing closed family of ordinals cofinal with t(a), where a, = 0.Foreachy < cf(t(a))weletW, = { f ~Via, < f ( a ) < a,+,}.Since cf(f(a)) > K by f~ X , and cf(a,) < K for limit y’s and also R is closed, W, = { W,ly < cf(t(a))} is a desired cover of V. Case 2: cf(t(a)) > K for all a < K. By Lemma 2.27 there exist anf < t in F and’a p < I satisfying {g E Vlf < g} c U,,. For S c K we denote W, = {g E Vlf(a) < g(a) for a E S and g(a) < f(a) for a E K - S}. If S # K, then tw, # t, hence W, = { W, I S c K } is a desired cover of V.
{n7<,
+
+
M.Arsuji
I08
Now we define W = { W E W,ICJ<
IC+
and W c U,forsomep < A},
then this is a desired refinement of 4. In fact, to see that W covers X , , take f i n X , . For any CJ < IC+ there exists a W, E W , including5 If T < 6,then, by condition (l), there exists a W , containing W,. For each a < IC we have t,(a) 2 t,(a), where ts = tw,, and so there exists a T= < K + such that t J a ) = t,(a) for all CJ > in K + . For r0 = sup{z,)a < K} < K + and for CJ with T,, < CJ < IC+ and W, 3 f we have t,, = t,. By condition (2), W, c U, for some p < I, and W, belongs to W . Obviously, W is made up of disjoint open sets by its construction and condition (3). Trivially, W refines 4. 0
2.29. Comment. Though it is an essential notion in the theory of product spaces, an application of “upper intersection” is not common. It has occasionally appeared explicitly and played an important role in the literature on product spaces (cf., e.g., Frolik [1960]and Isiwata [1964]). On the other hand, the notion of upper intersection plays an interesting role in another area of general topology. From a point of view of nonstandard analysis, Bernstein [1970] first defined and studied an 9-compact space, 9 a free ultrafilter on o,which was defined by use of an 9-limit point of a sequence of points {x,}. A point x is defined to be an 9-limit point of the sequence if x is in the set
namely, in our terms, an 9-limit point is a point in the upper intersection of the sequence. An 9-limit point is, if it exists, of course uniquely determined in Hausdorff space. 9-compactness is situated between compactness and countable compactness, and it is infinitely productive. Ginsburg and Saks [1975] generalize the notions of Bernstein. They call a point x an 9-limit point, 9 E Po - o,of the sequence {A, I n E o}of subsets of a space if x is in
n oxA,,
FEf
nsF
namely, in the upper intersection of the sequence. They study countable compactness and pseudocompactness of product spaces by making use of generalized 9-compactness. We also have interesting works of J. E. Vaughan on, specifically product spaces of, countably compact spaces by use of 9-limit points (cf., e.g., Vaughan [1984]).
Normality of Product Spaces I
109
3. The second and third of Morita’s three conjectures Before discussing Morita’s Conjectures I1 and 111, we show a few results on product spaces with a metric factor. Suppose that Mis a metrizable space with a a-locally finite base W = UW,, such that for every n a locally finite cover W,, of M has a star refinement a,,+, , a diameter of every member of which is less than l / ( n 1) with respect to some metric on M. Let W be the family consisting of members of W which are not discrete subspaces of M, and let W; = W‘ n W n. Let us show that we can inductively select a point p BE B for each B E a’in such a way that { pB1 B E W’} consists of distinct points. Well-order 49; = {litna}for each n and define an order nu < m/3 lexicographically. Suppose that we have already selected a desired point in En, for all na < m/3. Since Emsincludes an accumulation point which has a neighborhood meeting only finitely many members of U{W,,I n < m ) , Emsincludes a point p,,, which is not in {pen,I na < m g } . Thus we have a set
+
zo
=
{PeIBEW’J
of distinct points. We define an order on Z , by pel For any accumulation point z of A4 we denote 2, =
{PB
E ZOlZ E
< p,,
for El
3
B,.
B E @’}
and
9,=
I z is an accumulation point in M j.
It is easily seen that if M is not discrete, then a space with the property 9 ( Z o , 9,is )countably paracompact. 3.1. Theorem. Let M be a nondiscrete metrizable space. Ifthe product space X x M of a space X with M is normal, then X is a normal space with the property W Z o ,9,).
Proof. First, we show that Z , is a fit set. Let z be in 2, - 2,. For any p,, E Z , there exist an no with B, E W;, and an n, 2 no with St*(z, ail)c B,. and a neighborhood V of z such that {pel B E We find B, 3 z in U,,<,,l Wi}n V = 8. For everyp, in B, n V it holdsp, 2 p,,, and hence Z , is a fit part of Z, for z. Similarly for z E Z,, and thus Z, is fit. Next, we select a set Z, = { q B E BI B E W’}in the same way with Z , such that Z, n Z, = 8.Z, and 2, are similarly ordered, and Z, and Z, satisfy all the conditions required in Corollary 2.10. 0
M . Atsuji
110
In order to prove the converse of the theorem above, we shall use a lemma which is a very special case of a theorem given by Morita [1962].
3.2. Lemma. The property of being countably paracompact normal space is hereditary with respect to Fa sets.
Proof. The proof is not hard, so we only sketch it. First, we verify inductively that a set A which is a countable union of normal closed subspaces F, of a space Xis also normal. Next, if Xis normal and countably paracompact, then F, x I are normal closed subspaces of A x I by Theorem 2.21, and A x I is normal by the fact verified above, hence A is normal and countably paracompact by Theorem 2.21 again. 0 3.3. Theorem (Morita [1964]). If X is a normal space with the property 9 ( Z o ,To), then the product X x M of X with a nondiscrete metrizable space M is normal and countably paracompact. Proof. Let 0 = (0, I i E o}be a cover of X x M. We denote for B E 93 and
k
E
w
GB,k - U { L I L is an open set with L x B c O k }
and GB
=
U{GB,kl
‘We let G,, = GB for B U{G,,lz
E
E
O>.
93‘; then {G,, JpBE Z , } is increasing, and
BEW}
=
X
for any accumulation point z in M. By the property 9(Z,,, Z,), there exists an open Fa set HPBof X for each B in A?’ such that c G,, and
F,,
~ { H , , ) Z BE E a’}
=
X,
for any accumulation point z. We denote HB = HpBfor B E a‘ and HB = X or 0 for R F W - W according to B consisting of one point or otherwise; then we have HB c GBfor all B E 93 and U { H B l z ~ B ~ 9= 3 }X
for any point z in M. Since X is normal and countably paracompact, HB is also, normal and countably paracompact by the lemma above, and {GB.k n HB I k E w } is a normal cover of HB, and the cover {(GB,&
HB)
E w,
Normality of Product Spaces I
111
of H , x B is also normal. Since 9 is a-locally finite, X = (H,x B1 B E a} is a a-locally finite cover of X x M , and so it is a normal cover because H, and B are the cozero sets. {(GB,kn H,) x B I k E w } is, as stated above, a and normal cover of H, x B which refines the cover {(H, x B) n ok 1 k E a}, 0 is a normal cover by Lemma 1.9. 0 The definition of the property 9 ( Z o To) , is motivated by Characterization Theorem 4.1. in Przymusihski [19841, and one of fundamental notions of the theorem is essentially a basic cover of X x M with a special refinement defined by Morita [ 1963b], which is also one of main tools throughout Hoshina’s Chapter 4. 3.4. Definition (Morita [1964]). A space Xis said to be a P-space if, for any cardinal number K 2 1 and for every decreasing family {F,I a E r}, r = do, of closed subsets of X , there exists a family {GoI a E r}of open subsets of X such that F, c G, for each d E r and n{F,,.1 n E w } = 0 implies n{G,,. In E w } = 0 for every f E K”.
The following two theorems will be verified in Hoshina’s Chapter 4. As the converse of the theorems, Morita formed his conjectures. It is pointed out in Morita [1977] that Morita’s Conjecture I11 is affirmatively answered if his Conjecture I1 is, because, as is easily seen, any normal P-space is countably paracompact. Furthermore, it is known that every perfectly normal space is a P-space.
3.5. Theorem (Morita [1964]). In order that the product space X x M is normal for any metrizable space M , it is necessary and suficient that X is a normal P-space. Morita’s Conjecture 11. Let X be a topological space. X is metrizable if and only if the product space X x Y is normal for any normal P-space Y.
3.6. Theorem (Morita [1963b]). Let M be a metrizable space. In order that the product space X x M is normal for any countably paracompact normal space X , it is necessary and suficient that M is a a-locally compact space. Morita’s Conjecture In. Let X be a topological space. X is metrizable and a-locally compact if and only if the product space X x Y is normal for any countably paracompact normal space Y.
112
M. Atsuji
Morita’s Conjectures I1 and I11 are not completely solved yet. They are affirmatively answered under the set-theoretical assumption V = L (ChibaPrzymusinski-Rudin [1986]). However, they may be solved in ZFC because the assumption is used only in constructing an example; or they may have a counter example in some model for set theory, and thus Morita’s Conjecture I1 may be undecidable as Chiba, Prszymusinski and Rudin conjecture (cf. ibid.). So at present, we like to begin by mentioning two results on the conjectures obtained in ZFC by T. and K. Chiba. Since many important propositions on Morita’s Conjectures have too long proofs to cite in this chapter, we shall state them without proof. But it will be worthy to note that proofs of almost all key results heavily depend, directly or indirectly, upon the well-known example “H” of Bing [1951] or its modification in such a way that, under some assumption, if X is not metrizable, then X x Y is not normal for some perfectly normal space Y , the example H of Bing. The first remarkable step to Morita’s Conjectures I1 and I11 is the following theorem. 3.7. Theorem (T. and K. Chiba [1974]). Let X be a separable space. The following are equivalent: (a) X is metrizable. (b) X x Y is normal for any normal P-space Y . (c) X x Y is normal for any perfectly normal space Y .
The second large step to the conjectures will be this following theorem. 3.8. Theorem (K. Chiba [1980]). Let X be a closed image of an M-space. Then X is metrizable if and only if X x Y is normal for any normal P-space Y .
In the rest of this section we introduce the main results from ChibaPrzymusinski-Rudin [19861, which include interesting results not only on Morita’s conjectures, but also on related topics on perfectly normal spaces. 3.9. Definition. A Bing-type example H ( K ) , IC a cardinal > o,is a normal space With the following properties: (1) H ( K )contains a closed and discrete subset D, I D I 2 IC, consisting of functionsf: T + o for some T ;
Normality of Product Spaces I
113
(2) basic neighborhoods of elementf E D have the form B( f , F ) , where I; is a finite subset of T and B ( f , F ) n B(g, G ) # 8 iff1 F n G = gl F n G. The following proposition is fundamental in the paper referred to above. 3.10. Proposition. Suppose that (Y, F)is a nonmetrizable topological space and K = 19-I. There is a perfectly normal Bing-type example H ( K ) such that if Y x H ( K ) is normal, then there exists a monotone increasing family { Y, I a < o1} of sub.reis of Y such that
In particular, Y has an uncountable tightness.
The following example is given by BeSlagiC and Rudin [1985] under the assumption V = L. 3.11. Example ( V = L). For every cardinal 1there exists a collectionwise normal P-space X , having a monotone increasing open cover { U, I a < wI} with the property that if C,, is a closed, in X,,subset of U, for each u < o, and /3 < 1,then U{CasIa <
0 1 ,
B<
A}
+
3.12. Theorem ( V = L). Morita’s Conjecture I1 holds, namely, a space X is metrizable if and only if X x Y is normal for any normal P-space Y.
Proof. It suffices, by Theorem 3.5, to prove the “if” part. Suppose that X x Y is normal for any normal P-space X , and Y is not metrizable; then there exists a monotone increasing family { Y,la < o,}of subsets of Y satisfying relation (*) in Proposition 3.10. Let I YI < A. Index Z = u{Y,la < ol}= { y , l B < l}.Foranyyswetakea = min(y(Y,3ys)and rewrite y a p for y , . We define an order ap < yd lexicographically; then { y,, I a < wI,/3 < A} is primarily fit at a point y in the left-hand side of relation (*) in Proposition 3.10. Let X , and {Vala < ol} be the space and its cover given in Example 3.1 1. We let La, = X , - U,,, where U,, is the U , corresponding to y,,; then {L,, I yaSE Z } is the monotone decreasing closed family with the empty upper intersection at y by Lemma 2.6. Since the product X , x Y is normal, by Corollary 2.9 there exists an open family {DapIy,, E Z } with the empty intersection such that L,, t D,,,which is impossible. 0
114
M.Aisuji
3.13. Theorem ( V = L). Morita’s Conjecture I11 holds, namely, a space X is metrizable and a-locally compact if and only if X x Y is normal for any normal and countably paracompact space Y .
Proof. Suppose that the product space of X with any normal and countably paracompact space is normal. Then, since a normal P-space is countably paracompact, the product of X with any normal P-space is normal, so X is metrizable by Theorem 3.12, and hence X is a-locally compact by Theorem 3.6. Making use of a previous notation, we can state Theorems 3.12 and 3.13 in other words as follows. Let X be the class of all normal P-spaces (respectively all countably paracompact normal spaces); then, under V = L, N ( X ) = A = the class of all metrizable spaces (respectivelyN ( X ) = the class of all a-locally compact metrizable spaces). ‘On the other hand, as we have seen in many places above, perfect normality and being a normal P-space are closely situated each other on the problem of metrizability of a space X whose product X x Y is normal for any space Y with the properties. Therefore, it will be natural to pose the problem of characterization of N ( 9 ) , 9 the class of all perfectly normal spaces. Trivially, N ( 9 )2 A; and it seems that N ( 9 )is nearly A; but not the same as we shall see below. This problem is not answered yet, so we shall here introduce a little more detail on perfect normality. Let us recall the space Y in Ohta’s Example 2.18. As we point out there, the product X x Y is normal if and only if X is a normal space with the property 9 ( ~ : in~particular, ); it follows from Facts 2.14(12) that X x Y is normal for every perfectly normal space X , namely, Y belongs to N ( 9 ) . Meanwhile Y is obviously a regular nonmetrizable a-discrete space. This observation leads to the inequality N ( 9 ) # A. (This inequality was first disclosed by T. and K. Chiba [1974] giving another example.) In the meantime, Ohta’s example shows us an exciting other aspect of Morita’s Conjecture 11. Namely, if there exists some model for set theory in which every normal P-space has the property 9 ( Z )for any ordered set Z with I Z I = o,, then Ohta’s example will be a counter example for Morita’s Conjecture 11, and thus, together with Theorem 3.13, the conjecture becomes undecidable. Now, denote by c ( X ) and /(A’) the Souslin number and the Lindelof number of a space X. The following proposition and corollaries are a considerable improvement of results presented in T. and K. Chiba [1974], and in K. Chiba [1974, 1976, 19801.
Normality of Product Spaces I
115
3.14. Proposition. Every space X in N ( 9 ) is stratiJiable and w ( X ) = c(X) =
&(a.
3.15. Lemma. Let 4. be in N ( 9 )for every i < a.Then the product space n,,, X, x Z is perfectly normal for every perfectly normal space Z.
Proof. It suffices to show by induction that nicn X, x Z is perfectly normal fiJr every n < o (cf., e.g., Przymusinski [1984, Theorem 6.21). Assume that S = niG, X, x Z is perfectly normal; then T = X,,, x S is normal. Since X,,, is stratifiable, it has a o-discrete network ( A , , l n < o},A, = {M,,,1 c1 E Q,} a closed discrete family. Denoting L,, = U{ VI V is an open subset of S with M,, x V c G) for a given open set G c T, and observing that {Mnax L,, I n < o,c1 E a,} covers G, we can easily check that T is 0 perfectly normal. The following two corollaries are immediately derived from this lemma; note that a perfectly normal space is hereditarily normal.
3.16. Corollary. If X is in N ( 9 ) ,then X"' x Z is perfectly normal for every perfectly normal Z. 3.17. Corollary. N ( 9 )is countably productive and hereditary. 3.18. Corollary. Every space X in N ( 9 ) which is locally ccc or locally Lindelof or locally p-space is metrizable.
Proof. X is locally metrizable and paracompact by Proposition 3.14, and 0 .thus it is metrizable. The next is an immediate consequence of Proposition 3.10.
3.19. Corollary. Every space in N ( 9 )with countable tightness is metrizable. 3.20. Corollary. Every k-space X in .N(9) is metrizable.
1 I6
M . Atsuji
Proof. Let C be a compact subset of X. C is in N ( 9 )by Corollary 3.17, so it is metrizable by Corollary 3.18, hence X is sequential because X is a k-space, and thus the tightness of X is countable. 0
References Alas, 0. T. [I9711 On a characterization of collectionwise normality, Canad. Math. Bull. 14(1), 13-15. Alexandrof, P. and H. Hopf [I9351 Topologie I (Springer, Berlin). Arhangel'skii, A. V. [197I] On bicompacta hereditarily satisfying the Souslin condition. Tightness and free sequences, Dokl. Akad. Nauk SSSR 199, 1227-1230; English translation in: Soviet Math. Dokl. 12, 1253-1257. Atsuji, M. [I9771 On normality of the product of two spaces, in: General Topology and its Relations to Modern Analysis and Algebra, Part B, I V ; Proc. Fourth Prague Topology Symp., Prague 1976, 25-27. Benyamini, Y. and T. Starbird [I9761 Embedding weakly compact sets into Hilbert space, Israel J . Math. 23, 137-141. Bernstein, A. R. [I9701 A new kind of compactness for topological spaces, Fund. Math. 66, 185-193. BeslagiC, A. and M. E. Rudin [I9851 Set-theoretic constructions of nonshrinking open covers, Topology Appl. 20, 167-177. Bing, R. H. [I9511 Metrization of topological spaces, Canad. J . Math. 3, 175-186. Chiba, K. [I9741 On products of normal spaces, Rep. Fac. Sci. Shizuoka Univ. 9, 1-11. [1976] Two remarks on the normality of product spaces, Rep. Fac. Sci. Shizuoka Univ. 11, 17-22. [I9801 On the normality of product spaces, Math. Japon. 25, 559-565. [I9821 On the weak P-property of %products, Math. Japon. 27, 737-746. [I9841 On the weak 9-property, Math. Japon. 29, 551-567. [1985] Remarks on the weak @property of Z-product, Questions Answers General Topology 3, 1-9. [I9871 On the %property of a-products, Math. Japon. 32, 5-10. [I9881 A note on the weak @-property, to appear. Chiba, T. and K. Chiba [I9741 A note on normality of product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A 12, 165-173. Chiba, K., J. C. Przymusinski and M. E. Rudin [I9861 Normality of product spaces and Morita's conjectures, Topology Appl. 22, 19-32. Dieudonne, J. [1944] Une gknkralization des espaces compacts, J. Math. Pures Appl. 23, 65-76.
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Dowker, C. H. [I9511 On countably paracompact spaces, Canad. J. Math. 3, 219-224. Engelking, R. [19771 General Topology (Polish Scientific Publishers, Warsaw). Fleissner, W. G. [ 19841 The normal Moore space conjecture and large cardinals, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 733-760. Frolik, Z. [I9601 The topological product of two pseudocompact spaces, Czech. Math. J. 10, 339-349. Gillman, L. and M.Jerison [I9601 Rings of Continuous Functions (Van Nostrand, New York). Ginsberg, J. and V. Saks [I9751 Some applications of ultrafilters in topology, Pacific J. Math. 57, 403-418. Hoshina, T. 119841 Shrinking and normal products, Questiom Answers General Topology 2, 83-91. Ishikawa, F. [I9551 On countably paracompact spaces, Proc. Japan Acad; 31, 686-687. Isiwata, T. [I9641 Some classes of countably compact spaces, Czech. Math. J. 14, 22-26. Katuta, Y. [I9711 On the normality of the product of a normal space with a paracompact space, General Topology Appl. 1, 295-319. [ 1977al Paracompactness, collectionwise normality and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku See. A 13, 165-172. [1977b] Characterizations of paracompactness by increasing covers and normality of product spaces, Tsukuba J. Math. 1, 27-43. Kombarov, A. P. [I9721 On the product of normal spaces. Uniformity on Z-products, Dokl. Akad. Nauk S S S R 205, 1033-1035; English translation in: Sovier Math. Dokl. 13, 1068-1071. Michael, E. [I9531 A note on paracompact spaces, Proc. A M S 4, 831-838. Morita, K. [I9611 Note on paracompactness, Proc. Japan Acad. 37, 1-3. [I9621 Paracompactness and product spaces, Fund. Math. 50, 223-236. [1963a] On the product of paracompact spaces, Proc. Japan Acad. 39,559-563. [1963b] Products of normal spaces with metric spaces 11, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 8, 87-92. [I9641 Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. (19751 Cech cohomology and covering dimension for topological spaces, Fund. Math. 87, 31-52. [I9771 Some problems on normality of products of spaces, in: General Topology and its Relations to Modern Analysis and Algebra, Part B, I V ; Proc. Fourth Prague Topology Symp., Prague 1976, 296-297. (198I] Theory of Topological Spaces (Japanese) (Iwanami, Tokyo). Nagata, J. [ 19851 Modern General Topology (North-Holland, Amsterdam).
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Navy, C. L. [19811 Nonparacompactness in para-Lindelof spaces, Thesis, Univ. of Wisconsin. Negrepontis, S. [I9841 Banach spaces and topology, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 1045-1 142. Nobeling, G. (19541 Grundlagen der Analytischen Topologie (Springer, Berlin). Nogura, T. [I9761 Tightness of compact Hausdofispaces and normality of product spaces, J. Math. SOC. Japan 28, 360-362. Ohta, H. [I9811 On normal, non-rectangular products, Quart. J. Marh. Oxford 32, 339-344. [I9831 On a part of problem 3 of Y. Yasui, Questions Answers General Topology 1, 142-143. [1985/86] Katuta’s questions, normality of products, Quesrions Answers General Topology 3, 11 1-123. [1987] Private Communication to Y.Yasui. Przymusinski, T. C. [1984] Products of normal spaces, in: K. Kunen and J. E. Vaughan, eds., Handbook of SerTheoretic Topology (North-Holland, Amsterdam) 78 1-826. Rudin, [I9711 [I9781 [I9831 [I9841
M. E. A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. K-Dowker spaces, Czech. Math. J. 28(103), 324-326. Yasui’s questions, Questions Answers General Topo/ogy 1, 122-127. Dowker spaces, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 761-780. [I9851 K-Dowker spaces, in: I. M. James and E. H. Kronheimer, eds., Aspect of Topology, London Math. SOC. Lecture Note Ser. 93 (Cambridge Univ. Press,Cambridge) 175-193.
Starbird, M. [I9741 The normality of products with a compact or a metric factor, Thesis, Univ. of Wisconsin. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. A M S 54, 977-982. Tamano, H. [I9601 On paracompactness, Pacijic J. Math. 10, 1043-1047. [1962] On compactifications, J. Marh. Kyoto Univ. 1, 161-193. Telgarsky, R. [I9711 C-scattered and paracompact spaces, Fund. Marh. 73, 59-74. [1975] Spaces defined by topological games, Fund. Math. 88, 193-223. Tukey, J. W. [I9401 Convergence and Uniformity in Topology, Ann. of Math. Studies 2 (Princeton Univ. Press, Princeton, NJ). Vaughan, J. E. (19841 Countably compact and sequentially compact spaces, in: K.Kunen and J. E. Vaughan, eds., Handbook of Ser-Theoretic Topology (North-Holland, Amsterdam) 569-602. Yajima, Y. [1986] The shrinking property of Z-products, Questions Answers General Topology 4, 85-96.
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Yasui, Y. [1972] On the gap between the refinements of the increasing open coverings, Proc. Japan Acad. 48,8690.
[1983] On the characterization of the a-property by the normality of product spaces, Topology Appl. 15, 323-326. [I9841 A question of G. Gruenhage and E. Michael concerning the shrinkable open covers, Questions Answers General Topology 2, 124-126. [I9861 Some characterizations of a 8-property, Tsukuba J. Math. 10, 243-247. [1988a] Some remarks on the shrinkable open covers, Math. Japon., to appear. [l988b] Another characterization of some refinement of a monotone increasing open covering, to appear. Zenor, P. [1970] A class of countably paracompact spaces, Proc. AMS 42, 258-262.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 4
NORMALITY OF PRODUCT SPACES I1
Takao HOSHINA Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
Contents 1. Products of normal spaces with metric spaces. . . . . . . . . . . . . . . . . 2. Products of normal spaces with generalized metric spaces . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 140 158
In this chapter we shall begin with the normality of the product of a normal space with a metric space and discuss related results, and afterwards extend our concern to the case of the product of a normal space with one of certain generalized metric spaces. All spaces are assumed to be Hausdorff, and maps continuous and onto unless otherwise specified. N denotes the set of positive integers.
1. Products of normal spaces with metric spaces In general, the product X x Y of a normal space X and a metric space Y need not be normal. This was shown by Michael [I9631 by constructing a hereditarily paracompact space M , which is called the Michael line, such that M x P is not normal, where P is the space of irrationals (Nagata [1985, Example V.21). On the other hand, in [1964] Morita, by introducing the notion of P-spaces, established his theorem that for a product X x Y to be normal for any metric space Y, it is necessary and sufficient that Xis a normal P-space. Then the notion of P-spaces, as well as the method of his proof of this theorem, greatly influenced many later studies in the area of normality or paracompactness of product spaces. Then, in Morita [1963] the method
122
T. Hoshina
above was sharpened further to study the normality of the product of two given spaces, one of which is metric. On such a product the notion of basic covers was defined and played an essential role. In the present section, by introducing first the notion of basic covers as a central concept, we shall explain several results on the normality of products with a metrix factor and other related results. Let Y be a metric space. Then there exists a sequence {V,,} of locally finite open covers of Y such that each member of V,,has diameter < l/n. Hence, In E N} forms an nbd basis at each point y of Y. Let V,,= {St(y , f,,) {V,,IaER,,}. Letusput
W(aI,. . , , a,,) =
v,, n
* *
n V,,"
for a, E R,, . . . , a,, E a,,.
Let X be a space. Then, following Morita [1963], an open cover Y of X x Y is said to be a basic cover if Y has the form Y = {G(a,, . . . , a,,) x W(a,,. . . , a,,) I a, E R,,
v = 1 , . . . , n ; n N} ~ with open subsets G(a,, . . . , a,,) of X such that G(a,, . . . , a,,) c G(a,, . . . , a n , an+,)
f o r a , E R , , v = 1, . . . , n , n + 1.
(1
.a
Let Y be a basic cover in (1.1). If there exists a family {F(a,, . . . ,a,,)1 a, E Q, v = 1, . . . , n; n E N} of closed subsets of X such that (1.3) U{F(a,, . . . , a,,) x W a , , . . . , a,,) I a, E Q,,
v = 1, . . . , n ; n ~ N } = X x y , then we shall say that Y has a special refinement.
(1.4)
1.1. Lemma (Morita [1963]). Let X be a normal space. Thenfor a basic cover Y in (1.1) the following are equivalent: (a) Y has a special refinement. (b) There exists afamiry {F(a,, . . . ,a,,)I a, E R,, v = 1, . . . ,n; n E N} of open F;-subsets of X satisfying (1.3) and (1.4). (c) There exists a family {F(a,, . . . ,a,,) I a, E R,, v = 1, . . . ,n; n E N} of F,-subsets of X satisfying (1.3) and (1.4).
Normalily of Product Spaces II
I23
Proof. The implications (a)-(b) and (b)*(c) are obvious. Assume (c). Then there exist closed subsets C(a,, . . . , a,,; k) of X such that F(a,, . . . , a,,) = U?=,C(a,, . . . , a,,; k). If we put
< n, k
K(a,, . . . , a,,) = U{C(a,,. . . , ai;k)lj
Q n},
then K(al, . . . , a,,) is a closed subset and, clearly, K(al, . . . , a,,) c G(a,, . . . , a,,) by (1.2) and (1.3). Let (x, y ) be any point of X x Y. Then (x, y ) is contained in some F(a,, . . . , a,,) x W(a,,. . . , a,,), and x E C(a,, . . . , a,,; k) for some k. Let m = max{n, k}, and choose a , , + I E Q , , + l, . . . , a,EQ,sothatyE Fa,,j= n 1, . . . , m.Wethenhave
+
( x , y ) E K(al, . . . , a,) x WaI, . . . , a m ) *
Thus, {K(a,, . . . , a,,) x W(a,,. . . , a,,)Ia, E Q,, v covers X x Y. Hence, (a) holds.
=
1,
. . . , n;
n E N}
0
For results on normal covers we refer to the preceding chapter or often to Chapter 2: “Extensions of Mappings 11”.
1.2. Theorem (Morita [1963]). Let Xbe a normalspace and Y a metric space. Then a basic cover Y in (1.1) has a special rejbement ifl it is a normal cover. Proof. If Y has a special refinement, by Lemma 1 . 1 , there exist open F,-subsets F(a,,. . . , a,,) satisfying (1.3) and (1.4). Since Xis normal, each F(aI, . . . , a,,) is a cozero-set of X. Hence,
{F(a,, . . . , a,,) x W(a,, . . . , a,,)Ia, E R,,
#‘ =
v = 1,
. . . ,n ; n ~ N }
is a a-locally finite cozero-set cover refining Y. Hence, Y is normal. Conversely, suppose Y is normal. Then there exists a locally finite open cover A = {MA I A E A} of X x Y such that 2 = 13, E A} refines Y. Define M(a,, . . . , a,,; A) and F(al, . . . , a,,) by
{aL
Wa,,.. =
an;
A)
U{PI P open in X and P x W(a,, . . . , a,,) c MA},
..
9
an) = U{M(ai,
- .-
9
an; A l l
M(a,, . . . , a,,; A) c G(a,, . . . , a,,), A
E
A}.
Since M(a,, . . . , a,,; A) x W(a,,. . . , a,,) c MAand .A is locally finite, F(a,, . . . , a,,) is closed and, clearly, F(a,, . . . , a,,) c G(a,, . . . , a,,). To show (1.4), let (x, y ) be any point of X x Y. Then, for some A E A and
T. Hoshina
124
E R,, v = 1, . . . , n, we have (x, y ) E M,, HAc G(a,, . . . , a,,) x W(a,,. . . , a,,). Since (x, y ) E M Aand y E W(aI,. . . , a,,), we can choose an open nbd U of x and a, E R,, v = n + 1, . . . , m such that
a,
(x, Y ) E U x Wal, . . . , a,,
a,+l,
. . . , a,)
c MA.
We then have U c M(a,, . . . , a,,, a,,+,,
. . . , a,; A)
M(al, . . . , a,,, a,,+lr.. . , a,;
and
4
. . . , a,. a,,+l, . . . , a,) c M A . Hence (a1, . . . , a,,, a,+1, . . . , a,; 1) c G(a,, . . . , a,,); consequently, M(al, . . . , a,,, a.+l,. . . ,arn;4c F(aI,. . . , an, a,,+I,. . . ,a,). x W(aI,
Therefore,
Let Y be a metric space. Let A = { M AI I, E A} be an open cover of X x Y. For a, E R,, v = 1, . . . , n and 1 E A, we put G(aI, . . . an; 1) 9
= U{PlP open in X and
P x W ( a I ,. . . , a,,)
c
M A } , (1.5)
and
. . . , a,,; A) 11 E A}. {G(al,. . . , a,,) x W(a,,. . . , a,)la,
G(a,, . . . , a,) = U{G(a,,
(1.6)
E R,, Then it can be shown that v = 1, . . . , n; n E N} is a basic cover of X x Y and, moreover, satisfies the following property: for a:) E R,, v = 1, . . . , n, n E N,
n= I
u 00
+X
=
G(ay), . . . , a t ) ) .
n= I
If, in general, a basic cover Y in (1.1) satisfies (1.7), Y is said to be proper. We shall now prove the following theorem, which is due to Morita [1963] and will be fundamental in our discussions.
1.3. Theorem (Morita [ 19631). For a countably paracompact normal space X and a metric space Y the following are equivalent:
Normality of Product Spaces II
125
(a) X x Y is normal and countably paracompact. (b) Every basic cover of X x Y has a special refinement. (c) Every proper basic cover of X x Y has a special refinement.
Proof. (a)-(b): Since every basic cover Y is a a-locally finite open cover and since such a cover of a normal countably paracompact space is normal, by Theorem 1.2, Y has a special refinement. (b)*(c): This is obvious. (c)*(a): Assume (c). Let A = {Mk 1 k E N} be any countable open cover of X x Y. To prove (a), it suffices to show that 4 is normal. Define G(a,, . . . , a,,; k) and G(a,, . . . , a,,) as in (1.5), (1.6), where A = N. Then {G(a,, . . . , a,,) x W(a,, . . . , a,,)la, E R,, v = 1, . . . , n ; n E N} is a proper basic cover of X x Y. Therefore, by (c), there exist closed subsets F(a,, . . . , a,,) of X satisfying (1.3) and (1.4). Since Xis countably paracompact and normal and since F(a,, . . . , a,,) c G(a,, . . . , an), in view of ( I S ) , there exists a locally finite collection X ( a , , . . . , a,,) = {H(a,, . . . , a,,; k) I k E N} of cozero-sets of X such that H(a,, . . . , a,,; k) c G(a,, . . . , a,,; k)foreachk E N andF(a,, . . . , a,,) c U%(al, . . . , a,,). Let us put f l = {H(a,, . . . , a,,; k) x W(a,, . . . , a,,)Ia, E R,,
v = 1,. . . , n ; n , k ~ N } . Then X is a a-locally finite cozero-set cover of X x Y refining A since H(a,, . . . , a,,; k) x W ( aI ,. . . , a,,) c G(a,, . . . , a,,;
k) x W ( aI ,. . . , a,,)
c
Mk.
Thus, 4 is a normal cover of X x Y. Hence, (a) is proved.
0
The following theorem establishes the equivalence of normality and countable paracompactness for the product with a metric factor; the “if” part is due to Morita [1963a] (for his proof, see Ishii [1966]) and the “only-if” part due to Rudin and Starbird [1975]. We shall call this theorem Morita-RudinStarbird’s Theorem. 1.4. Theorem (Morita [1963a], Rudin and Starbird [1975]). Let X b e normal and countably paracompact and Y metric. Then X x Y is normal fi it is countably paracompact.
Proof. In either implication, it is sufficient to prove that X x Y satisfies Theorem 1.3(c).
T. Hoshina
126
To prove the "if" part, assume X x Y is countably paracompact. Let Y = {G(al,. . . , a,,) x W(a,, . . . , a,)la, E Q, v = 1, . . . , n; n E N} be a proper basic cover of X x Y. Let us put for each n E N G(al, . . . , v = 1,
. . . , n}.
Since { W(aI,. . . , a,) I a, E a,, v = 1, . . . , n} is locally finite, D,, is a closed subset of X x Y. Clearly, Dn2 D,,+,, and since Y is proper, we have I D,, = 8. Therefore, by assumption, there exist open subsets U,, of X x Y such that D,, c U,,, U,,,,c U,and flu,, = 8. Let us put, for a, E R,,v = 1, . . .,n,
n,"=
F(a,, . . . , a,,) = { x E XI({.}
x W(a I,. . . , a,,)) n
U,,= S } .
Then F ( a I , . . . , a,) is a closed subset of X, and (1.3) and (1.4) above are clearly satisfied. Hence, Y has a special refinement. To prove the "only-if" part we need a lemma, which is essentially due to Rudin and Starbird [1975]. 1.5. Lemma. Let 42 = { U,I 1 E A} be a a-locally finite collection of open sets of a space X such that each U, is not discrete. Then we can select two distinctpointsp, andq, from U,so that 1 it p implies { p,, q,} n {p,,, q,,} = 8.
Proof. Rewrite Q = UnaN %,, where %, = { V,,1 1 E A,,} is locally finite. Let A,, be well-ordered by < . For 1 E A,, and p E A,, define 1 < p by either n < m o r n = m,and 1 < p i n A,,. Let 1 E A, and assume, for each p < 1 with p E A,,, that p,, and q,, have been selected from U,,, so that the property of the lemma is true for p X and qr with 1' < 1. Let D = { p , , , q,,Ip < A}. Since D = {p,,, q,,;(pE A,,, n < m} v { p , , , q,,lp E A,,,, p < A } and since each %, is locally finite, D is discrete and closed. Hence, U, - D must be infinite and from it two distinct points p , and q, can be selected. Then the property of the lemma is satisfied for all p,,, q,, with p < 1. 0 h f of k r e m 1.4 (continued). ("only-if" part): Assume X x Y is normal. Define a subset of A, of a, x * - * x a,,for each n E N by A, = { ( a l , . . . , a,,) E 0, x . . * x
a,,l
W(aI,. . . , a,) is not discrete}.
Normality of Product Spaces II
127
By Lemma 1.5, for each (a,, . . . , a,,) E A,, there exist two distinct points p ( a l , . . . , a,,) and q(a,, . . . , a,,) of W ( a I ,. . . , a,,) such that (QI,
* *
3
an)
(PI, - *
9
+ (BI, . . -
Bm),
(a19. .
* { ~ ( a i ,-
Ir,) E A m
-
9
*
-
9
9
an),
a,,) E An, q(ai,
-.
7
a,,)}
. B,)} = 0. (1.8) Let Y = {G(a,, . . . ,a,,) x W(al,. . . ,a,,) I a, E Q, v = 1, . . . ,n; n E N} be a proper basic cover of X x Y. Let us put, for each n E N, n {P(Bl,
* *
I
Bm),
q(81,
*
*
9
An = U { ( X - G(a,, . . . , a,,)) x { p ( a , ,. . . , a,,)} I
(a,, *
* * 7
an) E A,,},
Bn = U{(X - G(a,, * (al, .
* * 9
an)) x {q(a,, *
* * 3
an)} I
- . , a,,) E An},
and put A = UA,,, B = UB,,. Since { W ( a l ,. . . , a,,)I(aI,. . . , a,,) E A,,} is locally finite, A,, is a closed subset of X x Y. Moreover, observe that A,, c D,,, where D,, = u { ( X G(a,, . . . , a,,)) x W(a,, . . . , a,,)la, E 9, n = 1, . . . , n}. And {D,,} is a decreasing sequence of closed sets with n,,D,, = 0 since Y is proper. Therefore, A is closed in X x Y. Similarly, B is closed in X x Y. Furthermore, from (1.8) it follows that A and B are disjoint. By assumption, there exist open subsets U and V of X x Y such that A c U , B c V and 0 n P = 0. Let us put, for (a,, . . . , a,,) E A,,, * * * 3
an)
= { X E Xl{x} x
E(aI, * =
and
* * 9
{XE
W ( a I , . . . , a,,) c X x Y - U},
an)
X l { x } x W ( a , , . . . , a,,) c X x Y - V},
Fl(a,, . . . , a,,) = D(a,, . . . , a,,) u E(a,, . . . , a,,).
Then D(a, , . . . , a,,) and E(a,, . . . , a,,) are closed, and so is Fl( a , , . . . , a,,). We have Fl(a,, . . . , a,,) c G(a,, . . . , a,) for ( a I ,. . . , a,,) E A,, and, for any (x, y) E X x Y with y nonisolated, there exists an (a,, . . . , a,,) E A,, such that (x, y) E Fl(a,, . . . , a,,) x W ( a I ,. . . , a,,). On the other hand, let I(Y) = { y E Y Iy is an isolated point}. Then, since I(Y ) is 'discrete, X x I( Y )is countably paracompact and normal. Applying Theorem 1.3(c) to the basic cover {G(a,, . . . , a,,) x ( W ( a l , . . . , a,,) n I(Y))Ia, E R,,v = 1,. . . ,n; n E N} o f X x I(Y),wecan takeclosed subsets
I28
T. Hoshina
F2(a1,. . . , a,,) of X such that
Finally, let us define F(a,, . . . , a,,) by
F ( a I , . . . , a,,) =
u Wa1, . . . ,
i= 1.2
F2(al,. . . , a,,)
an)
if ( a 1 ,. . . , a,,) E A,,, if ( a I , . . . , a,,) 4 A,,.
Then we have F(al, . . . , a,,) c G(a,, . . . , a,,), and U{F(al,. . . , a,,) x W ( a l , . . . , a,,)la, E R,, v = 1, . . . , n; n E N} = X x Y. Thus, Y has a special refinement. 0 Theorems 1.3 and 1.4 will play a central role to obtain further results. First we shall prove the theorem of Morita [ 19641mentioned in the introduction. Let m be a cardinal number. A space X is a P(m)-space if for any set R of indices with Card R (=cardinality of R) < m and for any collection {G(a,, . . . , a,,) I a, E R, v = 1, . . . , n; n E N} of open subsets of X such that G(al, . . . , a,,) c G(al, . . . , a,, a,,+I)for a, E R, v = 1, . . . , n, n 1, there exists a collection {F(a,, . . . , a,,)lavE R, v = 1, . . . , n; n E N} of closed subsets of X such that the conditions (a), (b) below are satisfied:
+
(a)
F(al, . . . , a,,) c G ( a I ,. . . , a,,) for a, E R, v = 1, . . . , n;
(b)
X = U,"=I F(a,, . . . , a,,) for any sequence (a,,) such that X = U,"=IG(al,. . . , a,).
A P-space is a P(m)-space for every m. A normal space X is a P(1)-space iff X is countably paracompact. By Morita [1964], for 2 < m < No,X is a P(2)-space iff it is a P(m)-space. If n < m, every P(m)-space is a P(n)-space. The theorem of Morita above now follows from the following. 1.6. Theorem (Morita [1964]). Xis a normal P(m)-space i f X x Y is normal for any metric space Y with weight w(Y) < m (m 2 No).
Proof. ("only-if part"): Assume that X is a normal P(m)-space. Then we first note that X is countably paracompact. Let Y be a metric space with w ( Y ) Z m. As before, Y has a sequence {V,,} of locally finite open covers V,,= { V,, I a E a,,} with Card R,, < m such that {St(y, V,)1 n E N} is a nbd basis at each point y of Y. Here, we may assume R = R,, for all n E N. Let
Normality of Product Spaces II
129
Y = {G(al, . . . , a,) x W(aI,. . . , a,)[ a, E R, v = 1, . . . , n ; n E N} bea proper basic cover of X x Y. Since G(a,, . . . , a,) c G(al, . . . , a,,, a,+1) for each a, E R, v = 1, . . . , n, n + 1 and X is a P(m)-space, there exist closed subsets F(al, . . . , a,,) of X such that (a) and (b) above are satisfied. Let (x, y ) E X x Y. Select V,,, E V,,for each n E N so that y E V,,,. Then y E W(al, . . . , a,), and since Y is proper, we have X = (J,"=IG(al, . . . , a,,). Hence, by (b), X = U =; ,F(aI, . . . , a,). Thus, for some a, E R, v = 1, . . . , n we have (x, y ) E F(al, . . . , a,) x W(aI, . . . , a,,), which shows that {F(a,, . . . , a,) x W(aI, . . . , a,) I a, E R, v = 1, . . . ,n; n E N} covers X x Y. Hence, in view of (a), Y has a special refinement. Thus, by Theorem 1.3, X x Y is normal. ("if" part): Assume the condition of the theorem is satisfied. Let {G(al, . . . , a,) I a, E R, v = 1, . . . , n; n E N} be any collection of open subsets of Xwith G(al, . . . , a,) c G ( a l , . . . , a,,, a,+1)with an index set R of Card R < m. Let B(R) denote the Baire space, that is, the set of all sequences a = (a,) of elements of R and a basic nbd of a = (a,) is defined by V(a,, . . . , a,) = = (pi) E B(R) I fli = al, . . . , 8, = a,}. B(R) is known to be a metric space with a metric d defined by
{a
Ilk if
d(a, 8) = for a = (ai), defined by
0
pi for i < k and
a, =
if ai =
8, for each
= (/Ii).B(R) has weight =
u
i
E
tlk
#
Bk,
N
m. Let S be the subspace of B(R)
m
S = { a = (a,) E B(R)l
G(al, . . . , a,) = X } .
n= I
Let B(al, . . . , a,,) = V(aI, . . . , a,,) n S. Note that n"k1V(c$), . . . , a(')) # 8 implies a?) = a!+') = = a?), i = 1, . . . , n. And since G(a,, . . . , a,) c G(al, . . . , a,,, a,,+,), we see that Q = {G(al, .
. . , a,)
x B(al,
. . . , a,)Ia,
E
R,
v = l,,..,n;nEN} forms a basic cover of X x S. Since X x S is normal, by Theorem 1.4, X x S is countably paracompact. Hence, by Theorem 1.3, there exist closed subsets F(al, . . . , a,) of X such that F ( ~ I* ,
9
-
a,) c G(al,
*
.
U{F(al, 01), x v = l,...,n;nEN} a
a
*
1
a
an),
.
an)lav E Q,
= XXS.
T. Hoshina
130
Then, clearly, (a) is satisfied. Suppose X = U:=IG(aI, . . . , a,,). Then a = (ai) E S. Let x be any point of X . Then ( x , a) E F(/l,, . . . , P,,) x B(P,, . . . , P,) for some PI, . . . , /I E,R. , Since a E B(/?,,. . . , B,,) implies PI = aI,. . . , P,, = a,,, we have x E F(al, . . . , a,,). Hence, X = U:=lF(al, . . . , a,,), which proves (b). Thus, X i s a P(m)-space. 0
Remark. The Michael line is not a P(2)-space. Answering a question in Morita [1964] for every m, n with KO G m < n Vaughan [1975] constructed an example of a P(m)-space which is not a P(n)-space. Since every perfectly normal space is a P-space (see the proof of (c)-(a) of Lemma 1. l), we have 1.7. Corollary (Morita [1964]). Let X be a perfectly normal space and Y a metric space. Then X x Y is normal. As another but useful one of P-spaces, we quote M-spaces defined also in Morita [1964], where M-spaces are characterized as those spaces which can be the pre-images of metric spaces under quasi-perfect maps, and hence they are both generalizations of metric spaces and countably compact spaces. As is known for paracompact spaces being M-spaces and p-spaces due to Arhangel’skii [1963] are identical. The effect of Theorem 1.3 works well in proving the following theorem, which characterizes metric spaces whose product with any countably paracompact normal space is normal. 1.8. Theorem (Morita [1963]). Let Y be a metric space. Then X x Y is normal for any countably paracompact normal space X iff Y is a-locally compact; that is, Y is the union of a countable number of locally compact closed subspaces.
u,?,
Proof. To prove the “if” part, let Y = Ci,where Ci is locally compact for each i E N. Let X be any countably paracompact normal space. Let B = (G(a,, . . . , a,,) x W(al,. . . , a,,)Ia, E R,, v = 1, . . . ,n ; n E N} be a basic cover of X x Y. Then for each i E N
%
= {G(al,. . . , a,) x
(W(al,. . . , a,,) n Ci)la,E R,,
v = 1, . . . , n ; n E N }
Normality of Product Spaces I1
131
is a basic cover of X x Ci. Since Ci is locally compact and metizable, X x Ci is normal. Therefore, by Theorem 1.3, there exist closed subsets &(al, . . . , a,,) of X such that
W I , . . , a,,) = W,, . . . , an), U{&(ai,
-
v = 1,
7
a,,) x
..., n
(w(al,-
-
7
a,,) n Ci)Iav E Q v ,
; n ~ N }= X x C i .
If we put F(al, . . . , a,,) = Ui",,&(a,, . . . , a,,), then F(a,, . . . , a,,) is an F,-subset of X and clearly (1.3) and (1.4) are satisfied. Hence, by Lemma 1.1, Y has a special refinement. Thus, X x Y is normal. To prove the "only-if"' part, assume to the contrary that Y is not o-locally compact. Then, by Stone [1962], there exists a complete metric space Z containing Y as its subspace, but not an F, in Z. Let us define a space 2, where 2 = 2 as a set and nbds of z E 2 are either those of z in 2 if z E Z - Y or {z} if z E Y. Then 2 is a paracompact Hausdorff space. Consider E = (2 - Y) x Y and F = {( y , y ) I y E Y } . Then E and F are closed in 2 x Y and disjoint. By Michael [1963], we see that E and Fcannot be separated by open sets. Consequently, 2 x Y is not normal. 0 The following is another useful application of Theorem 1.4. 1.9. Theorem (Rudin and Starbird [1975]). Let Y be a metric space and Z a compact space. Suppose that X x Y and X x Z are normal. Then X x Y x Z is normal.
Proof. We may assume Y is nondiscrete. Then, since X x Y is normal, X is countably paracompact (see the preceding chapter). Hence, by Theorem 1.4, X x Y is countably paracompact, and since Z is compact, (X x Y) x Z is countably paracompact. Since (X x Z) x Y = (Xx Y) x Z and X x Z is normal and countably paracompact, again by Theorem 1.4, (Xx Z) x Y = X x Y x Z is normal. 0 Let m be an infinite cardinal number. The following corollaries are proved by Theorem 1.9 above and Theorems 1.13 and 2.21 of the preceding chapter. 1.10. Corollary (Morita [1963]). Let Y be a metric space. Suppose X x Y is normal. If X is (respectively m-)paracompact, then X x Y is (respectively m-)paracompact.
132
T. Hoshina
1.11. Corollary (Okuyania [1967]). Let Y be a metric space. Suppose X x Y is normal. If X is (respectively m-)collectionwise normal, then X x Y is (respectively m-)collectionwise normal.
The above two results are easily extended to the following. 1.12. Corollary. Let Y be aparacompact M-space. Suppose X x Y is normal. is m-paracompact (respectively m-collectionwise normal), then X x Y is m-paracompact (respectively m-collectionwise normal ).
If X
Proof. By assumption, there exists a perfect map f:Y + T from Y onto a metric space T. Since 1, x f:X x Y + X x Tis perfect, X x Tis normal. In case T is discrete, Y is the discrete union of compact subspaces, and so the result is clear. In case T is nondiscrete, X is countably paracompact and X x T is m-paracompact (respectively m-collectionwise normal) by Corollaries 1.10 and 1.1 1. Since 1, x fis perfect, X x Y is m-paracompact (respectively m-collectionwise normal). 0 We now proceed to a further application of Theorems I .3 and 1.4. 1.13. Theorem (Rudin and Starbird [1975]). Let Z be a metric space. Suppose + Y is a closed map, then Y x Z is normal.
X x Z is normal. Iff: X
Proof. Assume X x Z is normal. We may assume that Z is nondiscrete. Hence, as before, X is countably paracompact and, by Theorem 1.4, X x Z is countably paracompact. Also note that Y is countably paracompact and normal. Let Y = {G(a,, . . . , a,) x W(a,,. . . , a,,)Ia, E Q, v = 1, . . . ,n; n E N} be any basic cover of Y x Z. Then { f - ' ( G ( a , , . . . , a,)) x W(a,, . . . , a,,)Ia,
v = I,
E Q,,
...,n ; n ~ N }
is a basic cover of X x Z. Hence, by Theorem 1.3, there exist closed subsets F(q,. . . , a,) of X such that F(al, . . . , a n ) c f - ' ( G ( a , , . . . , a,,)), U{F(a,,
.
* * 7
an) x W(aI, *
..
3
an)Iay
E
Qv,
v = 1, . . . , n ; n ~ N }= X x Z .
Theref6re, by considering the closed subset f ( F ( a , , . . . , an)) of Y, we see that Y has a special refinement. Thus, Y x Z is normal by Theorem 1.3. 0
Normality of Product Spaces I1
133
Theorem 1.13 is originally related to the following problem posed by Morita (cf. Morita [1961]): Let Z be a compact space. Let f : X -, Y be a closed map. Suppose X x Z is normal. Then must Y x Z be normal? Morita conjectured that the answer to this problem was affirmative, and indeed the positive answer was given afterward by Rudin [1975]. On the other hand, when drafting his paper [I9631 Morita further conjectured that in case Z was metrizable, the problem above is also affirmative and he noticed that the answer is yes if, in addition, X x Z is countably paracompact. Thus, Theorem 1.13 is a positive answer to this problem. The remaining part of this section will be devoted to Rudin’s result above as well as to results given recently by BeSlagiC [1986]. 1.14. Theorem (Rudin [1975]). Let Z be a compact space. Let f : X a closed map. X x Z is normal, then Y x Z is normal.
+
Y be
For the proof of this theorem we shall need several results, the first of which is proved in the preceding chapter. 1.15. Lemma (Morita [1975]). Let X be a space and Z a compact space with weight w ( Z ) < m. Let Y be a normal cover of X x Z. Then there exist a normal cover 42 = { U, I 1 E A} of X and a collection { 6. I 1E A} ofjinite open covers of Z such that { U, x V I V E V,, 1 E A} refines Y, where Card A < max{m, Card Y}. 1.16. Lemma (Starbird) (see Rudin [1975]). Suppose Z is a regular space. Then,for each a < w ( Z ) , there exist subsets A,, B,, U, and V , of Z such that (a) A, and B, are closed and A, c U, and B, c V,, (b) U, and V , are open and disjoint, (c) Vg < a, either A, Q U , or B, Q 5.
Proof. We define A,, B,, U, and V , by induction on a. Suppose that y < w ( Z ) and that A,, B,, U, and V , have been defined for all a < y . Since Card { U,},<, < w ( Z ) , { Ua}a
T.Hoshina
134
1.17. Lemma (Rudin [1975]). Let Z be a compact space. Suppose X x Z is normal; then X is w(Z)-collectionwise normal.
Proof. Let {FaI a < K} be a discrete collection of closed subsets of X,where K = w ( Z ) . Let A , , B,, U, and V , be subsets of Z defined in Lemma 1.16. Let us set D = U{F, x ( Z - U J l a < K } . C = U { F , x A,la < K } , Then C and D are disjoint closed subsets of X x Z since {Fa}is discrete, each Fais closed and .A, c U,. Since X x Z is normal, there exist disjoint open subsets G and H of X x Z such that C c G and D c H. Let us put for each a
Ha =
{X
E X ~ { Xx} ( Z
- U,)
c H}.
Since A, and 2 - U, are compact, G, and Ha are open subsets of X . Note that Ha x ( Z - U,) c H . G, x A, c G, Fa c G, n H a , Similarly, find disjoint open sets G' and H' of X x Y such that U { F , x B,la <
K} c
G',
U{F, x ( Z - V,)la <
K} c
H'
and define open subsets G: and H,' of X such that Fa c G: n H:,
G: x B, c G',
H,' x ( Z
-
V,)
c H'.
Let us now define for a < K
Ma
= G, n Ha n G: n H,'.
Then M ais an open subset of X and Fa c Ma.We show (Ma I a < K} is disjoint. Suppose not. Then, for some /? < a < K , Ma n M, # 8. Choose x E Ma n M,. By Lemma l.l6(c), either A, U, or B, b. Suppose A, - U, # 8 and y E A, - U,. Then
+
+
(x, y ) E (G, x A,) n (H, x ( X - Up))c G n H =
a contradiction. Similarly for the case B, - V, # must be disjoint.
8,
8. Thus, { M a l a <
K}
0
The following lemma is interesting and also necessary for proving Theorem 1.14; the proof however is highly involved and is therefore omitted.
Normality of Product Spaces 11
135
1.18. Lemma (Rudin [1975]). Let I be a cardinal > 2. Suppose that a space Y is u-collectionwise normal for all u < I , and Y is an open cover of Y of cardinality I which is refned by a hereditarily closure-preserving closed cover of Y. Then there exists a locally finite closed cover of Y that refnes Y .
We note that if, in particular, Y is I-collectionwisenormal, then the lemma is true. We shall now prove Theorem 1.14. Proof of Theorem 1.14. Assume that X x Z is normal. Z may be assumed to be nondiscrete. So Zcontains an infinite sequence {z,,} with an accumulation point z,. Then an F,-subset X x ({z,,} u { z o } ) is normal, and therefore X is countably paracompact. Hence, Y is normal and countably paracompact. Definecp:XxZ+ Y x Z b y c p = f x 1,. Let E and F be disjoint closed subsets of Y x Z. We put G = X x Z cp-'(E) and H = X x Z - q - ' ( F ) . Since X x Z is normal, the binary open cover {G, H} is normal. Hence, by lemma 1.15, there exist a normal open cover 4 = { U, I I E A} of X and a collection {Y,I I E A} of finite open covers of Z such that (0, x PI V E Y,, I E A} refines {G, H}, where Card A < w(Z). Since 4 is normal, there exists a locally finite closed cover 9 = {FAI I E A} of X such that FA c U, for each I E A. Let I E A and let us put V,, = { V E " Y ; ~ F ,V X c G } , "v,, = { V E Y , I F , X P c H } .
Since {FAx PI V E V,, I E A} refines {G, H}, we have Y, = Y,, u V,,. Let us put
W,, = { y E YI { y } x P c Y x Z - E, V E Y,,}, W,, = { Y E Y ~ { ~ ) XYPx C Z-F,VEY,~}. Then W,,is an open subset of Y x Z since P( V E Y,,)is compact and V,,is finite. Similarly for W,,. Observe that f (F,) c W,,n W,,. Therefore, if we put W,= W,, n W,,and W = { W,I I E A}, then W is an open cover of Y such that {W,x P I V E " V ; , I E A }refines { Y x Z - E , Y x Z - F } , and f ( 9 ) = { f(FJ 11 E A} refines W . (1*9) Since Xis w(Z)-collectionwise normal by Lemma 1.17, Y is also w(Z)-collectionwise normal. And sincef is closed,f(9) is a hereditarily closure-preserving closed 'cover refining W . By Lemma 1.18, W has a locally finite closed refinement { D , 11 E A} such that D, c W, for each I E A. Since Y is w(Z)collectionwisenormal and countably paracompact, there exists a locally finite
I36
T. Hoshina
open cover { M i I A E A} of Y such that Di c Mi and A?i c W, for each A E A. Therefore, if we put V = { M i x VI V E Yi,A
E
A},
V is a locally finite open cover of Y x Z such that {CI C E V} refines { Y x Z - E, Y x Z - F} by (1.9). Finally, let us put L = St(E, U). Then L is open in Y x Z , and we have E c L and t n F = 8 since V is locally finite. Thus, Y x Z is normal. 0
1.19. Corollary (Rudin and Starbird [1975]). Let T be a metric space and C a compact space. Let f : X + Y be a closed map. r f X x T x C is normal, then so is Y x T x C. Proof. Use Theorems 1.4, 1.7 and 1.14.
0
As is defined in Starbird [1974], let us denote by N (respectively 9)the class of all spaces Z having the following property: Let f:X -, Y be a closed map. r f X x Z is normal (respectively paracompact), then so is Y x Z . We have seen that every metric space, every compact space and every product of a compact space with a metric space are all in N ,and therefore it is easy to see that they are all in 8.On the other hand, it is known that every paracompact M-space is characterized as a closed subspace of a product of a metric space and a compact space (for the proof, see below). In view of this fact together with Corollary 1.19 it is natural to ask whether every paracompact M-space belongs to N,although we note that it belongs to 8 (Starbird [1974]). This question was asked in Rudin [1975a], but it is still unsolved. In connection with this BeSlagiC [ 19861 recently obtained the following result.
1.20. Theorem (BeSlagiC [1986]). Let Z be a paracompact M-space. Let f:X + Y be closed. r f X x Z is collectionwise normal, then so is Y x Z . To prove this theorem we need a lemma. Let Z be a paracompact M-space. Then there exists a perfect map g : Z + T from Z onto a metric space T. Let PZ be the Stone-tech compactification of Z . Define a map h : Z + T x PZ by h ( 4 = (g(4, 4, z E
z.
Then h is a homeomorphic embedding such that h ( Z ) is closed in T x PZ and g = p;h holds, where p r is the projection: T x PZ + T. The following lemma was proved by Starbird [I9741 (for the proof, see BeSlagiC [1986]); the proof is, however, too involved to write down.
Normality of Product Spaces I I
137
1.21. Lemma (Starbird [ 19741). Let Z be a closed subset of T x C, where T is metric, C is compact, and p T ( Z ) = T. r f X x T is normal, then X x Z is C*-embedded in X x T x C.
Here, p T is the projection: T x C + T. For the notion of C*-embedding, see Chapter 2: "Extensions of Mappings 11".
Proof of Theorem 1.20. The proof is a somewhat simple combination of those of Theorems 1.3 and 1.14. Assume that X x Z is collectionwise normal. By Corollary 1.1 1 , it suffices to prove that Y x Z is normal. Let g : Z + T be a perfect map, where T is metric. Since 1, x g :X x Z -+ X x T is perfect, X x T is normal. Here we may assume T is nondiscrete. For otherwise, Z would be a discrete union of compact subsets and the theorcm would be clear. Therefore, X is countably paracompact and, by Theorem 1.4, X x T is countably paracompact. Let h :Z -+ T x C be the embedding obtained above, where C = PZ. Let cp = f x l , : X x Z +
YXZ
and
I//
=
fx1TXl,:XXTXC+YXTXC,
Now let E and F be disjoint closed sets of Y x Z. Let us put G = Y x T x C - (1, x h)(E), H = Y x T x C - (1, x h)(F).
Since h ( Z ) is closed in T x C, G and H are open subsets of Y x T x C and {C, H } covers Y x T x C. Since cp-'(E) and cp-'(F) are closed and disjoint and X x Z i s normal, the open cover { X x z - c p - ' ( ~ ) , x x z - cp-'(F)} of x x Z is normal. Since X x 2 is homeomorphic to X x h ( Z ) , { X x h(Z) - (1, x h)(cp-'(E)h X x h ( Z ) - ( 1 , x Mcp-'(FN)
is also a normal cover of X x h ( Z ) . Since X x h ( Z ) is C*-embedded in X x T x C by Lemma 1.21, there exists a binary normal cover {G', H') of X x T x C such that G' n ( X x h ( Z ) ) c X x h ( Z ) - ( 1 , x h)(cp-'(E)),
H' n ( X x h ( Z ) ) c
x
x h ( Z ) - ( 1 , x h)(cp-'(F)).
Since X x h(Z) is closed in X x T x C, we may further put
G'
=
X x T x C - (1, x h)(cp-'(E)),
H'
=
X x T x C - (1, x h)(cp-'(F)).
T.Hoshina
138
Then observe that $-‘(G) =
G‘,
$ - ‘ ( H ) = H’.
Since {G’,H’} is normal, by Lemma 1.15 there exists a locally finite open cover 4’ = { U; I A E A} of X x T and a collection {V,11 E A} of finite cozero-set covers of C such that {U; x 91 V E b,A E A} refines {G’, H’}. Let A E A. Let us put VAl = { V E VAl U; x V,* = { V E V,IUi x
P c G’}, P c H’}.
Then V, = V,, u 95,. Let us put
U,, = {p E Y x T l { p } x U,, =
{PE
Y X Tl{p}x
V c G, V E V A I } , P c H,V E V , , , } .
Since P ( V E V,)is compact and VAis finite, UAI. and U,,, are open sets of Y x T. Observe that (fx lT)(U;) c U,, n UA2.Hence, if we put U, = U,, n U,,, then U, is an open subset of Y x T and we have that %! = { U, [ A E A} covers Y x Tand { U, x PI V E V,, A E A} refines {G, H}. Now for the metric space T, let us adopt the same notations used in the previous part of this section such as W ( a , , . . . , a,,), a, E R,, v = 1, . . . , n. Let us put
u(ai, * * * an; A ) = U{PI P open in Y and P x W ( a I ,. . . , a,,) c U,}, 3
U’(al, . . . , a,,; A) = U{PI P open in
X and P x W ( a I ,. . . , a,,) c U;},
and
M(a,, . . . , a,; =
4
U’(aI, . . . , a,,; A) n f - ’ ( U ( a , ,
. . . , a,,; A)),
(1.10)
and put
M(a,, . . . , a,,) = U { M ( a , ,. . . , a,,; A ) l A E A}.
(1.11)
We shall show that the collection A = { M ( a , , . , . , a,,) x W ( a I ,. . . , a,,)Ia, E R,,
v = 1,
...,n ; n E N }
is a ba’sic cover of X x T. Clearly, for a, E a,, v = 1, . . . , n, n + 1, M ( a I ,. . . , a,,) c M ( a , , . . . , a,,, a,,+,) holds. Hence, it remains to show that A covers X x T. Let ( x , t) be any point of X x T. Then ( x , t) E U; for
Normality of Product Spaces II
139
some A E A. Hence, there exist an open nbd P I of x and a, E R,, v = 1, . . . , n such that f E W ( a I ,. . . , a,,)and P I x W ( a I ,. . . , a,) c U;. On the other hand, (fx 1T)( U;) c U,, we have ( f ( x ) , t ) E U,. Hence, there exist an open nbd P2 off(x) and a, E a,,v = n + 1,. . . ,m such that t E W(a,, . . . , a,, a,+,, . . . , a,,,) and P2 x W(aI,. . . , a,,, a,,+,, . . . , a,,,) c U,. Then we have, by (l.W,
x
E
U’(a,, . . . , a,,; A) n f - ‘ ( U ( a , ,
c U’(aI,
. . an, an+,, . -
-
. . . , a,,
a,,+,, . . . , a,,,; A))
a,; A)
. . . , a,,, a,,+I, . . . , a,,,; A)) 1 M(a,, . . . , a,,, a,+,, . . . , a,,,; A).
nf-W(a,, =
Hence, A covers X x T, and A is a basic cover of X x T. Therefore, by Theorem 1.3, there exist closed subsets F(al, . . . , a,,) of X such that F(a,, . . . , a,,) c M ( a , , . . . , a,)
(1.12)
U{F(al,. . . , a,) x W a , , . . . , an)lav E Q,, v = 1, ...,n;
EN}
= X X
T.
(1.13)
Since 42’ is locally finite, so is { U’(al, . . . , a,,; A)l A E A}. Hence, {M(a,, . . . ,a,; A) 1 A E A} is locally finite. Therefore, by (1.1 l), (1.12) and the normality of X, there exists a closed subset F(al, . . . , a,,; A ) of X such that {F(al,. . . , a,; A) 11 E A} is locally finite, and F(al, . . . , a,,; A) c M(a,, . . . , a,; A) F(al, . . . , a,,) = U{F(a,, . . . , a,,; A)lA
(1.14) E
A}.
(1.15)
Then, sincefis closed, { f ( F ( a , ,. . . , a,,; A)) I A E A} is a hereditarily closurepreserving closed cover of the closed subset f ( F ( a , , . . . , a,,)) such that f ( F ( a l ,. . . , a,,; A)) c U(a,, . . . , a,; A ) by (1.10) and (1.14). Since Y is collectionwise normal, so i s f ( F ( a , , . . . , a,,)). Hence, applying the fact following Lemma 1.18 to the open cover {U(a,, . . . , a,,; A) n f ( F ( a , , . . . ,.,,))In E A} o f f ( F ( a , ,. . . , a,,)), we can take a locally finite closed cover {E(a,, . . . , a,,; A ) l A E A} of f ( F ( a , , . . . , a,,)) such that ,?(a,, . . . , a,,; A ) c U(a,, . . . , a,; A) for each A E A. Since Y is countably paracompact and collectionwise normal, there exists a locally finite collection {H(a,, . . . , a,,; A) I A E A} of cozero-sets of Y such that E(a,, . . . , a,; A) c H(a,, . . . , a,,; A) c U(a,, . . . , a,,; A).
140
T. Hoshina
Now we have constructed a collection 9 = { H ( a , ,. . . , a,,; A) x W ( a l ,. . . , a,,) x VI V E
6,
A E A ; ~ , E R , , v= I , . . . , n ; n ~ N } ,
each member of which is a cozero-set of Y x T x C, and it is easy to check that Y is a-locally finite, covers Y x T x C and refines {G, H}. Thus, 3 is a a-locally finite cozero-set cover, and so a normal cover of Y x T x C refining {G, H}. Hence, {G, H} is itself normal, and consequently, { Y x Z - E, Y x Z - F } is a normal cover of Y x Z . Thus, as is well known, E and F are separated by open sets. 0 A proof similar to the one above yields the following corollary.
1.22. Corollary (BeSlagiC [1986]). Let C be compact and M metric and let Z be closed in C x M . Suppose X is w(Z)-collectionwisenormal. Let f:X -, Y be closed. Then if X x Z is normal, so is Y x Z .
2. Products of normal spaces with generalized metric spaces In Section 1 we have discussed several normal products, where the notion of basic covers and Morita-Rudin-Starbird’s Theorem play an important and fundamental role. In general, for two normal countably paracompact spaces X and Y the countable paracompactness of X x Y need not imply its normality. For example, let X = w , and Y = wI + 1. As for the converse, it has been a still unsolved problem (see Rudin [1975a]), but under some set-theoretic assumptions counterexamples were constructed by Wage [ 19781 and recently by BeSlagiC [1985]. Therefore, in view of these facts, the MoritaRudin-Starbird’s Theorem is meaningful and of interest. In this section we shall find several conditions on X or Y for the product X x Y to allow equivalence between normality and countable paracompactness and we shall obtain some related normal products. First we note: since w , + 1 is compact, w , x ( 0 , + 1) shows that the above equivalence does not hold in general in case Y is compact. Hence, in view of Morita-Rudin-Starbird’s Theorem any possible approach to such an equivalence may lie in cases where at least one factor space is in a suitable class including metric spaces but not compact spaces. Along this direction we shall btgin our discussion with products whose one factor is a LaSnev space. A space which is the closed image of a metric space was characterized by LaSnev [1966]; such a space is simply called a LuSnev space. LaSnev spaces,
141
Normality of Product Spaces I1
which are a natural generalization of metric spaces, have been often discussed to study several “metric-like’’ properties. One of the successful studies of LaSnev spaces can be found in Leibo [1974], where the well-known KatEtovMorita coincidence theorem for metric spaces in dimension theory was extended to the case of an arbitrary LaSnev space. Let Y be a LaSnev space. Then, analogous to Theorem 1.3, we shall first obtain, after some preliminary results, a necessary and sufficient condition for a given X x Y to be normal and countably paracompact. Since Y is LaSnev, there exists a metric space Tand a closed mapf : T -+ Y . Then we first note that Y is paracompact and perfectly normal. Our method will be based on the following fact: Y can be expressed as Y = Yo u Ui, I u., where y. is discrete closed for each i 2 1, f - I ( y ) is compact for each y E Yo and Yo n (Ui,l KJ = 8. This was proved by LaSnev [1965] and is assured by the following (see Nagata [1985, Theorem VII.41). 2.1. Theorem. Let X be a normal M-space and f : X -+ Y a closed map. Then Y = Yo u Ui,l F , where is discrete closed for each i 3 1, and f - I ( y ) is countably compact for each y E Yo.
Since T is a metric space, there exists a sequence {%,} of locally finite open covers of T such that each member of 42, has diameter < l/n. Let %, = {U,,Ia E a,}. Let us put F,, = f ( u , , ) for c1 E R, and put 9, = {F,, I c( E O n } Since . @, is a locally finite closed cover of T andf:T -+ Y is closed, 9, is a hereditarily closure-preserving closed cover of Y . Moreover, the sequence (9 }, has the following property: Let y E Yo.Then g,is locally finite at y for each n E N, and for each open nbd W of y in Y there exists an n such that St(y, 9,c )W .
(2.1)
To see this, let y E Yo.Since @, is locally finite and f - ‘ ( y ) is compact, there exist an open subset G containing f - I ( y ) and a finite subset y of R, such that a $ y implies G n on,= 8. Then G # = Y - f ( T - G ) is a nbd o f y such that a $ y implies G X n F,,, = 8. Hence, 9, is locally finite at y . Let W be an open nbd of y . Since f - ’ ( y ) is compact, there exists an n such that l / n < e ( J - ’ ( y ) , T - f - ’ ( W ) ) ,where e is a metric on T . Then we have c W. St(f - I ( y ) , @,) c f - I ( W ) , which implies St( y , 9,) Now let us apply arguments given in Morita [1955]. Let R, be well-ordered. Let us define G,,, for each ct E R, by Gno = Int Fno,
G,, = Int F,, -
u Int
B
Fna.
142
T. Hoshina
Let 9,= {G,, I a E a,}. Then the following properties are satisfied: each G,, is open, and a #
B implies G,,
g, is locally finite at each point y
E
n Gns=
8,
Yo,
(2.2) (2.3)
r, = Int(U%'n)
(2.4)
In fact, G,, is open since 9, is hereditarily closure-preserving, and 9,is clearly disjoint. Property (2.3) follows from (2.1). To see (2.4), let y E Yo.We first prove y E Int(U{Int F,, I a E a,}).By (2.1) there exist an open nbd V of y in Y and a finite subset { a l , . . . , a,} of a, such that V c FnUI u * u F,,,,. Now, u F,,, u * * * u F,,,) is not empty. Then V, suppose that V, = V - (Int FnUI is a non-empty open set with V,c FnUl, that is, V, c Int F,,,, a contradiction. Similarly V c Int F,,, u Int Fnu2 u F,,, u * u FnUb, and finally we have V c Int F,,, u . . . u Int F,,,. Hence, y E Int(U{Int F,,la E a,}). Since U {Int F,, I a E Q,} = Ug,,, we have (2.4). Let us further put H,, = Int and H,= '{H,,,I a E a,}. Note that H,, = G,,, and that G,, n Gns= 8 implies H,, n Hns= 0.Hence, the above (2.2), (2.3) and (2.4) are also satisfied for H,. Let us put K(a,, . . . , a,) = HIUln . . . n Hnan for a, E Q, v = 1, . . . , n, and X, = {K(a,, . . . , a,,) I a, E Q,, v = 1 , . . . , n}. Then we also note that X, satisfies the following:
-
a
..
*
9
an)
+ (81,
* * * 9
a
Bn)
* K(a1, . . . > an) n K(B1, fin) = 0, 9,is hereditarily closure-preserving and is locally finite * * * 3
(2.2') (2.3')
at each point of Yo, Y, c Int(U2,).
(2.4')
For each point y E Yo, let us put L , ( y ) = Int St(y, 9,). By (2.3') and (2.4'), L , ( y ) is an open nbd of y. Let 9,= { L , ( y ) ( y E Yo}. Then the sequence {9, satisfies: }
&
c
UY,, for each n
E
N,
for each point y of Yoand each open nbd G of y in Y there exists an n such that St(y, 9,)c G .
(2.5) (2.6)
Clearly, (2.5) holds. For (2.6), let y E Yoand let G be an open nbd of y in Y. By (2.1) St(y, 5,)c G for some k. Then we have that L , ( y ) c G by (2.4') and that 2,refines F,. Again by (2.1), there exist an n > k such that St(y, 9,) c L , ( y ) . We shall show that St(y, 9,) c G. Suppose that
Normality of Product Spaces I1
143
s,),
y E L,,( y’). Then since L,( y’) c St( y’, we have y’ E St( y, 9,). Hence, y’ E Lk(y ) . Let y’ E K(&, . . . , B,). Then we have y E K ( f i I ,. . . , Bk) since ify $ K(fi,, . . . , Bk), we have K(a,, . . . , ak) n K(BI, . . . , Bk) = 0 for any K(a,, . . . , ah)with y E K(al, . . . ,ak), and hence, Lk(y ) n K(fiI,. . . , Bk) = 0, which contradicts y’ E L h ( y ) .Therefore we have St(y’, 2,) c St(y, 2k) and hence L,,(y’) c L k ( y ) . Thus we have proved St(y, Y,,)c L,(y). Hence, St(y, Y,,)c G. This proves (2.6). Now, from the fact above, we obtain the following theorem.
2.2. Theorem. There exists a normal sequence {V,}of locallyfinite open covers of Y such that for each point y of &{St( y, V,,)I n E N} is a local nbd base at y in the whole space Y.
Proof. Let 9,,be obtained above. Let us put H,, = U9,,. Since Y is perfectly normal, there exist open subsets Hnkof Y such that H,, = Ukm,,Hnkand A,,, c H,,, for k E N. Let us put, for n, k E N, d n k
= { L n H n k + l I L ~ I P , l u { y%- k } -
Then A,,,is an open cover of Y. Rewrite newly these Ank as A,,(n E N). Then by (2.5) and (2.6) we can easily see that for each point y of Yo {St(y, A,,)I n E N} is a local nbd base at y in Y . Using paracompactness of Y, take inductively a locally finite open cover V,,for each n E N such that V,, refines A, and V,,+ I is a star-refinement of V,. Then { V,,}has the desired 0 properties of the theorem. Now let {V,,}be the sequence of covers of Y obtained in the theorem KCl, above.LetV,, = {V,,IaER,,}.LetusputW(a,, . . . , a,,) = q a , n . . . n f o r a , E n , , v = 1, . . . , n. Let X be a space. Then a collection Y of open subsets of X x Y is said to be a basic semicover if Y has the form Y = {G(a,, . . . , a,) x W ( aI ,. . . , a,)Ia,
E
...,n ; n ~ N } with open subsets G(a,, . . . , a,,) of X and satisfies G(a,, . . . , an) c G h , . . . , a,, a,+l) f o r a , E n , , v = 1, . . . , n , n + 1 v = 1,
and
I
U{G(a,, . . . , an) x
W E , , . . - , a,) I a, E a,,
v = 1, . . . , n ; n E N }
I>
X x Yo.
a,, (2.7)
T. Hoshina
144
Let E be a closed subset of X x Y contained in X x Yo. Then the basic semicover Y in (2.7) is said to have a special refinement relative to E if there exist closed subsets F(a,, . . . , a,) of X such that F(a,, . . . , a,)
= G(al, . . . , a,,),
(2.10)
and U{F(a,,. . . , a , ) x W a , ,. . . , a , ) l a , ~ Q , , v = I,
...,n
; n ~ N 13 } E.
(2.1 1)
The following lemma can be proved similarly to Lemma 1.1, 2.3. Lemma. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo. For a basic semicover Y in (2.7) the following are equivalent : (a) Y has a special refinement relative to E. (b) There exisrs a family {F(a,, . . . , a,) I a, E R,,v = I , . . . ,n; n E N} of open F,-subsets of X satisfying (2.10) and (2.1 I). (c) There exists a family {F(a,, . . . , a,) I a, E R,,v = I , . . . , n; n E N} of F,-subsets of X satisfying (2.10) and (2.1 1).
The following is analogous to Theorem 1.2. 2.4. Theorem. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo.Then a basic semicover 3 ’ in (2.7) has a special reJinement relative to E i f there exists a a-locally finite collection Y of open subsets of X x Y such that E c U Y and 2 refines 9.
Proof. To prove the “only-if‘’ part, using Lemma 2.3(b), take a family {F(a,,. . . , a , ) ( a , E R,, v = 1, . . . , n; n E N} of open Fu-subsets of X satisfying (2.10) and (2.1 1). Then F(a,, . . . , a,) is a cozero-set of X and so is F(a,, . . . , a,) x W ( a , ,. . . , a,) of X x Y. Let L k ( a I ,. . . , a,) (k E N) be an open set of X x Y such that F(a,, . . . , a,) x W(a,, . . . , a,) = U k L k ( a l ,.. . ,a,) and &(alr . . . , a,) c LL+,(cq,. . . , a , ) for k c N. Since { W(a,, . . . , a,)lav E R,,v = 1, . . . , n } is locally finite, {L,(a,, . . . , a,)l a, E R,, v = I , . . . , n; n, k E N} is the desired a-locally finite collection of open subsets of X x Y. Conversely, let 9 = U Y k be a a-locally finite collection of open subsets of X x Y described in the theorem, where Yk= {Lkil1E A k } is locally finite. Put for a, E R,, v = 1, . . . , n, k E N and 1 E A Lk(a,, . . . an; 1) = U { P I P open in X a n d P x W(a,, . . . , a,) c Lki}, 9
Normality of Product Spaces II
145
and
Fk(al, . . . , a,,)
=
U{&(alr . . . , an; 41
L,(a,,. . . , a,,; 1) c G(a,, . . . , a,,), A E h k } . Since Ykis locally finite, Fk(a,, . . . , a,) is closed in X and Fk(aI,. . . , a,) c G(a,, . . . ,a,,). Let F(a,, . . . , a,,) = U k F k ( a l., . . ,a,,). Then F(a,,. . . , a,,) is an F,-subset of X and, in a similar way as in the proof of Theorem I .2, it can be shown that {F(a,, . . . , a,,) I a, E R,, v = I , . . . , n; n E N} satisfies (2.10) and (2.11). Hence, by Lemma 2.3, 9 has a special refinement relative 0 to E. Here, for later discussions let us consider several weak normality properties. A space X has property (6) if, for any open subsets U,, n E N, and any closed subset B such that 0, 0, n B = 8, U,, and B are separated by open subsets of X.A subset A of a space X is a regular Gb-set if A is written as A = Q, = Onwith some open subsets Unof X.Clearly, every zero-set is regular G b . According to Mack [1970] X is &normal if, for every pair of disjoint closed subsets one of which is regular G b , there are disjoint open subsets containing them. Normality implies property (6) and property (6) implies 6-normality. By Mack [ 19701, countable paracompactness implies &normality. This result is slightly refined as follows.
n,,
n,,
2.5. Lemma.
n,,
Every countably paracompact space has property (6).
Proof. Suppose Xis countably paracompact. Let U,, (n E N) be open subsets and Baclosedsubsetwith n B = 8. Then {X - On,X - Bin E N} is a countable open cover of X. Let V" be its locally finite open refinement. Consider G = St(B, V"). Then it is easy to check n,Un n G = 8. 0
n,,0,
Next for a product space X x Y we consider the following two properties: (*) For any point y of Y and any closed subset F of X x Y with F n (X x { y } ) = 8, Fand X x { y } are separated by open subsets of X x Y. (**) For any point y of Y and any open subsets U,, (n E N) of X x Y with X x { y } c U,,U,,, there exist open subsets V, of X x Y such that X x { y } c U,,V,,and c U,,foreachn~N.
<
2.6. Lemma. Each of the following holds. (a) Let Y be perfectly normal. I f X x Y is 6-normal, then it has property (*). (b) Property (**) always implies property (*). The converse is true if X is normal and countably paracompact.
T. Hoshina
146
Proof. (a) Since Y is perfectly normal, { y} is a zero-set of Y. Hence, X x { y ) is also a zero-set of X x Y. Thus, property (*) easily follows from &normality. (b) The first statement is clear. To prove the converse, assume X x Y has property (*) and Xis normal and countably paracompact. Let Un(n E N) be open subsets of X x Y with X x { y } c U,U,,. Since X x { y} is countably paracompact, there exist closed subsets F, such that X x { y } = UnFnand Fn c U, for each n E N. Since Xis normal, there exist disjoint open subsets G and H of X such that
Fn c G x { Y } ,
(XX{ Y } ) n (XX Y -
un)c
H x {Y}.
Then U, u (H x Y) is an open subset of X x Y containing X x { y}. By (*), take an open subset K of X x Y such that X x { y } c K and R c Unu (H x Y). Let V , = K n (G x Y). Then, clearly, we have F, c V , and c U,,. Thus, V , are the desired open subsets of X x Y.
0 Returning to the LaSnev space Y, we shall now prove the following theorem which is mentioned above. A basic semicover Y in (2.7) of X x Y is said to be proper if for a?) E a,, v = 1, . . . , n ; n ~ N
fi w(ap), . . . , a t ) ) n Y, z 8
n=
I
x
u ~ ( a p ). ,. . , W
=
at)).
n= I
2.7. Theorem. Let X be a countably paracompact normal space. Let Y be a Lainev space. Then the following statements are equivalent: (a) X x Y is normal and countably paracompact. (b) X x Y has property (*a) and any basic semicover has a special rejinement relative to any closed subset of X x Y contained in X x Yo. (c) X x Y has property (**) and any proper basic semicover of X x Y has a special refinement relative to any closed subset of X x Y contained in X x Yo. Proof. (a)=(b): Suppose X x Y is normal and countably paracompact. Clearly, X x Y has property (**). Let Y be a basic semicover in (2.7) and E a closed subset of X x Y with E c X x Yo.Then 9’= 9 u { X x Y - E} is a a-locally finite open cover. By (a), there exists a locally finite open cover X of X x Y such that 3 refines 9‘.Let 2’ = { H E Z 1 H n E # 0).Then A?’ is a a-locally finite open collection such that E c US’ and 3’ refines 9.Thus, by Theorem 2.4, 9 has a special refinement relative to E.
Normality of Product Spaces I1
147
(b)*(c): This is obvious. (c)*(a): Assume (c). Let A = { M k I k E N} be a countable open cover of X x Y . To show (a), it suffices to prove that there exists a o-locally finite open cover 2 ' of X x Y such that 2 refines A. Let y be any point of Uir I Then by property (**) there exist open subsets u y k O f X X Y such that X X { y } C U k u y k and U y k C h f k for each k E N.Let i > 1. Since Y is paracompact and Y, is discrete closed, there exists an open nbd V, for each y E such that { V , l y E y } is discrete. Let us put Hyk= v y k n ( X x V,) for each y E UialY,, and put
x.
r
Then we see that X I is a o-discrete open collection satisfying
(2.12) On the other hand, let us put, for a, E R,, v W E , ,
=
=
1, . . . , n and k E M,
. . . , a,,; k ) U { P I P open in Xand P x W ( a l , . . . , a,,) c
M k } ,
and put M(a,, . . . , a,,)
=
U(M(a,, . . . , a,,; k ) l k
E
(2.13)
N}.
Then it is easy to see that 9 = { M ( a , ,. . . , a,,) x W(a,, . . . , a,,) I a, E R,,
v = 1,
...,n ; n € N }
is a proper basic semicover of X x Y. Now apply (c) to 9 and the closed set E = X x Y - UH,which is contained in X x Yoby (2.12). Then there exist closed subsets F(a,,. . . , a,,) of X such that W a , ,. . . , a,,)
= M(a,, . . . , a,,),
U{F(al, . . . , an) x W a l , . . . , a,,)la,
v = 1, . . . , n ;IIEN}
2
(2.14) E Q,,
E.
(2.15)
Since X is countably paracompact and normal, by (2.13) and (2.14) there exists an open subset H ( a l , . . . , a,,; k) of X such that
F(a,,. . . , a,,) c U { H ( a , ,. . . , a,,; k)lk E N}, H(a,, . . . , a,,; k) c M(a,, . . . , a,,; k ) for each k
E
N.
(2.16)
T. Hoshina
148
Then we have H(a,, . . . , a,,; k ) x W ( a l ,. . . , a,,) c M k ,and therefore
Hz = { H ( a , , . . . , a,,; k ) x W ( a l ,. . . , a,,)Ia, E R,, v = 1, . . . , n ; n E N , k E N ) is a o-locally finite collection of open subsets of X x Y such that in view of (2.15) and (2.16) E c UXZand $z refines A. Now, H = Sl u H2is a o-locally finite open cover of X x Y such that 2 refines A. Thus, (a) holds.
0 With the aid of Theorem 2.7 we shall obtain several results on the normality of products with a LaSnev factor. The following theorem establishes the equivalence of normality and countable paracompactness of products with a LaSnev factor. 2.8. Theorem (Hoshina [ 19841). Let X be a normal and countably paracompact space and Y a LaSnev space. Then X x Y is’normal iff it is countably paracompact.
Proof. In either implication, it is sufficient to prove that X x Y satisfies (c) of Theorem 2.7; property (**) obviously holds and thus, only the latter condition of (c) is needed to prove. To prove the “if” part, assume X x Y is countably paracompact. Let 9 = { G ( a l ,. . . , a,,) x W ( a l ,. . . , a,,)[a, E R,, v = I , . . . , n ; n E N} be a proper basic semicover of X x Y and let E be a closed subset of X x Y contained in X x Yo.We shall show 9 has a special refinement relative to E. Let us put, for each n E N,
Ffl = U{(X - G(a1, v = 1,.
...
9
a,,)) x w(a1,
. . 1
9
a,,)Ia,
E Q,,
. .,n}.
Then F,, 3 F n + l ,and since { W(a,,. . . , a,,) I a, E Q,, v = 1, . . . , n } is locally finite, F,, is a closed subset of X x Y . Moreover, since S is proper, we have F, n ( X x Y,) = 8, n
(ui2,
that is, n,,F,, c X x q). Since X x Y is countably paracompact and X x Y, is a zero-set of X x Y which is disjoint from E, by Lemma 2.5, there exists an open subset U, of X x Y such that E c U,. and q n ( X x V , ) = 8. Then we have
Normality of Produci Spaces I1
149
ni,,
the latter implies q.n n,F, = 0. Thus, by Lemma 2.5 again, there exists an open subset K of X x Y such that
n F, = K,
RnE
=
8.
n
Then F,, - K is closed and {F,, - K l n E N} is decreasing and has empty intersection. Hence, since X x Y is countably paracompact, there exists I n E N} of X x Y such that n an increasing open cover H = {H,, (F,- K ) = 0 for each n E N. Let us put for a, E R,, v = 1, . . . , n
f‘(al, . . . , a,)
= {x E
XI { x } x W ( a I ,. . . , a,)
c
17,- K}.
Then F ( a I , . . . , a,) is a closed subset of X,and we have F(a,, . . . , a,) c G(a,, . . . , a,,). Moroever, one can easily check that
U { F ( a l ,. . . , a,) x W ( a , , . . . , a,)la, E a,, V =
1, . . . , n ; EN}
E.
3
Thus, Y has a special refinement relative to E. To prove the “only-if’’ part, assume X x Y is normal. Let us define a subset A, of Q, x . . . x R, for each n E N by
A, = {(a,, . . . , a,,) E R, x . . * x Q,I W ( a I ,. . . , a,) is not discretej. Then, by Lemma 1.5, for each (a,, . . . , a,j E A,, two distinct points u ( a l , . . . , a,) and v(a,, . . . , a,) can be selected !rom W ( a , , . . . an.! such that (,al.. . . , a,) E A,, (8,. . . . , 8,) E A m , ~
( a , , . . . , a,> # n {u(B,, . .
(B,, . . . . Em) = {#(aI, . . . . Z,J.
. B,,),
vtfi,, .
lj,c\t =
via,, . .
8.
. rn i -... ,
17’:
Let 3 = {Gi,a,, . . . , r,\r x Wta,, . . . . -,)la, E O,, v , . . . . n; r. &, be a proper basic semicover of A’ x Y and iet A be a ciosed subsei of .:. :.. ; contained in X x Y i . We shall show that % has a speciai reiinemeni remiv.. to A . Let us put, for n E N, -1
C;,
‘=
iJ((X - G(a,. . . . , a . , ) ~ x cu(a,. . . . , a,>ji jbi,
. . . , a,)
E
Anj,
T. Hoshina
I50
D,, = U { ( X - G(a1, -
* * 9
a,,)) x
{4%,. ,%)}I * *
(a,, . * . , a,,) E A,,} and put C = UC,, and D = UD,,. Since { W ( a l ,. . . , a,,)I ( a l , . . . , a,,) E A,,} is locally finite, C,, is a closed subset of X x Y. Moreover, we note that Cn c Fn =
U { ( X - G(al, . . . , a,,)) x W ( a l , . . . , a,,)Ia, E
v
= 1,.
9,
. . ,n},
and {F,,} is a decreasing sequence of closed subsets with OF,, n ( X x Yo) = 8 since Y is proper. Therefore we have C n ( X x Yo)= C n (X x Yo).Similarly, D n ( X x Yo) = b n ( X x Yo).And since C n D = 8 by (2.17), we have C n b n ( X x Yo) = C n D n ( X x Yo) = 8;
x).
that is, C n b c X x (UiLI Since A c X x Yoand X x Y is normal, there exist open subsets U,V and W of X x Y such that
C n b c W and P n A C - W c U ,
=
b - W c V
8, and
OnV=8.
Let us put, for ( a I ,. . . , a,,) E A,,,
D(al, . . . , a,,) = { x E XI { x } x W ( a I ,. . . , a,,) c
X x Y - (V u W ) } ,
E(a1, . . . a,,) = { x E X l { x } x W ( a l , . . . , a,,) c X x Y - ( V u W ) } 9
and
Fl(al,. . . , a,,) = D(a,, . . . , a,,) u E(a,, . . . , a,,).
Then Fl(a,, . . . , a,,) is closed. We have that
Fl(al,. . . , a,,)
c
G(al, . . . , a,,) for ( a I , . . . , a,,) E A,,,
and that, for any point (x, y ) E A such that y is a nonisolated point of Y , there exists an (a1,. . . , a,,) E A,,such that (x, y ) E Fl(al,. . . , a,,) x W(a,, . . . , a,,). On the other hand, let I( Yo) = { y E YoI y is an isolated point of Y } . Then X x I(Yo)is normal and countably paracompact and
iG(a1, * . a,,) x (W(a1, ' v = 1, . . . , n ; n E N } 3
* 9
a,,) n I(Y0))la"E
a,,
151
Normality of Produci Spaces 11
is a basic cover of X x I( Y,)in the sense of Section 1. Hence there exist closed subsets F2(a,,. . . , a,,) of X such that
Fz(aI, ... U{F*(a,,
*
a,,)
..
7
= G b l , . . . , a,,), a,,) x (Wa,, *
v = 1, . . . , n; n
Define
F(aI, . . . , a,,) =
E
*
9
a,,)
-f7
4 Y O ) ) l ~ "E
Q,,
N} = X x I ( & ) .
u eh,. . . ,
a n ) if (at,
i= 1.2
Fz(a1,
* * *
, @,I
. . . , a,,) E A,,,
if (al, . . . , a,) $ A,,.
We have then
Using Theorem 2.8 the same proof of Theorem 1.6 yields the following: in case Y is nondiscrete Lainev, Y contains an infinite convergent sequence, and so normality of X x Y implies countable paracompactness of X. 2.9. Theorem (Hoshina [1984]). Let Y be Lainev and Z compact. If X x Y and X x Z are normal. then so is X x Y x Z . 2.10. Corollary (Hoshina [ 19841). Let X be m-paracompact (respectively m-collectionwise normal) and Y Lainev. I f X x Y is normal, then it is m-paracompact (respectively m-collectionwise normal ).
Next we shall consider conditions on X under which the product X x Y with a Lagnev space Y is normal. In view of Theorem 1.6, a necessary condition is that Xis a normal P-space. However, that is not sufficient as is shown by the following example; detailed constructions of X and Y are already given in the preceding chapter and are omitted. A a-space is a regular space with a a-locally finite net (Okuyama [1967]). It is known that every normal a-space is perfectly normal and hence a P-space. 2.11. Example (T. Chiba and K. Chiba 1197411). A normal a-space X and a Lainev space Y such that X x Y is not normal.
. ,
.. . . I
152
T. Hoshina
Indeed, any necessary and sufficient condition on X has not been obtained yet. The following result may be a help to seek such a one.
2.12. Theorem. Let Y be a Lainev space. Let X be a normal P-space, more precisely, a countably paracompact normal space such that X x Yo is normal. Then X x Y is normal iflit satisjies (*). Proof. We have to prove the “if” part only. Suppose that X x Y has property (*) or, equivalently by Lemma 2.6, has property (**). Let Y = {G(a,, . . . , a,) x W(a,, . . . , a,)la, E R,, v = 1, . . . , n ; n E N} be a basic semicover of X x Y and E a closed subset of X x Y contained in X x Yo.Observe that Yo is metrizable and the collection {G(a,, . . . , a,) x (W(a,, . . . , a,) n Yo)I a, E R,, v = 1, . . . , n; n E N} is a basic cover of the product X x Yo.Since X x Yois normal, by Theorems 1.3 and 1.4, there exist closed subsets F ( q , . . . , a,) of X such that F(al, . . . , a,)
= G(a,, . . . , 4, U{F(a,, . . . , a,) x ( W a , ,. . . , a,) n yo)t a, E a,, v = 1, . . . , n ; n ~ N }= X x Y , .
Since E c X x Yo, the above obviously shows that Y has a special refinement relative to E. Thus, by Theorem 2.7, X x Y is normal. 2.13. Theorem (Chiba [1974]). Let X be a locally countably compact normal P-space and Y a Lainev space. Then X x Y is normal.
Proof. By Theorem 2.12 we only have to prove that X x Y has property (*). Let y E Y . Let F be any closed subset of X x Y such that F n (X x { y}) = 8. We shall find an open subset L of X x Y such that X x { y } c L and EnF=8. Let T be a metric space and f : T + Y a closed map. Let cp = 1, x f : X x T + X x Y . Then cp-‘(F) and X x f - I ( y ) are disjoint closed subsets of X x T, and since X x T is normal, there exists an open subset G of X x T such that X x f - I ( y ) c G and G n cp-’(F) = 8. Let x E X and take an open nbd U, of x such that D,is countably compact. Consider the projection p : 0, x T -+ T. Then p is a closed map since uxis countably compact and T is first countable. Hence, if we put V, = T - p(Dx x T - G ) , then V, is an open subset of T and we have f -I( y) c V, and U, x V , c G . Since f is closed, there is an open nbd W, of y such that f -‘(W,) c V,. Let us put K = U { U , x W,IXE X } .
Normality of Product Spaces 11
Then K is an open subset of X x cp-'(K) n cp-'(F) = 8 since cp-'(K) open subset H such that X x f - ' ( y ) way as above choose an open nbd U, x f - ' ( W i ) c H. Let us put
L = U{U, x
W,'IXE
153
Y and we have X x {y} c K and c G. Again by normality of X x T, c H and I? c cp-'(K). By the same W,' of y for each x E X such that
X}.
Then L is an open subset of X x Y with X x { y } c L and cp-'(L) c H. We will show E c cp(cp-'(K)).Suppose (x, z) .$ cp(cp-'(K)).Then ({x} x f - ' ( z ) ) n cp-'(K) = 8. Therefore, if we put A = {x' E X l z E W,.}, we must have x .$ U{ U,. I x' E A}. Take an open nbd 0, of x such that 0, is countably compact and 0,n U,, 1 x' E A}) = 8; the latter implies (0, x f -'( y)) n cp--'(K) = $4. Since A c cp-'(K), we have (0,x f-'(y ) ) n A = 8. Since 0, is countably compact, similarly as above we can find an open nbd Wofy such that (0, x f - ' ( W ) ) n A = 8. Since cp-'(L) c H, we have (0, x W) n L = 8. Thus, (x, y ) .$ E. Since cp-'(K) n cp-'(F) = 8, we have t n F = 8 as required. 0
(u{
2.14. Proposition. A space X is paracompact any regular space Y .
fi
X x Y has property (*)for
Proof. The "if" part of this proposition is essentially proved in Theorem 1.12 of the preceding chapter. The "only-if" part is easily proved. 0 With this proposition as well as with Theorem 2.12 we find that the product of a paracompact P-space and a LaSnev space is normal, and by Corollary 2.10 it is paracompact. But this is implied by a more general result below. In Nagami [I9691 the notion of X-spaces was defined. As is proved there, if Y is a X-space, then there exists a sequence {8,}of locally finite closed covers of Y, where 8,is written as 8,= {E(a,,. . . , a,)Ia, E R, v = 1, . . . ,n} for each n E N with a suitable set R of indices, and a cover {C(y) 1 y E Y} of countably compact closed subsets with y E C( y) for each y E Y such that
&,, . . . , a,)
= U{E(a,,
. . . , a,,
a,+I)Ia,+I E R},
for each y E Y there exists a sequence a', . . . , a,, . . . E R such that C( y) c V with V open implies C ( y ) c E(a,, . . . , a,) c V for some n.
(2.18)
(2.19)
It is known that all cr-spaces and all M-spaces are Zspaces. Since every LaSnev space is a a-space, it is also a Z-space.
T. Hoshina
154
2.15. Theorem (Nagami [1969]). Let X be a paracompact P-space and Y a paracompact X-space. Then X x Y is paracompact. Proof. Take {&,} and {C(y) I y E Y} as above. Let 9 be any open cover of X x Y. Let a" E R, v = 1, . . . ,n. Consider a pair ( V , U )of an open subset U of X and a finite collection C of cozero-sets of Y such that E(a,, . . . , a,) c U V and each U x V ( V E U )is contained in a set of 9.Denote all such pairs 3, E A@,, . . . , a,) and put W(a,,. . . , a,) = {Ui x VI V E f , , by (U,, 1E A(a,, . . . , a,)}. Let us put W = U { W ( a , , . . . , a,) 1 a, E R, v = 1, . . . , n; n E N}. Since Y is paracompact, each C( y) is compact. Hence, in view of (2.19), one sees that W is an open cover of X x Y and, by definition, W refines 9. F o r a , E R , v = 1 , . . . , nweputG(a,,..., a,)= U { U i . ( l ~ A (, a. .I. , a,)). Then G(a,, . . . , a,) is an open set of X , and since W(a,, . . . , a,) c W ( a I ,. . . , a,, a,,,) by (2.18), we have G(a,, . . . ,a,) c G(a,, . . . , a,, a,+,). Since X is a P-space, there exist closed subsets F(a,, . . . , a,) of X such that
c.),
F@,, . . . , a,) c G h , . . . , a,,),
u
G(a,, . . . , a,)
ns N
=
X
=-
u F(a,,. . . ,
(2.20) a,)
=
X.
(2.21)
nEN
Here we may assume F(a,,. . . , a,) c F(al, . . . , a,, a,+,). Since X is paracompact, in view of (2.20), there exists a locally finite cozero-set collection &'(aI, . . . ,a,) = {H(a,, . . . , an;L)11 E A(a,, . . . ,a,)} of X such that H(a,, . . . , a,; 1) c U,, 1E A and F(a,,. . . , a,) c U X ( a l , . . . , a,). On the other hand, by paracompactness of Y, take a locally finite cozero-set collection {L(a,, . . . , a,) I a, E R, v = 1 , . . . , n} of Y such that F(a,, . . . , a,,) c L(a,, . . . , a,). Let us now put
. . . , a,; 1) x ( V n L(al, . . . , a,))l V E C , 1E A(a,, . . . , a,); a, E R, v = 1, . . . , n; n E N}.
= {H(a,,
Then it is easy to check that &' is a a-locally finite cozero-set cover of X x Y and refines 9.Thus X x Y is paracompact. 0
Remarks. (a) Besides locally countable compactness or paracompactness any other (sufficient) condition on a normal P-space X,under which X x Y is normal for a LaSnev space Y, has not been found yet. In contrast to Corollary 1.7 and Example 2.11, perfect normality of X is not sufficient. In Chiba [1974] it is asked whether X x Y of a normal M-space and a LaSnev space Y is normal.
Normality of Product Spaces II
155
(b) In Hoshina [ 19871a necessary and sufficient condition on Xis obtained under which X x Y is normal for any closed image Y of a locally compact metric space. It is unknown to the author whether every LaSnev space belongs to the class JV due to Starbird [1974] (see Section 1); however, the following asserts that every Lahev space belongs to the class 8.
2.16. Theorem. Let Y be a Lainev space. Let f : X X x Y is paracompact, then so is X' x Y .
+
X' be a closed map. If
Proof. Let Y = {G(a,, . . . , a,) x W(a,,. . . , a,)lav E R,, v = 1, . . . , n; n E N} be any basic semicover of X' x Y and let E be any closed subset of X' x Y contained in X' x Yo.Then {f-'(G(al, . . . , a,)) x W(al, . . . , a,)lav
E
Q,,
v = 1, . . . , n ; n ~ N } is a basic semicover of X x Y and F = (fx 1, , - I ( E ) is closed in X x Y and contained in X x Yo. Since X x Y is normal, there exist closed subsets F(a,, . . . , a,) of X such that
F(a1, . . . , a,,) c f -'(G(al,
. . . , an)),
U{F(al, . . . , a,) x Wal, . . . , a,)lav E a,, v = 1, . . . , n ; n ~ N }3 F.
Then, sincefis closed,f (F(a,, . . . , a,,)) is a closed subset of X' and we have
f ( F ( a , , . . . , a,))
c G(a,,
...
a,,),
U{f(F(a,,..* 9 a n ) ) x W(al,...,an)Iav~Rv, v = 1, . . . , n ; n ~ N }2 E.
Thus, B has a special refinement relative to E. Observe X' x Y has property (*) by Proposition 2.14 since X' is paracompact. Therefore, by Theorem 2.7, X' x Y is normal. From Corollary 2.9 it follows that X' x Y is paracompact.
0 In case Y is an F,-metrizable space, Theorem 2.8 is also valid (Hoshina [1984a]);a space is F,-metrizable (Gruenhage [1980]) (= o-metric in the sense
T. Hoshina
156
of Nagami [1971]) if it is paracompact and is a countable union of closed metrizable subspaces. Lahev spaces and F,-metrizable spaces are mutually independent notions, but both included in the class of paracompact a-spaces. The author does not know whether Theorem 2.8 holds for paracompact o-spaces Y (cf. Hoshina [1983]). If, in addition, we assume several conditions on X besides normality and countable paracompactness, Theorem 2.8 may be true for other spaces Y. Indeed, in Hoshina [I9841 Theorem 2.8 was proved in case X is a normal P-space and Y a paracompact a-space; hence, the question above is affirmative if X is further assumed to be a P-space. As for the case Y being a closed image of a normal M-space the following theorem has been obtained. 2.17. Theorem (BeSlagiC and Chiba [1987]). Let X be a paracompact Jirst countable P-space and Y a closed image of a normal M-space. Then X x Y is normal fi it is countably paracompact. 2.18. Lemma (Hanai [1961]). Let f : X + Y be closed and Y paracompact. Suppose that for each y E Y every pair of disjoint closed subsets o f f ' ( y ) are separated by open subsets of X . Then X is normal. Proof. Let E and F be arbitrary disjoint closed subsets of X . Let y E Y . By assumption of the lemma, there exists an open subset U,' of X such that E n f"-'( y ) c U,'and ob'n (F n J ' ( y ) , = 8. Let F' = n F. Then F is closed and F' n f ' ( y ) = 0. Sincefis closed and Y is reguiar, there exists an open nbd 0 of y such that F' ' '(0i.r= 0. Put C: = U , f '((31. Then U, is open and we have E n J 'r y ) z L; and 6,n F = 8. Similarlq, we can take an open subset r ; oi' X such that J -'(yt - L', c 1: ano n E = 8. Then we note
u,'
~5
i
'( y )
y
[I,
-t
'b
15 n f ' ( y ) c O,, G , n F = QI,
-
F n f ' ( y >c V , ,
B ? E
= $9.
Since fis closed, there is an open nbd W , o f y such that f - I ( ) c U, LJ 1:. Then W ; I y t Y IS an open cover of L . l a ~ de iocally finite open cover [ L , , l yF Y ) such tha. 2, c El for each J' E t . N o w let us pui h = Ij(L', nf ' ( L , , ) , y E I.~).Theniriseasytocnec~thaiL: n H = @and c7 n I. = 8. Hence, x is normal c
Normality of Product Spaces I1
157
Proof of Theorem 2.17. By assumption there exist a normal M-space 2 and a closed map f:2 + Y. We first prove that X x 2 is normal (Chiba [1974]). Since Z is an M-space, there exists a quasi-perfect map g : 2 -+ T from Z onto a metric space T. Consider t,b = 1, x g : X x Z -+ X x T. Then it is not hard to see that t,b is a closed map because Xis first countable and g is quasi-perfect. By Corollary 1.10, X x T is paracompact. And since X is normal, every pair of disjoint closed subsets of t,b-'(x, y ) = {x} x g - ' ( y ) are obviously separated by open subsets of X x 2. Hence, by the lemma above, X x 2 is normal. Also we note X x Z is countably paracompact since I) is quasi-perfect and X x T is paracompact. Secondly we note that since a normal M-space is collectionwise normal, so is Y and, by Theorem 2.1, Y can be written as Y = Yo u Uf91 where is closed and discrete for each i 2 1 andf-'(y) is countably compact for' each y E Yo. Now, to prove the "only-if" part, assume X x Y is normal. Let 9 = { U, I n E N} be a countable open cover of X x Y with U, c U,, I . Since X is paracompact, by Proposition 2.14 and Lemma 2.6, X x Y has property (w). Therefore, by applying property (w) to each point y E Ufpl and using collectionwisenormality of Y, similarly as in the proof of (c)*(a) of Theorem 2.7, we can take a o-discrete collection V = UVnof open subsets, where V,, is discrete, such that X x (UfaI t U V and Q refines 9.Let us put F = X x Y - U V . Then Fis closed and contained in X x Yo. On the other hand, let cp = l X x f : X x 2 + X x Y. Then, since { q r 1 ( U n ) 1 E n N} is an open cover of X x 2 and X x Z is countably paracompact, there exists a closed cover {E,,ln E N} of X x 2 such that En c cp-'(U,) for each n E N. Now, since Xis first countable andf-'( y ) is countably compact for y E Yo, we note that the following holds for each n E N:
x,
x
x)
V(En) n F c
un.
Let 0,= U{ 7 1 Y E V,,}. Then the above arguments show that {D,,, cp(E,,) n F ( n E N} is a countable closed cover of X x Y that refines 9.Therefore, X x Y is countably paracompact since X x Y is normal. Conversely, assume X x Y is countably paracompact. Let E and F be any disjoint closed subset of X x Y. Then, similarly as above, there exists a a-discrete collection V = U k V kof open subsets of X x Y , where Vkis discrete, such that X x (UfrfY,)c U V and for each V E 9" either 7 n El = 8 or P n E2 = 0 holds. Let us put, f o r k E N a n d j = 1, 2,
q,k =
U{VEV l Y n
4 z S}.
158
T. Hoshina
Then we have, for k
ol,& n E2
E
N and j = 1, 2,
=
0
and El n 0 2 , k =
0.
On the other hand, since X x Z is normal, there exist disjoint open subsets GIand G2 of X x 2 such that cp-'(Ej) c Gj ( j = 1, 2). Let 4 = X x Y cp(X x 2 - Gj)( j = 1, 2). Then H I and H2 are open and disjoint, and by the same reason as above, we have
E, n (Xx 6 ) c Hj, j = 1,2. Let us put Lo = X x Y - R2and Lk = X x Y - i 7 2 , & , k 2 1. Then L,, are open and El - U k U 1 . k c L, (n = 0, 1, 2, . . .). Moreover, we have nnaOLn n E2 = 0.Hence, by Lemma 2.6, there exists an open subset Ul,oof
ol,o
X x Y such that El - U k a l Ul,&c U,,oand n E2 = 0.Similarly there exists an open subset of X x Y such that E2 - U k a I U2.k c U2,0and El n 022.0 = 0.Now we have chosen open subsets q,,, n = 0, 1, 2, . . . ; j = 1, 2 of X x Y such that
ul,,nE2 =
0,
EInO2,, =
0,
n=0,1,2,...
.
Thus, as is well known, E, and E2 are separated by open subsets of X x Y. Hence, X x Y is normal. 17 References Arhangel'skii, A. V. [1963] On a class of spaces containing all metric and all locally bicompact spaces, Soviet Math. Dokl. 4, 751-754. BeilagiC, A. [I9861 Normality in products, Topology Appl. 22, 71-82. (19851 A Dowker product, Trans. AMS, to appear. BeSlagiC, A. and K. Chiba [1987] Normality of product spaces, Preprint. Chiba, K. 119741 On products of normal spaces, Rep. Fac. Sci. Shizuoka Wniv. 9, 1-1 1. Chiba, T. and K. Chiba [1974] A note on normality of product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A . 12, 55-63.
Normality of Product Spaces 11
159
Gruenhage, G. [I9801 On the M3 =. M ,question, Topology Proc. 5, 77-104. Hanai, S. [I9611 Inverse. images of closed mappings, 11, Proc. Japan Acad. 37, 302-304. Hoshina, T. [I9831 Normality and countable paracompactness of product spaces, Questions Answers General Topology 1, 54-56. [I9841 Products of normal spaces with LaSnev spaces, Fund. Math. 124, 143-153. [1984a] Shrinking and normal products, Questions Answers General Topology 2, 83-91. [ 19871 Normality of product spaces, Questions Answers General Topology (Proc. Soviet-Japan Joint Symp. Japan) 5, 139-142. Ishii, T. [I9661 On product spaces and product mappings, J. Math. SOC.Japan 18, 166-181. LaSnev, N. [I9651 Continuous decompositions and closed mappings of metric spaces, Soviet Math. Dokl. 6, 1504-1506. [I9661 Closed images of metric spaces, Soviet Math. Dokl. 7, 1219-1221. Leibo, I. M. [I9741 On the equality of dimensions for closed images of metric spaces, Soviet Math. Dokl. 15, 835-839. Mack, J. E. [19701 Countable paracompactness and weak normality properties, Trans. A M S 148,265-272.
Michael, E. [1963] The product of a normal space and a metric space need not be normal, Bull. A M S 69, 375-376. Morita, K. [I9551 A condition for the metrizability of topological spaces and for n-dimensionality, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A 5, 33-36. [I9611 Note on paracompactness, Proc. Japan Acad. 37, 1-3. [I9631 Products of normal spaces with metric spaces 11, Sci. Rep. Tokyo Kyoiku Daiguku, Sec. A 8, 87-92. [1963a] Note on products of normal spaces with metric spaces, Unpublished manuscript. [I9641 Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. [I9751 t e c h cohomology and covering dimension for topological spaces, Fund. Math. 87, . 31-52. Nagami, K. [I9691 Z-spaces, Fund. Math. 65, 169-192. [I9711 Dimension for u-metric spaces, J. Math. SOC.Japan 23, 123-129. Nagata, J. [1985] Modern General Topology (North-Holland, Amsterdam). Okuyama, A. [I9671 Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 8,236-254. Rudin, M. E. [I9751 The normality of products with one compact factor, General Topology Appl. 5,45-59. [1975a] Lectures on Set Theoretic Topology, Reg. Conf. Ser. in Math., 23.
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Rudin, M. E. and M. Starbird [1975] Products with a metric factor, General Topology Appl. 5, 235-248. Starbird, M. [I9741 The normality of products with a compact or a metric factor, Ph.D. Thesis, Univ. of Wisconsin. Stone, A. H. [I9621 Absolute F,-spaces, Proc. AMS 13, 495499. Vaughan, J. E. [I9751 Nonnormal products of o,-metrizable spaces, Proc. A M S 51, 203-208. Wage, M. L. [I9781 The dimension of product spaces, Proc. Natl. Acad. Sci. USA 75, 46714672.
K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 5
GENERALIZED PARACOMPACTNESS
Yoshikazu YASUI Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka, 543 Japan
Contents
1.
2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . Characterizations of paracompactness . . Characterizations of submetacompactness Characterizations of metacompactness . . Characterizations of subparacompactness Shrinking properties . . . . . . . . . . Examples. . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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161 162 164 175 178 184 186 191 198
Introduction
The purpose of this chapter is to give some classes containing paracompact spaces and a rather detailed account of the characterizations of their classes. The class of paracompact spaces simultaneously contains two important classes: metric spaces and compact spaces. The concept of paracompactness was introduced by Dieudonni [1944] as a class which contains a class of compact spaces, and an important theorem (that is, every metric space is paracompact) was shown by Stone in 1948. After that, paracompactness and its associated concepts have become popular among topologists and analysts. In the last forty years, due to the introduction of locally finite or closurepreserving properties, many covering properties which are weaker than the property of paracompactness have been considered. In this chapter we consider some classes (subparacompact spaces, metacompact spaces, submetacompact spaces and shrinking spaces etc.) which
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seem to be very efficient and natural properties for studying the generalizations of paracompactness. This chapter can be read as a sequel to Nagata [1985, Chapter 51 but to make it self-containedI have recapitulated some of the main results announced in this book in some different format. A research had produced many results of various kinds concerning the covering properties. Many covering properties (for example paracompactness, metacompactness and subparacompactness etc.) are defined in terms of any open cover and so one of the purposes of this chapter is to characterize the covering properties in terms of any directed or any monotone increasing open cover (in Sections 2,3,4,5 and 6). In Section 7, we shall discuss the relations between the properties appearing in this chapter. The reader should be aware of several recent books on covering properties and applications. Among these are: Van Douwen [1980], Engelking [1977] and Nagata [1985]. This chapter lays emphasis on the relations among the characterizations of paracompact spaces and other covering properties. Although not very much knowledge is required to read this chapter, the reader is advised to consult Nagata [ 19851for some preliminary concepts and notions.
1. Preliminaries
Throughout this chapter, we assume the term space refers to a Hausdorff topological space. A mapping is a continuous onto function and the set { 1, 2, . . .} of the natural numbers is denoted by N. An ordinal is a set of all smaller ordinals and as a topological space, will have the order topology. Cardinal numbers are initial ordinals. The symbols w, w , and o2denote the first infinite ordinal, the first uncountable ordinal and the second uncountable ordinal, respectively. For a set A, the cardinality of A is denoted by I A I, and all the finite subsets of A are denoted by [A]. For a collection of sets 4, the family consisting of all finite unions of sets from 4 is denoted by 4'. For collections of sets 4 and Y and a set A, 4 A A denotes the collection ( U n A 1 U E 4 }and Q A Y the collection { U n V (U E 4 , V E Y } .If each element of 4 is contained in some element of Y , we shall say that 4 is a partial refinement of Y . If, moreover, 4 and Y are covers of X,then 4 is a refinement of Y . For a collection Q of subsets of a space X and a subset A of X,St(A, 4) denotes the set U{U I U n A # 8, U E 4}(if A = {x}, St({x}, 4) is simply denoted by St(x, 4)).The closure of A is denoted by C1 A and the collection {Cl U I U E Q} is denoted by C1Q.
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We say that a collection 4 is directed if %‘is a partial refinement of 4 ; that ! which is, if for any finite subcollection Y of 9, there is some element U of & contains UV. A collection 4 = { U,l a < t}, where t is an ordinal, is monotone increasing if U, c Upfor any a < fl < t. A collection Q of subsets of a space Xis said to be discrete (locallyfinite, locally countable) if for every x E X there is a nbd (=neighborhood) 0 of x such that the cardinality of {U E 4 1 0 n U # S} is at most one (resp. finite, countable). The collection Q is closure-preserving (resp. interior-preserving) if for any subcollection Y of 42, U(C1 VI V E V }is closed (resp. n(Int VI V E V }is open), where Int V denotes the interior of V. It is seen that a collection 4 is interior-preserving if and only if (X - U I U E 4 } is closure-preserving. A collection 4 of subsets of a set X is point-jinite (resp. point-countable) if each point x E X is contained in at most finitely many (resp. countably many) members of 4. A collection 4 is o-discrete (resp. 0-locallyfinite, etc.) if 4 can be expressed as 4 = U{%$ In E N} where each anis discrete (resp. locally finite, etc.) For later use we observe the following properties (they are easily verified and so we omit the proofs).
1.1. Lemma. Let 9 be a closure-preserving collection of closed subsets of a space X and A a closed subset of X. Then the collection 9 A A is closurepreserving .
1.2. Lemma. Let 4 be an interior-preserving collection of open subsets of a space X . Then the collection ( U Y 1 Y c 4 }(that is, all unions of all subcollections of 4 ) is also interior-preserving. The following properties are very useful later:
1.3. Lemma. Let 4 be a collection of open subsets of a space X and Y a point-finite open cover of X . Zf we let F ( U ) = { x E XISt(x, Y ) c U } for each U E 4, then the collection 9 = { F ( U )I U E 4 } is a closure-preserving collection of closed subsets of X. Proof. Let { U, I a E A } be any subcollection of 9 and any x # U{F(U,} 1 aEA}.SinceYispoint-finite,welet{V,, . . . , I / , } ~ ~ { V E Y J V}.Ifwe XE put Ai = { a E A I J( - U, # S} for each i = 1, . . . , n, then n (U{F(U,)( a E A i } ) = 8 for each i = 1, . . . , n and A = U { A i l i = 1, . . . , n}. So yl i = 1, . . . , n } is an open nbd of x which has the empty intersection with U(F(U,) I a E A}, that is, U{F(U,) I a E A } is closed. 0
n{
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2. Characterizations of paracompactness As stated above, we shall consider some classes of spaces as generalizations of paracompact spaces, and therefore it is unavoidable to study the characterizations of paracompact spaces. Morita [19621 introduced the notion of r-paracompactness in order to characterize the normality of the product space with a space and I' (where r is an infinite cardinal number and I' means the product space of r copies of Z = [0, 11). 2.1. Definition. Let r be an infinite cardinal. A space Xis t-paracompact if any open cover of X with cardinality < t admits a locally finite open cover as its refinement. If 7 = w, the notion of o-paracompactness is that of countable paracompactness. It is easy to see that the notion of a,-paracompactness is different from that of a,+,-paracompactness. In fact, for any ordinal a, the linearly ordered space a,+,is normal o,-paracompact but not w,+ ,-paracompact. r-paracompact spaces have many properties that are analogous to those of paracompact spaces. We will start with by stating the following characterizations of r-paracompactness. 2.2. Theorem (Morita [1962]). Let X be a space and r an infinite cardinal. Then the following are equivalent: (1)
X is r-paracompact.
(2)
X is countably paracompact and every open cover of X with cardinality < t has a a-locally finite open refinement.
Proof. (1)=42) Obvious. (2)~(1)Let9beanyopencoverofXwith(9~ < r a n d y = u{-V;,lnE N} a a-locally finite open refinement of 9, where each Ynis locally finite. If V , = uVnfor each n E N, then { V,,1 n E N} is a countable open cover of X , and therefore we have a locally finite open cover { W, I n E N} of X such that W, c V,, for each n E N. Then { W, n V J V E V,;n E N} is a locally finite open refinement of 02.Therefore Xis t-paracompact. It is interesting to be able to characterize the compactness of a space in terms of the existence of finite subcovers for all monotone increasing open covers of a space (Alexandrov and Urysohn [1929]). So we shall show the following characterizations of t-paracompact spaces in terms of monotone increasing (or directed) open covers:
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2.3. Theorem. Let X be a space and T an infinite cardinal. Then the following are equivalent: (1)
(2) (3)
X is T-paracompact. Every monotone increasing open cover of X with cardinality < T has a locally finite open refinement. Every monotone increasing open cover of X with cardinality < 7 has a 0-locallyfinite open cover Y of Xsuch that CI Y is its refinement.
Proof. (1)=>(2)Obvious. (2)*(3) Let 4 = { U, I a < t} be any monotone increasing open cover of X and Y a locally finite open refinement of 42, where we may express Y as {Val. < 7 } such that V, c U , for each a < z. If we let W, = u { O ( O is open in X , 0 n V, = 8 for any fl > a}, then W = { W,I a < t} is a monotone increasing open cover of X. Since C1 % n 5 = 8 for any p > a, we have: CI w, c x - U { y p > a }
= U{V,lB < a> = U{U,IP < a }
=
u,.
By (2), we have a locally finite open refinement d of W , and so CI d is a refinement of 4. (3)*(1) We shall prove this by transfinite induction on T. To begin with, we shall show that X is countably paracompact. Let 4 = {U,,ln E N} be any countable open cover of X . If we let 0,= U{& (i = 1, 2, . . . , n} for each n E N, then there exists a a-locally finite open cover Y such that CI Y is a refinement of { 0 , ln E N}, where we may express that Y = U{Y,,I n E N} and each Y,,= { V,,iI i E N} is locally finite with C1(V,,i) c Oifor each i E N. For each n, we put W, = K,j(i, j < n } . Then it is easily seen that { W,,1 n E N} is a monotone increasing open cover of X such that C1 W, c 0, foreachne N . L e t d = {Ul, U,, - CI W . - l l n 2 2}.Thenitisseenthatd is a locally finite open refinement of Q, and so Xis countably paracompact. Assuming that Xis A-paracompact for any Iz < T , we shall show that Xis r-paracompact. For this purpose let { U, I a < T } be any open cover of X . If 0,= U{U,l/3 < a } for each a .c T , then 8 = {O,(a < T ) has a a-locally finite open cover Y of X as in (3). Since X is countably paracompact, it is easily seen that there exists a locally finite open cover W = { W, I a < T} of X such that C1 W, c 0,. For each a < t (by the induction hypothesis), we have a locally finite open cover 9,of CI W, such that 9,is a partial refinement of {U,lB < a } , where local finiteness and openness of 9,mean in CI W,. Lastly we let
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d, = 9,A W,. Then dais a locally finite (in X) collection of open subsets of X which is a cover of W, and a partial refinement of 4 and therefore U{da 1 a < z} is a locally finite open refinement of 4. 0 2.4. Theorem. Let X be a space and z an infinite cardinal. Then the following are equivalent:
(1)
X is z-paracompact.
(2)
Every directed open cover of X with cardinality finite open refinement.
< z has a locally
(3)
Every directed open cover of X with cardinality finite closed refinement.
has a locally
Proof. (1)=.(2) Obvious. (2)*(3) Let 9 be a directed open cover of X with 1 9 I < z. Express 9 as 9 = { U, I a E D}, where D is directed and if a < /I (a, /I E D), then U, c U,. From the statement (2), we can take a locally finite open cover V = { V,I a E D } of X such that V , c U, for each a E D. For each a E D, we let W , = X U{Cl 5 I /I 4: a}. Since { W, I a E D } is a directed collection of open subsets of X such that C1 W, c 17, for each a E D, we shall show that { W, I a E D } is a cover of X.Let x E X.Then there is a finite subset (a,, . . . , a,,} of D such that X E n{Cl V,li = 1 , . . . , n} - U{Cl V,la # ai for any i = 1 , . . . , n } . Since D is directed, we have some a0 E D such that a. > ai for any i < n, and therefore x E Wq. From (2), we have a locally finite open refinement d of { W, I a E D } , Then the covering C1 d is a locally finite closed refinement. (3)=4) It suffices to show that (3) implies (2) by Theorem 2.3. We will use the method of the proof of Lemma 1 of Michael [1953]. Let { U, l a E D } be any directed open cover of X with ID1 < z and 9 a locally finite closed refinement of { U, 1 a E D}, where we may express 9 as 9 = {F, I a E D } such that Fa c U, for each a E D. For each A E [D], we let V ( A ) = X U{F, 1 a 4 A}. Since { V ( A )I A E [ D ] }is a directed open cover with cardinality < z, there is a locally finite closed refinement &' where we may express &' as &' = { H ( A )I A E [D]} such that H ( A ) c V ( A ) for each A E [D]. Let W, = U, - U{H(A))H(A) n Fa = 8} for each a E D. Since W = { W, I a E D } is an open cover of X such that W, c Uafor any a E D, it suffices to prove the local finiteness of W .For any x E X , we can choose an open nbd 0 of x such that {A E [ D ] ( On H ( A ) # S} is finite, and so we denote its finite set by { A l , . . . , A n } .For any a 4 U{Al, . . . , A n } ,we have H(AJ n F, c V ( A i ) n Fa = 8 and so H ( A i ) n W , = 8 for each i < n. Hence
Generalized Paracompactness
0 n W , c ( U { H ( A i )I i < n } ) n W , = elements of {W,laE U { A i l i < n } } .
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8.Therefore 0 meets at most many 0
2.5. Remark. In the proof of (2)*(3) in the above Theorem 2.4, we have proved that: (2)’ Every monotone increasing open cover 4 of X with cardinality < r has a locally finite open cover Y of X such that C1 Y is a refinement of 4, and therefore we have the following corollary. 2.6. Corollary. Let X be a space and r an infinite cardinal. Then thefollowing are equivalent: (1)
X is r-paracompact.
(2)
Every monotone increasing open cover 4 of X with I % I < t has a locallyfinite open cover Y of X such that C1 Y is a refinement of 4.
(3)
Every directed open cover 4 of X with 14 I < r has a locallyfinite open cover Y of X such that C1 Y is a refinement of 4.
If
T
= o in Theorem 2.3, the following well-known theorem holds.
2.7. Corollary (Ishikawa [ 19551). A space X i s countably paracompact ifand only iffor every countable increasing open cover { UnI n E N} of X,there exists an open cover { V ,I n E N} of X such that C1 V. c U,, for each n E N. As is well known the paracompactness can be characterized in terms of a
cushioned refinement (see Michael [1953]). Katuta [1977] characterized a t-paracompactness in the same terms which are used in order to obtain some results with respect to the normality of product spaces. We shall show the Katuta’s characterization. Let 4 and Y be collections of subsets of a space X:Then we say that Y is cushioned in 4 if we can assign to each V E Y a U( V) E 4 such that for every subcollection Y ’ of Y , Cl(UY’) c U { U ( V )I V E Y ’ } . 2.8. Theorem (Katuta [1977]). Let X be a normal space and t an infinite cardinal. Then the following are equivalent: ( I ) X is r-paracompact. (2) For every monotone increasing open cover 4 = { U, I a < A} of X with A < t,there exists a sequence { Y,I n E N} of collections of open sets of X such for that Y = U{YnI n E N} is a refinement of 4 and Ynis cushioned in Yn+, any n E N.
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Proof. (1)=+(2) Let 9 = {V,la < A} be any monotone increasing open cover of X with rl < T . Since Xis r-paracompact, there is a locally finite open refinement V of 9. Express V = { V,I a < A}. Since X is normal, there is a closed cover {FaI a < A } of X such that F, c V, for any a < A, and furthermore, for each a < A there is a sequence { & , I nE N} of open subsets of X such that F, c V,, c C1 V,, c V,n+l c V, for each n E N. If for each n E N, we let V, = { V,, I a < A}, then a sequence {V,I n } of open covers of X satisfies the statement (2). (2)=41) Let 9 = { V , I a < r } be any monotone increasing open cover of X. If we show that there exists a a-locally finite open cover V of X such that C1 Y is a refinement of 42, it follows X is r-paracompact by Theorem 2.3. At first we have some open refinement V = {V,1 n E N} of 9 such that V, is cushioned in V,+lfor each n E N, where we may assume that for each n E N, V, is indexed by T (that is, V, = { V,, I a < r } ) and V, is cushioned in V,+lwith respect to a < t (that is, for any A c r , Cl(U{ VnulaE A}) c u{V,,+lalaE A}). If for each n E N and each a < z, we let W,, = V,, Cl{U{V,+IsIp< a}),thenW = { W , , l n ~ N ( , < a r}isanopencoverofX such that C1 W is a refinement of 9. Therefore, to complete the proof, we shall show that { W,, I a < r } is locally finite in X. Fix n E N and x E X. If x is in U{V,+lula< r}, let a be the 1st of < T ~ X EV,+I,,}. Then V,,+lu n W,,, = 8 for any /3 < z with a < p and x$Cl(U{V,,IB < a}). Therefore an open set V,+lu - Cl(U{ V,,, I p < a}) is a nbd of x which intersects at most one member W,, of { WnsI fl < r } . I f x is not in U{ V,,+,, I a < r } , then 0 = X - Cl(U{ V,, I a < r } ) is a nbd of x which intersects no members of { W,,I a < r } . It completes the proof that { W,, I a < r } is discrete and therefore locally 0 finite (for any n).
{a
The above proof (1)*(2) of Theorem 2.8 shows that { W,, I CL < r } is discrete for each n E N , and therefore a normal space X is r-paracompact iff every monotone increasing open cover { U,I a < A} of X with A < T has a a-discrete open cover W such that CI W is a refinement of { U, I a < A}. In fact we have the following proposition. 2.9. Theorem (Katuta [1977]). Let X be a normal space and r an infinite cardinal. Then the following are equivalent: (I)
X is r-paracompact.
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169
(2)
For every monotone increasing open cover { U,I u < r} of X , there exists a 0-discrete open cover W of X such that Cl W is a refinementof {U,lu < T } .
(3)
For every open cover { U, I u < r} of X . there exists a a-discrete open cover W of X such that CI W is a refinement of { U, 1 a < r}.
Proof. It suffices to show that (1)*(3). Let 9 = { U, I u < r} be any open cover of X with cardinality < T and { V,Iu < r} a locally finite open refinec U, for each a < r. Since X is normal, there are a ment of 9 such that closed cover {F,I u < r} of X and a sequence { { W, I u < r} I n E N} of open covers of Xsuch that Fa c W, c CI W, c W,+ c V ,for any u < 'I. For each n E N and each u < r , we put On, = W,, - Cl(U{ Wn+lsI fl < a}). We leave the exercise of showing that if we put 0, = {On,I u < r} for each n E N, then U { U , l n E N} is an open cover of X such that CI (U{0,,,lnE N}) is a refinement of 9. To see that each 0, is discrete, let x be any point of X and a ( x ) the 1st of { a l x E W,,+l,}. For each u > a(x), it holds that Wn+lil(r)n On, = 8. For each u < u ( x ) , it holds that (X - Cl(U{ WnaIB < ~ ( x ) } ) n ) On, = 8. Therefore Wn+Iu(x) n (X - Cl(U { WnaI fl < ~ ( x ) } )is ) a nbd of x which intersects at most one member On,(x) of (0,. I a < r}. 0
,
2.10. Remark. (1) As mentioned above, a r-paracompactness was introduced by Morita [1962] in order to characterize the normality of a product space of its space and Z'. As is well known, the product space of a paracompact space and a metric space may fail to be normal; then, what is the necessary and sufficient condition for a normal space X in order that the product space of X with every space belonging to some class 9 of spaces be normal? We shall recall this problem in cases that its necessary and sufficient conditions are covering properties. A space Xis countably paracompact and normal iff X x (0,l/n} is normal, where (0, l/n} is the subspace of real line (Alas [1971]) iff X x Z is normal, where I is the closed unit interval (Dowker [1951]). A space X is countably paracompact and collectionwise normal iff X x u(X,) is normal, where u(X,) is the one-point compactification of a discrete space X , with I X , 1 = I X I , iff X x D" is normal, where D is two-point space with discrete topology (Alas [1971], Atsuji [1989]). (2) From Corollary 2.6 and Theorem 2.3, we have the following characterization of r-paracompactness:
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A space X is r-paracompact iff for every directed open cover Q of X with I Q I < T, there exists a a-locallyfinite open cover Y of X such that CI Y is
a refinement of Q. Since a space X is paracompact iff X is r-paracompact for any infinite cardinal T,the following theorem except (6) follows from the above theorems etc. 2.11. Theorem (Michael [1953, 1957, 19591, Mack [1967] and Junnila Let X be a regular space. Then the following are equivalent:
X is paracompact. Every open cover of X has a a-locally finite open refinement. Every monotone increasing open cover of X has a locallyfinite open refinement . Every monotone increasing open cover of X has a locallyfinite open cover whose closure is as a refinement. Every monotone increasing open cover of X has a a-locally finite open cover whose closure is as a refinement. Every monotone increasing open cover of X has a locally finite closed refinement. Every directed open cover of X has a locallyfinite open rejinement. Every directed open cover of X has a locallyfinite closed refinement. Every directed open cover of X has a locallyfinite open cover whose closure is as a refinement. Every directed open cover of X has a a-locally Jinite open cover whose closure is as a refinement.
Proof. It suffices to show that (6) implies (1). In order to show that X is paracompact, we shall show that by transfinite induction on 1:
(Pi)Every open cover Q with cardinality 1 of X has a locally finite closed refinement. Since it is easily seen that Xis countably paracompact, assuming that for some infinite cardinal T (T > w), every open cover Y of X with I Y I < r has a locally finite closed refinement and 43 any open cover of X with IQ I = r. If for each a < r, we let V , = U{V, 1 /3 < a}, then { V,I a < r} is a monotone
Generalized Paracompactness
171
increasing open cover of X and so there exists a locally finite closed cover { F , ( a < T} of Xsuch that F, c V, for any a c T. Since for each A < T, { U, I fi c A} A FA is an open cover of FA whose cardinality C T , there exists a locally finite closed refinement FA of { U, I fi < A } A FA(and hence .!FAis a locally finite collection of closed subsets of X).Hence U{.!FA I A} is a locally finite closed refinement of { U, I a c T} and so X is paracompact by the following theorem. The following characterizations are well known and so we give them without proof in order to show some contrast with another covering property.
2.12. Theorem (Michael [1957]). Let X be a regular space. Then the following are equivalent: (1)
X is paracompact.
(2)
Every open cover of X has a closure-preserving open refinement.
(3)
Every open cover of X has a closure-preserving refinement.
(4)
Every open cover of X has a a-closure-preserving open refinement.
2.13. Theorem (Michael [ 19591). Let X be a regular space. Then the following are equivalent: (1)
X
(2)
Every open cover of X has a cushioned open refinement.
(3)
Every open cover of X has a cushioned refinement.
(4)
Every open cover of X has a a-cushioned open refinement.
is paracompact.
2.14. Definition. Let d and LiJ be collections of subsets of a space X and x E X . We say that d is pointwise (local) W-refinementof 93 at x if there exists some W E [B] (and furthermore a nbd U of x) such that {A E d I x E A } ( { A E d I U n A # 8)) is a partial refinement of W . If we can choose %? to consist of a single element of W, then we say that d is a pointwise (local) star-refinement of W at x. If U d = UB and d is a pointwise (local) W-refinement of A? at each point of X , then we say that d is a pointwise (local) W-refinement of 9. These concepts were introduced by Worrell [1966b, 19681. Recall that a space Xis said to be fully normal if every open cover 42 of X has a pointwise star-refinement at any point x E X . It is known that a space
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X is paracompact iff X is fully normal (Stone [1948]). So we shall give a characterization of paracompactness which has a close connection with them of another covering property.
2.15. Theorem (Arhangel’skii [1961]). Let X be a regular space. Then the following are equivalent: ( I ) X is paracompact. (2) For every open cover 4 of X , there exists a sequence {4,, I n E N} of open covers of X such that for each x there is some n such that 4,,is a local star-refinement of 4 at x . The above sequence {%,, I n} of open covers of X is said to be locally starring in 4 .
Proof. (1)=42) It is clear because paracompactness implies full normality. (2)=41) Let 4 = {UrnI a E A } be an open cover of X and a sequence {q,, In E N} of open covers of X which is locally starring in 4. For each a E A and each n E N, we let C(rn,n) = U{O 1 0is open in X , St(0, 4,,)c V , } .Then it is seen that {CCrn,,,, I a E A , n E N} is an open cover of X and (C(rn,n) I a E A} is cushioned in 4 for each n E N. Therefore Xis paracompact by Theorem 2.13.
0 It is one of the important properties of paracompactness that all metric spaces are paracompact spaces. Stone was the first to prove directly the paracompactness of a metric space when he proved that every open cover of a metric space has a locally finite and a-discrete open refinement (Stone [1948]). After that, Michael [I9531 proved that if every open cover of a regular space X has a o-locally finite open refinement, then Xis paracompact and hence every metric space is paracompact because a regular space X is metrizable iff X has a a-discrete open base (Bing [1951]), or, iff X has a a-locally finite open base (Nagata [1950] and Smirnov [1952]). Furthermore, we may prove it by use of Arhangel’skii’s Theorem 2.15, that is, if for an open 0 is open, diam(0) < l/n and 0 c U for cover 4 of X , we let 4,,= (01 some U E 4 } ,then {@,,I n} is a locally starring in 4. Quite recently, Ohta [1987a] introduced a new class which contains a class of metric spaces and is contained in a class of M,-spaces (a regular space is said to be an MI-space if it has a a-closure-preserving open base (Ceder [1961]). The Ohta’s class is hereditary and countably productive (it is an open question whether a class of M,-spaces is closed hereditary or not).
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I73
In his paper, he introduced a new concept of a collection of subsets, which is called FCP, and he proved a new characterization of a paracompactness in terms of FCP.
2.16. Definition. A collection 9 of subsets of a space X is called to be finitely closure-preserving (FCP) in X (Ohta [1987a]) if for any subcollection 9’ of 9and any point x E X , there are a nbd 0 of x and a finite subcollection 9“ of 9‘such that 0 n (UP’) c Cl(U9”). It is easy to see that a local finiteness implies the FCP property and FCP property does a closure-preservingproperty. The following lemma is useful to prove Theorem 2.18. The proof is left to the reader. 2.17. Lemma (Ohta [1987b]). r f {Vala< 7 } is a monotone increasing collection of subsets of a space X , then the following are equivalent: (1)
(U,(a <
(2)
For any a. < T and any x E X , there are a nbd 0 of x and some a , < aosuch that 0 n (U{U,lB < a o } ) c C1 Ua,.
T}
is FCPin X .
2.18. Theorem (Ohta [1987b]). Let X be a regular space. Then the following are equivalent: (1) X is paracompacr. ( 2 ) For every monotone increasing open cover 43 = { iJaI a < T } of X , there exists a monotone increasing and FCP o-Den cover { V ,1 a < T} of X such that C1 V , c C,.for each a < T . Proof.
I‘ 1:*(2) Obvious. i 2 ) 2 ( 1 ) Let 42 = {Cu 1 ct .< T’. be any monotone increasing open cwer 01‘ = V ,I a < r } a monotone increasing and FCD open zovx of A’ A’. and such that C1 1.: c LT- for any o! < 7 If fot each Q : : ae let F = C1 V , - J{Cl C/,:o < a1.then.F = !F,ln < 7 , isacoverofXsuzh that C1 Fa c i‘,for any a < r. To finish the Droof it suffices to show thai F,I CI < T ; is locally finite by Theoreni 2.1 i . For t h ~ spurpose let x be any point of X-,a(Of the first of { a I x E 6 [ and H , = &, . Then we have thar ti,>
is an open nbd of x and (0;
Hc, n F,
=
8 for any y > ~ ( 0 )
By Lemma 2.17, there are an open nbd H , of x and some a( 1 ) < a(0) sucn thatH, n (U{l$Ib < a(O)}) c Cl(l&). Foranypwitha(1) . /3 < a(O),we
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have H I n l$ c H I n C1 c H I n Cl(U{yly < < /3}) = 8. Hence (C1 l$ - U(C1 (1)
H I n F, =
8
for any /3 with a(1) <
/3
/3}>
and so H I n
< a(0).
By repeated use of Lemma 2.17, there are an open nbd H,, of x and a sequence {a(n)ln}such that a(1) > a(2) > a(3) > . . . and:
(n)
H,, n FB =
8 for any /3 with a(n) < /3 < a(n -
1).
n{&.
Therefore a(n) = 0 for some n. If we let H = I i < n } , then H i s a nbd of x which does not meet the elements of 9 i = 0, 1 , . . . , n}. Hence we complete the proof that 9 is locally finite. 0 The last characterizations of a paracompactness are obtained in terms of an interior-preservingcollection. They do not appear frequently but we want to show them in order to show the contrast with other covering properties. 2.19. Theorem (Junnila [1979a]). Let X be a space. Then the following are equivalent : (1) X is paracompact. (2)Every interior-preserving directed open cover of X has an interior-preserving open local star-refinement. ( 3 ) Every interior-preserving directed open cover of X has a a-closurecovers X . preserving closed refinement 9 such that { Int F I F E 9) (4)Every directed open cover of X has a closure-preserving closed refinement 9 such that { Int F I F E 9} covers X .
Proof. (1)*(4) and (4)*(3) Obvious. (3)*(2) The first step is to note that X is countably paracompact by use of Theorem 2.3. Let 4 = {U,,In} be an increasing open cover of X. Then there is a a-closure-preserving closed refinement 9 of 4 such that {Int F I F E 9} covers X because a monotone increasing open cover is intenorpreserving and directed. In this place we may express 9 as 9 = U{9,, I n } where for each n, 9,, = {F,,,lm E N} such that F- c V, for any m and FnI c Fn2 c . . . . Then it is clear that {U,, - U { F h p 1 I i< n - 1}1n E N} is a locally finite open refinement of Q. To actually see that X satisfies the condition (2),let Q be any interior-preserving directed open cover of X and so there is a a-closure-preserving closed refinement 9 = U{9,,ln}(where each F, is, closure-preserving) such that { Int F I F E F}covers X. If for each n, we let C,, = u{Int F I F E F,, then },{C,,I n } is a countable open cover of X and so we have a locally finite open cover { V ,I n } of X with V , c C,, for
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175
each n E N. Since for each n, H,, = {Cl V , n F I F E Sn} is a cover of V ,and a closure-preserving collection of subsets of X by Lemma 1.1 and {CI V ,I n} is locally finite. U { S f l1 n } is a closure-preserving closed refinement of 4 such that { Int H 1 H E Hn, n} covers X. Therefore 4!l has an interior-preserving open local star-refinement by Lemma 4.2 which will be proved later on. (2)=41) By Lemma 4.2 and the preceding part of (3)*(2), Xis countably paracompact and so by Theorem 2.2, it suffices to show that every monotone increasing open cover of X has a o-locally finite open refinement. This condition follows from Lemmas 4.2 and 4.4.
3. Characterizations of submetacompactness A class of O-refinable spaces (= submetacompact spaces) was introduced by Worrell, Jr. and Wicke [1965]. This class is useful in the theory of covering properties as well as in the theory of generalizations of metric spaces. “8-rejinable” is called to be “submetacornpact” by Junnila [1978]. Recall the following definition of submetacompact spaces.
3.1. Definition. A topological space X is said to be submetacompact if for every open cover 4 of X , there exists a sequence {4,,I n} of open refinements of 9 such that for each x E X there is some n such that ord(x, %,) is finite, where ord(x, 4,,)denotes the cardinality of { U E 4,,I x E U } . Furthermore, the above sequence {a,,1 n> is called to be &sequence or 8-rejinernent of 4. Secondly we shall define a concept which is useful for the submetacompactness, subparacompactness and others.
3.2. Definition. Let 4 be a cover of a space X,{4”I n } a sequence of covers is a pointwise (resp. local) of X and x a point of X. A sequence {4,,11} W-refiningsequencefor 9 at x if there exists some n such that %, is a pointwise (resp. local) W-refinement of 9 at x. If a sequence {en I n} is a pointwise (resp. local) W-refining sequence for 4 at x for any x E X, then {%,,In} is said to be a pointwise (resp. local) W-rejining sequencefor 4 (see Worrell [1966a]). In the above definition, we can strengthen by adding that each a,,is a refinement of 4. By Worrell[1967], the next characterization of &sequence (which is useful to submetacompact spaces and subparacompact spaces, etc.) was given in terms of a pointwise W-refining sequence. We shall give it without proof.
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3.3. Proposition (Worrell [1967]). For an open cover 4 of a space X , the following are equivalent: (1) 4 has a &sequence of open refinements. (2) There exists a sequence {4,I n} of open refinements of 4 such that for each x there is a sequence { t(n)I n} of natural numbers such that 4tr(n+l) is a pointwise W-refinement of a,(,)at x for any n E N. By use of Proposition 3.3, we can prove the following theorem which is worth contrasting with Theorem 2.15.
3.4. Theorem (Worrell [19671). A space X is submetacompact if and only if every open cover of X has a pointwise W-refining sequence by open covers of x. Proof. “If” part: Let 4 be any open cover of X . By hypothesis we have a pointwise W-refining sequence W , for 4. By repeated use of hypothesis, we have a countable collection W, of open covers of X for each n E N such that for each W E W,, there exists a countable subcollection W‘ of W, such that W’ is a pointwise W-refining sequence for W . Then {W,In} satisfies the condition (2) of Proposition 3.3, and so 4 has a &refinement. “Only if” part: We can prove similarly by use of Proposition 3.3. 0
In Theorem 2.1 1, we can characterize paracompactness in terms of monotone increasingcovers, that is, it is enough to find the locally finite refinement for any monotone increasing open cover only. A similar statement is true for submetacompactness as follows:
3.5. Theorem (Junnila [1978]). A space X is submetacompact if and only if every monotone increasing open cover of X has a &refinement of open covers. Proof. It suffices to prove only the “if” part. This proof is based on Junnila [1980]. For each cardinal K , we set (P,) as follows:
(P,):
Every open cover of X with cardinality K has a pointwise W-refining sequence by open covers.
We shall prove the statements (P,) by the transfinite induction on K. Since it is clear for finite cardinal IC,we assume that for infinite cardinal K, (PI) holds for any il < K. Let 4 be any open cover of X with cardinal K and so we may express 4 as 4 = { 17I, a < K}. If for each a < K, we let V. = U{U,1 B < a}, then { V ,I a < IC} has a &sequence {V,I n } of open covers of X . For each n,
Generalized Paracompactness
I77
we let H,, = {x I ord(x, Vn) is finite} and a(x, n) the 1st of {a I St(x, Vn)c V,} foreachx E H,,. Ifforeacha < Kandeachqlet P,,,= ~ { V .V,l E V Q V,}, then a,,, = { U, I fl < a } u {Pan}is an open cover of X with cardinality < K and so a,, has a &sequence { Wa,,kI k} of open covers of X . For each n, k E N and each x E H,,, we can select some member W,,(x) of W,(x,n),,kwith x E W,k(X) and we set Gnk(x)= Wn,(x)n . . n W,k(X) n (n{vlXE VE~,,}).IfWelet%,,k = {Gnk(X)IXEHn}U {{Xlord(x,~,,)2 k + l}}, then the sequence {9,,, I n, k } of open covers of X will be a pointwise W-refining sequence of%. For any point x of X , we select some n with x E H,,. If we let A = {the 1st of {a I V c K} 1 x E V E V,,}, then A is finite. For each a E A, there exist some k(a) E N and some finite subcollection W ( a ) of a,,, such that { W I x E W E W&,} is a partial refinement of W(a). Let R = U{R(a) A 4 I a E A } and k = max{ord(x, V,,), k(a) I a E A}. Then it is seen that {G 1 x E G E gnk}is a partial refinement of a finite subcollection 9l of 4.
17 As the last characterization of submetacompactness, we shall have it, as well as that of paracompactness, in terms of refinements for interior-preserving open covers of a space (see Theorem 2.19) or a-closure-preservingclosed refinements (see Theorem 2.12).
3.6. Theorem (Junnila [1974]). For a space X , the following are equivalent: (1)
X is submetacornpact.
(2)
Every interior-preserving directed open cover of X has a a-closurepreserving closed rejinement .
(3)
Every directed open cover of X has a a-closure-preserving closed rejinement.
= { U, I a E A} be any directed open cover of X and {an In} a &sequence of open covers of X for 9. If for each n, k E N, let Fnk= {XIord(x, 4,,)4 k} and @,,k = { U E %,,I U n Fnk# s}, then {Fnk I n, k } is a closed cover of X and @,,k is a cover of &k . Furthermore, ord(x, %,,k) 4 k for any X E F n k . 1 n } ) be an enumeration of a collection {a,,,I n, k } Let {V,,I n} (resp. { Y,, (resp. {FnkIn,k}) indexed by N such that V,,covers H,, for any n. Then W = { V - U{H, I i < n} I V E V,,}is a point-finite open refinement of %. If for each a E A, we let F, = {x E XISt(x, W ) c U , } , then {Fala E A} is a closed cover of X by the direction of 4. Therefore {F, I a E A} is a closurepreserving closed refinement of 4.
Proof. (1)*(3) Let 4
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Y. Yusui
(3)*(2) Obvious. (2)=4 1) Since a monotone increasing open cover is interior-preserving, it suffices to show that every interior-preserving open cover 9 has a 8-sequence of open refinements from Theorem 3.5. Since Qfis an interior-preserving directed cover of X by Lemma 1.2. 9'has a a-closure-preserving closed refinement 9, where we express 9as 9 = U { F nI n} and each 9" is closurepreserving. If for each n~ N and each X E X , we let K,x= n { U E 9 l x E U } ~ { F 9E" I x $ F}, then K,xis an open nbd of x and Vn= { <.xIx E X} is interior-preserving by Lemma 1.2. Also we can see that {Vn 1 n} is a pointwise Since each Vnis an interior-preserving open cover, W-refining sequence of 9. we have a pointwise W-refining sequence of Yn,each of which is interiorpreserving. By repeated use of the above argument, we have a countable collection V of interior-preserving open refinements of 4 such that for any x E X and Y E V, there exists some W E V which is a pointwise W-refinement of V at x. Hence 9 has a 8-sequence of open refinements. This completes the proof.
3.7. Remark. Bennett and Lutzer [I9721 introduced the generalization of submetacompact spaces, which is called to be weakly O-refinable spaces, that is: A space Xis weakly 8-rejinable if for every open cover 9 of X there exists a sequence {a"1 n} of collections of open subsets of X such that (1) each a,, is a partial refinement of 9 and (2) for each x E X , we have some n with 1 < ord(x, 9,, <)o. Normal and weakly 8-refinable spaces are not aiways 8-refinable (see Example 6.1). But in a class of perfect spaces (that is, every open set is Fg),weak 8-refinability is equivalent to O-refinability (Bennett and Lutzer [1972]). Though weak 8-refinable spaces have many interesting and important properties, we shall omit to discuss them (refer to Bennett and Lutzer [1972], and De Caux [1976], etc.).
4. Characterizations of metacompactness Metacompactness was introduced by Arens and Dugundji [ 19501. They showed that a countably compact space is compact iff it is metacompact. On the other hand, Bing [ 19511 introduced the pointwise paracompact spaces (which are equivalent to metacompact spaces) and he showed that metacompact developable spaces are not necessarily paracompact. One of the important and interesting results on metacompact spaces is the theorem that every
179
Generalized Paracompactness
collectionwise normal metacompact space is paracompact, which was proved independently by Michael [1955] and Nagami [1955]. The techniques of the proof used in that papers are useful for the study of the covering and other properties. In this section we shall give the characterizations of metacompact spaces. A metacompactness is often called a weak paracompactness. Let us begin with the definition of metacompactness. 4.1. Definition. A space X is metacompact if every open cover of X has a point-finite open refinement. By the definition of submetacompact space, it is clear that every metacompact space is submetacompact (the converse is not true; see Section 6). Before giving the characterizations of metacornpact spaces, we shall show the next lemma, which is powerful for the study of covering properties. This lemma, essentially due to Worrell [1966a], is a weak form of a result from Junnila [1979a]. Our proof is due to Burke [1984]. 4.2. Lemma (Junnila [1979a]). Zf{%, I n E N} is a sequence of open covers of a space X such that @,+ I is a pointwise (resp. local) W-rejinement of 42, for any has a u-point-finite (resp. a-locally finite) open refinement. n E N, then Proof. We shall prove the lemma only for pointwise W-refinement. We express as {U,la < T}. If for any U E u{%,ln = 1, 2 , . . .}, we let 6 ( U ) = the 1st of { a < T ~ cU U,} and W, = { W E % , I ~ ( W )= 6 ( U ) if U E and W c U } for each n = 2, 3, . . . . To show that U{W,I n = 2, 3, . . .} is a cover of X , let x be any point of X. Since for each we let a, = sup{6( U )I x E U E a,}. n > 1, %, is a pointwise W-refinement of Then it holds that a2 2 uj 2 * . . and so there is some: k > 1 such that a k = U k + l = U k + 2 = * * * . Since %k+2 is a pointwise W-refinement of % k + l , of @ k + l such that { U E %k+2 I x E U } is a there is a finite subcollection partial refinement of where we may assume that x E U for any with 6(Uo) = a k . For UE From a k + l = a k , we have some U, E any U E %k with U, c U , we have ak = 6(Uo) < 6 ( U ) < ak and so 6(Uo) = 6 ( U ) , that is, U,, is a member of wk+l.Therefore W, I n > l} is a cover of X . If we let V,, = U { W E W,16(W) = a } , for each n > 1 and eacha < T , V ,= {V,,Ia}foreachn > 1 a n d V = U { V , l n > 1 } , t h e n V is an open refinement of In this place it suffices to show that Y, is point-finite for each n > 1. To show that, for any n > 1 and any x E X , we define A = { a < T 1 x E V,,}. For each a E A, we select some W, E W, such that x E W, and 6(W,) = a.
u{
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180
Since Wnis a pointwise W-refinement of %,+, , there is a finite subcollection 9 of 4 - ,such that { W,Ict E A} is a partial refinement of 9. Hence, if W, c U with U E 9, then ct = 6( W,) = 6 ( U ) E 6(9)which is finite, and so
0
A is finite.
The next characterizations of metacompact spaces are considered the modifications of paracompact spaces (see Theorem 2.15) and submetacompact spaces (see Theorem 3.4). 4.3. Theorem (Worrell [1966a], Junnila [1979b]). For a space X , the following
are equivalent: (1)
X is metacornpact.
(2)
Every open cover % of X has a point-finite refinement Y such that x E Int(St(x, Y ) )for any x E X .
(3)
Every open cover of X has an open pointwise W-refinement.
Such a cover “fof (2) is called to be a semi-open cover Proof. (1)=-(2) Obvious. be any open cover of X and I - a point-finite semi-open (2)*(3) Let refinement. For each I/ E I’ we select any member U ( V ) of @ with V c U( V ) .Iffor each x E X , let Yx, = Int(St(x, Y 7 ) n U( V )I x E V E $“}), then t i r , is an open nbd of x. It suffices to show that -w^ = { Qx)IxE X is a pointwise W-refinement of %. Let x be any point of X and I”, . . . . V” 1 be all members of %* which contain Let 1; be any member of -lI whicri contains x, then x E b , , z Sti p, I 1 and so ix, y : c I/‘ for some < n. and SO T{,, z h/(V”i. So { 1, v ) i x e I;,) E W ) is a partial refinemen1 of : LT(V‘) I i ( 3 b g1 1 This Drool IS due (0 Burke [1Q84].Let % be any open co\?r oL. X By ixnima 4.2. we niiw 9 a-point-finite open refinement Y’ = { *, I n : a1 %, where eacb 7 ” IS point-finite. If for each n. we let I., = (11 I/ I i E , L! . . . LJ 1 then there is an open pointwise W-refinement % of { 1;,.Jn, by repeated us: of Lemma 4.2. Since { V , , J d is increasing. E, = ( x E XI Sttx, %”) c 1 : : is a crosed set which is contained in Y :For each nl and {&,Ina I: A cover of A . If for eacri re, we define C, by el, = i l - LJ;€~IK < n\ C E r n i . then [ J ( U J n }is a point-finite open refinement of 9. J
(n{
),
2
821
~
-
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181
In order to discuss the next characterizations of metacompactness, it is necessary to prove the following lemma which is needed for the discussion of the interior- and closure-preservations and pointwise W-refinements, etc., which were used in the proof of the previous Theorem 2.19. 4.4. Lemma (Junnila [ 1979a, 1979b1). For an interior-preserving open cover Q of a space X,the following are equivalent:
(1)
4' has a closure-preserving closed rejinement (resp. furthermore whose interior is a cover of X ) .
(2)
4 has an interior-preserving open pointwise (resp.local) W-rejinement.
(3)
4' has an interior-preserving open pointwise (resp. local) starrefinement.
Proof. The following proof is due to Burke [1984]. (1)=42) Let 9be a closure-preserving closed refinement of 9'.If for each X E X , we define P,' as n { U l x E U E @ } - u { F l x $ F , F ~ 9 } then , { V,I x E X } is an interior-preserving open cover of X by Lemma 1.2. To see that { V, I x} is a pointwise W-refinement of 4, for each x E X we select any Fo E 9 with x E Fo and a finite subcollection of 4 such that Fo c U{U 1 U E 4 0 }Then . { V, I x E K.} is a partial refinement of @., ( 2 ) 4 1) Let Y be an interior-preserving open pointwise W-refinement of 4. If for each G E Q', let P ( G ) = { x I St(x, Y ) c G}, then { P ( G )1 G E 4'} is a closed refinement of a', and so we shall show that { P ( G ) JG E 4'} is closure-preversing. Let { G , 1 a E A } be any subcollection of Q', where we can express G , as G , = UQu for some finite subcollection Qu of Q (for each a). In order to show that U{P(G,)(a E A } is closed, let x be any point of X - U { P ( G , )I a}. Since Y is a pointwise W-refinement of 4, there exists a finite subcollection { U , , . . . , U,,}of 4 such that { V I x E V E Y }is a partial refinement of { U , , . . . , U,,}, where we may assume that x E U , for any i = 1, 2, . . . , n. For each u, there is some V , E Y such that x E V , and V, c t G, and so, there is some i ( a ) < n with V, c U,(,). Therefore 0= Ui(,)I a} is an open nbd of x such that 0 n P(G,) = 8 for any a, that is, U { P ( G , )I a} is closed. We shall leave the other parts of this lemma to the reader. (The proof is similar to the above.) 0
n{
The next characterizations of metacompact spaces are modifications of Theorems 2.11 (3) and 2.12 (3) for paracompact spaces and Theorems 3.5 and 3.6 for submetacompact spaces.
Y. Yasui
I82
4.5. Theorem (Junnila [1979a], Sconyers [1970]). For a space X , the following are equivalent:
(I)
X is metacompact.
(2)
Every directed open cover of X has a closure-preserving closed refinement.
(3)
For any open cover 9 of X , 9 ‘has a closure-preserving closed refinement .
(4)
Every monotone increasing open cover of X has a point-finite open refinement.
Proof. (1)=42) Let 9 be any directed open cover of X and express 42 as 9 = { U, I a E D},where D is directed. Furthermore, let Y be a point-finite If for each U E 4 ,we define W(V )as {x I St(x, Y ) c V } , open refinement of 9. then it is clear that { W ( U )I U E 9}is a closed cover of X such that W ( U ) c U for any U E 9, and so it suffices to show that for any D‘ c D, U{ W(U,)la E D’)is closed. For this prupose, let x be any point of X - U{ W ( U , )I a E D’}. Since *Ir is point-finite and St(x, Y ) U , for any a E D’,there is a finite subcollection { V,, . . . , V , } of Y with x E for each i < n such that for each a E D’ we have some i(a) < n with K(,) Q U,, which means that I&) n W(V,) = 8. Therefore V = Fl i = I , . . . , n} is an open nbd of x such that V n (U{ W(U,)I a E D’})= 8. (2)0(3) Obvious. (2)*(4) Let 9 be any monotone increasing open cover of X . By Lemma 4.4, there is a sequence {9, = 9, a2,423, . . .} of open covers of Xsuch that 9,,+, is an interior-preserving open pointwise W-refinement of 9,for any n. On the other hand, by Lemma 4.2, there is a a-point-finite open refinement V = U{V, I n } of 9,where each Y, is point-finite. If for each n, we let V , = U{UYkI k < n } , then { V ,I n} has a closure-preserving closed refinement IF, I n } by condition ( 2 ) such that F, c V , for any n. Then it is clear n = 1, 2, . . .} is a point-finite open that { V - U{eI i < n - l} I V E Yn, refinement of 9. (4)=4 1) We shall prove this by transfinite induction on the cardinality of a given open cover of X . Since Xis countably metacompact, we assume that for some uncountable cardinal T, every open cover of X whose cardinality is less than T has a point-finite open refinement. Let 4 be an open cover of X with cardinal T and so express it as 42 = { U, 1 a c r}. If V, = U( U, I /I< a } for each a < ‘I, then { V , I a c r} has a point-finite open refinement W = {W,la c z} such that W, t V , for any a c z. For each a, we let
+
n{
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9, = {U,lfl < a } u {U{W,lfl > a}}.Then9uisanopencoverofXwith cardinality < T and so there is a point-finite open refinement F, of Ya.Let Fu'= {TE FulT c U, for some j3 < a } for each a < T and d = { W, n TI T E Fu',a < T}. Then it is easily seen that d is a point-finite open refinement of 43. This completes the proof. 0
The next characterizations of metacompact spaces which were obtained by Junnila [1979a] are modifications of Theorems 2.19 and 3.6. Theorem 4.6 is useful to obtain the following analogue of Remark 2.10 (see Fletcher and Lindgren [1972]):A completely regular space Xis metacompact iff every open cover of X x B ( X ) has an interior-preserving open refinement, where B ( X ) is the Stone-(5ech's compactification of X . 4.6. Theorem (Junnila [1979a1). For a space X , the following are equivalent:
(1)
X is metacompact.
(2)
Every interior-preserving directed open cover of X has a closurepreserving closed refinement.
(3)
Every interior-preserving directed open cover of X has an interiorpreserving open pointwise star-refinement.
Proof. (1)*(2) Theorem 4.5. (2)*(3) Theorem 4.4. (3)*(1) We shall show this by use of Theorem 4.5. Let 4 = { Vula < T} be any monotone increasing open cover of X and an interior-preserving open pointwise star-refinement. Since 4; is interior-preserving and directed, there is an interior-preserving open pointwise star-refinement a2of 42;. By repeated use of Lemma 4.4, we have a sequence {a,,1 n } of open covers of X such that is a pointwise W-refinement of 4, and therefore we have a o-point-finite open refinement of 4 by Lemma 4.2. If for each n, we let W, = U{U"w;(i < n } , then there is an open pointwise star-refinement 9 of { W, I n } . If for each n, we let F, = {x I St(x, 9') c W.}, then it is easily seen that { W - Fn-lI W E W,; n = 1, 2, . . .} is a point-finite open refinement of 43, (where we assume that Fo = 0). 0 As is well known, spaces with the closure-preserving covers by compact sets are not necessarily paracompact (Potoczny [1972]), but it is easily seen that such spaces are metacornpact by the above Theorem 4.6.
4.7. Corollary (Katuta [1974], Potoczny and Junnila [1975]). I f a space X has a closure-preserving cover by compact subsets, then X is metacornpact.
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5. Characterizations of subparacompactness Subparacompactness is one of the important and interesting properties in general topology. Its property was introduced by McAuley who called it as F,-screenability (McAuley [ 19581). He showed that every semi-metric space is F, -screenable and every collectionwise normal F,-screenable space is paracompact. On the other hand, Arhangel’skii [I9661 studied a class of spaces with a a-discrete network (afterwards, every space of this class was named a a-space by Okuyama [1967]). To begin with, we shall give the definition of subparacompactness:
5.1. Definition. A space Xis said to be subparacompact if every open cover of it has a a-discrete closed refinement. This definition will be considered as a generalization of paracompactness by Theorem 2.1 1 (2). Since the following theorem is well known (Burke and Stoltenburg [1969], Coban [I9691 and Junnila [1978]), its proof can be found in many papers (Burke [I9841 and Nagata [1985], etc.) and will therefore be omitted.
5.2. Theorem (Burke, [1969], Junnila [ 19781). The following are equivalent for a space X : (1)
X is subparacompact.
(2)
Every open cover of X has a a-locallyfnite closed refinement.
(3)
Every open cover of X has a a-closure-preserving closed refinement.
(4)
Every open cover of X has a a-cushioned refnement.
From the above Theorem 5.2, we have the following corollary. 5.3. Corollary. r f a space X has a countable closed cover by paracompact sets, then X is subparacompact. We have the following characterizations of subparacompectness if we take a side view of Theorem 2.15. The second condition of Theorem 5.4 was introduced by Arhangel’skii [ 19661 who called it a-paracompact.
5.4. Theorem (Burke [1969], Burke and Stoltenburg [1969], Coban [1969]). The following are equivalent for a space X :
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(1) X is subparacompact. (2) For every open cover 4 of X there is a sequence {%,,In E N} of open covers of Xsuch thatfor any x E X there is some n E N such that 9, is pointwise star-rejinement of 4 at x. (3) Every open cover 4 of X has a sequence {%,, I n E N} of open refinements of 4 such that for each x E X , there is some n E N with ord(x, 4,,)= 1. The above sequence {@,,I n } of the condition (2) is called to be a pointwise starring in 4.
Proof. (1)-(3) and (3)-(2) Obvious. (2)=4 1) Let 4 = { U, I a E A } be any open cover of X and a sequence {%,, 1 n } of open covers of X which is pointwise starring in 4. For each a E A and n, we let U,, = {x I St(x, a,,) c U,}.Then it is seen that F,, = { U,, I a } is cushioned in 4 and U { F nI n} is a closed refinement of 4. 0 From Theorem 5.4, we have the following corollary.
5.5. Corollary (McAuley [ 19581). Every Moore space is subparacompact. It is easily seen that paracompactness is stronger than subparacompactness. For example Niemytzki space (see Engelking [1977] or Nagata [1985]) is subparacompact (by Corollary 5.3 or 5.5) and is not paracompact (by nonnormality of its space). In Section 6, we shall give a Moore space which is not metacompact (Example 6.8). Finally, we shall have a characterization for subparacompactness analogous to Theorems 2.19, 3.6 and 4.6. 5.6. Theorem (Junnila [ 19781). A space X is subparacompact if and only if every interior-preserving open cover of it has a a-closure-preserving closed refinement.
Proof. The “only if” part is obvious and so we shall prove the “if” part. By Theorem 3.6, Xis submetacompact. In order to show that Xis subparacompact, it suffices to show that for any discrete collection {F,I a E A } of closed sets of X , there exists a mutually disjoint collection {G, I a E A } of G,-subsets of X such that F, c G, for any a E A (see Theorem 7.15). If for any a, we let Ha = X - U{FBI fl E A , fl # a}, then J f = {HaI a } is an interior-preserving open cover of X and so Z has a a-closure-preserving closed refinement 9.
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Therefore, we may express 9 as 9 = U ( 9 , I n } such that each 9, is closurepreserving. If for each a E A and n E N, we let Can= X U{FE 9 , l F n Fa = then Can is an open set which contains Fa. Therefore, to complete the proof, it is sufficient to show that n{G.. I n } n (n{G,, I n } ) = 0 for any a, /? E A with a # /?.Let a, /? E A with a # /?, and any x E X. Since 9 is a cover of X,there are some i E N and some F‘ E q.with x E F‘. Here we may assume F’ n Fa = 0 without loss of generality, which means x E Cai 3 (-){Gun1 n } . Therefore (-){Can In } n
a},
(n{GB,In>>=
0.
6. Shrinking properties Normality of a space X is that for every binary open cover { U , , U z }of X , there is a binary closed cover { F , , F2} of X such that F, c U, for each i. Generally speaking, it is well known that a space is normal iff every pointfinite open cover { UaI a E A } of the space has a closed refinement {FaI a } such that Fa c Ua for each a E A iff every point-finite open cover { U, 1 a E A } of the space has an open refinement { V,1 a } such that CI V, c Uafor each a E A. One of the purposes of this section is to generalize the above properties. 6.1. Definition. An open cover { U a l aE A } of a space X is said to be shrinkable if there is an open cover { V ,I a E A } of X such that CI V, c Uafor each a E A . A space Xis shrinking if every open cover of the space is shrinkable.
In the above definition, we can say that a space X is shrinking iff for any open cover { U, I a E A } of X , there is a closed cover {FaI a E A } of Xsuch that F, c U, for any a E A. Every paracompact space is shrinking and by Corollary 2.7, Xis countably paracompact iff every countable increasing open cover of X is shrinkable. Furthermore, Chiba [ 19841 proved that every perfectly normal space or subparacompact and normal space is shrinking. Afterwards, Chiba’s result was generalized as follows. 6.2. Theorem (Yasui [1984]). A submetacompact space is shrinking if and only if it is normal. Proof. We shall prove “if” part: Let 42 = { UaI a E A } be any open cover of a normal and submetacompact space X and { VnI n E N} a &refinement of 42, where we may assume that Vn= {<.la E A } with c Ua for any
<,
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E A and any n E N. If for each n, we let Y, = { y I ord( y , V,)is finite}, then by a lemma of Yasui [1984], there exist open subsets Hnni( i E N) of X such that C1 Hnni c V,, for any i and any a, and Y, c u{HnaiIa,i } . Therefore, {Ifnni I a, n, i } is a cover of X. Therefore, every open cover of X is shrinkable 0 by Beilagik [ 19861.
a
Fleischman [ 19701 proved that every linearly ordered space is shrinking, and therefore, because every subspace of a linearly ordered space is a closed subspace for some linearly ordered space (see Nagata [1985]), we have the following theorem. 6.3. Theorem (Fleischman [ 19701). Every subspace of a linearly ordered space is shrinking.
6.4. Remark. There are several results for the shrinking properties of C-products. Let us recall the C-product which was introduced by Corson [1959]. Let {X,lA E A} be a collection of spaces, and X = n X , the product space of { X i I A} and f = (fi), E X. If we let C = { X I {A Jx, # A } is at most countable}, then the subspace C of X is called to be the C-product of spaces X , (A E A) with the base point f E X . Rudin [1983a] and Donne [I9851 showed the generalization of Rudin’s Theorem, that is: A C-product of paracompact M-spaces is shrinking iff it is normal. And quite recently Yajima [I9861 proved that a C-product of strong C-spaces is shrinking iff it is normal, where a space X is a strong C-space (Nagami [1969]) if there exist a cover X of X by compact sets and a o-locally finite collection 9of closed subsets of Xsuch that for any K E X and any open set U with K c U , there is some F E 9 with K c F c U . As mentioned above, a space is countably paracompact iff every countable increasing open cover of it is shrinkable, and so we shall consider the following definition. 6.5. Definition. A space X has a property 9 (resp. a property 93) if for every monotone increasing open cover { U, I a < T} of X , there is an open cover { V,la < T} of X such that C1 V , c U, for any a (resp. furthermore { V ,I a < T} is monotone increasing). A property was introduced by Zenor [ 19701who showed that a separable regular space is Lindelof iff it has a property 93 and every countably compact space with a property 33 is compact. On the other hand, a property 9 was introduced by Yasui [I9721 who called it a weak %property. In Remark 2.10, we noted that a normal space Xis countably paracompact iff X x l i s normal.
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Y. Yasui
Afterward, the issue of the existence of normal but not countably paracompact spaces arose and so any normal but not countably paracompact space soon become known as a Dowker space. Dowker space was constructed without using any special set-theoretic assumptions by Rudin [ 19711. Afterward, a generalization of a Dowker space was given as K-Dowker space, that is, a space which is normal and there exists a monotone increasing open cover of it which is not shrinkable (see Rudin [I9781 and Atsuji [1976]). From the above circumstances we shall give a new name as a property 9. Therefore a normal space X has a property 9 iff X is not a K-Dowker space for any K. We have the following trivial implications. 6.6. Proposition. Paracompactness able paracompactness.
=-property 93 =-property 9 * count-
Of the above implications, none of the converses hold (see Rudin [ l983b, 1984, 19851 and Yasui [1972]). With respect to a property W,we have the following characterizations which are modifications of those of paracompactness (see Theorem 2.12). 6.7 Theorem (Tani and Yasui [1972]). For a space X , the following are equivalent
(I)
X has a property W.
(2)
Every monotone increasing open cover of X has a cushioned open refinement.
(3)
Every monotone increasing open cover of X has a a-cushioned open refinement.
Proof. (1)=42) and (2)*(3) Almost obvious and so we shall omit. (3)*( 1) Let 4 = { 17I, a < T} be any monotone increasing open cover of X and V = Vn1 n E N} a a-cushioned refinement of Q, where each “v;, is cushioned in Q and we may express it as V, = {V,,la < T } and Cl(U{ V,. I a E A } ) c U{U, I a E A } for any subset A of z and any n. Since X is countably paracompact by the condition (3), there is a locally finite open cover { W, I n E N} of X such that W, c V,, I a } for any n. If for each a, we let 0, = U{ W, n (U{ KsI fl < a } ) In}, then {O,l a } is a monotone increasing open cover of X such that CI 0, c U, for any a. 0
u{
u{
As a characterization of a property W with use of terms of any open cover (not necessarily monotone increasing), we recently have the following.
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6.8. Theorem (Yasui [1986]). For a space X , the following are equivalent: (1) X has a property W . (2) For any monotone increasing open cover { U, I a < t} of X , there exists an open cover { V ,I a < t} of X such that (a) V , c U, for each a < t and (b) for each x E X , there exist a nbd 0 of x and some a. < T such that 0 n ( U { V , l a 2 a,}) = 0. (3) Every infinite open cover 4 of X has an open refinement Y such that each x E X h a s a nbd 0 of x such that the cardinality of { V E Y l O n V # $4) is less than 1 % I. Proof. (1)0(2) See Yasui [1986]. (3)*( 2) Obvious. (1)*(3) Let 4 be any open cover of X and Q express as { U, 1 a < t},where T is a minimal ordinal number whose cardinality is equal to I4 1. By Yasui [1986], there is an open cover Y = {GSl/3 < a; a < T} of X such that (i) KS c U, for any a, B with /3 < a, and (ii) each x of X has a nbd 0 and some a, such that 0 n (U{V,sl/3 < a ; a, < a } ) = 0.Then {V,,lB < a, a < t} is a desired cover. 0
With respect to Remark 2.10, it is well known that a regular space X is paracompact iff X x Y is normal for any compact space Y iff X x I' is normal for any cardinal t iff X x /3(X)is normal, where, B ( X ) is the StoneCech's compactification of X (Morita [1962], Tamano [1960, 19621, and Atsuji [ 19891). For the case of a property 9,there is no compact space Y such that X has a property 9 iff X x Y is normal (Rudin [1983b, 19851). But we have the following characterization: 6.9. Theorem (Yasui [1983]). For a normal space X , X has a property W if and only if X x I, is normalfor any cardinal t,where I, is the set of all ordinals < t with the topology { U c I, 1 t $ U or I, - U has a cardinality less than T}.
Proof. Suppose that X x I, is normal and {U,la -= t} is a monotone increasing open cover of X . Define H = U { ( X - U,) x { a } I a < t} and K = X x {t}. Since H and K are closed and disjoint in X x I,, there are disjoint open sets W and V of X x I, with H c W and K c V. For each a < t, define V, = { x i 0 x { B l a < /3 < t} c Vfor some nbd 0 of x}. Clearly V , is open and C1 V, c U,. Furthermore, { V,I a < t} is a monotone increasing cover of X . Conversely, suppose that Xis a normal space with a property a,and H and K are disjoint closed subsets of X x I,. By Starbird [1974], we may assume
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that H = X x { t } . For each a < 7 , let U, = U { U l U is open in X and (U x {/I1 a < /I < 7 } ) n K = 0}. Since { U, I a } is a monotone increasing open cover of X , there is a monotone increasing open cover { V , ( a < T} of XwithCIV=c U,.IfweletW= U{W,x{/IIa < / I } l a < ~ } , t h e n W i s an open set containing H . It is easy to check that K n C1 W = 8; thus X x Z, is normal. 0 When we concern with Theorems 2.3,2.18, 3.5 and 4.5, etc., which are the characterizations of covering properties in terms of monotone increasing covers, we have the following question: “If every monotone increasing open cover of X is shrinkable, then is every open cover of X shrinkable?”. The answer is “No” by BeilagiE and Rudin [1985] (see Theorem 7.5). The next is a sufficient condition in order that a space is shrinking, and such a space is Bing’s example C etc., for example. We shall omit the proof. 6.10. Proposition (Yasui [1985]). Let {A, B } be a disjoint cover of a normal space X such that each x of A is isolated in X and B is a discrete subspace of X . Then X is shrinking.
Next to the shrinking, we shall consider the para-Lindelof spaces, where a space is para-Lindelof if every open cover of it has a locally countable open refinement. The purpose of dealing with the para-Lindelof spaces is to introduce a Navy’s space (Navy [1981]). Navy had constructed a space which is a para-Lindelof, locally compact and normal space which is not collectionwise normal (and so not paracompact). We shall give a characterization of para-Lindelof spaces which is a modification of Theorem 4.3. 6.11. Theorem (Burke [1980]). A space X is para-Lindelif ifand only ifevery open cover of Xhas a locally countable rejinement Y such that x E Int(St(x, Y ) ) for any x E X .
Proof. It is sufficient to prove “if” part. Let 9 be any open cover of X and Y a locally countable refinement of 9 such that x E Int(St(x, Y ) )for any x . Furthermore, let W be an open cover each member of which meets at most countably many elements of Y and d a locally countable refinement of W such that x E Int(St(x, d)) for any x. If for each V E Y ,let us select some U ( V ) of Q with V c U ( V ) , and let G ( V ) = Int(St(V, d ) )n V ( V ) ,then Y = {C(V )I V E Y }is a locally countable open refinement of Q. 0
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7. Examples Throughout the above sections we discussed, for the most part, the characterizations of the covering properties. Implications among their properties are shown in the following diagram: subparacompact
I
dco~:t 1I para-Lindelof
+ countably paracompact 1 property 9
submetacompact
\
+ normal \
shrinking
\property 9 1
/
countably paracompact All the implications will be induced by the definitionsor their characterizations with some exceptions. In this section, we shall give the non-trivial implications, the examples in which the implications are not reversible and the sufficient conditions under which two different classes are coincident. At first we shall mainly give the subspaces of Bing's example G (Bing [1951]). For completeness, let us review Bing's example G: Let P be uncountable, d a collection of all subsets of P,D = (0, l } and F the product spaces of 19 I-copies of D , that is, F = {flf: 22 + D}. For each x E P, leth E F a s follows:f,(Q) = 1 ifp E Q andfp(Q) = 0 otherwise. and p E P, let We let a subset Fo as { f , I p -P}. ~ For each 43 U ( p , 43) = {YE F(f(R) = f , ( R ) for any R E W}.Let us describe a nbd basis 9(f) o f f € F as follows:
7.1. Example (Burke [1974]). not submetacompact.
If 1 P 1 > 2", then F (= Bing's example G ) is
Proof. Let% = { U ( p , { p } ) l p P~} u { F - F0}and{9,,!,n}anysequence of open refinements of 9, where we may assume that for each n, %, = { U ( P ,W ( , n ) )I P E P} u {{f}If€F - Fo} for some W ( p . n ) E [$I* For each p E P, let A,i = { R E W ( p , n ) I n E N, P E R } , Ap2 = { R E W ( p , n ) I n E N, P 4 R } and A, = API u A,, (hence A, = U{W,,,,, In}), Then there exists a subset P' of P such that API n A,, = 8 for any p, q E P' ( p # q ) and 1 P' 1 > 2" by
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Theorem I. 1 of Burke [ 19741. Now we define & E F as follows: S,(R) = 1 if and only if R E U { A , , J pE P’}.Then& E U(p, W(p,nl)for anyp E P’,that is, ord(fo, 9,,)2 I P’I > a. 0 If we let K = Fo u { f F~l f - ’ ( l ) is at most countable} be a subspace of F ( = Bing’s example G), then K is not metacompact by the similar proof of the above Example 7.1 (Burke [1974]) and it was shown by Hodel that K is meta-Lindelof, that is, every open cover of K has a point-countable open refinement. Later we shall show that every subspace of F ( = Bing’s example G ) is shrinking and so F is shrinking and not metacompact. The second example shows that subparacompact spaces and metacompact spaces are not coincident.
7.2. Example (Burke [1974]). Let F be a Bing’s example G with I PI > 2” and Y = { f Flf(R) ~ = 1 for at most two many elements of one-point sets R} a subspace of F. Then Y is normal and metacompact, but not subparacompact.
Proof. Every open cover 9 of Y has a refinement Y which is expressed as { W p , W,,)) n YIP E p } u {{f}I f € y - Fo} with {PI E g(,). Then ord(f(Y)) < 3 for anyfE Y, which shows the metacompactness of Y. To see that Y is not subparacompact, it suffices to show that 9 = { V ( p , {p}) ( pE Y} u { Y - Fo} does not have a &refinement. Let {a,,In} be any sequence of open refinements of 9. Without loss of generality, we may assume that 9,,= {U(p, W,,,,,) n Y l p E P} u {{f}I f € Y U(WP, g ( p , n ) ) Ip E P}},where { P} E W ( p , n ) for any P E P. For each P E P, wedefine A,, = { R E W ~ , , , , ) I ~ E N , ~ E R {} R, A EW ~ ~( ,=, , ) I ~ E N , ~ # R } and A, = A,, u A p 2 .Then there is a subset P‘ of P such that A,, n Aq2 = 0 for anyp, q E P‘ ( p # q) and 1 P‘ I > 2” by Theorem 1.1 of Burke [I 9741. We select two distinct pointsp and q of P’and define& E Y as follows:&(R) = 1 iff R E A,, u A q , . Then it is seen that ord(&, a,,)= 2 for any n, that is, Y is not subparacompact by Theorem 5.4. 0 The existence of a subparacompact space which is not metacompact is well known (for example Niemytzki’s plane (see Nagata [ 19851)). The third example is normal and submetacompact but not metacompact or subparacompact.
7.3. Example (Burke [1974]). Let 1 PI > 2”, {p,,ln E N} be a sequence of distinct points in P,9o= {{p}Ip E P},9, = { { q ,p , , . . . ,pn}I q E P} and
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193
X, = { f Flf-'(l) ~ n 9,, is at most finite} u {fp,, . . . ,fp,}. Then a subspace X = u { X ,I n E N} of F (= Bing's example G ) is normal and submetacompact but not metacompact or subparacompact.
Proof. Any open cover of X, has an open refinement $'- such that V = {U(P, W,,,)) n XIP E PI u {{fllf~ X - U { W , W(,JIP E PI}, where we may assume that {p}, {p, p , , . . . ,p,} E W,p, for any p E P. Since V is point-finite, each X , is closed and metacompact and so X = u{XnIn}is submetacompact. On the other hand, the space Y of Example 7.2 is a closed subspace of X, and so X is not subparacompact. We shall show that an open cover % = { U ( p , { p } ) n X ( p E P} u {X - F,} does not have a point-finite open refinement. Let W be any open refinement of@, where we may assume that W = { U ( p , W,,,) n X l p E P} u {{f}I f € X - F,}. Since, for each p E P, there is some n, E N such that W(,, n i 2 n,}) = 0,there are some subset P' of P and some n E N such that 1 P ' J > 2" and W,,, n i 2 n}) = 0 for any p E P'. If for each p E P', we let A,, = { R E W(,,Ip E R } and AP2 = W,,- A,,, then there is a subset P" of P' such that I P" 1 > 2" and A,, n Aq2 = 0 for any p, q E P" with p # q by Theorem 1.1 of Burke [1974]. DefinefE F as follows: f ( R ) = 1 iff R E IJ{ApI Ip E P"}.ThenfE X n V ( p , W,,,) for any p E P", which shows that W is not point-finite at f. 0
(u{QiI
(u{$]
7.4. Example. Every subspace of the space F (=Bing's example G ) is shrinking. If 1 PI is uncountable, then F does not have a property 93.
Proof. Since every subspace of F is shrinking by Proposition 6.10, we shall show that F does not have a property 93. Without loss of generality, we may assume that P = o,and so F,, = {f,I a < o,}. If for each a < wI, we let Ha = { f a l a < < ol},then { H a l a < ol} is a monotone decreasing closed collection with the empty intersection. Let { U, I a < o,} be any monotone decreasing open collection with Ha c U, for any a < 0,. If for we select some W,,, E [9] such that V(a,R,,,) c U,, then by the each a < a,, Sanin's lemma (see Juhasz [1970]), there exist some n, some subset P' of P and some W' = { R , , . . . , R,} E [9] such that I W,,,I = n for any a E P', I P' I = w , and {W,,, - 9' I a E I"} is mutually disjoint. Furthermore, we have a subset P" of P' with cardinality o,such that P" c Ri or P" n Ri = 0 for any i < m. Then we can define g c F as follows: g ( R ) = 1 if R = Ri 3 P" for some i, g ( R ) = f , ( R ) if R E W,,, - W' for some a E P" and g ( R ) = 0 otherwise. Then it is seen that g E U(a, W,,,) c U, for any a E P".By the monotone decrease of { U, I a < a,}, g E n { V, I a < o,}.
0
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Y. Yasui
The above example shows that a property W is stronger than a property 9. Another example of such a space is o,(Yasui [1972]). In the shrinking properties, the natural questions arise: If every monotone increasing open cover of X is shrinkable, then is every open cover of X shrinkable? It is answered negatively under some set-axiom as follows. Let ic be an infinite cardinal and E = { a E ~'(cofinalityof a = ic}. O + + ( E ) holds in L. We omit the proof.
7.5. Example ( O + + ( E )(Beilagii: ) and Rudin [1985]). There is a space A such that A is T2,ic-ultraparacompact and collectionwise normal and every monotone increasing open cover of A has a clopen shrinking, but there is an open cover having no closed shrinking. Rudin [1983b and 19841 showed that a Navy's space S does not have a property a.It was already known by Navy [1981]that S is para-Lindelof and countably paracompact but not paracompact (Theorem 6.8):
7.6. Example (Navy [1981], Rudin [1983b and 19851). A Navy space S has a property W but is not paracompact. To see that a property 9 is stronger than the countable paracompactness, we choose an increasing sequence of regular cardinals {A, Ia < ic] for which A: = A,, where ic is an uncountable cardinal. Then Xis the subset of the box product 0 {(A, + 1) I a < rc} consisting of precisely those functions f for which there is a B < K such that K
7.7. Example (Rudin [1978]). Let ic be an uncountable cardinal. Then a ic-Dowker space X is normal and countably paracompact and it has a monotone increasing open cover indexed by ic which is not shrinkable (and so it does not have a property 9). In Example 7.2, we described the space which is metacompact but not subparacompact. It is seen that the space is not locally compact, whose property is useful for the study of covering properties and so we shall give the existence of a space which is locally compact and metacompact but not subparacompact. The proof is left to the reader.
+
+
7.8. Example (Burke [1964]). Let X = (a2 1) x (a2 1) - {(a2, 02>}. LetL = { a } x (w2 + 1)andK = (02 + 1) x {a}.Topologized Ysuch that
Generalized Paracompactness
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the points in w2 x w2are isolated, the points (a, w 2 )have a nbd basis by sets of the form La - A (where A is finite) and the points (02, a ) have a similar nbd basis, with Ka replacing La.Then Xis locally compact and metacompact but not subparacompact. The above space X has some subspace which is interesting with respect to covering properties as follows:
7.9. Example (Burke [1964]). Let Y = (w2 + 1) x (w u { w 2 } )- {(w,, w 2 ) } be a subspace of Example 7.8's space X.Then Y is locally compact, metacompact and subparacompact but not paracompact.
Proof. Since Y is a closed set of X (=space of Example 7.8), Y is locally compact and metacompact. Also Y can be expressed as a union of countable members of closed paracompact subspaces and so Y is subparacompact by Corollary 5.3. It is easily seen that Y is not normal and so not paracompact. 0 Every Moore space is subparacompact by Theorem 3.4, but the next example shows that we cannot have the fact with subparacompactness replacing metacompactness. The proof is easy. 7.10. Example (Gillman and Jerison [1960]). Let d be a maximal collection of countably infinite subsets of N such that d is almost disjoint and Y(N) = d u N topologized as follows: the points of N are isolated in Y(N) and the points A of d have a nbd basis by sets of the form {A} u ( A - F) (where F is finite). Then the Moore space Y(N) is not metacompact. The above examples show that classes of paracompact spaces, subparacompact spaces, metacompact spaces and submetacompact spaces do not coincide with each other and so we shall discuss the conditions under which the above classes coincide. For example collectionwise normality is well known as one of such conditions and so we shall have the following without proof: 7.11. Theorem (Michael [ 19551, Nagami [1955], McAuley [1958], Worrell and Wicke [1965]). For a collectionwise normal space X, the following are equivalent : (1) X is paracompact. ( 2 ) X is subparacompact. (3) X is metacornpact. (4) X is submetacornpact.
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Next we shall consider the question: under what conditions is subparacompactness equivalent to metacompactness? For this purpose we shall define collectionwise &normality and prove some lemmas.
7.12. Definition. A space Xis said to be collectionwise 6-normal if for any discrete collection { F, 1 a E A} of closed subsets of X , there exists a mutually disjoint G,-sets’ collection {G,la E A } such that F, c G, for any a E A (Kramer [ 19711, Junnila [ 19801).
7.13. Lemma (Junnila [ 19811). Let X be collectionwise &normal, { F, 1 a E A } a discrete closed collection and { U,I a E A } an open collection with F, c U,for any a E A. Then there exist a a-discrete closed collection 9 and a Gs-set H such that U{F,I a E A } c H c U S and 9 is a partial refinement of { un I a E A } . Proof. Let {G, I a E A } be a mutually disjoint collection of G,-sets such that Fa c G, = n{G,,,In E N} c U,,where each G,,, is open and U,,3 G,, 3 G,, 3 . . . with G,, n F, = 8 for any fl # a (for any a E A). If we let G,, = U{G,, I a } for each n, then there are G,-sets H, = n{Hnk I k E N} and J,, = n{J,,, I k E N} (where all H,,, and Jnkare open) such that U{F,I a } c H,, and X - C,,c J,,. Let %,,k = {G,,Ia} u {Jnk}, c,,,= {xlord(x, grin) = I } and 9 , k = {G,,, n G,,, I a E A} for each n, k E N. Finally we let 9 = 1 n, k} and H = n{H,,1 n E N}, then 9 and H are desired.
u{&
7.14. Lemma (Junnila [1980]). Let Q be an open cover of a collectionwise &normal space X and X(42) = { x E XI ord(x, 42) isfinite}. Then there exists a a-discrete closed collection 9 such that X(Q) c UP and 9 is a partial refinement of Q. Proof. Let X , = {xlord(x, Q) < n} for each n. Then X(Q) = U{X,,In}. By repeated use of Lemma 7.13, we have a-discrete closed collections 9, and is a partial refinement of Q, for G,-sets H,, such that X , c H,, c U 9 , and 9,, each n. Then U{9,,ln}is desired. 0 Under the above lemma, the following theorem is easily seen: 7.15. Theorem (Boone [1973]). A space X is subparacompact if and only if X is collectionwise &normal and submetacompact.
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Next we shall consider a necessary and sufficient condition under which submetacompactness is equivalent to metacompactness. Smith and Krajewski [I9711 gave such a condition for the first time and a few years later, Boone [1973] gave a condition weaker than that used by Smith-Krajewski. To see this we shall prove the following fact which will be useful in a Boone’s theorem: 7.16. Proposition (Gittings [ 19741). Every submetacornpact space is countably metacompact.
Proof. Let { U, I n E N} be any countable open cover of a submetacompact space X and {@n,ln}a &refinement of it. Let V, = U,and V , = U, n St(X - ( U , u . . . u Un-l), A { @ i l i < n - I } ) for any n 2 2. Then { V ,I n E N} is a point-finite open refinement. 7.17. Theorem (Boone [1973]). A space X i s metacornpact ifand only i f X is submetacompact and every discrete collection of closed subsets of X has a point-finite open expansion.
Proof. The “only-if” part is easily seen and so we shall only prove the “if” part. Let {%,In} be a &sequence of an open cover 9 = { U, I a E A}. If for n, k E N, let x , k = { X I ord(x, a,,) < k } , then 4, is closed and X = U{XnkI n, k } . By repeated use of the given condition, there exist point-finite open collections “v;,, for n, k such that “v;,k is a partial refinement of 9 and a cover of Xnk - x n k - l . By Proposition 7.16, we have a point-finite open cover { W,k I n, k } of X such that W,, c u v n k for any n, k. Then it is seen that { W,k n VI V E V , k ; n , k E N} is a point-finite open refinement of %. The above Theorems 7.15 and 7. I7 characterize subparacompactness and metacompactness in terms of an expansion. In the case of paracompactness, the following theorem is known (the proof is left to the reader). 7.18. Theorem (Krajewski [1971]). A space X is paracompact if and only if X is submetacornpact and every locally finite collection of closed subsets of X has a locally finite open expansion.
Studying whether or not a topological property preserves under some class of mappings is one of the important and interesting problems. It is obvious that all classes of spaces appearing in this chapter do not preserve under the continuous mappings and so we shall consider the class of closed mappings.
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The next theorem follows easily from the definitions or the characterizations (Theorems 2.12 (2), 3.6 (3), 4.5 (2) and 5.2 (3)) because the closure-preserving property preserves under any closed mapping. 7.19. Theorem (Michael [1957], Worrell [1966b], Burke [1964], Junnila [1978]). Let f be a closed mapping from a space X onto a space Y . Zf X is paracompact (resp.metacompact, subparacompact, submetacompact, shrinking), then Y is so, respectively. Though paracompactness preserves under closed mappings, it is known that t-paracompactness does not (Zenor [1969]). But the domain space which was constructed by Zenor does not satisfy the complete regularity. Recently, Ohta constructed such an example in the class of Tychonoff spaces as follows. 7.20. Example (Ohta [1983]). There exists a closed mapping from a
Tychonoff space X with a property 9 onto a Tychonoff space Y which is not countably paracompact (and so does not have a property 9).
Proof. Let S be the subspace of all P-points of w 2 , 2 = {(a, 8) I /3 E S , /3 < a < w 2 } a subspace of the product space (a2+ 1) x S and X = Z x o.Then 2 has a property 9 (and so countably paracompact). Define Y by a quotient space obtained from X by collapsing the set {((02 /3),,n) I n < o}to a point j? for each fi E S. Since each is a P-point, it is easily seen that the quotient mapping f:X + Y is closed. Let us set F,, = f ({((/!I, /3), m ) I /3 E S , m < n } )for each n. Then {F, I n} is a sequence of closed subsets of Y with empty intersection. If G,,is an open set of Y containing F,,for each n, then C1 C,intersects {j? I /3 E S}at a cofinal subset, and so n{Cl G,,In} # 8.By Corollary 2.8, Y is not countably paracompact (and so does not have a property 9). 0
B
Though the countable paracompactness and the property 9 do not preserve under closed mappings, it is clear that in a class of normal spaces, their properties preserve under them.
References Alas, 0. T. [1971] On a characterization of collectionwise normality, Canad. Math. Bull. 14, 13-15. Alexandroff, P. S. and P. Urysohn [1929] Memoire sur les espaces topologiquescompacts. Verh. Akud. Wetensch. Amsterdam 14, 1-96.
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Arens, R. and J. Dugundji [1950] Remark on the concept of compactness, Portugal Math. 9, 141-143. Arhangel’skii, A. V. [I9611 New criteria for paracompactness and metrizability of an arbitrary T , space, Dokl. Akad. Nauk SSSR 141, 13-15. [I9661 Mappings and spaces, Russian Math. Surveys 21, 115-162. Atsuji, M. (19761 On normality of the product of two spaces, General Topology IV, Proc. Fourth Prague Topology Confer., Prague, B25-27. [I9891 Normality of product spaces I, Chapter I11 of this volume. Bennett, H. R. and D. J. Lutzer [I9721 A note on weak 0-refinability, Gen. Topology Appl. 2, 49-54. BeSlagiC, A. (19861 Normality in products, Topology Appl. 22, 71-82. BeSlagiC, A. and M. E. Rudin (19851 Set theoretic constructions of non-shrinking open covers, Topology Appl. 20, 167-177. Bing, R. H. [I9511 Metrization of topological spaces, Canad. J . Math. 3, 175-186. Boone, J. R. [I9731 A characterization of metacompactness in the class of O-refinablespaces, Gen. Topology Appl. 3, 253-264. Burke, D. K. [I9691 On subparacompact spaces, Proc. Amer. Math. SOC.23, 655-663. [1974] A note on R. H. Bing’s example G, Topology Confer. VPZ, Lecture Notes Math. 375 (Springer, New York) 47-52. [ 19801 Para-Lindelof spaces and closed mappings, Topology Proc. 5, 47-57. [I9841 Covering properties, in: K.Kunen and J. E. Vaughan, Eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam). Burke, D. K. and R. A. Stoltenberg [I9691 A note on p-spaces and Moore spaces, Pacific J. Math. 30,601-608.
De Caux, P. [I9761 A collectionwise normal weakly O-refinable Dowker space which is neither irreducible nor realcompact, Topology Proc. 1, 67-77. Ceder, J. [I9611 Some generalizations of metric spaces, Pacific J. Math. 11, 105-125. Chiba, K. [I9841 On the weak E-property, Math. Japon. 29, 551-567. Coban, M. M. [I9691 On u-paracompact spaces, Vestnik Moskov Univ. Ser. I, Math. Mech., 20-27. Corson, H. H. [I9591 Normality in subsets of product spaces, Amer. J. Math. 81, 785-796. Dieudonn6, J. [I9441 Une generalisation des espaces compacts, J. Math. Pures Appl. 23, 65-76.
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Donne, A. L. [1985] Shrinking property in Z-products of paracompact p-spaces, Topology Appl. 19.95-101. van Douwen, E. K. [I9801 Covering and separation properties of box products, in: G. M. Reed, Ed., Surveys in General Topofogy (Academic Press, New York) 55-129. Dowker, C.H. [1951] On countably paracompact spaces, Canad. J. Math. 3, 219-224. Engelking, R. [1977] General Topology (Polish Scientific Publishers, Warszawa). Fleischman, W. M. (19701 On coverings of linearly ordered spaces, Washington State University Topology Confer., March 1970, 52-55. Fletcher, P and W. F. Lindgren [1972] Transitive quasi-uniformities, J. Marh. Anal. Appl. 39, 397405. Gillman, J. and M. Jerison [1960] Rings of Continuous Functions (Van Nostrand, Princeton, NJ). Gittings, R. F. [1974] Some results on weak covering conditions, Canad. J. Marh. 26, 1152-1 156. Hodel, R. E. A note on separability and meta-Lindel6f spaces, unpublished. Ishikawa, F. [1955] On countably paracompact spaces, Proc. Japan Acad. 31, 686-687. JuhAsz, I. [I9701 Cardinal Funcrions in Topology (Math. Centrum, Amsterdam). Junnila, H. J. K. [ 19781 On submetacompactness, Topofogy Proc. 3, 375-405. [ 1979al Metacompactness, paracompactness and interior-preserving open covers, Trans. Amer. Math. SOC.249, 373-385. [ 1979bl Paracompactness, metacompactness and semi-open covers, Proc. Amer. Marh. SOC.73, 244-248. [1980] Three covering properties, in: G. M. Reed, Ed., Surveys in General Topology (Academic Press, New York) 195-245. KatEtov, M. [1958] Extension of locally finite coverings (in Russian), Colloq. Marh. 6, 145-151. Katuta, Y. [I9741 On spaces which admit closure preserving covers by compact sets, Proc. Japan Acad. 50, 826-828. [1977] Characterizations of paracompactness by increasing covers and normality of product spaces, Tsukuba J. Marh. 1, 27-43. Krajewski, L. [1971] Expanding locally finite collections, Canad. J . Marh. 23, 58-68. Kramer, T.R. [1973] A note on countably subparacompact spaces, Pacific J. Math. 46, 209-213. Mack, J. [1967] Directed covers and paracompact spaces, Canad. J. Math. 19,649-654.
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McAuley, L. F. [19581 A note on complete collectionwise normality and paracompactness, Proc. Amer. Math. SOC. 9, 796-799. Michael, E. [I9531 A note on paracompact spaces, Proc. Amer. Marh. Soc. 4, 831-838. [I9551 Point-finite and locally finite coverings, Canad. J. Math. 7, 275-279. [1957] Another note on paracompact spaces, Proc. Amer. Math. SOC.8, 822-828. [I9591 Yet another note on paracompact spaces, Proc. Amer. Math. SOC.10, 309-314. Morita, K. [19621 Paracompactness and product spaces, Fund. Marh. 50, 222-236. Nagami, K. [19551 Paracompactness and screenability, Nagoya Math. J. 8, 83-88. [1969] Z-spaces, Fund. Math. 65, 169-192. Nagata, J. [I9501 On a necessary and sufficient condition of metrization, J. Insr. Polytech., Osaka City Univ. 8, 93-100. [I9851 Modern General Topology (North-Holland, Amsterdam). Navy, K. [19811 Paralindelofness and paracompactness, Thesis, University of Wisconsin, Madison. Ohta, H. [I9831 On a part of problem 3 of Y. Yasui, Questions Answers Gen. Topology 1, 142-143. [ 1987al Well-behaved subclasses of M , -spaces, in preparation. [1987b] Letter. Okuyama, A. [1967] Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 9, 236-254. Potoczny, H. B. [1972] A non-paracompact space which admits a closure-preserving cover of compact sets, Proc. Amer. Math. SOC.32, 309-31 1. Potoczny, H. B. and H. J. K. Junnila [I9751 Closure-preserving families and metacompactness, Proc. Amer. Math. Soc. 53,523-529. Rudin, M. E. [1971] A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. [I9781 K-Dowker spaces, Czech. Math. J. 28(103) 324-326. [1983a] The shrinking property, Canad. Math. Bull. 26, 385-388. [1983b] Yasui’s questions, Questions Answers Gen. Topology 1, 122-127. [I9841 Dowker spaces, in: K. Kunen and J. E. Vaughan, Eds., Handbook of Set-theoretic Topology (North-Holland, Amsterdam). (19851 rc-Dowker spaces, in London Math. SOC.Lecture Note Series 93 (Cambridge University Press, Cambridge) 175-195. Sconyers, W. B. I19701 Metacompact spaces and well-ordered open coverings, Notice Amer. Marh. SOC.18,230. Smirnov, Yu. M. [1951] On metrization of topological spaces, Amer. Math. SOC.Transl. Ser. l(8) 63-77. Smith, J. C. and L. Krajewski [I9711 Expandability and collectionwise normality, Trans. Amer. Math. SOC.160, 437-451.
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Starbird, M. [I9741 The normality of products with a compact or metric factor, Ph.D. Thesis, University of Wisconsin. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. Amer. Math. SOC.54,977-982. Tamano, H. (19601 On paracompactness, Pacific J. Math. 10, 1043-1047. [I9621 On compactification, J. Math. Kyoto Univ. 1, 161-193. Tani, T. and Y. Yasui [I9721 On the topological spaces with the 93-property, Proc. Japan Acad. 48, 81-85. Worrell, J. M. W., Jr. [1966a] A characterization of metacompact spaces, Portugal Math. 25, 171-174. [1966b] The closed continuous images of metacompact topological spaces, Portugal Math. 25, 175-179. [I9671 Some properties of full normalcy and their relations to Cech completeness, Notices Amer. Math. SOC.14, 555. [I9681 Paracompactness as a relaxation of full normalcy, Notices Amer. Math. SOC.15, 661. Worrell, J. M. W., Jr. and H. H. Wicke [I9651 Characterizations of developable spaces, Cunad. J. Math. 17, 820-830. Yajima, Y. [I9861 The shrinking property of 8-products, Questions Answers Gen. Topology 4, 85-96. Yasui, Y. [I9721 On the gaps between the refinements of the increasing open coverings, Proc. Japan Acad. 48, 8 6 9 0 . [I9831 On the characterization of the B-property by the normality of product spaces, Topology A&. 15, 323-326. [I9841 A note on shrinkable open coverings, Questions Answers Gen. Topology 2, 143-146. [I9851 Some remarks on the shrinkable open covers, Math. Japon. 30, 127-131. [ 19861 Some characterizations of a 93-property, Tsukuba J. Math. 10, 243-247. &nor, P. [ 19691 On countable paracompactness and normality, Prace Math. 13, 23-32. 42, 258-262. [I9701 A class of countably paracompact spaces, Proc. Amer. Math. SOC.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 6
THE TYCHONOFF FUNCTOR AND RELATED TOPICS
Tadashi ISHII Chiba Institute of Technology, Narashino, 275 Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Tychonoff functor. . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Product spaces and the Tychonoff functor . . . . . . . . . . . . . . . . . . 3. w-Compact spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A space X such that r ( X x Y ) = r ( X ) x T ( Y ) for any space Y . . . . . . . . . 5. A space X such that r(X x Y ) = T ( X ) x r ( Y ) for any k-space Y . . . . . . . . 6. Products of w-compact spaces . . . . . . . . . . . . . . . . . . . . . . . I. w-Paracompact spaces and the Tychonoff functor . . . . . . . . . . . . . . . 8. A generalization of Tamano's theorem. . . . . . . . . . . . . . . . . . . . 9. Rectangular products and w-paracompact spaces . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
203 204 201 209 215 218 22 1 224 230 234 241
Introduction It seems that the term of the Tychonoff functor was first used by Morita [1975]. The origin of the Tychonoff functor is the complete regularization of general topological spaces. It is now recognized, from the categorical viewpoint, as a covariant functor from the category of topological spaces and continuous maps into itself. The Tychonoff functor t is the reflector from the category above to the full subcategory of Tychonoff spaces. The purpose of this chapter is to present the basic theories of the Tychonoff functor and related subjects, for example, w-compact spaces (due to Ishii [ 1980a]), w-paracompact spaces (due to Ishii [ 19841) and those applications to rectangular products in the sense of Pasynkov [1975]. The reader should be aware of several recent articles containing results on the Tychonoff functor and related subjects. Some of them are: Pupier [1969],
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T. Ishii
Morita [1975, 19801, Oka [1978], Hoshina and Morita [1980], Ishii [1980a, 1980b, 19841, Pasynkov [1975, 19801. The starting point is a work of Pupier [I9691 who showed that if X is a locally compact Hausdorff space, then t ( X x Y) = t(X)x t ( Y ) for any topological space Y. After that, Morita proved that for topological spaces X and Y, t ( X x Y ) = r ( X ) x t( Y) if and only if any cozero-set of X x Y is a union of cozero-set rectangles of X x Y (cf. Hoshina and Morita [1980], Morita [1980]), and gave another proof of Pupier’s result. Further, by using Morita’s result above, Oka [I9781 proved that for a Tychonoff space X the converse of Pupier’s result is valid. For the case of X being a general topological space, Ishii [1980al found the necessary and sufficient condition for X to satisfy the condition that z ( X x Y ) = z ( X ) x t ( Y ) for any topological space Y by introducing the concept of w-compact spaces. The concept of w-paracompact spaces follows naturally that of w-compact spaces and is a generalization of paracompact Hausdorff spaces at the same time. It is interesting that every w-paracompact space X satisfies t ( X x Y) = t ( X ) x t( Y) for any Tychonoff space Y. A characterization of w-paracompact spaces in terms of product spaces is investigated in Section 8, which parallels to Tamano’s theorem for paracompact Hausdorff spaces. For this purpose the concept of w-normal spaces is introduced. Applications of w-paracompact spaces to rectangular products are stated in Section 9, which contains some generalizations of Hoshina and Morita’s theorems [1980]. For some facts, the proofs of which are omitted, the reader is referred to Hoshina’s chapter on extension of mappings I1 and Atsuji’s chapter on normality of product spaces I. All spaces in this chapter are assumed to be general topological spaces, and for regular spaces the T,-axiom is not assumed, unless otherwise specified. By a Tychonoflspace we mean a completely regular T,-space. Letters N, I and R denote the set of positive integers, the closed unit interval and the real line respectively. The cardinality of a set S is denoted by card S or I S I. The reader is referred to the texts of Nagata [I9851 and Engelking [1977] for standard terminologies and theorems in general topology.
1. The Tychonoff functor
Let us recall that a subset A of a space X is called a cozero-set if A = { x I f ( x ) > 0) for some continuous mapf: X -,I, and the complement of a cozero-set is called a zero-set.
The Tychonoff Functor and Related Topics
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For a space X , let I" be the set of all continuous maps cp : X + I and consider a continuous map t x :X + P ( X ) from X to the product space P ( X ) = n{Z, I cp E I"} defined by tx(x) = (cp(x))E P ( X ) , where I, = I for any rp E I". Let us put t ( X ) = t x ( X ) c P ( X ) . Then for a continuous map f : X + Y we have a continuous map t ( f ) : t(X) + t( Y ) by defining t( f ) ( t ) to be the point of t( Y )whose $-coordinate is the $ ofcoordinate o f t E t ( X ) , where $ E I " , and the diagram Xf.
Y
z ( X ) .o!,T ( Y ) is commutative. Thus t is a covariant functor from the category of spaces and continuous maps into itself. Obviously t(X) is a Tychonoff space, and as is easily seen, t X :X + t(X) is a homeomorphism if X is a Tychonoff space. Hence we call t the Tychonof functor. It is the reflector from the category above to the full subcategory of Tychonoff spaces. 1.1. Proposition. Any continuous map f from a space X into a Tychonof space Y is factorized through r ( X ) such that f = go tXfor some continuous map g : z ( X ) + Y , where g is determined uniquely by$
1.2. Proposition. For any cozero-set G of a space X , tx(G) is a cozero-set of z ( X ) with t;'(t,(G)) = G. Proof. Let G be a cozero-set of X such that G = { x ) g ( x )> 0) for some continuous map g : X + I. Since by Proposition 1.1 there is a continuous map h : t(X) + Z with g = h 0 tx, it holds that ti'({tE
z ( X ) ( h ( t )> 0 } )
=
G,
which implies that z,(G) = { t E t(X) I h(t) > 0 } and tx'(tx(C)) = G. Thus t,(G ) has the required properties. 1.3. Definition. A subset A of a space X is called t-open if A is a union of cozero-sets of X and the complement of a z-open set is called a t-closed set.
Clearly the union and the finite intersection of r-open sets are r-open and hence the intersection and the finite union of t-closed sets are t-closed. A subset P of a space Xis r-open if and only if P = s;'(Q) for some open set Q of t ( X ) .
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T. Ishii
1.4. Proposition. Let A be a subset of a space X . If for any cozero-set U of A there exists a z-open set V of X such that U = V n A , then ?,(A) = ? ( A ) , that is, ?,(A) is homeomorphic to ? ( A ) . Proof. By Proposition 1 . 1 , the restriction map T, I A :A + ?,(A) of z, to A is factorized through ? ( A ) such that zxI A = g 0 zA for some continuous map g : ? ( A ) --+ ?,(A). We shall verify that g is a homeomorphism. Sinceg is onto, it sufficesto see that g is a one-to-one open map. Let t E ?,(A) and u E g - ' ( t ) . Suppose there is a point x, E A such that x, E (?I, A ) - ' ( t ) - z;'(u). Then there is a cozero-set U of A such that x, E U and z;'(u) E X - U, since ?;I@) is z-closed in A. By assumption we can take a z-open set V of X with U = V n A. Since z;'(z,(V)> = V and t E z,(V), we have (zxl A)-'(t) c U , which is a contradiction. Therefore g is one-to-one. To see that g is an open map, let G be a cozero-set of ? ( A ) . By assumption, for a cozero-set t;I(G) of A there exists a z-open set W of X with z;'(G) = W n A. Since r x ( W ) is an open set of T ( X ) such that t x ' ( ~ , ( W ) )= W, it is easily seen that g(G) = z,(W) n ?,(A), which shows that g(G) is an open set of ?,(A). Thus g is a homeomorphism of ? ( A ) onto ?,(A). 0 A subset S of a space X is called z-embedded in X if any zero-set of S is represented in the form S n 2 for some zero-set 2 of X and is called C*-embedded in X if every continuous map f : S 3 I has a continuous extension 3:X --t I. Clearly, a C*-embedded subset of a space X is z-embedded in X . 1.5. Corollary. I f a subset A of a space X is z-embedded (or C*-embedded) in X , then we have ?,(A) = ? ( A ) .
1.6. Definition. A collection 9 of subsets of a space X is called r-locally finite in X if for any point x of X there exists a cozero-set neighborhood of x which intersects only a finite number of members of 9. A collection 9 of subsets of a space X is z-locally finite if and only if z x ( F ) = { ? , ( A ) / A E F}is locally finite in z ( X ) .
1.7. Proposition. For any locallyfinite cozero-set cover 4 of a space X , there exists a locallyfinite cozero-set cover V of t ( X ) such that z ; ' ( V ) refines %, where t;'(V) = {t;'(V)I V E Y } .
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The Tychonof Functor and Related Topics
In other words, any locally finite cozero-set cover of a space X admits a T-locally finite cozero-set refinement. To prove the proposition, we need the following lemmas.
1.8. Lemma. Let {FaI a E R} be a collection of zero-sets of a space X . Ifthere exists a locally finite collection { U, I a E Q} of cozero-sets of X such that FaE U, for each a E 0,then U{F, I a E a} is a zero-set of X . 1.9. Lemma. For any locallyjnite cozero-set cover { U, Ia E R} of a space X , there is a locally finite zero-set cover { F, 1 a E R} of X such that F. c U,for each a E R. As for the proofs of these lemmas, see Chapter 2, Lemma 1.3 and Theorem 1.4.
Proof of Proposition 1.7. Let 4 = { UaI a E R} be a locally finite cozero-set cover of a space X. Then by Lemma 1.9 there exists a zero-set cover { F, I a E R} of X such that F, c U, for a E R. For each a E R, take a continuous mapf,:X + I such that F, = f.-'(l) and U, = { x l f . ( x ) > o} and let
V,
= {.If.(.> >
f}
and K, = { x I f . < x >2
$1.
Then V, is a cozero-set and K, a zero-set in X such that F, c V , c K, c U, and X = U { V , l a E R } . Let us put U ( x ) = X - U { K , ~ X E X K- , } for each point x of X. Since U {K, I x E X - Ka} is a zero-set of X by Lemma 1.8, U ( x ) is a cozero-set of X , and {a I U ( x ) n V, # S } is clearly a finite set. Therefore Y = {T( V , )I a E R} is a locally finite cozero-set cover of ?(A'), and it is obvious that T; ' ( V )refines 4 , completing the proof. 0 We notice that Proposition 1.7 also follows from Proposition 1.1 and the following result: for any normal open cover 4 of a space X there exists a continuous map f : X + T of X onto some metrizable space T and a normal open cover V of T such that f - ' ( Y )refines 4 (see Chapter 2, Theorem 1.1 and Morita [1970]). 2. Product spaces and the Tychonoff functor Let { X , I a E R} be a family of spaces and let us put X = ll X , . For brevity we denote by T, the natural map T~~ : X , -, ? ( X u )for each a E R. By Proposition 1.1, for the product map f = n ~ ,X: + n r ( X , ) there is a continuous X )nt(X,) such that f = g o T x . map g = T ( ~ ~ T , ) : T (+
T. Ishii
208
2.1. Proposition. For any family of spaces { X uI a n 7 ( X a )is one-ro-one and onto, where X = H X , .
E
R}, the map g : r ( X ) +
Proof. Clearly the map g is onto. To show that g is one-to-one, assume that for a point t = (1,) E nt(X,)there are two different points u and v of r ( X ) such that g(u) = g(a) = t. Let x E r;'(u) and y E r ;' (v ). Then there is a continuous map h : X + I such that h ( x ) # h( y ) . Hence if we put x = ( x u ) and y = (y,) in X , it is shown that
h(x, x for some
(ZU)U#,)
# h(Y, x
(Zu).+,)
B E R and some point (z,),,,
. . . , E Q we have
h((x,))
= K Y ,
E
n{X,Ia # p } . If not, for ao, a,,
x (XU),#,)
= h ( ~ ux, Y,, x ( x a ) a # , , u , )
=
* *
3
so that by the axiom of choice (or the well-ordering of R and transfinite induction) we have h((x,)) = h(( y,)), which is a contradiction. Therefore, it holds that 7,(xS) # 7,( yo), implying that (r,(x,)) # (t,( y,)) in n t ( X , ) . Since (g07x)((xu))= ( 7 u ( x u ) ) and ( g 0 7 , ) ( ( ~ a ) = ) ( 7 a ( ~ u ) ) 9 we have ( g 0 7 x ) ((x,)) # (g 7*)(( y m ) )that , is, g(u) # g(a), which is a contradiction. Thus g is one-to-one. 0 0
If the map g = z(lI7,) : 7 ( r I X , ) + n 7 ( X m )above is homeomorphic, we put 7(nX,) = rI7(XU).
2.2. Definition. Let {Xu I a E Q} be a family of spaces and U,, a cozero-set of X U i ,where ai E Q, i = 1, . . . , n. Then a cozero-set U,, x . . . x Uunx n{X,I a # a ' , . . . , a,,} of nX, is called a cozero-set rectangle. 2.3. Theorem. For any family of spaces {XuI a E R}, thefollowing conditions are equivalent. (1) 7(nX,) = nT(X,). ( 2 ) For any cozero-set G of X = nX,and for any point x = ( x u )E G there exists a cozero-set rectangle A = U,, x * * * x Uunx n{X,I a # a ' , . . . , a,,} in X such that x E A c G. Proof. (1)+(2). Let G be a cozero-set of X with x = (x,) E G. Since 7,(G) is a cozero-set of 7 ( X ) with r;'(t,(G))= G by Proposition 1.2 and the map g = r(n7,): 7 ( n X , ) + n r ( X , ) is a homeomorphism by the assumption,
The TychonoffFunctor and Related Topics
209
( g 0 7,)(G) is an open set of n r ( X , ) containing g(7,(x)), and hence there is a cozero-set rectangle B = V,, x . . x V," x n { r ( X , )1 a # a,, . . . , a,} of the Tychonoff space l I t ( X , ) such that g(z,(x)) = (lI7,)(x) E B c (go 7,)(G). Therefore, letting U,, = r;'(V,,), i = 1, . . . , n, then A = U,, x . . . x U,. x l I { X aI a # a,, . . . , a,} is a cozero-set rectangle of X with x E A c G . (2)+(1). To show that the map g = r(nz,) : 7 ( X ) + n r ( X , ) is a homeomorphism, it suffices to verify that g is an open map, that is, for any cozero-set G of X , ( g 7,)(G) is open in n z ( X , > .Now let G be any cozero-set of X and let z E ( g 0 7,)(G) and x E ( g 0 z X ) - ' ( z ) . Since x E G , by the assumption there exists a cozero-set rectangle A = U,, x . . . x Uanx n{X,I a # a,, . . . , a,} of X such that x E A c G . Let us put V,, = 7,,(U,,),i = 1, . . . , n. Then B = V,, x . . . x 5, x II{r,(X,)la# a ' , . . . , a,} is a cozero-set rectangle of n z , ( X , ) such that z E B c ( g o T,)(G), from which it follows that (gor,)(G) is open in H7,(X,). 0 0
As a corollary of Theorem 2.3, we have the following result (Hoshina and Morita [ 19801). 2.4. Theorem. For two spaces X and Y the following conditions are equivalent. (1) 7 ( X x Y ) = 7 ( X ) x 7 ( Y ) . (2) For any cozero-set G of X x Y and f o r any point ( x , y ) E G there exists a cozero-set rectangle U x V of X x Y such that ( x , y ) E U x V c G .
3. w-Compact spaces
In this section we introduce the concept of w-compactness and study its basic properties. For a subset A of a space X , the 7-closure of A , C1,A in notation, means the set of points x in X such that any cozero-set neighborhood of x intersects A. If A = Cl,A, A is 7-closed in X . 3.1. Definition. A space Xis called w-compact if any open cover { U, I a E Q} of X contains a finite subfamily { U , , , . . . , U,"} such that X = Cl,(U,, u * . . u U,").
A Hausdorff space X is called H-closed if X is a closed subspace of every Hausdorff space in which it is contained, and it is known that a Hausdorff space X is H-closed if and only if every open cover { U, I a E Q} of X contains
T. Ishii
210
-
a finite subfamily { U , , , . . . , U,,} such that Oulu u Dan= X (see, for example, Engelking [1977]). Hence very H-closed space is w-compact as well as every compact space. However there exists a w-compact space which is not a H-closed space.
1x1
3.2. Example. There exists a regular TI space X with > 1 such that every continuous map f : X + R is constant (Novak [1948], Hewitt [1946], Herrlich [1965]). Such a space Xis not H-closed but w-compact. In fact, if we assume that X is H-closed, X is regular H-closed and hence it is compact, contradicting that any continuous map f : X + R is constant. It is obvious that X is w-compact. 3.3. Proposition. For a space X , the following conditions are equivalent. ( I ) X is w-compact. ( 2 ) Zfa family { A , } of closed sets of X is closed under thejinite intersection and each A , contains a nonempty cozero-set of X , then n A , # 8. ( 3 ) Zfa family {P.} of z-open sets of X has thejinite intersection property, then OFa # 8. Proof. It is obvious that (2) is equivalent to (3). (l)+(3). Let {PaI a E R} be a family of z-open sets of X with the finite intersection property. Suppose that OF, = 8.Then by (I), for an open cover {X - pala E R} of X,there exists a finite subset { a I , . . . , a,} of R such that
X
= Cl,(U{X - F , , ( i = I ,
. . . ,n } ) .
But this is a contradiction, since n{P,,I i = I , . . . , n } is a nonempty z-open set. Therefore we have nFa # 8. (3)+(1). Let { U, 1 a E R} be any open cover of X. Suppose that, for any finite subset { a l , . . . , a,} of R, X - Cl,(u{ Ua,l i = 1, . . . , n } ) # 8. Putting P ( a l , . . . , a,) = X - Cl,(u{ Uu,l i = 1,
. . . , n}),
the family {P(a,, . . . , a,) 1 aI,. . . , an E R, n E N} of z-open sets of X has the finite intersection property. Hence by (3) we have
n{ml,. . . , a,) l a 1 ,. . . , a, contradicting X =
u{U,I a
E
E
n, n E N}
#
8,
R}. Thus X is w-compact.
3.4. Proposition. Zfa space X is w-compact, then t(X) is compact.
0
21 I
The TychonoffFunctor and Related Topics
Proof. Let { V ,I a E n}be any open cover of r ( X ) . For each point t of ?(A/), take a(t) E R with t E K(,) and an open set W; of r ( X ) such that t E W; c q t K(,).Since Xis w-compact, for an open cover { r j l ( 1 t E r ( X ) } of X , there exists a finite subset {tl, . . . , t , ) of r ( X ) such that
w)
X = C I T ( U { T i 1 ( ~ , )=l i 1, . . . , n}) = U{cl&i'(W;,))p = 1,
. . . , n}.
Noting that CIT(~i'(W;)) = r i l ( q ) for t E ?(A'), we have r ( X ) i = 1, . . , n) c U{ V,(,,)I i = 1, . . . , n}. Thus r ( X ) is compact.
=
U{q,:,l
0
3.5. Proposition. There exists a regular Hausdorfspace X such that X is not w-compact but z ( X ) is compact. Proof. We prove the proposition by using an example (due to Tychonoff [1929]) of a regular Hausdorff space which is not Tychonoff. Let wo be the first countable ordinal and wIthe first uncountable ordinal. We denote by W(wi 1) the space of all ordinals a 6 mi with the interval topology, where i = 1, 2. Let us put
+
+ 1) x W ( 0 , + 1)
s
= W(w1
P
= {(a, w o ) l a < ol} and
-
(01, oo),
Q = {(a1, n)ln c w o } .
For each n c coo, let S, be the copy of the space S and p, a homeomorphism of S onto S,. In the topological sum US, of {Sn},we identify a point ~ 2 m I -( P ) with C P , ~ ( Pfor ) P E P and a point ~ 2 r n ( q )with ( ~ 2 m + l ( q ) for 4 E Q. By this identification we have a quotient space Y , which is locally compact Hausdorff. Let X be a space obtained by adding a new point 5 to Y and introducing the topology in X as follows: the base at 5 is the sets Y - U{p,(S)l 1 < j < n}, n < wo,and the base at x # 5 is the same as in Y. Then Xis a regular Hausdorff space which is not Tychonoff. Further it is shown that z ( X ) is compact but X is not w-compact. In fact, noting that for any continuous map g : W ( o , + 1) + I there exists < wIsuch that g(a) is constant for each a 2 (see, for example, Engelking [1977]), it is easily seen that for any continuous map f : X 4 I there is Po c w1 such thatfis constant on the set U~=I{pn((a, w,))l a 2 Po}. Hence every cozero-set of X containing has to include a set of the form
u(X where
\j
I=
I
Yu
pi(^) u i = I pi(T(y, ki)),
T(y, k ) = {(a, n) E S 1 a 2 y, n >, k } for y c w 1 and k <
0,.
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T. Zshii
Therefore any cozero-set cover of X has a finite subcover and so z ( X ) is compact. To see that X is not w-compact, we consider the family n < o,,} of closed sets of X , which is closed under the {cp,(T(tl,n))l tl < o,, finite intersection and each cp,(T(a,n)) contains an isolated point (that is, a cozero-set), while ncp,(T(a, n)) = 8. Consequently, X is not w-compact.
0 3.6. Proposition. A z-closed subset of a w-compact space is not always w-compact. We shall prove the proposition by an example. 3.7. Example. We introduce a topology e in the closed unit interval I as follows: the base at x # 0 is the &-neighborhoodsU,(x), E > 0, and the base at x = 0 is the sets U,(O) - A, E > 0, where A = { l / n l n E N}. Then the space X = (I,e) is H-closed and hence it is w-compact. But the z-closed subset S = A u {0} of X is not w-compact, because it is a countable discrete subspace of X . 3.8. Proposition. Let X be a w-compact space. Then every z-open set P of X has the w-compact closure.
3.9. Lemma. Let U be a cozero-set of a space X . Then every cozero-set V in U is also a cozero-set of X . Proof. Let f : X -+ I be a continuous map such that U = { X I f ( x ) > 0} and g : U -+ I a continuous map such that V = {x E U l g ( x ) > O}. Let us define a map h : X -+ I as follows: h(x) = f ( x ) g ( x )
for x
E
V,
and h(x) = 0 for x
E
X - V.
Then it is easily seen that h is continuous over X and V = {x1 h(x) > 0}, showing that V is a cozero-set of X . 0 Proof of Proposition3.8. Let P be a t-open set of a w-compact space X . Let { Q,} be a family of z-open sets of P with the finite intersection property. Then by Lemma 3.9, P n Q, is a non-empty z-open set of X for each a. Since the family { P n Q , } has the finite intersection property and Xis w-compact, we have n P n Q, # 8 and hence n Q , # 8. Thus P is w-compact.
The Tychonoff Functor and Related Topics
213
3.10. Proposition. The continuous image of a w-compact space is also
w-compact.
The above proposition is easily seen, and so the proof is left to the reader. A mapf : X + Y of a space X to another space Y is called a 2-map (Frolik [1961]) if it is continuous and f ( Z ) is closed for any zero-set Z of X . As for 2-maps, see also Isiwata [1967, 19691 and Noble [1969a, 1969bl. 3.11. Theorem. For a space X , the following conditions are equivalent.
(1)
X is w-compact.
(2)
The projection nY: X x Y
(3)
The projection K : X x Y + Y is a Z-mapfor any Hausdorflspace Y with at most one nonisolated point.
+
Y is a Z-map for any space Y .
Proof. (1)+(2). Let X be a w-compact space and Y an arbitrary space. Let 2 be a zero-set of X x Y such that Z = f - ' ( O ) for some continuous map f : X x Y + I. Suppose that n y ( Z )is not closed in Y . Then there is a point yo of Y such that yo E n Y ( Z ) - n y ( Z ) .Sincef (x, y o ) > 0 for x E X , we have inf{f (x, y o )I x E X } = a > 0; this follows from the compactness of 7 ( X ) . For each point (x, y o ) E X x Y , take an open neighborhood U, x V, of ( x , y o ) in X x Y such that f ( u , v ) > $ a for (u, v ) E U, x V,. Since { U, I x E X } covers the w-compact space X , there exists a finite number of points x,, . . . , x, of X such that X = U{Cl,U,,I i = 1, . . . , n } . Putting V = n{VJi = 1 , . . . , n } , we have Cl,(U{U,, x V l i
=
1 , . . . ,n } ) n 2 =
0,
because f ( u , v ) 2 + a for (u, v ) E Cl,(U,, x V ) , i = 1, . . . , n.
Hence it holds that ( X x V ) n Z = 0, contradicting the fact that yo E n Y ( Z ) .Therefore n y ( Z )is closed in Y . (2)+(3). This is obvious. (3)+(1). Suppose X is not w-compact, that is, there exists a family 9 = {PaI a E R} of 7-open sets of X with the finite intersection property and n { P uI tl E R} = 0,where we may assume without losing generality that 9is closed under the finite intersection. We introduce an order in R such that a < B means P, 3 Psand construct a space Y = R u { t}by adding a new point 5 to R and introducing a topology in Y as follows: each point of R is open and the base at is the sets U, = { /? E R I a < j} u { t},a E R. Then
214
T. Ishii
Y is a Hausdorff space with only one nonisolated point. Now for each a E R, take a cozero-set G, and a nonempty zero-set Fa in X such that F, c G, c P,. Then there is a continuous map g,:X + I such that g,(x) = 1 for x E X - G, and g,(x) = 0 for x E Fa. Let us define a map g : X x Y4Iinsuchawaythat g(z) = g,(x)
for z
=
(x, a) E X x a,
a E R,
and g(z) = 1 for z = (x, <) E X x
5.
Then it is easily seen that g is continuous. Hence if we put Z = g-'(0), it is a zero-set of X x Y and n y ( Z )is not closed in Y,since n y ( Z ) = Y - {<}. Thus X is w-compact. 0 Finally, we shall describe a theorem that is more precise than Theorem 3.11.
3.12. Theorem. For a space X the following conditions are equivalent. (1)
X is w-compact.
(2)
For any space Y and for any zero-set Z of X x Y, n y ( Z ) is a zero-set of Y.
(3)
For any Hausdorff space Y with at most one nonisolated point and for any zero-set Z of X x Y, n y ( Z )is a zero-set of Y.
The theorem above is an immediate consequence of the following lemma and Theorem 3.1 1.
3.13. Lemma. Let X be a w-compact space and Y an arbitrary space. Then for any continuous m a p f : X x Y + Y, the map g : Y + I defined by g(Y)
=
inf{f(x, Y ) l X E G Y Y ) }
is continuous. Proof. Let yo be any point of Y and E any positive number. For any point (x, y o ) E X x Y, take an open neighborhood U, x V, of (x, y o ) in X x Y such that If(u, v ) - f(x, yo)I c a e for (u, w ) E U, x V,. Then for an open cover { U' I x E X} of X, there is a finite number of points xI,. . . ,x, of Xsuch that X = U{Cl, Ux,I i = 1, . . . , n}, since X is w-compact. Let us put V = n{V,,li = 1, . . . , n}. Then it holds that If(x, y ) - f(x, y o )I
< 38
for x E X and y E V .
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The TychonoffFunetor and Related Topics
In fact, any point x of X is contained in some C1, Uxk,1 for y E V we have If(x,Y )
- f ( ~ k ,
.YO)(
< $8
< k < n, and hence
and If(x, yo)
-f(xk,
yo)[
< $&,
which implies that I f ( x , y ) - f ( x , y o ) ! < $ 8 . Consequently, from y E V it follows that 1 g( y ) - g( y o )I < E , showing the continuity of g : Y + I. 0
3.14. Remark. As for an open Z-map, Frolik [1961] proved the following result: Let f : X + Y be an open 2-map of a Tychonoff space X onto a Tychonoff space Y. If g:X4 R is continuous and if g,(y) = inf{g(x)l x E f - I ( y)}(g'( y ) = sup{g(x) I x E f - I ( y ) } ) is a real number for every point y E Y , then gi: Y 4 R (resp. g" : Y + R) is continuous. However his result is valid for the case of X and Y being general spaces. Hence by use of this result, Lemma 3.13 is also derived as a corollary of Theorem 3.11 (1
)+(a.
4. A space X such that r(X x Y ) = z ( X ) x r ( Y ) for any space Y The following theorem gives a characterization of a space X such that z ( X x Y ) = z ( X ) x z ( Y ) for any space (or any regular T I space) Y .
4.1. Theorem. For a space X the following conditions are equivalent. (1)
z ( X x Y ) = z ( X ) x z ( Y ) f o r any space Y.
(2)
t(X x Y ) = r ( X ) x r ( Y ) f o r any regular T , space Y.
(3)
For each point x of X , there exists a cozero-set neighborhood U of x such that 0 is w-compact.
For the proof of the theorem, we need two lemmas. 4.2. Lemma. Let U be a cozero-set of a space X whose closure is not wcompact. Then there exists a Hausdorffspace Y with only one nonisolatedpoint yo and a continuous map f 1 X x Y + I such that f ( z ) = 1 for z
E
( X x yo) u ( ( X - U )x Y ) ,
and f-'(o)n(Uxy>Z
B foryEY-yo.
The above lemma is shown in a similar way as Theorem 3.11 (3)+(1).
T. lshii
216
4.3. Lemma. There exists a regular TI k-space X containing two diferent points a and b such that g(a) = g(b)for every continuous map g : X + R.
Proof. Let X , P,Q , Snand qnbe the same as in the proof of Proposition 3.5, where n runs through all integers. In the topological sum US, of {Snln = 0, f 1, . . .} we identify a point q z n - , ( pwith ) qzn(p)forp E P and for q E Q . By this identification we have a a point qzn(q)with q2n+1(q) quotient space Y, which is locally compact Hausdorff. Let X be a space obtained by adding two different new points a and b to Y and introducing a topology in X as follows: the bases at a and b are the sets
respectively, and the base at y E X - a u b is the same as in Y. Then X is a regular TI space and the k-ness of X follows from the facts that each point y E X - a u b has a compact neighborhood and that each of two points a and b has a countable local base. Moreover it holds that g(a) = g(b) for every continuous map g : X --+ R, since any cozero-set of X including the point a has to contain a set of the form
where = {(a, n) E
T(y, k)
SI a 2 y , n 2
k} for y < o,and k <
Thus the space X has the desired properties.
w,,.
0
The fact that the space Xabove is a k-space will be used in the next section. Proof of Theorem 4.1. (1)+(2) is obvious. (2)+(3). Suppose that there exists a point x, of Xsuch that the closure of any cozero-set neighborhood of x, is not w-compact. Let { U,1 ct E Iz} be a base of cozero-set neighborhoods of x,. By Lemma 4.2, for each a E Iz there exists a Hausdorff space Y, with only one nonisolated point y, and a continuous map f,: X x Y, --t Z such that f,(z)
=
1
for z E ( X x y,) u ( ( X - U,) x Ye),
and
f,-'(O)
n (U, x y )
z0
for y
E
y, - y,.
Let 2 be a regular T Ispace containing two different points a and b such that g(a) = g(b) for every continuous map g : Z -+ R; such a space exists by
The Tychonoff Functor and Related Topics
217
Lemma 4.3. Let 2,be the copy of Z for each a E R and cp, a homeomorphism of Z onto 2,. In the topological sum Y , u Z , , we identify a point 6, = cp,(b) with y,. By this identification we obtain a quotient space T, which is regular T I .Let us define a space Y as a quotient space of the topological sum (JT, obtained by identifying every point a, = cp,(a) E Z , , a E R; Y is clearly regular T I .Let q: T, - Y be the quotient map and put 5 = q(a,) for every a E R. Then it is shown that there is a cozero-set G of X x Y including (xo. 5 ) for which there is no cozero-set rectangle U x V with (xo, 5 ) E U x V c G . To see this, we define a map g: X x Y -+ I in such a way that
u
g(z) = 1
for z E X x ( Y - u ( Y , - y,)),
and g(z) =
f , ( z ) for z
E
X x y, y
E
Y,
- y,.
Then g is a continuous map and G = {(x, y ) I g(x, y ) > 0} is a cozero-set of X x Y with (xo,5 ) E G . Let U x V be any cozero-set rectangle of X x Y containing (x,,, 5 ) and take /? E R such that U, c U . Since 5 E V , we have 6, = cp,(b) E Vfor each a E R and hence V n ( Y , - y,) # 8. On the other hand, we have fL'(0)n (Us x y ) # 8 for y E Y, - y,, so that g-'(0) n (Us x V ) # 8. Therefore U, x V is not contained in C,which implies that z ( X x Y ) # 7 ( X ) x z ( Y ) by Theorem 2.4. Thus (3) follows from (2). (3)-+(1). Let Y be an arbitrary space. Let us take a cozero-set G of X x Y and a point (xo,y o ) E G . If we put G ( y o ) = { x E X I (x, y o ) E C}, it is a cozero-set of X with xoE G( yo).As is easily seen by Proposition 3.8, there is a cozero-set neighborhood U of xo whose closure is w-compact and is contained in G ( y o ) . Since 6 x Y - G is a zero-set in 6 x Y , x y ( 6 x Y - G ) is a zero-set of Y by Theorem 3.12. Hence, if we put V = Y - x y ( 6 x Y - G ) , it is a cozero-set of Y with yo E V and so we have (xo,y o ) E U x V c G . Thus it holds that z ( X x Y ) = z ( X ) x z( Y ) by Theorem 2.4, completing the 0 proof. As a corollary of Theorem 4.1, we have the following. 4.4. Corollary. Let X be a Tychonoffspace. Then thefollowing conditions are
equivalent. ( 1 ) X is locally compact. (2) z ( X x Y ) = t ( X ) x z ( Y ) f o r any space Y .
Corollary4.4 (1)+(2) and (2)+(1) are due to Pupier [I9691 and Oka [1978], respectively.
T. Ishii
218
5. A space X such that r ( X x Y ) = r ( X ) x r( Y ) for any k-space Y
We shall begin with a characterization theorem of pseudocompact spaces in terms of product spaces. A space X is called pseudocompact if every continuous map f:X + Iw is bounded. As is easily seen, a space Xis pseudocompact if and only if for any decreasing sequence { U , } of nonempty # 0. cozero-sets of X we have
no,,
5.1. Theorem. For u space X the following conditions are equivalent.
(I)
X is pseudocompact.
(2)
The projection n,: X x Y k-space Y.
(3)
The projection n,: X x Y Y = W ( 0 , 1).
+
+
Y is a 2-map for any HausdorfT
+
Y is a Z-map for the space
is obvious. Let X be a pseudocompact space and Y any Hausdorff k-space. Let Z be a zero-set of X x Y such that Z = f -'(O) for some continuous map f : X x Y + I. To prove that n r ( 2 ) is closed in Y , it suffices to show that n,(Z) n K is closed for any compact set K of Y. Hence we may assume that Y is itself a compact Hausdorff space. For such a space Y, we have t(X x Y) = t(X) x T( Y) by Corollary 4.4, so that there exists a continuous map g : r ( X x Y) -+ I such that f = g o t x x , . Let us put 2, = g - ' ( 0 ) and denote by i , the identity map of Y onto itself. Since (rx x i , ) - ' ( Z I )= Z , it remains to show that n , ( Z , ) is closed in Y. Noting that the product z(X)x Y of a pseudocompact space r ( X ) with a compact space Y is pseudocompact (see, for example, Engelking [ 1977]), by Glicksberg's theorem (Glicksberg [1959]; see also Nagata [1985]) we have / I ( t ( X ) x Y ) = / I ( t ( X ) ) x Y , which implies that every continuous map h :t(X)x Y + I has a continuous extension over B ( r ( X ) ) x Y, where P ( r ( X ) )is the Stone-Cech compactification of t(X). Therefore the map g has a continuous extension 2 : / I ( t ( X ) ) x Y + I. If we put Z , = g-'(O), then we have n,(Z,) = n,(Z,) and n y ( Z 2 )is closed in Y by Theorem 3.1 1. Thus ny(ZI)(and hence n , ( Z ) ) is closed in Y. (3)+(1). Suppose Xis not pseudocompact. Then there exists a decreasing sequence { U,,} of nonempty cozero-sets of X such that non= 0. Take a nonempty zero-set Z,, of X contained in U, and a continuous mapf, : X + I such that 2, = f - ' ( 0 ) and X - U,, = f - ' ( l ) . Let us define a map f : X x W(o,+ 1) + I as follows:
Proof. (2)+(3) (1)+(2).
f ( x , n)
= fn(x)
for x E X and n < o,,,
The Tychonoff Functor and Related Topics
219
and f ( x , o,,)= 1 for x
E
X.
Then it is easy to see thatfis continuous. Further, letting Z = f -'(O), n y ( Z ) is not closed in Y = W(w, 1). Thus (3) implies (I), completing the proof. 0
+
As a more precise result than Theorem 5.1, we have the following theorem.
5.2. Theorem. For a space X the following conditions are equivalent. (I)
X is pseudocompact.
(2)
For any Hausdorf k-space Y and for any zero-set Z of X x Y , ny(Z) is a zero-set of Y.
(3)
For the space Y = W ( o , + 1) andfor any zero-set Z o f X x Y , ar(Z) is a zero-set of Y.
The above theorem is an immediate consequence of Theorem 5.1 and the following lemma. 5.3. Lemma. Let X be a pseudocompact space and Y a Hausdorf k-space. Then for any continuous map f : X x Y + I , the map g : Y + I defined by g ( y ) = inf{f(x,y)JxEn;'(y)} iscontinuous.
Proof. Since the lemma is reduced to the case of Y being a compact Hausdorff space and X being a pseudocompact Tychonoff space, it is derived from Glicksberg's theorem and Lemma 3.13. 0 We notice that Lemma 5.3 is also derived from Theorem 5.1 and Frolik's theorem mentioned in Remark 3.14. Now we prove a theorem characterizing a space X such that 7 ( X x Y ) = r ( X ) x 7 ( Y ) for any Hausdorff k-space Y.
5.4. Theorem. For a space X the following conditions are equivalent. (1)
7(X
x Y ) = 7 ( X ) x r ( Y ) f o r any Hausdor-k-spaces Y .
(2)
7(X
x Y)=
(3)
For each point x of X there exists a cozero-set neighborhood U of x such that 0 is pseudocompact.
7(X)x
7 ( Y )for any regular T , k-space Y .
T . Ishii
220
To prove the theorem, the following lemmas are needed.
u
5.5. Lemma. Let U be a cozero-set of a space X such that is not pseudocompact. Then there exists a continuous map f : X x W(w, + 1) -+ Isuch that f(z)
=
1 for z E ( X x
0,)
u ( ( X - U ) x W ( o , + l)),
and f - ' ( O ) n (U x n) #
0
for n <
0,.
Since the above lemma is shown by the similar way as in the proof of Theorem 5.1 (3)-+(1), the proof is left to the reader.
5.6. Lemma. Let U be a cozero-set of a space X such that 0 is pseudocompact. If V is a cozero-set of X contained in U, then is also pseudocompact.
v
Proof. Let {Gn>be a decreasing sequence of non-empty cozero-sets of p. Let us put H,, = G,, n V for each n. Then {H,,}is a decreasing sequence of nonempty cozero-sets of X by Lemma 3.9. Since D is pseudocompact and H,, c U for each n, we have nClOH,, # 0 and hence nClVG,, # 0,showing that is pseudocompact.
v
Proof of Theorem 5.4. (1)+(2) is obvious. (2)-+(3).Suppose there is a point xo of X such that any cozero-set neighborhood of x, does not have the pseudocompact closure. Let { U, I a E R} be a base of cozero-set neighborhoods of x,. Since 0, is not pseudocompact for each a E R, by Lemma 5.5 there exists a continuous map f , : X x Y, + I, where Y, = W(w, + 1) for each a E R, such that f , ( z ) = 1 for z
E
(X x
0,)
u ((X- U , ) x Ya),
and
f,-'(O) n (u, x n) z 0 for n < w,. Let Z be a regular T, k-space mentioned in Lemma 4.3 which contains two different points a and b such that g(a) = g(h) for every continuous map g : Z + R. Let Z , be the copy of the space Z and cp, : Z Z , a homeomorphism from Z onto Z , , a E R. Let us denoted by T, the space obtained from the topological sum Y, u 2, by identifying h, = cp,(b) with w, E Y,. We now define the space Y as a quotient space of the topological sum U{T,(a E R} obtained by identifying every a,, = q,(a), a E R. Let q be the quotient map and put 5 = q(a,) for each a E R. Since the topological sum Y, u Z , is a regular T , k-space, so is the quotient space T,, and hence the -+
The Tychonoff Functor and Related Topics
space Y is a regular T I k-space as well as the topological sum Let g : X x Y + I be a map defined by g ( z ) = 1 for z
E
X x U(Y - (Y,
-
22 1
u{T , I a
E
Q}.
w,)),
and g ( z ) = f , ( z ) for z E X x n , n
E
Y, -
0,.
Clearly g is continuous, and G = { ( x , y ) l g ( x , y ) > 0 } is a cozero-set of X x Y with (x,, 5 ) E G. Further it is easily seen that there is no cozero-set rectangle U x V of X x Y such that (x,, 5 ) E U x V c G, showing that r ( X x Y ) # t ( X ) x r ( Y ) . Thus ( 2 ) implies (3). (3)+( 1). This is derived from Lemma 5.6 and Theorem 5.1 by the similar way as in the proof of Theorem 4.1 (3)+(1) and so the proof is left to the 0 reader. Theorem 5.4 yields the following result due to Pupier [I9691 as a corollary.
5.7. Corollary. I f X is a space such that r ( X )is locally compact, then we have r(X x Y ) = r ( X ) x r ( Y )f o r any Hausdorfk-space Y . Proof. Let r ( X ) be locally compact. By Theorem 5.4 it suffices to see that each point x of X has a cozero-set neighborhood U such that 0is pseudocompact. Let x E X and t = t x ( x ) .Then there exists a cozero-set neighborhood V of t E r ( X ) such that P is compact. Let us put U = t x ’ ( V ) .Then U is a cozero-set of X with x E U and C1, U = z;’( F), and further every continuous map g :X + R is bounded on 0as well as C1, U,since g = h t xfor some continuous map h : z ( X ) + R. From this it easily follows that 0 is pseudocompact. 0 0
6. Products of w-compact spaces
This section is mainly concerned with the products of w-compact spaces. 6.1. Theorem. Let { X uI a E Q} be a family of w-compact spaces. Then the product IIX, is also w-compact and it holds that z ( n X , ) = llz(X,). 6.2. Lemma. Let X,, i = 1, . . . ,n, be any spaces and {P;, I A E A} a maximal family of z-open sets of the product X = II{X,.u,li = 1, . . . , n } with thefinite intersection property. Z f x , E n{Clx,ni(Pj,)I1 E A} for i = 1, . . . , n, then (XI,
where x i :X
3
. . . , x,)
E
npi I 1
X, is the projection.
E
A},
222
T. Ishii
Proof. We shall prove the lemma by induction for positive integers n. In case n = I , it is obviously valid. Suppose the lemma is true for n = k - 1, k > 2. To show that it is valid for n = k, let A = { P i 1 I E A} be a maximal family of r-open sets of X = II{xI i = 1, . . . , k} with the finite intersection and xi E n{Cl,ni(P,)II E A} for i = I , . . . , k. Let us denote by K the projection from X to X’ = ll(X,l i = 1, . . . , k - 1). Then it is easy to see that IT(&) = {.(Pi) 1 I E A} is a maximal family of t-open sets of X’ with the finite intersection property and xi E n{Cl,n,:(7z(Pi)) I I E A} for i = 1, . . . , k - I , where nj is the projection from X’ to X,. Hence by the assumption for the case of n = k - I , we have (XI,
...,
E
n{ci,.A(P,) I I E A}.
To show that X
= (XI,
.
* *
xk) E n{p$
E
A},
let us take an arbitrary open neighborhood UI x * . . x Uk of x in X. Then it is obvious that Qj.= ((17, x . . x Uk-1) x Xk) n Pi # 8 for each I E A. Since A is closed under the finite intersection, {Qil I E A} has the finite intersection property and so does { z k ( Q i ) l I E A}. Hence for each I E A, nk(Qi) u n , ( A ) has the finite intersection property. As is easily seen, nk(Qi) is a 7-open set of xk for I E A. Therefore we have ITk(Qi) E K k ( A ) for I E A, since q ( A )is a maximal family of t-open sets of Xk with the finite intersection property. Consequently, we have xk E Cl,Q, for each I E A, from which it follows that +
(ul x
’ ’ ’
x Uk-1 x
uk)
n Pi #
8
for 1 E A. Thus it holds that ( x I ,. . . , xk) E n{pi A E A}, completing the induction.
Proof of Theorem 6.1. Let X,, a E R, be w-com: act spaces and 9 any family of t-open sets of the product X = l l X , with the finite intersection property. Then by Zorn’s lemma there exists a maximal family A = { P i I I E A} of r-open sets of X having the finite intersection property and Let us denote by IT, the projection from X to X,. Since for each containing 9. a E R, X, is a w-compact space and {ta(PA) I A E A} is a family of t-open sets of X, with the finite intersection property, there exists a point x, of X, such that x, E n{Cl,n,(P,) I I E A}. Let us put x, = ( x , ) E X . Then it remains to n;’(U,) be an arbitrary show that x, E n { P i l I E A}. To see this, let open neighborhood of x o , where U, is an open neighborhood of xz, in X,, for each i. Let us put X’ = ll{X,,I i = 1, . . , , n} and let A be the projection
The Tychonoff Functor and Related Topics
223
from X to X’. Since {n(Pi)1 I E A} is a maximal family of z-open sets of X’ with the finite intersection property, by Lemma 6.2 we have (Xnp
f
..
9
E fI{C1,,n(P,)
X,J
II
E
A},
which implies that / n
\
that is,
n n,‘(Ui) n Pi # 8 n
for I
E
A.
i= I
Thus x, E (){Pi 11 E A}, showing the w-compactness of X = n X , . We next prove the latter half of the theorem. Denoting by z, for brevity, the natural map zx, of a space X, to z(X,), for the product map nz, : X --t n z ( X , ) there exists a continuous mapg :z ( X ) + n z ( X , ) such that nz,= g o z x . By Proposition 2.1, the map g is one-to-one and onto. Since X = nX, is w-compact, z ( X ) is compact by Proposition 3.4, and hence the map g is a homeomorphism of z ( X ) onto nz(X,). Thus we have r(IIX,) = n z ( X , ) , completing the proof. 0 We next describe an application of Theorem 6.1. The following theorem is reduced to Mibu’s theorem (Mibu [1944]) for the case of X,, a E R, being compact Hausdorff spaces.
6.3. Theorem. Let { X,I a E R} be a family of w-compact spaces. Thenfor any continuous map f:nX, -, Z, there exists a countable subset Q, of R and a continuous map g : n ( X , I a E Q,} + Z such that f = g 0 IC, where IC :nX, -, n{X,I a E Q,} is the projection. Proof. Let us put X = n X , and let f:X -, Z be any continuous map. Then there exists a continuous map g :z ( X ) + Z such that f = g z x . Since X,, a E R, are w-compact, we have z ( X ) = n z ( X , ) by Theorem 6.1 and z(X,) are compact. Hence by Mibu’s theorem there exists a countable subset Q, of R and a continuous map h : n { r ( X , ) I a E Q,} + Zsuch that g = h 0 n’, where n‘ : n z ( X , ) -, n { z ( X , ) I a E Q,} is the projection. If we denote by n the projection of X to X , = n { X , I a E Q,}, we obtain 0
f = g o t , = (hoa’)ot, = =
h 0 (zx0
0
Z)
h o ( ~ ’ o ~ , )
= (h 0 t x o0)71.
Therefore the map g = h , ,z :n { X , 1 o! E R,} that f = g n, completing the proof. 0
0
+
Z is a continuous map such
0
224
T. Ishii
7. w-Paracompact spaces and the Tychonoff functor In this section we introduce the concept of w-paracompact spaces, which is a generalization of both w-compact spaces and paracompact Hausdorff spaces, and study the basic properties of those spaces and the relationship between those spaces and the Tychonoff functor. Recall that any normal open cover of a space admits a locally finite cozero-set refinement and that any a-locally finite cozero-set cover of a space is a normal open cover (as for the proofs of these facts, see Chapter 2, Theorem 1.2). 7.1. Definition. A space X is w-paracompact if for any open cover { U ,I 1 E A} of X there exists a locally finite cozero-set cover { V ,I a E R} of X such that for each a E R there is a finite subset {,u(l), . . . , p(n)} of A such that V , c C17((J{U,(i)li= 1 , . . . , n}). A family { Aj,11 E A} of subsets of a space X is called weakly Cauchy (due to Corson [ 19581) if for any normal open cover { U, I a E R} of X there is b E R such that { U s } u { Aj,1 il E A} has the finite intersection property. The following proposition is a natural extension of Corson’s theorem that a Tychonoff space X is paracompact if and only if for any weakly Cauchy family { Aj , 1 1 E A} of subsets of X we have A ,I 1 E A} # 0.
n{
7.2. Proposition. A space X is w-paracompact if and only if for any weakly I 1 E A} of T-open sets of X we have n { F j ,1 1 E A} # 0. Cauchy family {P;.
Proof. Necessity. Suppose there is a weakly Cauchy family {Pj,I 1 E A} of z-open sets of X with n { F j I, I E A} = 0. If we put U, = X - Fj, for any 1 E A, { U ,I il E A} is an open cover of X with (Cl, U , ) n Pi = 8. Since X is w-paracompact, there exists a locally finite cozero-set cover { V , I a E R} of X such that for each a E R there is a finite subset { ~ ( l )., . . , p(m)} of A with
V , c CIT((J{U,(i)li= 1 , . . . , m}). Take /? E R such that { 5 ) u {Pj.11 E A } has the finite intersection property. Then, since V, c CIT(lJ{Uv(i)l i = 1, . . . , k}) for some v(l), . . . , v(k) E A, we have V, n n{Py(i)li= 1 , . . . , k} = 0, which is a contradiction. This proves the “only-if”-part. Sufficiency. Let { Uj.11 E A} be any open cover of X . In case X = U{Cl, U,(i)I i = 1 , . . . , n} for some A( l), . . . ,1(n) E A, it suffices to take the
225
The Tychonoff Functor and Related Topics
space X itself as a locally finite cozero-set cover { V ,1 a E f2} of X satisfying the required condition. In the case that X - u{Cl, Uj,(,) I i = 1, . . . , n } # 0 for any finite subset {A(l), . . . , A(n)} of A, if we let G(A(l), . . . , A(n)) = X - U{ClrUj.(i)li= 1,
. . . ,n } ,
and
9
= {G(A(l),
. . . , A(n))I L(l), . . . , A(n) E A, n E N},
then 9 is a family of r-open sets of X with the finite intersection property, while n{G(A(f), . . . , A(n))lA(l), . . . , A(n) E A, n E N} = 0. Hence 9 is not weakly Cauchy by our assumption, so that there is a locally finite cozero-set cover { V , I a E f2} of X such that each V , is disjoint from some Therefore for each a E R there are p( I), finite intersection of elements of 9. . . . , p ( k ) E A such that V, n G(p(l), . . . , p ( k ) ) = 0,showing that
V , c Cl,(u{Up(i)li= 1, . . . , k}). Thus X is w-paracompact, completing the proof.
0
7.3. Proposition. I f a space X is w-paracompact, then t ( X ) is paracompact. But the converse is not true in general. More precisely, even when r ( X ) is compact, X is not always w-paracompact.
To show this proposition, the following lemmas are needed. 1.4. Lemma.
For a space X , the following conditions are equivalent.
(1)
r ( X ) is paracompact.
(2)
For any weakly Cauchy family { P,.I A have (7{C1,Pj,(AE A} # 0.
E
A} of r-open sets of X , we
Since the above lemma is shown by the similar way as in the proof of Proposition 7.2, the proof is left to the reader. The following lemma is easily seen.
7.5. Lemma. I f a space X is w-paracompact and r ( X ) is compact, then X is w-compact. Proof of Proposition 7.3. The first part is an immediate consequence of Lemma 7.4. To prove the second part, let X be a regular Hausdorff space
226
T. Ishii
which is not w-compact but r ( X ) is compact, as is shown in Proposition 3.5. Then by Lemma 7.5, X can not be w-paracompact, which completes the proof. 17
By Proposition 7.3,any w-paracompact Tychonoff space is paracompact. However, even for regular Hausdorff spaces, w-paracompactness does not imply paracompactness. In fact, let X be a regular Hausdorff space with I XI > 1 such that every continuous mapf :X + R is constant. Such a space Xis w-compact and hence is w-paracompact, but it can not be paracompact, since it is not even a Tychonoff space. 7.6. Proposition. Let X be a w-paracompact space. Then,for any r-open set P of X , .p is w-paracompact. However a r-closed set (and hence a closed set) of X is not w-paracompact in general.
To prove the second part, we use a result due to Stone [I9481 as a lemma. 7.7. Lemma.
The product NK'is not normal.
Proof. We only sketch the proof of the lemma. Let A,, i = 1,2, be the set consisting of all { x n }E N" such that for any positive integerj # i we have xi. = j for at most one 2. Then A , and A, are closed and disjoint, and further it is shown that those sets are not separated by disjoint open sets. Thus NK1 is not normal. 0 Proof of Proposition 7.6. The first part of the proposition is easily seen, so that the proof is left to the reader. To show the second part, let X be a w-compact space mentioned in Example 3.7. Then by Theorem 6.1 the product XI is w-compact and hence is w-paracompact. But the r-closed subset (A u {O})K1 is not w-paracompact, is not normal and hence is not since A u {0}is homeomorphic to N and NK1 paracompact. This completes the proof. 0 7.8. Proposition. For a space X the following conditions are equivalent. (1)
X is w-paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(2)
X is w-paracompact and r ( X ) is locally compact.
The Tychonoff Functor and Related Topics
221
(3)
r ( X ) is paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(4)
There exists a locallyjnite cozero-set cover { U, I a E Q} of X such that 0, is w-compact for any a E R.
Proof. (1)+(2). Let x E X and t = r x ( x ) .Let U be a cozero-set neighborhood of x such that 0 is w-compact. Then r x ( U ) is a cozero-set of r ( X ) by ( )is compact by Proposition 3.10, which implies that Proposition 1.2 and t X 0 r x ( U ) is an open neighborhood o f t whose closure is compact. (2)+(3). We shall prove only the latter half of (3). Let x E X and t = r x ( x ) .Since r ( X ) is locally compact, there is a cozero-set neighborhood V of t such that P is compact. Let U = t i ’ (V ) and let {Pi I I E A} be any family of ?-open sets of 0 with the finite intersection property. Then Pi n U, I E A, are nonempty r-open sets of X by Lemma 3.9 and the family of those sets has the finite intersection property. Hence {rx(Pj.n U ) l I E A} is a family of open sets of t ( X ) with the finite intersection property and rx(Pi n U ) c V c V for each 1 E A. Since P is compact, we have n { r x ( P i n U ) ( 1E A} # 8, that is, n{Cl,(P,. n U ) l I E A } # 8, from which it follows that { P i n U I I E A} is a weakly Cauchy family of r-open sets of X. Therefore by the w-paracompactness of X we have n{-lL E A} # 8, showing that n{pi,llE A} # 8. Thus 0 is w-compact. (3)+(4). This is obvious. (4)+(l). We prove only that X is w-paracompact. Let { U, 1 a E Q} be a locally finite cozero-set cover of X such that each 0, is w-compact and let { I 1 E A} be any open cover of X. Since { 0, n 11 E A} is an open cover of a w-compact space 0,, there are p(I), . . . , p ( k ) E A such that
v,
v.
0,
=
CITE(U{O,n
c ( ii)=~ I , . . . , k } ) ,
where CITameans the .r-closure in 0,. Noting that CITaAc CIJ subset A of X , we have
for any
17~c CIr(u(OEn c(i)li= 1, . . . , k}) c Clz(U{<(i)li = 1,
. . . ,k } ) .
Consequently X is w-paracompact, completing the proof.
0
The following theorem gives the necessary and sufficient conditions for a space X to be w-paracompact in terms of the Tychonoff functor.
228
T . Ishii
7.9. Theorem. For a space X the following conditions are equivalent. (1)
X is w-paracompact.
(2)
T ( X ) is paracompact and r ( X x Y ) = r ( X ) x r ( Y ) for any Tychonofl space Y.
(3)
r ( X ) is paracompact and r ( X x Y ) = r ( X ) x T ( Y )for any Hausdorflspace Y with at most one nonisolated point.
The proof of Theorem 7.9 is based on the following result. 7.10. Theorem. For a space X the following conditions are equivalent. (1)
r ( X x Y ) = t ( X ) x r ( Y )for any Tychonoflspace Y.
(2)
r ( X x Y ) = t ( X ) x T ( Y )for any Hausdorffspace Y with at most one nonisolated point.
(3)
For any family {Pj,11 E A} of r-open sets of X with the finite intersection property which is closed under the finite intersection, n { C l ,Pj.11 E A} # 8 implies n{pj,11 E A} # 8.
Proof. (1)-(2) is obvious. (2)+(3). Suppose there is a family {Pi.I 1 E A} of r-open sets of X with the finite intersection property which is closed under the finite intersection and forwhich n{C\Pill E A} # 0but n{FilA E A} = 0. Letx, E n { C / P i ) I E A}. We denote by 9 = { Qa I a E R} a family of subsets of X each of which is in the form Pi. n U ( x o ) ,1 E A, where U ( x , ) is any cozero-set neighborhood of xo.Then 9is a family of r-open sets of X with the finite intersection property and I a E R} = 8. We now define an order in R such that a < b means Qs c Q, and construct a space Y = R u { <} by adding a new point to R and introducing a topology in Y as follows: each point of R is open and the u {<}, a E R. Then Y is a Hausdorff base at is the sets { B E R l a < /I} space with only one nonisolated point. For each a E R, take a nonempty zero-set F, and a cozero-set G, in X such that Fa c G, c Q,, and let g,: X + Z be a continuous map such that g i ' ( 0 ) = Faand g;'(l) = X - G,. Let us define a map g : X x Y + I as follows:
n{e,
r
<
g ( x , a ) = g,(x) and g(x,
r)
= 1
for (x, a ) E X x R,
for (x, 5 ) E X x
<.
Then g is clearly continuous. Further, if we put G = {(x, y ) Ig(x, y ) > 0 } , then G is a cozero-set of X x Y with (xo,5 ) E G and we have U x V - G # 0
The Tychonoff Functor and Related Topics
229
for any cozero-set rectangle U x V of X x Y containing (xo,0,contradicting (2) by Theorem 2.4. Thus (2) implies (3). (3)+(1). Suppose there exists a Tychonoff space Y such that r ( X x Y ) # r ( X ) x r ( Y ) . Then by Theorem 2.4 there is a cozero-set G of X x Y and a point (xo, y o ) E G such that any cozero-set rectangle of X x Y containing (xo, y o )is not contained in G. Let F be a zero-set and H a cozero-set in X x Y such that (xo,y o ) E H c F c G. Let us put P ( U , V) = n,(U x V - F) for any cozero-set rectangle U x Vof X x Y with (xo,y o ) E U x V , where nx is the projection of X x Y to X. Then it is easily seen that the sets P ( U , V ) are r-open in X and x,, E Cl,P(U, V). Since the family of such sets P ( U , V) has the finite intersection property and is closed under the finite intersection, by (3) there is a point x, of X such that xI E P ( U , V) for every cozero-set rectangle U x Vcontaining (xo, yo). We notice that (xl, y ) E H; otherwise, x, E X - P ( U , V) for any cozero-set neighborhood U of xo such that 0 c H ( y o ) = { x E XI (x, y o ) E H} and any cozero-set neighborhood V of yo. Hence there exists an open neighborhood A x B of (xI , y o ) contained in H , where we may assume that B is a cozero-set of Y , since Y is a Tychonoff space. But for any cozero-set neighborhood U of x,, the set P(U, B) does not contain any point of A , that is, P(U, B) n A = 8. Therefore we have xI E X - P ( U , B), which is a contradiction. Thus (3) implies (l), completing the proof. 0
Proof of Theorem 7.9. (1)+(2). Let X be a w-paracompact space. Since r ( X ) is paracompact by Proposition 7.3, we prove only the latter half of (2). Let {P;, I I E A} be a family of r-open sets of X with the finite intersection property and n{Cl,P, I I E A} # 8, which is closed under the finite intersection. Then it is clear that such a family is weakly Cauchy, so that we have n{Pil 3, E A} # 8 by the w-paracompactness of X. Therefore by Theorem 7.10 it holds that r ( X x Y ) = r ( X ) x r ( Y ) for any Tychonoff space Y. (2)+(3). This is obvious. (3)+(1). Assume that X satisfies the condition (3). Let {P;. I I E A} be any weakly Cauchy family of z-open sets of X , where we may assume without loss of generality that the family is closed under the finite intersection. Since r ( X ) is paracompact and { r x ( P j ,1)I E A} is a weakly Cauchy family of open sets of r ( X ) , we have n{r,(PJ 13, E A} # 8 by Corson’s theorem, which implies that n{Cl,PiI I E A} # 8. Hence by the latter half of (3) and Theorem 7.10, we have n { P , 11 E A} # 8. Thus Xis w-paracompact, completing the proof.
0
T. Ishii
230
8. A generalization of Tamano’s theorem
In this section we establish a characterization theorem of w-paracompact spaces in terms of product spaces, which parallels Tamano’s theorem (Tamano [19601) for paracompact spaces. For this purpose we introduce the concept of w-normal spaces.
8.1. Definition. A space Xis called w-normal if for any disjoint sets A and B of X such that A is z-closed and B is either z-closed or the closure of a z-open set, there exists a r-open set U of X with B c U c 0 c X - A. Let A and B be the disjoint sets of a w-normal space X with the properties mentioned above. If we put GI = X - A, GI is a z-open setand by the above definition there is a r-open set G(0) such that B c G(0) c G(0) c G I .Since G(0) c G I , there is a z-open set G ( $ ) such that (al)
-
-
G(0) c G ( $ ) and G O ) c G I .
Further, we can take z-open sets G(1/2’) and G(3/22) such that
-
(a’)
G(O) c G(1/2’), G(1/2’) c G O ) ,
Go) c G(3/2’), G(3/2’) By induction for n such that (a,)
E
c GI.
N, we can define r-open sets G(k/2”), k = 0, 1, . . . , 2 ” ,
0 < k < k‘
< 2“ * G(k/2“) c G(k’/2“).
Hence, as in the proof of Urysohn’s lemma (Urysohn [1925]), we can define a map f:X -+ Z such that f(x) = inf{k/2”Ix E G(k/2”)} for x
E
GI,
and f(x) = 1 for
X E
A = X - GI.
Then it is shown that f is a continuous map such that f(x) = 0 for x E B and f(x) = 1
f o r x e A.
Thus we obtain the following proposition.
8.2. Proposition. For a space X the following conditions are equivalent.
(1)
X is w-normal.
The Tychonoff Functor and Related Topics
23 1
(2)
z ( X ) is normal and for any z-open sets P and Q of X such that 0 c P, there is a t-open set U of X such that 0 c U c 0 c P.
(3)
Any disjoint sets A and B of X with the same properties as in Dejinition 8.1 are completely separated, that is, there exists a continuous map f : X + I such that f ( x ) = 0 for x E B and f ( x ) = 1 for x E A .
8.3. Proposition. normal.
There exists a non-w-normal space X such that r ( X ) is
Proof. Let X be a space mentioned in the proof of Proposition 3.5, that is, a regular Hausdorff space not being Tychonoff constructed by Tychonoff [1929]. As is shown in the proof of Proposition 3.5, Xis not w-compact but t ( X ) is compact. To see that Xis not w-normal, let us consider the one point set {t}which is z-closed and a z-open set q I ( S - (P u Q)). Then the point t is not contained in q , ( S ) = q l ( S - (Pu Q)) but there does not exist a continuous map f : X --t I such that f ( < ) = 1 and f ( x ) = 0 for x E q , ( S ) . Thus X is not w-normal, while t(X)is normal. 0 8.4. Proposition. Every w-paracompact space is w-normal.
Proof. Let X be a w-paracompact space. Since r ( X ) is paracompact by Proposition 7.3, it is normal. Let P and Q be t-open sets of X such that Q c P. For each point x of &, there is a cozero-set neighborhood C ( x ) of x such that Cl,G(x) c P, and hence we can consider an open cover $ = {G(x),X - Q I X E Q } OfX. In case Y has a finite subcover, we have
&
c U { G ( x i )I i = 1, c
. . . , n}
CI,(U{G(xi)li = 1, . . . , n } ) c P
for some points xI, . . . , x, E Q. Hence if we put U = U { G ( x i )I i = 1, . . . , n}, then U is a cozero-set of X such that Q c U c D c P, since
0
c
c1,u.
In case Y has no finite subcover, since Xis w-paracompact, there exists a locally finite cozero-set cover $2 = { U, I a E R} such that each U, is contained in the t-closure of some finite union of members of 9. By Proposition 1.7, Q admits a z-locally finite cozero-set refinement, so that we may assume that Q is itself z-locally finite. Let us put R’ = {u E R I U, n Q # @} and W, = U, n Q for tl E R’. Since { W, I u E R’} is a locally finite collection of
232
T. Ishii
r-open sets ofXand Q = U{ W,l u E a},we have Q = U{ R l u E a}.Further, for each u E Q’, W, is contained in the t-closure of some finite union L, of members of {G(x)JxE Q}. Let us put V, = U,, n L, for u E 0’. Since { V,I u E Q’} is r-locally finite in X and W, c C1, V,, we have
Q
= U { ~ ~ U € Rc ’ }U{Cl,V,~UEQ’} = Cl,(U{ V,lu
E
Q’}) c P .
By the normality of r ( X ) , there exists a cozero-set U of X such that cl,(U{V,~uER’})c so that
0c
uc
0
c
P,
U c 0 c P. Thus X is w-normal by Proposition 8.2.
0
8.5. Theorem. The product of a w-paracompact space with a w-compact space is w-paracompact.
Proof. Let X be a w-paracompact space and Y a w-compact space. Then by Theorem 4.1 we have r ( X x Y) = t ( X ) x r ( Y ) . Since r ( X ) is paracompact and t ( Y ) is compact, r ( X ) x r ( Y ) (=?(A’ x Y)) is paracompact (see, for example, Nagata [1985]). On the other hand, by Theorems 4.1 and 7.9, for any Tychonoff space Z we have T((X X Y ) X
z)
=
T(Y
X
(xX z)) =
= r(Y)x
r(X)x
z
T(Y)X
= r(Xx
T(xX z ) Y) x
z.
Thus X x Y is w-paracompact by Theorem 7.9, completing the proof.
[7
By the above theorem and Proposition 8.4 the product of a w-paracompact space with any w-compact space is always w-normal. The following theorem shows that the converse is also true. 8.6. Theorem. If for a space X the product X x /3( Y ) is w-normal for any Hausdorfspace Y with at most one nonisolatedpoint, then Xis w-paracompact.
To prove the theorem, we need a lemma which is due to Morita [ 19751 for the case of Y being compact Hausdorff. 8.7. Lemma. Let X be a space and Y a compact Hausdor-(or w-compact) space. Then for any normal open cover Y of X x Y , there exists a normal open cover 9 = { U, 1 u E Q} of X such that for a suitable collection {V,I u E Q} of jinite cozero-set covers of Y, the collection ( U , x V I V E V,,u E Q} is an open rejinement of 8.
233
The Tychonof Functor and Relaied Topics
As for the proof of Morita’s theorem for the case of Y being compact Hausdorff, see the chapter: Normality of product spaces I(Section 1). For the case of Y being w-compact, the lemma is a consequence of Morita’s theorem, Theorem 4.1 and Proposition 1.7, which will be used in the next section. Proof of Theorem 8.6. Let 9 = { P i I 1 E A} be any weakly Cauchy family of t-open sets of X . By Proposition 7.2, it suffices to show that (){pi 11 E A} # 8, where we may assume without losing the generality that 9 is closed under the finite intersection. Now suppose the contrary. Let us introduce an order in A such that 1 < p means Pj: 3 P, and, as in the proof of Theorem 7.10, construct a Hausdorff space Y = A u {<} with only one nonisolated point 5, where 5 is a new point and the base at 5 is the sets (1 E A 11 2 p } u { <}, p E A. Since each A E A is isolated in Y, so is in B( Y); this follows from the fact that Y is C*-embedded in B ( Y ) . Hence U{Pi x , I11 E A} is a t-open set of X x B( Y ) and its closure A = CL,,,,(U{P,. x 111E A>>
is disjoint from a t-closed set B = X x <. Therefore, by the w-normality of X x B ( Y ) , the binary open cover 92 = {X x B ( Y ) - A , X’x B ( Y ) - B} is normal, so that by Lemma 8.7 there is a locally finite cozero-set cover 4‘2 = { U, I a E Q} of Xsuch that, for a suitable collection {Vm I c1 E Q} of finite ( the collection { U, x V I V E Y,, a E Q} is an open cozero-set covers of /IY), refinement of Y. Since 9is weakly Cauchy, there is /? E Cl such that { U p >u 9 has the finite intersection property. Let V be an element of Y, with E V . Then we have (Us x V ) n B # 8, which implies that (Us x V ) n A = 8. But this contradicts the fact that { U p } u 9 has the finite intersection property. Thus we have n{Fj.1 1 E A} # 8,completing the proof. 0
<
Theorems 8.5 and 8.6 yield the following lemmas. 8.8. Corollary. For a space X the following conditions are equivalent.
(I)
X is w-paracompact.
(2)
X x Y is w-normalfor any w-compact space Y
(3)
X x /J(Y) is w-normalfor any Hausdor-space Y with at most one nonisolated point.
8.9. Corollary. For a Tychonoj” space X the following conditions are equivalent.
(1)
X is paracompact.
T. Ishii
234
(2)
X x Y is w-normalfor any w-compact space Y.
(3)
X x B ( Y ) is normal for any Hausdorfspace Y with at most one nonisolated point.
9. Rectangular products and w-paracompact spaces
The concept of rectangular products was introduced by Pasynkov [ 19751 as a generalization of Fproducts due to Nagata [1967]. 9.1. Definition. The product X x Y of two spaces X and Y is called rect-
angular if any finite cozero-set cover of X x Y is refined by a a-locally finite open cover of X x Y which consists of cozero-set rectangles.
As for rectangular products, Pasynkov [1975] proved an interesting result that, if X x Y is a rectangular product of Tychonoff spaces X and Y, then the inequality dim(X x Y) f dim X
+ dim Y
holds, where dim X means the covering dimension of a space X (as for the covering dimension, see, for example, Nagata [19831). After that, Hoshina and Morita [1980] improved his result to the case of X and Y being general spaces by showing the following theorem.
9.2. Theorem. The product X x Y of spaces X and Y is rectangular if and only i f r ( X x Y ) = 7 ( X ) x r ( Y ) and 7 ( X ) x r ( Y ) is rectangular. Proof. Since the “if”-part easily follows from Proposition 1.2, we prove the “only-if”-part. Assume that X x Y is rectangular. To show that 7 ( X x Y) = 7 ( X ) x 7 ( Y ) , let G be a cozero-set of X x Y with ( x o , y o ) E G. Taking a zero-set F of X x Y such that (x,,, y o ) c F c G, Y = {G,X - F} is a binary cozero-set cover of X x Y. Hence there exists a a-locally finite open refinement W of Y consisting of cozero-set rectangles. Let U x V be a cozero-set rectangle of W with ( x o , y o ) E U x V. Then we have U x V c G, so that 7 ( X x Y) = r ( X ) x r ( Y ) by Theorem 2.4. We next prove that 7 ( X ) x 7( Y) is rectangular. Let Q = { Ql i = 1, . . . , n} be a finite cozero-set cover of 7 ( X ) x 7 ( Y ) ( = T ( X x Y)). Since r i ; y(4Y) = y ( U , )I i = 1, . . . , n} is a finite cozero-set cover of X x Y, by the rectangularity of X x Y it is refined by a o-locally finite open cover -Ir = U/”=,5
{T~A
235
The Tychonofl Funcror and Related Topics
<
of X x Y consisting of cozero-set rectangles, where for each j , = { U,, x y a l a E Q j } is a locally finite collection of cozero-set rectangles of X x Y. If we take a nonempty cozero-set rectangle G,,, x Hi,, and a zero-set rectangle F,, x Kja in X x Y such that Gj, x Hja c
4ax
Kju c
qax ya,
then it is shown that {q,x 4 a I a E Q j } is t-locally finite in X x Y. In fact, noting that, for any subset Sofa,, (J{F,, x 4, I a E S } is a zero-set of X x Y by Lemma 1.8, the set W(z) = X x Y -
U{F,, x qalz E X
x Y -
F,,
x Kja,a E
a,},
which is defined for each point z E X x Y, is a cozero-set neighborhood of z and intersects only finitely many members of {Gja x HjuI a E Q j } . Thus {Gj, x HjaI a E a,} is t-locally finite in X x Y. Now, for each j and a, let us take a sequence of nonempty cosero-set rectangles {Gjakx HjakI k E N} and a sequence of zero-set rectangles (4.k X 4 . k I k E N} in X X Y such that Gjak
4ak
4 a k
4 a k
qa
vu,
and
qaX ya
= u{Gjak X
4 . k
Ik E
N}.
Then by the above argument, q k = {Gjmkx HjukI a E a,}is r-locally finite in X x Y and hence u { q k lj, k E N} is a o-t-locally finite open refinement of ti:,,(a).Thus admits a o-locally finite open refinement y($k) lj, k E N} consisting of cozero-set rectangles t,(Gjak) x tY(Hjak),a E aj,j,k E N, of t ( X ) x t ( Y ) , completing the proof. 0
u{t,,
9.3. Theorem. A space X i s w-paracompact ifand only i f X x Y is rectangular for any Hausdorflspace Y with at most one nonisolated point. Proof. Necessity. Let X be a w-paracompact space and Y a Hausdorff space with at most one nonisolated point. Then we can show that any cozero-set cover of X x Y admits a locally finite open refinement consisting of cozeroset rectangles, which implies that X x Y is rectangular. Let Y = { G; I 1E A} be any cozero-set cover of X x Y. In case Y has no nonisolated point, we have nothing to prove, and so we may assume that Y has only one nonisolated point 5. For each point (x, 5 ) E X x 5, take 1(x) E A with (x, 5 ) E Gi(,). Since r ( X x Y) = r ( X ) x Y by Theorem 7.9, there exists a cozero-set rectangle U, x V, such that (x,
5 ) E ux
x
v, = Gl(,,.
236
T. Ishii
By the paracompactness of T ( X )the cozero-set cover { U, I x E X} of Xadmits a locally finite cozero-set refinement { H , I a E R}. If for each a E Q we take x(a) E X such that H , c Uxca,.then YP = {Ha x I a E R} is a locally finite collection of cozero-set rectangles of X x Y covering X x <. Let {F, I a E R} be a locally finite zero-set cover of X such that F, c H , for any a E R. Since each V, is open and closed in Y , {F, x K(,)I a E R} is a locally finite collection of zero-set rectangles such that F, x P‘,(,) c Ha x <(,), and hence by Lemma 1.8, F = u { F , x K(.) I a E R} is a zero-set of X x Y. For each point y # in Y , let us put W ( y ) = (X x Y - F ) n (X x y). Since W( y) is a cozero-set of a w-paracompact space X x y, W( y) is also w-paracompact by Proposition 7.6. Therefore there exists a locally finite cozero-set cover dy= {A, I y E r v }of the space W(y ) which refines a cozero-set cover {G, n W ( y ) l AE A} of W ( y ) , and 91y = {A, n W ( y ) ( yE r,,}is a locally finite collection of cozero-sets of X x y covering W ( y ) . Finally let us put
<
3-
= Jf
u
U{qY z
r, y
E
Y}.
Then it holds that 3- is a locally finite refinement of 9 consisting of cozero-set rectangles of X x Y. In fact, it is clear that X refines Y and X is locally finite at every point (x, y) E X x y , y # 5. For each point (x, 5 ) E X x 5, take an open neighborhood N ( x ) of x which intersects only finitely many members of { H , I a E R}, say H a , , . . . , Ha,, and put V
=
n{V,(,,)Ii= 1 , . . . , n}.
Then it is easily seen that an open neighborhood N ( x ) x Vof (x, 5 ) intersects only finitely many members of X . This proves the “only-if”-part. Suficiency. Assume that X x Y is rectangular for any Hausdorff space Y with at most one nonisolated point. Then by Theorem 9.2 it holds that r ( X x Y ) = 7 ( X ) x Y. Hence by Theorem 7.8 it remains to show that 7 ( X ) is paracompact. Now let 92 = { U i 12 E A } be any cozero-set cover of X , where we may assume that 92 has no finite subcover. For each point x of X , let us take 2(x) E A with x E UiCx,. Then there is a zero-set Fi(,) and a cozero-set Ei(,) in X such that x E Ei(,, c FicX,c Uicx,.Let r be the set of all nonempty finite subsets of X , and for y = {x,, . . . , x,} E r let us put
V,
=
U{Ui(.r,)li= 1 , . . . ,n},
K,
=
u { F i ( x , ) p= 1, . . . , n}.
Then V, is a cozero-set and K, a zero-set in X with 4 c V,. We introduce an order in r such that y < 6 means y c 6 and construct a Hausdorff space
The Tychonoff Functor and Relaled Topics
231
Y = r u { <} with only one nonisolated point 5 , where 5 is a new point and the base at is the sets W, = {a E rib 2 y} u {t},y E r. n ( ( X - 4) x y) = 8 for x E X and y E I?, { ( X - 4) x Since (E,,,, x y I y E r}is a discrete collection of cozero-sets of X x Y. Hence by Lemma 1.8, Z = U { ( X - 5 ) x y I y E r} is a zero-set of X x Y. Letting P = U { ( X - K , ) x y I y E r},by the rectangularity of X x Y , a binary cozero-set cover {P,X x Y - Z } admits a a-locally finite open refinement Q = Up,9?, consisting of cozero-set rectangles, where 9?, = { L , x MmjI LY E SZ,} is a locally finite collection of cozero-set rectangles. Let us put
<
9
= {L,l5€Maj,LYERj,~E~}.
Then 9 is a a-locally finite cozero-set cover of X and each L, E 9 is contained in some V, = U{Uj.(x,)li = 1, . . . , n } ; this follows from the fact that ( L , x MaJ)n 2 = 8 for L, E 2’. Therefore, if we consider a cozero-set cover WQJ = { L , n U,,(v,) I i = 1, . . . , n } of L,, then W = U {WaJI LY E a,,j E N} is a a-locally finite cozero-set refinement of 42. Thus T ( X )is paracompact, completing the proof. 0 As a corollary of Theorem 9.2, we obtain a result due to Hoshina and
Morita [1980]. 9.4. Corollary. A Tychonof space X is paracompact if and only if X x Y is
rectangular f o r any Hausdor-space Y with at most one nonisolated point.
The following theorem gives a characterization of a space X such that X x Y is rectangular for any space (or any Tychonoff space) Y. To prove it, we make use of a method employed by Ohta [1978] in proving that, for a Tychonoff space X , X x Y is C-embedded in X x p ( Y ) for every Tychonoff space Y if and only if X is locally compact, where p ( Y ) is the topological completion of a Tychonoff space Y , that is, the completion of Y with respect to the finest uniformity consisting of all normal open covers of Y. A subspace S of a space X i s called C-embedded in X if every continuous map f : S -, R has a continuous extension3: X -, R (as for the uniformity, see, for example, Nagata [1985]). Recall that for any normal open cover 42 of a Tychonoff space Y there is a normal open cover Y of p( Y ) such that Y n Y = { V n YI V E Y } refines 42; equivalently, any normal open cover of Y is extended to a normal open cover of ,u( Y ) .
T. Ishii
238
9.5. Theorem. For a space X the following conditions are equivalent. (1)
X is w-paracompact (or t ( X ) is paracompact) and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(2)
X x Y is rectangular for any space Y.
(3)
X x Y is rectangular for any Tychonofspace Y.
To prove the above theorem, we need the following lemmas.
9.6. Lemma. I f a space X is w-compact, then X x Y is rectangular for any space Y. More precisely, any normal open cover of X x Y admits a locallyfinite open refinement consisting of cozero-set rectangles of X x Y. The above lemma is an immediate consequence of Lemma 8.7. The following lemma is concerned with a regular cardinal. An initial ordinal I is called regular if there is no a < I which is cofinal with I , and a cardinal n is called regular if the initial ordinal I with I I I = n is regular (see, for example, Engelking [1977]). For a regular ordinal a,,we consider the space W(w,) = { I l I < w,} with the interval topology.
9.7. Lemma. Let w, be a regular ordinal. Then the space W(w,) has the following properties: (1) Any locally Jinite open cover (JiY of W(w,) has a finite subcover { U , , . . . , U,,},and there is yo < w, such that { y E W(w,) I y 2 y o } c U, for some k , 1 < k < n. (2) For every continuous map f:W(o,) + Z there is y , < w, such that f(r) = f ( y 1 ) f o r any Y 2 YI. Since the lemma above is similarly proved as in the case of W ( w , ) ,the proof is left to the reader.
Proof of Theorem 9.5. (1)+(2). We first notice that in (1) the w-paracompactness of X is replaced by the paracompactness of t(X)by Proposition 7.8. Let X be a space satisfying the condition (1). Then by Proposition 7.8 there exists a locally finite cozero-set cover { G , I a E R} of X such that G, is w-compact for each a E R. Let Y be an arbitrary space and %! = { U,I i = 1, . . . , n } a finite cozero-set cover of X x Y. For each a E R and i = 1 , . . . , n, let Hi,= n (G,x Y ) .Then Xu = {Z&I i = 1, . . . , n} is a finite cozero-set cover of G, x Y. Hence by Lemma 9.6 there exists a locally
239
The TychonoffFunctor and Related Topics
finite open cover X, consisting of cozero-set rectangles of G, x Y which refines X,. If P x Q E X,, then P n G, is a cozero-set of X by Lemma 3.9. Therefore X, n (G, x Y) = {(P x Q)n (G, x Y))P x Q E is a locally finite collection of cozero-set rectangles of X x Y which covers G, x Y, so that U{Xan (C, x Y) I a E a}is a locally finite open cover of X x Y consisting of cozero-set rectangles of X x Y which refines 42. Thus X x Y is rectangular. (2)+(3). This is obvious. (3)+(1). Assume that X x Y is rectangular for any Tychonoff space Y. Then by Theorem 9.3, X is w-paracompact. Now suppose that there is a point x, of X such that for any cozero-set neighborhood U of x,, 0 is not w-compact. Let { U, I I E A] be a base of cozero-set neighborhoods of xo. Then by Lemma 4.2, for any I E A there exists a Hausdorff space Y, with only one nonisolated point 5, and a continuous map hi : X x Y, + Z such that
x}
(X x t i ) u ( ( X
h,(z) = 1 for z
E
h,'(O) n ( V , x p )
z8
-
U , ) x Y,),
and forp E Yi
- ti,
where each Y, is an ordered set and the base at ti is the sets { q E Yi - t i 1 4 2 P} u ti},^^ Yi - ti* Let n be a regular cardinal greater than sup{I Y, 1 11 E A} and w, the initial ordinal of n. For each I E A, put Z ( I , 1) = W ( o , 1) x Y, and let Z ( I , 2) be the copies of W ( o , + 1) x W(wo + 1). By identifying a point (y, 5,) E Z(A, 1) with a point ( y , wo)E Z(A, 2) for each y < w,, we obtain a quotient space Z, and a quotient map cpi from the topological sum Z ( I , 1) u Z(A, 2) to Z,. Let Z be a quotient space obtained from the spaces Z,, I E A, by identifyingevery point cpi(wa,B ) E Z,, I E A, for each B 6 wo, where (a,, B) E Z ( I , 2), and let i,b: UZ, -, Z be a quotient map. Put e, = cp,, z,, = ei(o,,ti) for A E A and Y = Z - z,,. Then Y is clearly a Tychonoff space and it is shown that Z c p(Y). To see this, it suffices to show that for any locally finite cozero-set cover Y of Y there exists a locally finite cozero-set cover W of 2 such that W n Y = { w n Y l W E W } r e f i n e s V . L e V = {V,IoEX}beanylocally finite cozero-set cover of Y. For I E A, y < a,,p E Y;. - ti.and n < wo,let us define a subset
+
$ 0
A ( 5 71 P) = ((6, q ) l y < 6 G
ma,
P < 4 G ti} -
(ma,
ti)
of Z ( I , 1) and a subset
B(I, Y, n)
=
((7,
P)IY < 6 G
ma,
n < B G wo} -
(ma,
wo>
T. Ishii
240
of Z(A, 2). We denote by C(A, y(A), p(A), n(A)) the set Qi(A(A,y(A), p(A)) u B(A, y(A), n(A)) in Y. Then by Lemma 9.7 and the fact that n > sup{ 1 Y, I 11 E A}, for a fixed A, E A there is VboE V such that C@O, Y ( A 0 h P@O),
n(A0))
KO
for some y(&) < o,, p(Ao) E Y;, - tC and n(A,) < w,, from which it follows that for any A E A, there is y ( A ) < o,and p(A) E Y, - 5;. such that c ( A 7
Y ( 4 , P ( 4 , n(A0)) = KO.
W, = U{C@,y ( 4 , p ( 4 , n(AoNIA E A} and W , = V, n (Y - W,), Z. Then W, is an open and closed set of Y contained in V,,, so that V' = { W,, W,la E Z} is a locally finite cozero-set refinement of V . If we consider the set Po= W, u {z,}, @, is open and closed in Z and W = {I&, W, 1 0 E Z) is a locally finite cozero-set cover of 2 such that W n Y refines V , showing that Z c p(Y). Now let us define a map h : X x Y -, Z such that Let
CJ E
h ( x , v)
=
hi(&
h(x, y )
=
I
and
4 for Y
for y
E
=
@;.(Y, 4 E e;,(Z(A,1))
e,(Z(A,2))
-
-
zo,
z,.
Then it is obvious that h is continuous. Hence we can consider a binary cozero-set cover {G, H } of X x Y such that G
=
{( x , Y ) l h ( X , Y ) > 01 and H
=
I<.,
y)lh(x, y ) <
f}.
Since by the assumption X x Y is rectangular, there is a a-locally finite open cover 9 = UE,gi of X x Y consisting of cozero-set rectangles which refines the binary cover {G, H } of X x Y, where gi = {Gsi x Hail6 E Ai} is locally finite in X x Y. Since S ( x o ) = {HaiIx,E Gai, 6 E Ai, i E N} is a normal open cover of Y, there is a normal open cover S of p ( Y ) such that S n Y refines S ( x o ) . For each 6 and i, let H$
=
~ ( y-) Clp(y)(Y - Hai).
Then we have (x,,, z o ) E GEkx HZ for some k E N and E E A k .Take a cozeroset neighborhood Up of x , with Up c GEkr p E A. Then the open set H 2 of p( Y) with zo E HZ contains a set C ( p , y(p), p ( p ) , n(p)) for some y ( p ) < o,, p ( p ) E Y,, - 5, and n ( p ) < w,. Since h ( x , y ) = 1 for any y = e,,(y, 5,) E Q p ( Z ( p ,l)), we have GEkx H e k c G. On the other hand, it holds that h-'(Oj n (upx e P ( A ( p ,YW P(P)))
+ 0,
which is a contradiction. This completes the proof.
0
The Tychonoff Functor and Related Topics
24 1
As a corollary of Theorem 9.5, we obtain a theorem of Hoshina and Morita [1980].
9.8. Corollary. For a Tychonof space X the following conditions are equivalent. ( I ) X is locally compact and paracompact. (2) X x Y is rectangular for any space Y . ( 3 ) X x Y is rectangular for any Tychonofspace Y . 9.9. Remark. Concerning the relationship between rectangular products and topological completions, Hoshina and Morita [ 19801 proved that (a) for any Tychonoff spaces X and Y , X x Y is rectangular if and only if p ( X x Y ) = p(X)x p ( Y ) and p ( X ) x p ( Y ) is rectangular. On the other hand, Ohta [I9781 proved that (b) for a Tychonoff space X , p ( X x Y ) = p(X)x p( Y ) for any Tychonoff space Y if and only if X is locally compact. Using (a), (b) and Corollary 9.4, Hoshina and Morita [I9801 obtained Corollary 9.8. For further information about rectangular products, see Nagata [ 19671, Pasynkov [1975, 19801, Hoshina and Morita [1980], Ohta [1981, 19831, K. Tamano [1982] and others.
References Corson, H. H. [I9581 The determination of paracompactness by uniformities, Amer. J. Math. 80, 185-190. Engelking, R. [I9771 General Topology (Polish Scientific Publishers, Warszawa). Frolik, Z. [I9611 Applications of complete families of continuous functions to the theory of Q-spaces, Czech. Math. J . 11, 115-133. Glicksberg, I. [I9591 Stone-Cech compactifications of products, Trans. Amer. Math. SOC.90, 369-382. Herrlich, H. [I9651 Wann sind alle stetigen Abbildungen in Y konstant?, Math. Zeitschr. 90, 152-154. Hewitt, E. [I9461 On two problems of Urysohn, Ann. Math. 47, 503-509. Hoshina, T. and K. Morita [I9801 On rectangular products of topological spaces, Topology Appl. 11, 47-57. Ishii, T. [1980a] On the Tychonoff functor and w-compactness, Topology Appl. 11, 175-187.
242
T. Ishii
[1980b] Some results on w-compact spaces, Uspehi Mat. Nauk 35,61-66; Russian Math. Surveys 35, 71-77. [ 19841 On w-paracompactness, Math. Japonica 29, 847-858.
Isiwata, T. [I9671 Mappings and spaces, Pacific J. Math. 20, 455480. [I9691 2-mappings and C*-embeddings, Proc. Japan Acad. 45, 889-893. Mibu, Y. (19441 On Baire functions on infinite product spaces, Proc. Imp. Acad. Tokyo 20, 661-663. Morita, K. [I9701 Topological completions and M-spaces, Sci. Rep. Tokyo Kyoiku Daigaku 10, 271-288. [I9751 Cech cohomology and covering dimension for topological spaces, Fund. Math. 87, 31-52. [I9801 Dimension of general spaces, Surveys in General Topology (Academic Press, New York). Nagata, J. [I9671 Product theorems in dimension theory I, Bull. Acud. Polon. Sci.Ser. Sci. Math. Astronom. Phys. 15, 439-448. [ 19831 Modern Dimension Theory (Herdermann Verlag, Berlin). [ 19851 Modern General Topology (North-Holland, Amsterdam). Noble, N. [1969a] Products with closed projections, Trans. Amer. Math. Soc. 140, 381-391. [1969b] A note on z-closed projections, Proc. Amer. Math. SOC.23, 73-76. Novak, J. [I9481 A regular space on which every continuous function is constant, fasopis P k t . Mat. F ~ s73, . 58-68. Ohta, H. [I9781 Local compactness and Hewitt real compactifications of products, Proc. Amer. Math. SOC. 69, 339-343. [I9811 On normal non-rectangular products, ,Quart. J. Math. Oxford 32, 339-344. [I9831 Rectangular products with a metric factor, Questions Answers Gen. Topology 1, 57-61. Oka, S. [I9781 The TychonotT functor and product spaces, Proc. Japan Acad. 54, 97-100. Pasynkov, B. A. [I9751 On the dimension of rectangular products, Soviet Math. Dokl. 16, 344-347. [I9801 On the dimension of topological products and limits of inverse sequences, Dokl. Acad. Nauk. 254, 596-601. Pupier, R. [ 19691 La completion universelle d’un produit d’espaces completement reguliers, Publ. Dept. Math. Lyon 6, 75-84. Stone, A. H. [I9481 Paracompactness of product spaces, BUN. Amer. Math. SOC.54, 977-982. Tamano, H. [I9601 On paracompactness, Pacific J. Math. 10, 1043-1047. Tamano, K [I9821 A note on E. Michael’s example and rectangular products, J. Math. SOC.Japan 34, 187-190.
The Tychonoff Functor and Relafed Topics Tychonoff, A. [I9291 Uber die topologische Erweiterung von Raumen, Mufh. Ann. 102, 544-561. Urysohn, P. [I9251 Uber die Machtigkeit der zusammenhangenden Mengen, Math. Ann. 94, 262-295.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 7
METRIZATION I
Jun-iti NAGATA Symposium of General Topology, Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka, 543 Japan
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metrizability in terms of g-functions. . . . . . . . . . . . . . . . . . . . . Metrizability in terms of base of point-finite rank . . . . . . . . . . . . . . . Metrization in terms of hereditary property . . . . . . . . . . . . . . . . . Some other aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245 245 251 259 265 271
Introduction The primary purpose of this chapter is to supplement Sections VI.1 and VII.3 of J. Nagata’s book [1985]. That is, we are going to discuss here some of the significant results of metrization theory that were not treated in detail there. Note that all spaces are at least T , throughout this chapter, and X denotes a space. Also note that z ( X ) denotes the topology of a space X,and N the set of all natural numbers. Nagata’s book [1985] will be quoted frequently with the abbreviation “a]. Thus, e.g., p a , Theorem VI.21 means Theorem VI.2 of that book.
1. Metrizability in terms of g-functions 1.1 Definition. A map g : N x X .+ T ( X )is called a g-function if x E g(n, x) for every x E X and n E N. Sometimes we may assume simply that g ( n , x) is a (not necessarily open) nbd (= neighborhood) of x . In that case we call g a g-function (n.n.0.).
246
J . Nagata
1.2. Theorem (Nagata [1986/87]). A T,-space X is metrizable ifl X has a g-function (n.n.0.) such that (i) for every x E X and its open nbd P, there is an n E N for which
x 4 [ U M n , Y ) lY E X - P}I-, (ii) for any subset Y of X
=
U{g(n9 Y ) l Y
E
Y}.
Proof. Since necessity of the condition is obvious, we prove only its sufficiency. For each (n, m) E N x N we define Un,(x) and Wnm(x)as follows: If x 4 [U{g(m,4 I z E { Y I x $ g(n9y>}-}I- holds, then Unrn(X)
X
=
{ Y I x $ g ( n ,Y > } - ,
-
(1.1)
Wnm(x) = X - [U{g(niz)Iz E X - Unrn(x)}I-.
(1.2)
Otherwise, Unm(X) = Wnm(X)=
x.
By condition (ii), Unm(x) is an open nbd of X. It is obvious that Wnm(x) is also an open nbd of x. Now suppose that x, is a fixed point of X and let Y
{ x E XIX, $
=
z
U,,(X)},
=
{ x E Xlx,
E
W,,(X)}.
Then, for each x E Y, xo E X - Unm(x).Hence it follows from ( I .2) that g(m, x,) n Wnm(x)= Suppose x
E
0
for every x E Y .
2 and Wnm(x)# X ; then x,
W X O )
=
Wnm(x).Put
E
x - { Z I X O $ g(m, .I>-.
Note that, by (ii), W ( x o )is an open nbd of xo. Further we claim that
x-
W(X0) = { z I xo 4 g(m, 4 3
To prove the inclusion
3,
g(m, P) c X -
-
{ Y I X 4 g(n, Y>>let p
E
Wnm(x)
=
X - Unm(x).
X - Un,(x). Then
$ xo
follows from (1.2), and thus P E
{ZIXo4g(m,z)} = {zIxO$g(m9z)l-?
proving the inclusion. Therefore, W ( x o )c Un,(x) for every x
E
Z.
(1.3)
Metrization I
241
Now, put ~m(X0) = g(m, xo)
WXO)
Kn(X0).
Then Vm(x,) is a nbd ofxosuch that xo 4 U , ( x )implies V,,(xo) n Vm(x) = 8 by (1.3), and x, E V,(x) implies V,,,,,(xo) c U,,,,,(x) by (1.4). Finally, assume that P is an open nbd of x E X.Then, by (i), for some n E N, x 4 [U{g(n,Y ) l Y E X - PH-. (1.5) By (i) and (ii), there is an m E N for which x 4 [U{g(m,4 1 2 E { A x 4 g k Y>}-Il-. Then, by (l.l), Um(x) = X - { y l x $ g ( n , y ) } - c P, because y $ P implies g(n, y) # x by (lS), and thus Y E {YlX 4 g(n, y > > - = X - U"rn(X).
This proves that {U,(x) In, m E N} is a nbd base of x. Therefore, by p a , Theorem VI.21, X is metrizable. 0 Theorem 1.2, like ma, Theorem VI.21, implies many metrization theorems as its corollaries, which will be seen in the following. 1.3. Corollary (Borges [1983]). A T,-space X is metrizable i f l X has a g-jiunction (n.n.0.) satisfying (i) for each x E X and its open nbd P there is an n E N for which
(ii) {g(n, x ) I x E X } is closure-preserving for each n.
Proof. Necessity of the condition is almost obvious. To prove sufficiency, simply note that g(n, x) in the corollary satisfies the condition of Theorem 1.2. 0
1.4. Corollary. A T,-space X is metrizable ir X has a g-function (n.n.0.) satisfying (i) for every x E X and its open nbd P there is an n E N for which
x 4 [U:g(n, Y ) I Y (ii) y E g(n, x) implies x
E
x - PI]-3
E g(n,
y).
248
J . Nagata
Proof. Necessity is obvious. Sufficiency follows directly from Theorem 1.2.
0 It is well-known [Na, Theorem VI.251 that a T,-space X i s M 3 iff X has a g-function satisfying (i). Thus the symmetry (ii) of g represents the difference between M3-spaces and metrizable spaces.
1.5. Definition. A map cp: X x 9 ( X ) --., [0, co), where 9 ( X ) denotes the collection of all closed subsets of X , is called an annihilator provided cp(x, F ) = 0 holds iff x E F. rp is called monotone if ~ ( xF, ) < cp(x, G ) whenever F 13 G. cp is called continuous if cp(x, F ) is continuous in the variable x for every fixed F. cp is additive if, for any 9’ c 9(X), cp(x,
US’)
= inf{cp(x, F ) 1 F E S’}.(’)
1.6. Corollary. A TI-space X is metrizable ifl X has a monotone continuous c S ( X ) , inf{cp(x, F ) IF E S’] is a conannihilator cp such that f o r any 9’ tinuous function of x.
Proof. To see necessity, put
cp(x, F )
=
d ( x , J‘) = inf{e(x, Y ) l Y E F } ,
where e is a metric of X . To prove sufficiency, put g(n, x) = ( y E Xlcp(y, {x}) < l h ) , n
E
Nx E X.
Then it is easy to see that g is a g-function satisfying (i), (ii) of Theorem 1.2. Thus X i s metrizable. 0 1.7. Corollary (Nagata [1957]). A T,-space X is metrizable i f l X has a continuous annihilator rp such that for any 8’ c S ( X ) , inf{rp(x, F ) I F E S’} and sup{cp(x, F ) I F E 9’) are continuous functions ofx. Proof. Necessity is obvious. Assume X has an annihilator that satisfies the said condition. Then we define II/: X x 9 ( X ) + [0, co) by
$(x, F )
=
SUP{~(X F’) , I F c F‘
E
9(X)}.
’ More generally, cp can be defined for any collection Y(X)of closed subsets of X,and then for Y(X).But in the present section we consider only annihilators for the collection of all closed sets. See Suzuki, Tamano and Tanaka [1987] for annihilators and metrization. cp is called an annihilator
Metrization
I
249
Then $ is a monotone continuous annihilator. Define g: N x X s(n9 x ) = I Y
E
XI
$(Y9
+
z ( X ) by
(4) < l/n}.
Then g is a g-function. Suppose P is an open nbd of x. Then, for each z E X - P,$(x’, P‘) < +(x’, { z } )for all x’ E X . Suppose $ ( x , P‘) > l/n. Then there is a nbd U of x such that +(x’, P‘) > I/n for all x’ E U . Thus U n g(n, z ) = x
8 for all z E X
4 [U{g(nt ,912 E
-
P ; that is,
P ‘ l l - 1
which is condition (i) of Theorem 1.2. Suppose x 4 U{g(n. z ) I z E Y } . Then $ ( x , { z } ) 2 l / n for all z E Y . Thus for each z E Y we can select F, E 9 ( X ) such that z E & and ~ ( x&) , > l/(n 1). Then
+
inf{q(x, & ) l z
E
Y}2
1
n + 1’
Thus x has a nbd U such that inf{cp(x’, & ) J z E Y } > 0 for all x’ E U. Therefore, U n Y =
8 proving (ii) of Theorem
1.2. Hence X is metrizable.
0 1.8. Corollary (Isiwata [1987]). A T,-space X is metrizable fz continuous additive annihilutor.2
Proof. This follows directly from Corollary 1.6.
X has a
0
1.9. Corollary (Zenor [1975/76]). A T,-space X is metrizable i f X has a monotone annihilator cp such that cp(x, { y } ) is continuous in the variable x for eachjxed y E X and also in the variable y for eachjxed x E X .
Proof. Put
’T. lsiwata [ 19871 studied various metrizability conditions in terms of annihilator for the collection of regular closed sets.
250
J . Nagata
Then g is a g-function. It is easy to see that g satisfies (i) and (ii) of Corollary 1.4. Thus X is metrizable. 0 1.10. Definition. A collection Q = {VelaE A} of subsets of X is called hereditarily closure-preserving if, for any choice of subsets V, of U, for each a E A, { 6 I a E A} is closure-preserving.
The following is a direct generalization of Nagata-Smirnov’s Theorem m a , Theorem VI.31. 1.11. Corollary (Burke, Engelking and Lutzer [1975]). A regular space X i s metrizable i f it has a o-hereditarily closure-preserving base 9 = 9,,,i.e., 9 is a base and each 9,,is hereditarily closure-preserving.
u,“=,
Proof. Since “only-if” is obvious, we shall prove “if”. Suppose x is a nonisolated point of X. Then, for each n E N, Q,,(x) = { U E enI x E U } is finite. Because, otherwise there are distinct UI, U,,. . . , E Q,,(x). Note that { x } = for some open sets i E N. Let ct: = Ui A i E N. Then, for each i E N, x $ - K+,and M( c Ui, while
n,:
v,
i= I
v,
w+,
i= I
This contradicts that a,, is hereditarily closure-preserving. Thus Q,,(x) is finite. Now we define a g-function (n.n.0.) g by
To prove (i) of Theorem 1.2, suppose P is an open nbd of x . Then, for some n and U E Qn, x E U c 8 c P. Thus
u{
To prove (ii), let x $ g(n, y ) I y E Y } for a subset Y of X. Then, for every y E Y, x $ g(n, y ) , which implies that, for some U E 9 , , ( x ) , 0 # y . Thus, if x is not an isolated point, U 1 U E Q,,(x)} is an open nbd of x that is disjoint from Y. If x is an isolated point, { x } is a nbd of x disjoint from Y. Hence, x $ y, proving (ii). Therefore, by Theorem 1.2, X is metrizable.
n{
25 1
Metrization I
1.12. Definition. A collection 2.d of pairs of disjoint subsets of X is said to separate two sets U and Vif for any pair (x, y) of points with x E U and y E V there is a pair (B, 8)E W such that x E B and y E 8.8) 1.13. Corollary (Hung [1980]). A TI-space X is metrizable fi there is a collection {(Bt,B,)1 5 < a } of pairs of disjoint subsets of X , where a is aJixed ordinal number of countable coJinality4 that satisJies (i)for each 8 < a, n { X - ~ , ( x B,, E 5 < 8) is a n6d of each X E X , (ii) for each x E X and its open nbd U there is an open nbd V of x such that for some 3/ < a, { ( B e ,fit) 15 < 8) separates U and V.
Proof. To see necessity of the condition, note that {(S,/,(x), S2/,(x))I x E X , i E N} obviously satisfies the condition, where S,(x) denotes the c-nbd of x. To see sufficiency, define a g-function (n.n.0.) by g(n, x) =
n{x - B t \ x E B,, 5
< a,},
n E N, x
E
X.
Suppose U is an open nbd of x. Then there is an open nbd V of x, and an n E N such that { ( B t ,Bt) I < a,} separates X - U and V. Now, for each x E V and y E X - U,there is a 5 < a, such that x E Bt and y E Bt. Thus x 4 g(n, y). Hence, V n g(n, y) = 8, and hence
<
proving (i) of Corollary 1.4. Now, suppose x # g(n, y). Then there is a pair ( B c ,Bt) with 5 < a, satisfying y E B,, x E Bt. Hence g(n, x) $ y proving (ii) of Corollary 1.4, and accordingly X is metrizable. 0 This theorem looks somewhat similar to [Na, Theorem VI.21, but some theorems could be derived from the present theorem more easily. For example, one can see how easily Nagata-Smirnov's Theorem follows from the present one.
2. Metrizability in terms of base of point-finite rank The concept of rank (defined in the following) is especially useful in dimension theory since it has been proved that a metric space X has dim < n 'Note that in the following argument (B, B ) is not an ordered pair. Thus x E B, y E that either of the members of the pair contains x and the other one contains y. 4There is a sequence a, < a2 < . . . of ordinal numbers with sup a, = a.
B means
252
J . Nagata
iff i t has a base of rank < n (see Nagata [1983]). If metrizability of Xis not assumed, then how close is a space with a base of finite rank to a metrizable space? This question was studied by several mathematicians and especially by A. V. Arhangel'skii [1963], G . Gruenhage and P. Zenor [I9751 and G. Gruenhage and P. Nyikos [1978]. In the present section we are going to introduce Gruenhage-Nyikos' theory in this aspect. 2.1. Definition. Let be a collection of subsets of a space X . Two members U , V of @ are called independent if U Q V and V Q U . g is called independent if any distinct two members of 93 are independent. Let x E X ; then we
put g ( x ) = { B E 93 I x E B } . Now rank,@ is the maximum cardinality of the independent subcollection of g ( x ) , and rank 9 = supfrank,gIx E X } . If r a n k y g < 00 for each x E X , then @ is of point-Jinite rank. Moreover, a subcollection g'of 9t7 is called a chain in 9 if any two sets chosen from 9' are dependent (i.e., not independent). Further we denote by & the collection of all sums of maximal chains in B. In the forthcoming discussions we need the following lemma. 2.2. Lemma (Dilworth [ 19501). If P is a partially ordered set such that every subset of n -k 1 elements of P is dependent while at least one subset of n elements is independent, then P can be expressed as the sum of n disjoint totally ordered sets.
Proof. Omitted. See the original paper.
0
2.3. Corollary. /f rank.,B < n for a collection 99 of subsets of X and goc B ( x ) , then gocan be expressed as the sum of at most n chains. (Note that n denotes a natural number in the present section.) 2.4. Proposition. Suppose rank,,g
< n and goc
@(x). Then lBOl< n.
Proof. We give a rather sketchy proof. Suppose U , , . . . , gkare maximal chains in gosuch that UU,, . . . , UUk are distinct elements of d$.Select the maximal elements (sets) from {UU,, . . . , UVk} to denote them by { Uq,, . . . , U q k ,}. Then select maximal elements from { uqk,+, , . . . , Ugk} to denote them by {uqk,+,, . . . , U q k , } . Continue the same process to decompose {UU,, . . . , u U k } to the sum of {UV,, . . . , (JUkl},{ U U k , + , , . . . , UUk2},. . . , {UUkk,+,, . . . , UCe,}. For each pair U i , W,( 1 < i, j < k )
253
Merrization I
satisfying u W l
+ u'%;we select a point p,,
E
UW,
W,' = (B E W, I B 3 p,, for all j with
-
UW,
UT. For each i we put $
UW,},
1
< i < k.
Now, for each i with 1 < i < k , , we fix B, E W,'. Then, for each i with k, 1 < i < k , and for every j with 1 <j < k, and UW, t UT, we can select B, E W,' satisfying B, 4 B, because 59, is a maximal chain and UW, 5 Uq. Let B, = max{B,,I 1 < j < k , , UW, c UW,}. For each i with k, 1 < i < k, and for every j with 1 < j < k, and UW, c UW,, we can select B,, E W,' satisfying B,, Q B,. Then put
+
+
B, = max{B,,I 1
<j
d k,, UW, c
UT}.
Continuing this process we obtain B , , . . . , Bk E B. Then it is easy to see that { B l , . . . , Bk} is independent and x E B,. Hence, k < n, proving the proposition. 0
of=,
2.5. Corollary. Suppose B is a collection of point9nite rank. Then point-finite.
b
is
0
Proof. Obvious.
2.6. Theorem (G. Gruenhage and P. Nyikos [1978]). Zfa TI-spaceX has a base B of point-finite rank, then X is metacompact.
Proof. For each x E X we put
W(x,Q) = ( B E B I X E B C U f o r s o m e U E Q } , b ( x , 42) = the collection of all sums of maximal chains in g ( x , a), &(Q) =
U(d(X,
42) 1 x E X } .
Then B(42)is point-finite by Corollary 2.5. From each V E b(42)that is not a singleton we select a fixed point x( V ) E V and some B( V ) E B(x(V ) , 42) such that B(V) c V. For each y E V with y # x(V) we define g ( y , V ) = ( B E B ( y 94211 B c
-
{x(J'>}},
b(y , V) = the collection of all sums of maximal chains in B( y , V). Then, by Corollary 2.5, B ' ( V )= {B(V)} u [U{~'(Y, U1.Y E V , y
z
X(V)jl
254
J. Nagata
is a point-finite open cover of V. Let
W‘ = {all singleton members of &(42)} u [U {a( V )I V E &(%) and V is no singleton}], w” = { B E W I I B c Uforsome
U E ~ } .
Then it is obvious that W is a point-finite collection such that W < 9.Thus it suffices to show that w” covers X. Let y E X and let V be a minimum element of &( y , 42). If V is a singleton or y = x( V), then y is contained in some element of W . Hence, assume that V is no singleton and y # x ( V ) . Suppose V is a maximal chain in W(y, V). Then there is a maximal chain V‘ in W ( y , 42) such that V‘ =) V. Since UV’ E & ( y , 42) and V is minimum, either UV’ = V or UV’ c$ V holds. Thus there is B‘ E V’ such that B c B’ for all B E V. Because, otherwise UV’ = UV c V - { x ( V ) } , which is a contradiction. Hence, y E UV c B‘ c U for some U E 42 and hence, y E UV E W“.This proves that w” covers X , and thus X is metacompact.
0 2.7. Theorem (Gruenhage and Nyikos [1978]). I f a compact T2-spaceXhas a base W of$nite rank, then X is metrizable.
Proof. Let rank B < n and X = Y u Z, where Y is the set of all isolated points of X and Z = X - Y. Then Z is compact and T 2 .For each B E 93 with B # 0 we define that W ( B ) = { B ’ E W I B c B’}.
Then, by Corollary 2.3, we can decompose W ( B )as W ( B ) = W l ( B )u . . . u B n ( B ) ,
where each W i ( B )is a chain. Now we select subcollections
9, j E N of a such that
is a minimum cover of Z ; if j < k, U E q?, V E %: and V c U, then c U ; if j < k, U E qo, and V E 42:, then U c t V. The construction of such q o is easy because of compactness of 2, so it is left to the reader. Then we define that W
421 =
0 %$?, j= I
M,
= {x E
21every nbd of x contains some element of
Metrizarion I
255
Then M I is obviously a closed set. Further we define, for each U, U’ E a,,
+
K(U, U’) = U { B E L ? ~ ~ ( U ) ~UB’ } , 1 ,< i “u; = { K ( U , U’)IU, U ’ E ~ 1, ,< i
< n,
< n}.
It is obvious that V,is countable. Furthermore, we claim that V,is a base for the points of M I in X . Let x E M I and P be an open nbd of x (in X ) . Select B, E g such that x E B, c P . Select U, E 42, such that U , c B,, which is possible because of the definition of M I . Select x, E U , such that x , # x. Select B2 E g such that x E B2 c B, - {x,}. Select U2E 42, such that U, c B2. Select x2 E U2 such that x2 # x. Select B3 E L?d such that x E B3 c B2 - {x2}.Continuing this process we get sequences of open nbds of x, P 3 B, 3 B2 3 . . 3 Bn+,and U , , . . . , Un+,satisfying
-
a343
U , ~ 4 2 , ,i = 1,
...,n +
1
and Bi+ *
q-,,i = 2 , 3 , . . . , n +
1.
Then
u n
B , , . . . Bn+, E g(un+,)= 9
gi(un+l)*
i= I
Thus for some i we have Bj and Bk with j < k satisfying Bj, Bk E L?di(Un+,). Now x E Bk c K(Un+,,V,) follows from Bk $ V,. On the other hand, V(Un+,,q) c Bj can be proved as follows. If B E B i ( U n + ,and ) B V,, then B Bj E L?di(Un+,), which implies B c Bj, because L?di(Un+,) is a chain. Thus, K(Un+,,V,) c Bj. Therefore, x E V(Un+,,q) c P, proving our claim. Let #? c o,and assume that, for every a c #?, Ma,Vaand 427,j E N have been defined in such a way that
+
+
for each j
E
N,
42; is a finite subcollection of a,
{MaI a < #?}is an increasing sequence of closed sets, where Ma = { x
E
2 1 every nbd of x contains some element of = U { y ’ I a ’ < a, j E N}}, (2.1)
Vais a countable base for the points of M in X
(Let Mo = 0, 6 = {0}.) Then we define that
M, = {x For U E
E
Z I every nbd of x contains some element of a,= U { q l a < # ? , j N}}. ~
U G l y and U’ E U ; , q ’
with a’
c #?, we define open sets
J. Nagata
256
F(U, U ’ ) = U { B E ~ , ( U )Ip~ B U ‘ } and
y ( U , U ‘ ) = K(U, U ’ ) - M u , 1
< i < n.
Put V, =
{ y(u,
m
U’)l U E
m
u y , u y’, U’ E
j= I
where u’
j= I
< B; 1
I
.
Then V, is countable. We claim that V, is a base for the points of M , in X. Let x E M, and suppose that 8’ is the first ordinal such that x E M F .Further, let P be an open nbd of x. Select B , E 93 such that x E B, c P . Select U , E @r such that U , c B, (see (2.1)). Select x, E U , such that x, # x. Select B2 E such that x E B2 c B, - {x,}. Select U2 E @fl. such that U, c B2. Continuing the same process we get sequences of open nbds of x:
P
3
B,
1
B,
I>
’
3
B,+, and
U , , . . . , U,+,
such that B i 3 U,,
i = 1,
. . . , n + 1 and
B i P Q - , , i = 2 , . . . , n + 1.
Further we may assume without loss of generality that if U, E U,?,@$, U,. E U,?,@;‘ (u, u‘ < 8’) and i < i’, then u < u’. Because if every nbd of x contains an element of ?ao with uo < u, then x E MEby (2. l), contradicting that u < j3’. Thus there is a nbd P ‘ of x that contains only elements of q ’ with a’ 2 u, which means that one can select U, satisfying the above additional condition. Now, since B,, . . . , B,,, E .%?(U,+,)= u:=,.%?,(Un+,), there are iand j , k withj < ksuchthatBj,BkE&?i(U,+,).ThenxE y(Un+,,V,) c Bj c Pcan be proved as before. Since x 4 Mufor all u < /I’and U,,+ E U,? quo for some M o < S‘,
,
,
proving our claim. (Note that K ( U , + , , V,) E @, because of the above additional condition satisfied by { U,}.) Next we define @ f , i E N as follows. First note that M, is G6in X (and accordingly in 2, too). Because there are only countably many finite subcollections of V, that cover M,, and M, is the intersection of the sums of such collections. Thus we put Z - M , = Urn“=,F, for closed sets F,, m E N. For
257
Metrization I
each m E N we define a minimum cover 49; of F, u . . . u F, by elements of 9 such that ,; m’ < m and V c U, then if U E %,$, V E %
u
i f U e q f l u (m ’ < m %;.)andVE%{,
Pc
U , (2.3)
t h e n U Q V.
(2.4)
In fact, we can select %; as follows. Suppose Urn.<,%,$ have been defined. Then, for each x E F , u * u Fm,select B, such that
x E B,
E
B, c
n
9, B, n M ,
i
0,
=
I
u %is that contain x .
all members of
m‘<m
Further B, can be chosen to satisfy the following condition. From each member U of Urn,,,%$ we pick an x ( U ) E U such that x ( U ) # x. Then we select a B, that contains no x ( U ) . Further note that, since x $ M,, x has a nbd that contains no element of %., Thus we can choose a B, that contains no element of %., Now, select a minimum cover 49; of the compact set F, u u F, from { B , I x E F, u * . * u F,}. Then (2.3) and (2.4) are satisfied. Now we claim that Z = U{M,la < o,}. Assume the contrary: then there is an x E Z - U { M a l a < o,}. Let
%(x)
=
i
UlXE VE
(J
%,
a<wI
I
.
By Corollary 2.3,
%(x)
=
%,(x) u . . . u 4&(x),
where each %i(x) is a chain. Put
T, = { a l a < o,, ai(x)n %; %i(x) n % ,; #
z 0,
0 for some distinct m
and m‘
I
.
Observe that for each tl c o,there is an i such that T, 3 a. Because from (2.4) it follows that %(x) n 42; # 0 for infinitely many m’s. That is, U;=, T, = { a I tl < w , }. Hence, for some i, is cofinal in (cx 1 c1 < o,}. Then for such an i we have fY@i(x) =
n{oIu E % i ( X ) }
(2.5)
by (2.3), (2.4) and the fact that %i(x) is a chain. Suppose U E %; n %i(x),
U’ E % ;; n e i ( x ) and
U
2
U’;
J. Nagara
258
then by (2.4) either a < a' or a = a' and m < m'. (Because %, is a minimum cover, a = a' and m = m' cannot happen.) Thus %i(x) is well-ordered by the inclusion, and no countable subcollection is cofinal in %;(x). Thus we may Put %i(x)
=
{U~IY < w,},
where UyXI U,. whenever y c y'. U, I y < o,} is countably compact. To this Now we can show that Z end, let
n{
{ z i ( i EN] c Z -
n U,. Y
Suppose zi E Z - U,,, i E N. Then { z i } c Z - U,, for y = supiyj. Since Z - U, is compact, {q} has a cluster point z E Z - U, c Z - n,Uy, proving that Z - oyU, is countably compact. From Theorem 2.6 it follows that Z - nYU,is metacompact. Hence, by [Na, V.4.D) and V.4.F)], Z - nyUyis compact. On the other hand, (2.5) and (2.4) imply that {Z - D I U E %;(x)} is an open cover of Z U,, which contains no finite subcover. This contradiction shows that Z = U{M,la < ol}. Next, let us show that Y" = U,<, Y", is point-countable at each x E Z. Suppose x E M. Then the only elements of Y" that contain x are of the form v ( U , U') - M B , where 1 < i < n, U E %$,U' E %!,' fi' < fi < a. Obviously, there are only countably many such elements. Thus the compact T,-space Z has a point-countable base. Hence, by m a , Theorem VII.61, Z is metrizable. On the other hand, by A. MiiEenko's lemma [Na, VII.3.B)I there are at most countably many minimum covers of Z by members of V . Hence Z is Gdin X. Let X - Z = q.,where each Hi is closed in X. Note that each Hi is metrizable because it consists of isolated points only. Thus, by m a , Theorem VI. 131, X is metrizable.
nY
u?,
2.8. Corollary. A compact T,-space X with a base B of point-finite rank is metrizable. Proof. Let X, = { x E XI rank,$3 < n}. Then X, is a closed subspace with a base of rank < n. Hence, by Theorem 2.7, X, is metrizable. Thus, by [Na, Theorem VI. 131, X is metrizable. 0 2.9. Corollary. A locally compact T,-space with a base of point-finite rank is metrizable.
Metrizarion I
259
Proof. By Theorem 2.6 and Corollary 2.8, there is a point-finite open cover 9 = {U,laE A} of X such that each Omis compact and metrizable and accordingly separable. Thus each U, meets at most countably many Us’s. Therefore Ukm_,Stk(Uol, 9)= Stm(U,, %) is a sum of countably many separable metrizable sets. Hence, by “a, VI.4.C)], Stm(Uu,%) is metrizable. Since X is a discrete sum of St” (U,, a),it is metrizable. 0 By use of the theorem of dimension theory mentioned at the beginning of this section, we get the following corollary. 2.10. Corollary. A locally compact T,-space is metrizable and has dim iff
it has
a base of rank
< n.
2.11. Example. The space X in “a, Example V.71 is obviously a developable nonmetrizable space with a base of rank 2. 3. Metrization in terms of hereditary property
A. V. Arhangel’skii initiated the study of metrizabilityin terms of hereditary properties, which opened a new aspect for metrization theory. The purpose of this section is to take a quick view of Arhangel’skii-Balogh’s theory in this aspect. Throughout the present section every space is at least regular’.
3.1. Definition. A space X is called an Fpp-spaceif every subspace of X is paracompact and M. 3.2. Definition. A sequence 42, ,e2, . . . of open covers of X is said to satisfy wA-condition if x,, E St(x, %J, n E N implies that the point sequence {x,,} has a cluster point (see “a]). 3.3. Proposition. Every perfectly normal Fpp-spaceX is Jirst countable.
Proof. By [Na, Theorem VII.3-Corollary I], there is a perfect mapffrom X onto a metric space Y. Then there is a normal sequence {qi 1 i E N} of open covers satisfying wA-condition and nr=,St(x, 9,,) = f - I ( y ) for each y E Y and x E f -’( y ) (see the proof of [Na, Theorem VII.31). For each x E X there are open sets G,,, n E N, such that {x} = G,,. Then, select open sets W,,
or=,
5By “regular” we mean regular and TI.
260
J . Nagara
n E N, such that x
E
W , c G, n St(x, @,I,
W,+,c
W,.
Now we claim that { W, I n E N) is a nbd base of x. Because otherwise, there is an open nbd U of x such that W, c l U for all n E N. Select x, E W, - U . Then {x,} has a cluster point x’ E X - U because of the wA-condition. On the other hand, x’ E W, = {x}, which is a contradiction. Thus X is first countable. 0
or==,
3.4. Proposition. Let X be compact T2 and first countable. Then X has at most 2°-number of separable compact subsets.6
Proof. Every separable compact set is of the form {x, I n E N} for some {x,} c A’. Hence the cardinality of the collection of such sets is at most I X 1“’ d (2”’)” = 2”, where we have used Arhangel’skii’s Theorem [Na, Theorem VIII.6-Corollary] to derive I XI G 2”. 3.5. Proposition. Let X be a compact T,-first countable space with I XI > w . Then X has a nonempty compact set with no isolated point. Proof. Let K
=
{x
E
XI every nbd of x has cardinality > w } .
Then K is obviously closed and thus compact. If K = 8, then X is covered by finitely many open sets each of which is countable. Thus I XI 6 w , which is impossible; hence, K # 8. Assume x is an isolated point of K. Then there is a nbd U of x such that 0 n K = {x}. Suppose { U, In E N} is an open nbd base of x. Each y E 0 - U, has a nbd V( y ) with I V ( y ) I < w because y fji K. 0 - U,, is covered by finitely many V( y)’s, and hence, I 0 - U, I < w . Hence I 0 - {x} I d w , and thus I 0 I d w , which is impossible. Therefore, x is not an isolated point of K. 0 3.6. Proposition. Every compact T,-space X with no isolated point has a countable subset S such that 1 1 2 2”.
Proof. Select distinct points x(O),x(1) E X and open nbds U(O), U(1) of x(O), x ( l ) respectively such that U(0) n U(1) = 8. Select distinct points x ( i , 0), x ( i , 1) E U ( i ) , i = 0 , 1, and open nbds U(i, 0 ) and U(i, 1) of x(i, 0 ) 6We use w to denote the least countable ordinal as well as the countable cardinal No.
Metrization
I
26 1
- and x(i, I ) respectively such that U(i, 0) n U(i, 1) = 0. Continue the same process to select x ( i , , i2, . . . , i k ) E X , i , , . . . , ik = 0, 1 such that x ( i , , . . . , i k , 0 ) # x ( i l , . . . , i k , I ) and open nbds U(i,, . . . , ik, 0 ) and U(i,, . . . , i k , 1) of x ( i , , . . . , i k , 0 ) and x ( i , , . . . , i k , 1) respectively such that U(i,, . . . , ik, 0 ) n U(i,, . . . , ik, 1) = 0. Then S = { x ( i l ) , x ( i l , iz).. . . l i , , i2, . . . = 0, I } satisfies the desired 0 condition.
If X is a compact TZ-first countable space with I XI > w , then it contains a countable subset S such that I S I = 2".
3.7. Proposition.
Proof. By Proposition 3.5, X contains a nonempty compact set K with no isolated point. By Proposition 3.6, K contains a countable set S with I S I 2". By Arhangel'skii's Theorem, I S 1 < 2", and thus I SI = 2'". 0 3.8. Proposition (Balogh [1981]). I f X is compact T, andfirst countable, then there are disjoint subsets A , B of Xsuch that X = A v B andsuch that neither A nor B cmtains an uncountable compact set.
Proof. Let X be the collection of all separable compact subsets of X with cardinality 2". By Proposition 3.4, X = {K,IO < a < T}, where T is the least ordinal with cardinality 2". Pick x,, yo E KOwith xo # yo. Assume x,, y, E K, have been selected for all a < /3 in such a way that x,, y,, a < jl, are all distinct. Since 1 K,( = 2" and 1 {x,, y,(a < 0) 1 < 2", we can select x B , y g E KB such that x,, y , , a < jl, are all distinct. Then let A = {x,Ia < T},
B = X - A.
Now, let K be an uncountable compact subset of X.Then, by Proposition 3.7, K contains a separable compact subset S with IS1 = 2". Then K =I = K, 3 x, for some a. Since x, 4 B, B p K. On the other hand, since y , E K, n ( X - A ) c K n ( X - A ) , A K. This proves Proposition 3.8. 0
+
3.9. Proposition (Balogh [ 19811). Every first countable Fpp-space X is the sum of countably many metrizable subspaces.'
Proof. Since Xis paracompact and M , there is a perfect map f from X onto a metric space Y. Then, for each y E Y , f-'(y ) is compact and first 'It is known that this proposition is true for any Fp,,-space, see Balogh [1981].
262
J . Nagata
countable. Thus, by Proposition 3.8, there are disjoint subsets A, and By of f - ' ( y ) such thatf-'( y ) = A, u By and such that neither A,. nor By contains an uncountable compact set. Let A = U{A,lY
E
0, B
= U{B,lY
E
y>.
Then, since A and B are paracompact and M, there are a perfect map g from A onto a metric space P and a perfect map h from B onto a metric space Q. Note that we may assume g--'(p) c f - ' ( y ) for each p E P and for some y E Y , and also h-'(q) c f - ' ( y ) for each q E Q and for some y E Y. In fact, we have a normal sequence {aI, a,, . . .} of open covers of X satisfying the wA-condition such that f - ' ( y ) = n:=,St(x, a,,) for each y E Y and x ~ f - l ( y ) Then . we consider a normal sequence {q, V,, . . .} of open St(x, V,,) covers of A satisfying the wA-condition such that g-'(p) = for each p E P and x E g-'(p) and such that V,, < a,,(see the proof of "a, Theorem VII.31). Now it is obvious that g-'(p) c f - ' ( y ) holds for every p E P and for some y E Y. The same argument is valid for B, Q and h. Thus g - ' ( p ) c A n f - ' ( y ) = A, for each p E P and for some y E Y. Hence, Jg-'(p)1 < o.Similarly, 1 h-'(q)l < o for each q E Q.Therefore we put
n:=,
g-'(p)
= {PI42,.
h-'(q)
=
P,
=
. .I, P
(41, 42, . .
{PilPEP},
E p,
qE Qi
Q,
= {qilqE Q } .
Now the restriction of h to P, is a one-to-one continuous map of P, into P@). Thus we can easily see that P, has a point-countable p-base. Since P, is T, and M, by [Na, Theorem VII.71, it is metrizable. Similarly, each Qi is also metrizable. Since X = UEl P, u Qi, the proposition is proved. 0 3.10. Proposition. Let X be a T,-M-space. If X = u."=lD,,for dense metrizable subspaces D,,, n E N, then X is metrizable.
u,"=
Proof. Since D,, is metrizable, it has a 0-disjoint base a,, = I where each is a disjoint collection of open sets of 0,.Extend each element U of *anrn to 0 = X - D, - U . Then Grim = { 0 I U E %,,m} is a disjoint collection of open sets of X. Now we can prove that 4 = UnmJn=,@,,mis a p-base of X as follows. Let x, y E X , x # y . Suppose x E 0,.Then there is a nbd W ofy in X such that W $ x. Select U E a,,such that x E U, U n W = 8.Then it is obvious 'Generally, a space X is called subrnefrizable if there is a one-to-one continuous map from
X onto a metric space.
Metrization I
263
that x E 0 $ y proving that @ is a p-base of X. Since 4 is a-disjoint, it is point-countable. Thus, by [Na, Theorem VII.71, X is metrizable.' 3.11. Proposition. Every first countable F,-space X has a dense metrizable subspace. Proof. By Proposition 3.9, X = U,"=,M, for metrizable subspaces M,, m E N. Each M, has a a-locally finite base { U, I tl E A}. Thus, if we pick x, E U,, tl E A, we obtain a a-discrete subset { x aI tl E A}, which is obviously dense in M,,, . Thence we have a a-discrete set D = Uz I Di, which is dense in X, and where each Di is discrete (in itself). For each x E Di we denote by {V,,(x) In E N) a countable open nbd base of x in D such that U,,,(X)n Di = {x}. Now = {Q,,(x)lx E Di}is an open cover of Q,, = &(x) I x E D i } . Since q, is paracompact, there is a a-locally finite open cover of V , such that "y;, < sin. Note that "v;, is a point-finite open collection in D. Then Y = I q,, is a point-countable open collection in D. Now we claim that V is a p-base of D. Let x, y E D , x # y . Assume x E Di.Then there is an n E N for which Uin(x)$ y . Since U,,,(x) is the only element of %in which contains x, if x E V E Cn,then V c L(,,(x). Thus y 4 V, proving that Y is a point-countable p-base of D. Since D is T2 and M , by [Na, Theorem VII.71, D is metrizable. Thus the proposition is proved.
u{
<,,
uz=
Actually, Arhangel'skii [ 19771 and Balogh [ 19791 proved that every Fppspace has a dense metrizable subspace. This fact follows from Proposition 3.1 1 and Ismail's [1978] theorem that every Fpp-spacecontains a dense open first countable subspace." 3.12. Proposition. If X is T2 and M and if X = zable subspaces 4, i E N, then X is metrizable.
Ugl&.for
e..
closed metri-
e.
Proof. Each 6.has a a-locally finite closed network Then Uz I is a o-locally finite closed network for X , that is, X is a a-space. Thus, by p a , Theorem VII.5-Corollary 21, X is metrizable. 0
3.13. Theorem (Balogh [1981]). A topological space X is metrizable fz it is a perfectly normal Fpp-space. 'Actually, Michael and Rudin [I9771 proved that if X = U,"l X, is regular and if each X, has I 2,has a u-disjoint base. a a-disjoint base, then "The theorem is proved there under a more general condition.
264
J . Nagata
Proof. It is obvious that every metrizable space is perfectly normal and F,. To prove the converse, suppose X is a perfectly normal Fpp-space.Then, by X,, Proposition 3.3, X is first countable. Thus, by Proposition 3.9, X = for metrizable subspaces X,,, n E N . Assume that X itself is not metrizable. Then, by Proposition 3.10, there is an n E N such that X,, is contained in no dense metrizable subset of X. Let U, = X - Z,,.Now, by Proposition 3.1 1, U,, contains a dense metrizable subset M. On the other hand, since X is perfectly normal, U,, = UEI F, for closed sets F,, i E N of X. Put
u,"=,
Mi = M n 6 ,
i E
N,
Y,
=
M , u X,,, i E N ,
Y = MuX,,.
(Jzx.
Then Y, is closed in Y, and Y = I Since Y is T, and M, if every Y, is metrizable, then Y itself is metrizable by Proposition 3.12. However, since Y is a dense subset of X containing X,, it cannot be metrizable. Thus Y, is not metrizable for some i. On the other hand, Y, = Miu X, is a discrete sum of two metrizable sets Mi and X,,, and hence is metrizable. This contradiction proves that X is metrizable. 0 3.14. Corollary (Arhangel'skii [1973]). A regular space X is separable metrizable fi every subspace of X is Lindelof and M . Proof. Let U be an open set of Xwhich satisfies the said condition. For each U.Since U is Lindelof, { V ( x ) ( xE X} has a countable subcover, that is, U is an F,-set. Since X is obviously normal, it is perfectly normal. X is obviously an Fpp-space, too. Hence, by Theorem 3.13, X is metrizable. cl
xE U there is an open nbd V ( x ) such that V ( x ) c
3.15. Example. Let X be an uncountable discrete space and a ( X ) be Alexandroff's one-point compactification of X . Then a ( X ) is an Fpp-spacebecause each subset of a ( X ) is either discrete or compact. However, a ( X ) is not metrizable because it is not separable.
Balogh [I9811 showed that the existence of a first countable nonmetrizable Fpp-space is consistent with and independent of the usual axioms of set theory. Benett and Lutzer [I9801 proved that a GO-space is metrizable iff it is F,.Okuyama and Hodel [ 19761 proved that a regular wb-space X with a point-countable p-base is separable metrizable if every subset of X satisfies
ccc.
Metrizarion I
265
4. Some other aspects In the present section we are going to discuss metrizability of spaces satisfying certain special conditions like separability, Lindelof and local connectedness. Throughout the section, X denotes a T ,-space.
4.1. Definition. Let 9 ( X ) be the collection of all closed sets of X. Then, for finitely many open sets U ,, . . . , U,,we define
v 9
=i
u k
F E ~ ( X ) I F C U . , U , n F # O f o r i = 1, . . . , I=,
kI
.
If we take all subsets of 9 ( X ) of this form as a base of a topology, then we obtain thejinite topology (or Vietori’s topology) of 9 ( X ) . If X has an annihilator cp which is continuous on X x 9 ( X ) , where 9 ( X ) is endowed with the finite topology, then cp is called a CPN-operator and X a CPN-space (continuously perfectly normal space). A map $ : {(x, y , F) E X x X x 9 ( X ) 1 y 4 F } + [0, 11 is called a CCR-operator if $ ( y , y , F ) = 1,
$(x, y , F) = 0
for x E F.
and if $ is continuous on {(x, y , F) E X x X x 9(X)ly4 F}. Then X is called a CCR-space (continuously completely regular space). It is obvious that every CCR-space is completely regular, and every CPNspace is perfectly normal and CCR. Because if cp is a CPN-operator, then $(x, y, F) = min
is a CCR-operator.
CCR-spaces and CPN-spaces were first studied by Zenor [1972, 19761, and his results were further developed by Gruenhage and others. We are going to introduce here some of Gruenhage’s [1976] results. 4.2. Proposition. Every separable CCR-space X is j r s t countable.
Proof. Suppose $ is a CCR-operator of X and D a countable dense subset of X. Let H be a finite subset of X and suppose x, y E X , x # y 4 H. Then Put U,(x, y , H ) = { z E X I $ k Y , H u { z } ) < l/n}, n E N.
J . Nagata
266
It is obvious that Un(x,y, H) is an open nbd of x. Suppose x, E Xis not an isolated point. Now we can show that Q(xo) = {U,(x,, y , H ) l y E D, H i s a finite subset of D, x, # y 4 H , n E N} is a nbd base of x,. Let W be an open nbd of x,. Assume no member of %(x0) is contained in Then let ( W - {x,}) n D = { p 1 7 p 2., . .}. Pick z , E U l ( x o , p I8) , n (IVY n D. Pick z2 E U2(x0,p 2 , { z I } )n n D. Generally,
m.
(wy
Zn E U n ( X 0 ,
Pn,
IZI7
...
7
Zn-,})
n
(myn D.
Now, put Z = { z , , z 2 , . . .}-. Then 2 c W' # x,. Hence, t,b(x,, x,, Z) = 1. Thus there are nbds U(x,) of x, and V ( Z ) of 2 such that p E U(x,) and p 4 F E V ( Z ) implies t,b(xo,p, F) > +. Suppose V ( Z ) = ( U , , . . . , U , ) , zi, E U , , . . . , zC E U,. P u t j = max{i,, . . . , ik}. Then for any zj+,, . . . , zI we have { z , , . . . , z j , . . . , z I } E V ( Z ) . Now, select I such that I > j , 1 > 2 and p, E U(x,), which is possible because x, is a cluster point of { p I , p 2 , . . .). Then I,&, PI, { z I ,. . . , zj, . . . ,z I } ) > t . On the other hand, zI E U,(x,, pI, { z , , . . . , zI-, }). Hence, p / , (21, . * * , Z/}> < 1/1 < 8, which is a contradiction. Therefore, our claim is proved, and accordingly X is first countable. 0 W
O
,
4.3. Theorem (Gruenhage [ 19761).
Every separable CCR-space X is metriz-
able,
Proof. The argument is somewhat similar to the proof of Proposition 4.2. By use of the symbols used there, we claim that Q = { Un(x,y, H ) I x, y E D, H is a finite subset of D, x # y 4 H , n E N} is a base for.'A Suppose W is an open nbd of a fixed point x, of X. Since there are at most countably many isolated points, it suffices to prove x, E U c P for some U E Q, assuming that x, is not an isolated point. To this end, we assume the contrary. Let W I> W, I> W2 2 . . . be a nbd base of x, by use of Proposition 4.2. Select p , , q, E W, n D with p , # q, such that x, E U l ( p , ,q l ,8). This is possible because t,b(xo,q , , {x,}) = 0 and thus I&,, q , , {x,}) < 1 ifp, is sufficiently close to x,. By the assumption we can pick
(m)'
zI E Ul(pI,q l , 8) n n D. Then select p2, q2 E W2 n D with p2 # q2 such that xo E U2(P2, 42, {ZI)).
This is possible because $(xo, q2, {z,,x,}) = 0 and thus JI(p2,q2, {z,, x,}) < f if p 2 is sufficiently close to x,. Then we can pick
z2
E
U2(P2, 42, { Z I } ) n Wt)'n D-
261
Metrization I
Generally, we pick p,, q, E W,n D with p,, # q,, such that xo E
Un(Pn, q n r {zlr . . *
9
zn-I
1)
and Z,
. . . ,Zn-,})
E Un(Pn,q n , {ZI,
n (p)'n D .
Now put Z = { z I ,z2, . . .}-. Then, since x, 4 Z, +(xo, x,, Z) = 1. Hence, there are an n E N and a basic nbd V(2)"" of Z such that x, x' E W, and x' 4 H E V ( Z ) imply #(x, x', H ) > $. k t { z I ,. . . , zk} E V ( Z ) and m > max(n, k , 2). Then, since z, E U,(p,, q,, ( z I , . . . , z , , - ~ } ) , +(pmi
qm, {zi,
9
Zm})
< I/m <
3.
On the other hand, since p,, q, E W, c W, and { z I ,. . . , z,} E V ( Z ) , +(Pm,
q m , {zli
...
9
zm}>
>
3,
which is a contradiction. Thus we have proved that 42 is a countable base for X. Therefore, X is metrizable. 0 4.4. Theorem (Gruenhage [ 19761). The following conditions are equivalent: (i) X is separable metrizable, (ii) X is a CPN-space satsifying CCC, (iii) X is a Lindelof CPN-space, (iv) X is a separable CPN-space.
Proof. (i)*(ii) is obvious. (ii)*(iv):
Generally, if for every open cover 42 of X there are U I , U 2 ,
. . . E 42 such that ui"pI V, = X , then we call X a weakly Lindelif space. It is
almost obvious that every space satisfying CCC is a weakly Lindelof space. Thus we assume that X is a weakly Lindelof space with a CPN-operator cp. Now, for each n E N,we define a countable subset Z,, = {z(n,i) I i E N} o f X as follows. In the following we use the symbol U,(F) = { x E X l d x , F ) <
114,
(4.1)
where F E 9 ( X ) . Then U,(F) is an open nbd of Fin X. Thus { U,({z})I z E X} is an open cover of X. Since X is weakly Lindelof,
"Namely V ( 2 )is of the form ( U , , . . . , Uk).
268
J . Nagara
for countably many points z(l, i ) , i e N. Then put Z, = {z(l, i ) l i c N}. Assume Z , , . . . , Z,- have been defined. Then, for each finite subset F of Z, u . * . u Z , - , we consider an open cover { U,(F u { z } )I z E X} of X. Since X is weakly Lindelof,
,
[6 i= I
U n ( F u { z i ( ~ ) } ) ] -=
x
for countably many points z , ( F ) , i E N. Now, put { z , ( F )I F is a finite subset of Z , u . . . u Z n - , ,i c N} = { z ( n , j ) l jN} ~ = Z,,. We claim that Z = u,",,Z, is dense in X. Assume the contrary and x 4 2. Then cp(x, 2) > I/n for some n E N. Thus there are a nbd U ( x ) of x and a basic nbd V ( 2 )of 2 such that x'
E
U ( x ) and F
E
V ( 2 ) imply cp(x', F ) > I/n.
(4.2)
Suppose z , , . . . , zk E Z and F = { z , , . . . , z k } E V ( 2 ) . Further, assume z, E Z,,,,i = I , . . . , k and m = max{n,, . . . , n k , n} + 1. Then
and hence
U ( x ) n Um(Fu { z i ( F ) } )# Pick x'
E
8
for some i.
U ( x ) n Um(Fu { z i ( F ) } ) Then, . by (4.l), cp(x', F u { z i ( F ) } )< l/m < I/n.
On the other hand, since x'
E
U ( x ) and F u { z i ( F ) }E V ( Z ) , by (4.2),
~p(x',F u { z i ( F ) } )> l/n.
This contradiction proves that z = X. Hence, X is metrizable. (iv)*(i) follows from Theorem 4.3,because every CPN-space is CCR. (iii)=-(i):Every Lindelof space is weakly Lindelof. By the previous argument, every weakly Lindelof CPN-space is separable. Thus Xis separable and accordingly metrizable by Theorem 4.3. (i)=(iii) is obvious. 0 Gruenhage [ 19761 showed that a CPN-space is not necessarily first countable, and a first countable CPN-space is not necessarily metrizable. He also showed that a Lindelof CCR-space is not necessarily metrizable. Zenor [I9761 proved that X is metrizable iff it is a CCR-wb-space iff it is a CCR-LaSnev space.
Metrization I
269
Recently, the normal Moore space problem is being studied mainly in relation with set theory, which is an aspect not to be discussed here. But we are going to show a result of Reed and Zenor [1974], which is among a few interesting results obtained without assuming any set-theoretical hypothesis.
4.5. Proposition. Let X be T,,Jirst countable, locally compact, locally connected and connected. Then I XI < 2'". Proof. To each x E X we assign a compact nbd O ( X ) and a connected nbd V ( x ) such that V ( x ) c O(x). Then, by Arhangel'skii's Theorem I V ( x ) l < I o(x) I < 2" [Na, Theorem VIII.6-Corollary]. Now we define an open set V , for each a with 0 < a < o1as follows
6
= V ( x o ) for a fixed point xo E
X.
Assume V , have been defined for all a < b. Then define
Now it is obvious that { Vp10 < /3 < o,} is an increasing sequence consisting of connected open sets. It is also easy to prove I VpI < 2" for every < o, by use of induction on /3 and [Na, VIII.2.B)I. Now put V = U,<, V,. Then V is open. We can prove that V is closed as follows. Suppose x E P. Let W, 3 W, 2 . . . be a nbd base of x. Then cr/; n V,, # 0,i E N. Hence, for a = supla,we obtain cr/; n V , # 0 for all i E N. Hence, x E V2 c V , that is, V is closed. Since X is connected, and V is a nonempty clopen set, V = X. 0 Thus 1x1 = I Ua<,, V,l < 2". 4.6. Proposition. Let X be a normal dcvelopnble space satisfying I XI < 2 . Then X is submetrizable, i.e., X can he mapped on a metrizahle space by a one-to-one continuous map. Proof. Let % be a given open cover of X. Then we can show that there is a continuous map g from X into a metric space Y such that, for each y E Y , g - ' ( y ) is contained in some member of %. Since X is subparacompact "a, V1.8.B) and D)], there is a a-discrete closed cover U,2,FI that refines 92, where each 9, is a discrete collection. Since I.c1< 2'" by the assumption, there is a one-to-one mapf; from 9,into the real line R. Let 93 = { B , , B,, . . .} be a countable collection of subsets of R such that for all distinct points x and x' of R there is a B, such that x E B, $ x'. (Suppose, e.g., that 93 is a countable base of R.) Then, for each (i, j ) E N x N we consider a copy Z,, of [O, I ] to put Y
=
n {J,l
i , j E N}.
J . Nagata
210
Now, for each ( i , j ) E N x N we put
C,
=
U { F E E.lJ(F)E B,}.
Then Gll is a closed set of X . Since Xis perfectly normal, there is a continuous map g, : X + such that g; ‘(0) = G,. Then we define a continuous map g : X + Yby d x ) = { g l , ( x ) l i , j E N},
XEX.
Now, suppose y E g ( X ) and x E g - ’ ( y ) . Assume x E F E 9,. Then g ( x ) = y = { y , I i, j E N} satisfies y,, = 0 for a l l j E N such thatf;(F) E El. Suppose x’ E X - F. If x’ # UR, then g , ( x ’ ) # 0 for any i E N. Hence, g ( x ) # g ( x ’ ) . If x’ E F‘ E PIand F’ # F, then there is E, E G9 such thatf;(F) E El, f;(F’) # El. Thus, g,(x) = 0, g,(x’) # 0. Namely, g ( x ) # g ( x ’ ) . Therefore, g - ’( y ) c F, which is contained in some member of 9, proving our claim. Now, we denote by {qlr 9*,. . .} a development of X. Thus for each k E N, there is a continuous map gk from X into a metric space Yk such that for each y E Yk, g;’( y ) is contained in some member of 9 k . Then define cp:X + llF=, Yk by d x ) = { g k ( x ) l k E N},
x
E
x.
Obviously, cp is a one-to-one continuous map of X onto the metric space cp(X). Namely, X is submetrizable. 0 4.7. Proposition. Let X be normal, developable, locally compact, locally connected and connected. Then X is metrizable. Proof. By Proposition 4.5, I X I < 2“‘. Hence, by Proposition 4.6, X is submetrizable. Thus there is a one-to-one continuous map from X x X onto Y x Y, where Y is metric. Therefore, A = {(x, x ) l x E X } = W,for countably many open sets W,, n E N of X x X , because Y x Y has the same property. Now, let 9, = { U I U is an open set of X such that U x U c W,}. Then &, is an open cover of X such that for any distinct points x and y of X there are n E N, a nbd U ( x ) of x and a nbd V ( y ) of y satisfying St(U(x), @,,) n V ( y) = 8. To see it, select an n E N such that ( x , y) 4 W,. Then there are nbds U ( x ) and V ( y ) of x and y respectively such that U ( x ) x V ( y ) n W, = 8. Let U E ~ , then , ; U x U c W,. Hence, if U n U ( x ) # 8 and U n V( y) # 8, then, for x’ E U n U ( x ) and y’ E U n V( y). we have ( x ’ , Y’) E W x ) x V ( y ) n K.
n:=,
Metrization I
But this is impossible, and hence U n U ( x ) # Namely, St(U(x), “91,) n V ( y ) = 0.
27 1
0 implies
U n V( y ) =
0.
In the following we assume <% ,,! without loss of generality. We may also assume that each member of %n is connected because X is locally connected. Now, suppose C is a compact nbd of a fixed point x, of X . Let W be an arbitrary open nbd of x, with W c C. Suppose W 3 U , 3 U, =I . . * be a countable nbd base of x, where ach Unis connected. Then, as proved in the above, for any y # x,
there is an n E N for which y
4 St(Un, %,,). (4.3)
Now, to show St(U,,, %,,) c W for some n, we assume the contrary. Then, since St(Un, a,,)is connected, St(Un,q,,) n (C - W ) #
0, n E N.
Since C - W is compact, we have m
y
E
st(un, @), n (c -
w )# 0.
n= I
Then x, # y E St(Un, a,,)for all n E N, which contradicts (4.3). Hence our claim is proved. Therefore, by Arhangel’skii’s Theorem [Na, Exercise VI.21, X is metrizable. 0
4.8. Theorem (Reed and Zenor [1974]). Every locally compact, locally connected, normal, developable space X is metrizable. Proof. X is decomposed into the discrete sum of clopen components C,, a E A. Since each C, satisfies the condition of Proposition 4.7, it is metrizable. Since X is a discrete sum of metrizable sets, it is metrizable. 0 Chaber and Zenor [1977] generalized the above result as follows. Every rim-compact, locally connected, normal, developable space is metrizable, where we call a space rim-compact if each nbd of each point contains an open nbd whose boundary is compact. References Arhangel’skii, A. V. [I9631 Ranks of systems of sets and dimensionality of spaces, Fund. Math. 52, 257-275.
J . Nagata
272
[I9731 On hereditary properties, General Topology Appl. 3, 39-46. [I9771 On left separated subspaces, Vesrnik Moskov Univ. Ser. Marh. Meh. 5, 30-36.
Balogh, 2. [I9791 On the structure of spaces which are paracompact p-spaces hereditarily, Acta Marh. Arad. Sci. Hungar 33. 361-368. [I9811 On the metrizability of F,-spaces and its relationship to the normal Moore space conjecture, Fund. Math. 113, 45-58. Benett, H. R. and D. Lutzer [ I9801 Certain hereditary properties and metrizability in generalized ordered spaces, Fund. Moth. 107, 71-84. Borges, C. R. [ 19831 Expansions of closure-preserving collections and metrizability, Math. Japonica 28.
67-7 I .
Burke, D.. R. Engelking and D. Lutzer [ 19751 Hereditarily closure-preserving collections and metrization, Topology Appl. 1 I, 275279. Chaber, J. and P. Zenor [I9771 On perfect subparacompactness and a metrization theorem for Moore spaces, Topology Proc. 2, 401-407. Dilworth, R. P. [I9501 A decomposition theorem for partially ordered sets, Ann. Marh. 51, 161-166. Gruenhage. G. [I9761 Continuously perfectly normal spaces and some generalizations, Trans. A M S 224. 323-338. Gruenhage, G. and P. Nyikos [I9781 Spaces with bases of countable rank, Topology Appl. 8, 233-257. Gruenhage. G. and P. Zenor [I9751 Metrization of spaces of countable large basis dimension, Pacific J Marh. 59,454460. Hung, H. [I9801 An alternative to Bing’s generalization of Urysohn’s metrization theorem, Topology A&. 11, 275-279. Isiwata, T. [I9871 Metrization of additive K-metric spaces, Proc. A M S 100, 164-168. Ismail, M. [I9781 A note on a theorem of Arhangel’skii. General Topology Appl. 9, 217-220. Michael, E. and M. Rudin [I9771 Another note on Eberlein compacta, Pacijic J . Marh. 72, 497499. Nagata, J . [I9571 A contribution to the theory of metrization, J . Ins/. Polyrech. Osaka Ciry Univ. 8, 185-192. [ 19831 Modern Dimension Theory, revised and extended edition (Heldermann, Berlin). [I9851 Modern General Topology, second revised edition (North-Holland, Amsterdam). [ 1986/87] Characterizations of metrizable and LdSnev spaces in terms of g-function, Questions Answers General Topology 4, 129-139.
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Okuyama, A. and R. Hodel [I9801 Cardinal functions and a metrization theorem, Math. Japonica 21, 61-62. Reed, C. and P. Zenor [I9741 Pre-images of metric spaces, Bull. A M S 80, 879-889. Suzuki, J., K. Tamano and Y. Tanaka [ 19871 K-metrizable spaces and stratifiable spaces, Questions Answers General Topology 5, 167-1 7 1. Zenor, P. [I9721 On continuously perfectly normal spaces, in: Proc. Univ. Oklahoma Topology Conf 1972, 334-336. [I9761 Some continuous separation axioms, Fund. Math. 90, 143-158.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 8
METRIZATION I1
Yoshio TANAKA Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spaces which contain a copy of S, or S, . . . . . . . . . . . . . . . . . . . Spaces dominated by metric subsets . . . . . . . . . . . . . . . . . . . . . Spaces with o-hereditarilyclosure-preserving k-networks . . . . . . . . . . . . Spaces with certain pointcountable covers . . . . . . . . . . . . . . . . . . 5. Quotient s-images of locally separable metric spaces. . . . . . . . . . . . . . 6. K-Metrizable spaces and 6-metrizable spaces . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. 2. 3. 4.
275 276 284 287 291 301 304 31 I
Introduction We are familiar with many fundamental metrization theorems; for example, we have Nagata-Smirnov’s metrization theorem that a regular space is metrizable if and only if it has a a-locally finite open base, and others, especially metrization theorems on Moore spaces, M-spaces, and other generalized metric spaces. These metrization theorems are given in Nagata [1985, Chapters VI and VII] and in Chapter 7 by Nagata in terms of bases, open covers, or g-functions, etc. Now, let X be a space, and V be a family of subsets of X . We recall basic concepts associated with %?. A space X has the “weak topology” with respect to V if F c X closed in X if and only if F n C is relatively closed in C for every C E V. The topology of any quotient space of a metric space is precisely the weak topology with respect to the family of its metric subsets. A cover B is a “k-network” for X if, for any compact K and open U with K c U,there exists a finite 8’c B with K c US’ c U . LaSnev spaces and
216
Y. Tanaka
certain quotient spaces of metric spaces can be characterized by means of k-networks. For a family 4 of closed sets of X,a nonnegative real-valued function cp: X x 4 --* R is an “annihilator” for 4 if cp(x, F ) = 0 if and only if x E F. Stratifiable spaces, rc-metrizable spaces, and 6-metrizable spaces, etc., can be characterized by means of annihilators. In this chapter, we survey metrization theory in terms of weak topologies, k-networks, or annihilators, etc. We generalize some of the fundamental metrization theorems, and give metrization theorems on spaces dominated by metric subspaces, spaces with certain point-countable covers, certain quotient spaces of metric spaces, and generalized metric spaces such as M-spaces, rc-metrizable spaces, etc. We assume that all spaces are Hausdorff, and all maps are continuous and onto.
1. Spaces which contain a copy of S, or S,
Sequential spaces are precisely the quotient images of metric spaces. The class of sequential spaces contains FrCchet spaces and strongly Frkchet spaces. These spaces play an important role in metrization theory. The sequential fan S, and the Arens’ space S, are canonical examples of sequential spaces. Using these concrete spaces S, and S,, we analyse the gaps among sequential spaces, FrCchet spaces and strongly FrCchet spaces. 1.1. Definition. We recall the canonical quotient spaces S, and S,. Let Lo = {a,l n E N} u {co} be an infinite sequence with limit point co. Here N denotes the set of natural numbers. For each n E N, let L , be an infinite convergent sequence containing its limit point a,,. Let L be the topological sum X{L,,ln E N}. S, is the quotient space obtained from L by identifying all the limit points a,, in L with co. S, is the quotient space obtained from the topological sum Lo and L by identifying each a, E Lowith the limit point a,, E L. Then the collection of sets oftheform{co} u {a,,(n 2 m} u {U,,ln 2 m } , where U,,isaneighborhood of a, in L,,, is a neighborhood base ( = local base) of {a}in S,. 1.2. Definition. A space X is called Frtchet (= FrCchet-Urysohn) if, for every A c X and every x E 2,there exists a sequence {x,,ln E N} in A converging to the point x . A space X is called strongly Frtchet if, for every decreasing sequence { A , [n E N} with x E n{A, I n E N}, there exists a
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sequence {x, I n E N} converging to the point x with x, E A,. First countable spaces are strongly Frechet, and strongly Frechet spaces are Frechet. The concept of strongly Frechet spaces was introduced by Siwiec 119711. Michaei [ 19721 used the terminology “countably bi-sequential” instead of ”stronglk Frechet”. 1.3. Proposition. Every closed image X of a j r s t countable space is Frechet.
Proof. Let f:S + X be a closed map with S first countable. Let X E A in X . Choose B c S such that f(B) = 2.Then f -‘(x) n B # 0, so choose h E f -‘(x) n B. Then there exists a sequence {b, 1 n E N} in B converging to h. Thus { f ( b , ) ( nE N} is a sequence in A converging to x. Hence. X is Frechet. C 1.4. Definition. Let X be a space. A subset LI of X is called sequentiall-yopen if each sequence in X converging to a point in I? is eventually in [ I . A space Xis called sequential if every sequentially open subset of X is open in X. First countable spaces are sequential. Sequential spaces are precisely the quotient images of metric spaces (Franklin [1965]). We note that a space Xis sequential if and only if U t Xis open in X whenever LI n C is relatively open in C for every compact metric subset C of X . Here we can replace “open” by “closed”. A space X is called a k-space if we replace “compact metric” by “compact”. Sequential spaces are k-spaces. k-spaces are precisely the quotient images of locally compact spaces (Cohen [I 9541). 1.5. Proposition. Each of the bfollowingimplies that X is sequential. (1) X is a Frkchet space, ( 2 ) X is the quotient image of a j r s t countable space, ( 3 ) X is a k-space in which every point is a Gar (4) X is a CW-complex.
Proof. Let U be a sequentially open set in X. We show that U is open in X. For ( l ) , suppose that U is not open in X. Then there exists a sequence in X - U converging to a point in U . This is a contradiction. Then U is open in X. For (2), let f:S --t X be quotient with S first countable. Sincef - ‘ ( U ) is sequentially open in S, f - I ( U ) is open in S. Hence, U is open in X. For (3), let K be a compact set of X. Since each point x of K is a Gsin K , there exists a sequence { V,I n E N) of open sets in K such that c V , and { x} = V ,I n E N}. Since K is compact, { V ,I n E N} is a neighborhood base of x in K. Then K is first countable. Thus K n U is open in K . But X is a
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k-space. Then U is ouen in X. For (4), note that G is open in X if and only if G n P is open in P for every closed cell P in X. But, since 2 is metrizable, each U n t is open in 2. Then U is open in X. (For the details about CW-complexes, see Whitehead [ 19491.) 0 1.6. Proposition. ( I ) S, is a FrPchet space. But S, is not strongly FrPchet, thus any strongly FrPchet space contains no copy of S,. ( 2 ) S2is a sequential space. But S, is not FrPchet, thus any FrPchet space contains no copy oj-S,.
Proof. We note that no sequence A in S, - { co}or S, - Lo converges to =3 if A n L, is at most finite for each n E N, and that the strongly Frechetness ,md the Frechetness are hereditary. Hence, it is easy to show that (1) and (2) Zold. 0
1.7. Definition. A space Xis called a c-space (= space of countable tightness) A. then x E C for some countable C c A . First countable spaces, and hereditarily separable spaces are c-spaces. :f whenever x E
1.8. Proposition. Every sequential space
X is a c-space.
Proof. Let x E 2, and B = U{clC c A is countable}. Let D c B be zountable. Then D c for some countable C c A. Hence, b c B. This snows tnat B n E is closed in E for every countable E c X. Especially, 5 n C is closed in C for every convergent sequence C in X. Since X is sequential, B is closed in X . Hence, x E A = B = B. Then x E c for some countable C c A .
c
Now. S,* (respectively S,) can be considered as { 0 0 ) u (N x N) (respectively ( 0 0 ) u { a n l nE N} u (N x N)). Let F be the family of all functions from N into N. For f ’ F, ~ put S( f) = ( 0 0 ) u { ( n , rn) Irn 2 f ( n ) } , and u { a , ( n E N}. Then T ( f ) = H u {(n, rn)Irn 2 . f ( n ) ) , where H = {a} { S (f ) If E D ] (respectively { T(f ) If E F } ) is a neighborhood base of 00 in S, (respectively H in S,). For f, g E F, we define f < g if and only if {n E N I f @ )> g(n)} is finite.
1.9. Proposition (Nogura and Tanaka [1988]). Let X be a regular c-space. Then X contains a copy of S, (respectively S,) if and only if it contains a closed copy of S, (respectively S2).
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Proof. The “if” part is clear. So we prove the “only-if” part for the case of S, (for the case of S,, the proof is similar). Since every subset of Xis a c-space, we can assume that X contains S, = {a}u {a,ln E N} u (N x N) as a dense subset. Since each T ( f ) is homoemorphic to S,, it is sufficient to show that some T ( f ) is closed in X . To show this, suppose that T(f)is not closed in X for any f E F. Let A ( f ) = S, - T(f).We first show co E U { A ( f ) - A ( f ) l f ~F}. Let U be a neighborhood of co in X . Choose an open set V in X such that 00 E V c F c U . Put M = { n E N la, # V } . Then M is a finite set. Put W = V u {a, I n E M } u U{T, I n E M},where T, = {n} x N. Then W n S, is a neighborhood of H in S,. Choose T ( g ) such that H c T ( g ) c W n S,. Then T ( g ) - T ( g ) is a nonempty subset of F, so let x E T ( g ) - T(g). Choose a neighborhood 0 of x and an h E F such 0 n T(h) = 8. Since = X , x E 0 c S, - T(h). Then, x E A(h) - A(h); hence, U n (A(h) - A(h)) # 8. Thus, co E U { A ( f ) - A ( f ) l f ~F}. Since Xis a c-space, there exists {x,ln
E
Nl = U { A o- A(f)IfE
F}
with co E {x,,ln E N}. For each n E N, choose f, E F such that A ( f , ) A ( f , ) 3 x,. ChoosefE F such that f > f, for all n E N. Since each A ( f , ) is contained in A(f) except finitely many points, A(f,) - A(f,) c A ( f ) . Let G be a neighborhood of co in X such that G n S, = T ( f ) . Then G n {x,ln E N} c G n A(f) = 8; hence, co 4 { x , ( n E N}. This is a contradiction. Hence, X contains a closed copy of S,. 0 From Propositions 1.8 and 1.9, we have the following corollary. 1.10. Corollary. Let X be a regular, sequential space. Then X contains a copy of S, (respectively S,) ifand only if it contains a closed copy of S, (respectively
SdThe following proposition was observed in Siwiec [ 19751. 1.1 1. Proposition. Let X be a FrPchet space. Then X i s strongly FrPchet ifand only if it contains no copy of S, . When X is regular, the preJx “closed” can be added to “copy”. Proof. The “only-if” part follows from Proposition 1.6(1), so we prove the “if” part. Suppose that X i s not strongly FrCchet. Then there exist a point x E X and a decreasing sequence {A, n E N} such that x E A, - A, for each n E N, but no sequence {x, I n E N} with x, E A, can converge to the point x.
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Since x E A,, there exists a sequence C , in A converging to x. Put in = 1. There exists an i, > in such that C , n Ai, is finite. Put K I = C , - A i l . By induction, for each n E N, we can choose a sequence K,, in Ain-l(in > i n - , ) converging to the point x such that K,,, n K, = 8 if m # n. Put X,, = { x} u U { K ,I n E N}. Then each K,, contains at most finitely many nonisolated points in Xn. Suppose that some Kn0contains infinitely many nonisolated points a,,, (m & N) in Xn. For each m E M, there exists a sequence L, in X - {a,,,} converging to a,,,. Let X , = U{L,,,lm E N}. Since {a,Im E N} converges to the point x , x E XI - {x}. Then there exists a sequence L in XI - { x } converging to x . But we can assume that L,,, c U{K, 1 n 2 m } for each m E M ,because each L, n K, is finite. Then L meets infinitely many K,,. Thus there exists a sequence { x,, I n E N } converging to x with x, E A,,. This is a contradiction. Then each K,, contains at most finitely many nonisolated points in Xn. So we can assume that each point except x is isolated in Xn.Hence, to show that X,, is homeomorphic to S,, it is sufficient to show that, for each finite F,, c K,,, x $ U{F,I n E N}. Suppose that x E U{F,In E N}. Then there exists a sequence { y,, I n E N} converging to x with y, E A,. This is a contradiction. Then Xn is homeomorphic to S,. Hence Xcontains a copy of S,. The latter part follows from Proposition 1.10.
0 1.12. Corollary (Olson [ 19741). Every regular, countably compact FrPchet space is strongly FrPchet. 1.13. Proposition. Let f:X + Y be a closed map such that X is normal. I f X below holds. When X isfirst countable, ( I ) and ( 2 ) are equivalent. ( I ) Y contains no closed copy of S,. ( 2 ) Every Bf - I ( y ) is countably compact, where B denotes boundary. is sequential, then (1)+(2)
Proof. To show that (1)+(2) holds, suppose that some B F - ' ( y ) is not countably compact. Since X is normal, there exists a discrete closed set { x,, I n E N} o f Bf - I ( y ) and a discrete open collection { U,,I n E N} in X with x, E U,. Since y E f ( U , ) - { y } , y is not isolated inf( U,,).But, as in the proof of Proposition 1.5 (2), Y is sequential, hence it is a closed setf(U,,) of Y. Thus there exists asequence C,, = { ynmI m E N} inf( U,,) - { y } converging to y. Since C,, c f (U,)for each n E N, { C,,I n E N} is hereditarily closure preserving. Thus, for each i E N, there exist only finitely many C,, each of which contains infinitely many points of Ci. Thus we can choose a collection { Ei I i E N} of infinite sequences in Y converging t o y such that E, c Cn(i), and
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Ei n E, = 8 if i # j. Since {Ei1 i E N} is hereditarily closure preserving in Y, it is easy to show that S = { y } u {Eil i E N} is a closed copy of S,. This is a contradiction. Hence, every Ef - ' ( y ) is countably compact. Next we show that (2)+(1) holds when Xis first countable. Suppose that Y contains a closed subset S,,, = { co} u U{T, I n E N}, where T, converges to 00 4 T,. Since each f -'(T,) is not closed in f -'(T, u {a})there , exists a sequence A, inf -'(T,,)converging to x, Ef -'(a) Since . f -'(T,) nf -'(a) = 8, x, E Ef -'(a)But . since Ef -'(co) is countably compact, the sequence { x , I n E N} has an accumulation point x E Ef -'(a) Let . { V, I m E N} be a neighborhood base of x in X . Then there exists a sequence {a, I m E N} such 1). Since the sequence {a, I m E N} that a, E V, n A,,(,) with n(m) < n(m converges to x, the sequence { f(a,) Im E N} converges to co. This is a contradiction. Hence, Y contains no closed copy of S,. 0
+
1.14. Corollary. Let f : Y + Y be a closedmap such that X is aparacompact, j r s t countable space. Then Y contains a closed copy of S, if and only if every Ef y ) is compact.
In the following theorem, the equivalence of (I), (2), and (4) is due to Michael [1972] (Morita and Hanai [1956], and Stone [1956] proved that this equivalence holds under Y being first countable). 1.15. Theorem. Let f : X
Y be a closed map such that X is a metric space. Then the following are equivalent. (1) Y is metrizable. (2) Y is strongly Frtchet. (3) Y contains no closed copy of So,. (4) Every Ef - ' ( y ) is compact.
Proof. (1)+(2) is obvious. (2)+(3) follows from Proposition 1.6 (I). (3)+(4) follows from Corollary 1.14. For the proof of (4)+(1), for example, see Nagata [1985, p. 2781. O Recall that a space Xis LaSnev if it is the closed image of a metric space. The following corollary is due to Harley [ 19721. 1.16. Corollary. Let X x Y be a Frtchet space such that X and Y are nondiscrete LaSnev spaces. Then X x Y is metrizable.
Proof. We prove that Xis metrizable. Since Y is nondiscrete, there exists an infinite convergent sequence C including its limit point. Suppose that X is
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not metrizable. Since Xis LaSnev, by Theorem 1.15, X contains a closed copy S of S,. Obviously, S x C contains a copy of S,. Thus X x Y contains a copy of S,. This is a contradiction by Proposition 1.6 (2). Hence X is metrizable. Similarly, Y is also metrizable. Thus X x Y is metrizable 0 1.17. Remark. More generally, when X x Y is a k-space with X and Y LaSnev, Tanaka [ 19791 proved that the following assertion holds under the continuum hypothesis (CH): X x Y is a k-space if and only if one of the following holds: ( I ) X and Y are metrizable. (2) X or Y is locally compact metrizable. (3) X is the countable union of locally compact metrizable, closed sets X, such that F c Xis closed whenever F n X, is closed for each n E N, and Y has also the same property.
Gruenhage [1980] showed that the above assertion is equivalent to a certain set-theoretic axiom weaker than (CH). (If X = Y, then the assertion holds without (CH).) 1.18. Proposition (Franklin [1967]). Let X be sequential. Then X is FrPchet ifand only if it contains no subspace which, with the sequential closure topology (i.e., all sequentially open sets are open), is a copy of S,.
Proof. The “only-if” part follows from Proposition 1.6 (2), so we prove the “if” part. Suppose that Xis not Frkhet. For A c X , let A’ be the set of all limit points of sequences in A. Then there exists a B c X with 8 # B‘. Since B‘ is not closed in X, there exists a sequence {x, 1 n E N} in B‘ converging to a point x E l? - B‘ such that the x,, are all distinct and x, # B. Let { V,,I n E N} be a collection of pairwise disjoint open sets of X with V,, 3 x,. For each n E N, there exists a sequence L, = {x,,lm E N} in B n V,, convergingtox,,.PutM = {x} u {x,ln E N} u u { L , I n E N}.Wenotethatno sequence of x,,’s converges to x. Thus, if M has the sequential closure topology, then M is a copy of S,. That completes the proof. 0 1.19. Lemma. Let C = { x, I n E N} be a sequence in X converging to x such that the x, are all distinct and x 4 C. If X - { x} is normal, then there exists a sequence { W,,I n E N} of open sets of X satisfying (1) and ( 2 ) below: (1) W, 3 x, for each n E N. ( 2 ) I f K , is a compact set of W, for each n E N, a n d i f K = U { K , l n E N}, then R c K u {x}.
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Proof. Let { V ,I n E N} be a collection of pairwise disjoint open sets in X such that V , 3 x , for each n E N . Let V = U { V , ~ ~ E Nand } , let D = P - ( V u {x)). Then C and D are disjoint closed sets of X - {x}. Since X - {x> is normal, there exists an open set W of X such that W =I C and W n D = 0. Let W , = V , n W for each n E N. Then { K i n E N} satisfies (I). We show that (2) holds.
RnV,
= K n V , n V , = €?,,nV, = K , n V ,
Hence, R n V = K. But R Thus
I?
=
Hence, (2) holds.
En
t
W, so R n D
P c ( E n V ) u {x}
=
0,thus R n P c
= K,.
V u {x}.
= K u {x}.
0
The following result for the hereditarily normal case is due to Kannan [19801. 1.20. Proposition. Let X be a sequential space. Suppose that X is a regular space in which every point is a Gator a hereditarily normal space. Then X is Frkchet fi and only if it contains no closed copy of S, .
Proof. The “only-if” part follows from Proposition 1.6 (2), so we prove the “if” part. Suppose that X is not Frkhet. Then X contains the subspace M = {x} u U{F,,InE N}, where F, = {x,} u L,, as in the proof of the “if” part of Proposition 1.18. If X is a regular space in which every point is a Gs,then there exists a sequence { V,ln E N} of open sets of X such that E+, t V, and {x) = V ,I n E N}. Without loss of generality, we can assume F, c V , for each n E N. Then it is easy to show that M is closed in X . If X is hereditarily normal, then, by Lemma 1.19, there exists a sequence { W, I n E N} of open sets of X such that each F, is assumed to be a subset of W , , and if F = U{F,InEN},thenPc F u {x}.ThenR = M;hence,Misclosedin X. ‘Therefore,M is closed in X for any case. Then M is sequential, hence has the sequential closure topology. Thus M is homeomorphic to S,. Then X contains a closed copy of S,. This is a contradiction. 0
n{
The following example, due to Franklin [ 19671, shows that the assumption of X in the previous proposition is essential. 1.21. Example. A compact sequential space (hence, it contains no closed copy of S, and no S2), which is not Frkchet.
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Proof. Let 9” be an infinite maximal pairwise almost disjoint collection of infinite subsets of N. Here, A, B c N are almost disjoint if A n B is at most finite. Let I,+ = { a FF[ E 9} u N with points of N isolated and neighborhoods of wF these subsets of @ contain oFand all but finitely many points of F; see Gillman and Jerison [1960, p. 791. Since each F u {aF} is compact, I,+ is locally compact. Let @* = I,+ u {co} be the one-point compactification of @.Then I,+* is compact. Any finite sequence in F converges to oF,and any infinite sequence in {oFIF E S}converges to co. Also, no sequence N’ in N converges to co, and if N’ converges to oF,then N’ - F is finite. Hence, any sequentially open set of I,+* is open in I,+*. Then @*is sequential. Clearly, 00 E N, but no sequence in N converges to 00. Hence, I,+* is not FrCchet. 0 The following theorem, due to Tanaka [ 19831, follows from Propositions 1.11 and 1.20. 1.22. Theorem. Let X be a sequential space. Suppose that X i s a regular space in which every point is a Gg,or a hereditarily normal space. Then the following are equivalent: (1) X is a strongly Frkchet space. (2) X contains no copy of S, and no S,. (3) X contains no closed copy of S, and no S,.
2. Spaces dominated by metric subsets The topology of a sequential space can be reconstructed by means of the “weak topology” determined by the cover of its metric subsets (= metric subspaces). Let X be a space, and let V be a cover of metric subsets of X.We consider metrizability of X dominated by %? in terms of the weak topology. 2.1. Definition. Let X be a space, and V be a cover (not necessarily closed or open) of X. X is said to be determined by V if A c X is closed in X whenever A n Cis relatively closed in C for each C E V. Here we can replace “closed” by “open”. We use “Xis determined by W’ instead of the usual “X has the weak topology with respect to V”. If V is an open cover of a space X , then X is determined by %?. We note that a space is a k-space (respectively, sequential space, c-space) if and only if it is determined by the cover of all compact subsets (respectively, compact metric subsets, countable subsets).
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2.2. Definition. Let X be a space, and let V be a closed cover of X . Xis said to be dominated by W if the union of any subcollection W of W is closed in X , and the union is determined by W. We use “X is dominated by V’ instead of “X has the weak (hereditarily weak, or Whitehead weak) topology with respect to V” used in some literature. A family { A , l a E A } of subsets of a space is said to be hereditarily closure-preserving, if U{B,I a E A’} = u { & I a E A ’ } for any A’ c A and B, c A, for each a E A ’ . If W is a hereditarily closure-preserving closed cover of X , then X is dominated by W. We remark that if a space is dominated by V , then it is determined by V , but the converse does not hold. The following proposition is routinely verified, so we omit the proof. 2.3. Proposition. (1) Let X be determined by a cover { X, I a}. ZfX, c Y, c X for each a, then X is determined by { Y, I u } . ( 2 ) Let X be determined by a cover {XuI a}. Zf each X, is determined by a cover {XaBI/3},then X i s determined by (XasIa,8 ) . (3) Let f : X + Y be a quotient map. If X is determined by a cover V , then Y is determined by f ( % ) = { f ( C ) I C E %}. (4) Let W = { X , I u } be a cover of a space X . Then X is determined by V if and only if the obvious map f : C{Xu1 a } + X is quotient. 2.4. Proposition. ( I ) If X is determined by a closed cover {X,In E N} of metric subsets (respectively compact metric subsets), then X is dominated by metric subsets (respectively compact metric subsets). (2) Every C W-complex is dominated by compact metric subsets. Proof. ( I ) Let Y, = U { X ,I i < n } for each n E N. Then X is dominated by a closed cover { I n E N} of metric subsets (respectively compact metric subsets). (2) Every CW-complex is dominated by the collection of its finite subcomplexes, each of which is compact metrizable. 0 As is well known, every space dominated by metric subsets is paracompact, perfectly normal (Morita [1953], Michael [1956]). For the proof of the following, see Borges [1966, Theorem 7.21. Concerning stratifiable spaces, see Chapter 10 by Tamano, or Nagata [1985, VI.81.
2.5. Proposition. Let X be dominated by a closed cover of metric subsets. Then X is stratifiable (hence X is paracompact, perfectly normal).
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2.6. Lemma. Suppose that X is a Frkchet space dominated by a closed cover {X,la< ? } . L e t Y, = X, - U { X a l b < a }foreacha < y . Then{Y,Ia < y } is a hereditarily closure-preserving closed cover of X .
Proof. Suppose { Y, 1 a < y} is not hereditarily closure preserving. Then, for each a, there exists a closed set Fa such that Fa c Y , and F = U{F,1 a < y} is not closed in X . Let x E E - F. Then there exist a sequence { x,, I n E N} in F converging to x and a collection {F,Jn E N} such that x, E Fan and a, < a,,+,. For each n E N, there exists a sequence L,, in X,, - U { X a (/? < a,,} converging to x,,. Let L = U{L,,ln E N} - {x}. Then x E t.Hence, there exists a sequence T in L converging to x. Let C = u { X a n I nE N}. Then T c C. Moreover, T n L,, is finite for each n E N and L,,, n Xan = 8 if m > n, thus each T n Xanis finite, hence closed in Xan.Then T is closed in X , so x E T. This is a contradiction. Thus { Y, I a < y} is hereditarily closure 0 preserving. 2.7. Proposition (Tanaka and Zhou [1984]). Let X be dominated by metric subsets. I f X is Frkchet, then X is LaSnev.
Proof. Let X be dominated by {X,la< y} of metric subspaces. Let Ya = Xa - U{X,ljl < a } for each a < y. Let X’ = Z{Y,Ia}, and let f : X’ + X be the obvious map. Then, by Lemma 2.6,fis a closed map. Since X’ is metric, X i s LaSnev. 0 The following example shows that the Frkchetness of X in the previous proposition is essential. 2.8. Example.
A countable CW-complex which is not LaSnev.
Proof. Let X be the quotient space obtained from the topological sum of countably many triangles, Aa,b,c,,, by identifying all segments with one segment. Then Xis a countable CW-complex. Since Xcontains a closed copy of S,, X is not Frkchet by Proposition 1.6 (2). Hence, X is not LaSnev by Proposition 1.3. 0 2.9. Lemma. Let X be dominated by a closed cover V = { Xa I a } of X.I f X is jirst countable, then for each x E X , x E Int U 9 for somejinite 9 c V. Proof. Suppose that x # Int U 9 for any finite 9 c V. Let { V ,I n E N} be a neighborhood base of x in X . Then there exist a sequence {x,,I n E N} in X
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and a subcollection {Xa(n)In E N} of % ?! such that xn
E
K n X a ( n ) - U{xa(i) Ii E
N}
with x, # x . But, {xnln E N} n Xa(,) is finite for each n E N. Then { x , In E N} is closed in X . This is a contradiction. 0
2.10. Theorem (Tanaka and Zhou [1984]). Let X be dominated by metric subspaces (respectively compact metric subspaces). Then the following are equivalent: (I) X is metrizable (respectively locally compact, metrizable). (2) X isJirst countable. (3) X contains no closed copy of S, and no S,. Roof. (1)+(2) and (2)-+(3)are obvious, so we prove (3)+( 1). By Proposition 2.5, Xis a regular space in which every point is a Gd,while Xis a sequential space which contains no closed copy of S, and no S,. Thus, by Theorem 1.22, X is strongly Frkchet. Then, by Proposition 2.7 and Theorem 1 . I 5, X is metrizable. The parenthetical part holds by Lemma 2.9. From Theorem 2.10 and Proposition 2.4, we have the following corollary. The result for case ( I ) is due to Franklin and Thomas [1977].
2.11. Corollary. Let X satisfy properties (1) or (2) below. Then X is locally compact metrizable if and only if it contains no closed copy of S, and no S,. ( I ) X is determined by a countable cover of compact metric subspaces. (2) X is a C W-complex.
3. Spaces with a-hereditarily closure-preserving k-networks We give a characterization of LaSnev spaces (=closed images of metric spaces) by means of “k-network”, and apply it to metrization theory.
3.1. Definition. A cover 9’ of a space X is called a k-network for X if, for any open set U in X and any compact set K c U, there exists a finite 9’c B such that K c (JY c U.Such collections have played a role in &,-spaces (i.e., regular spaces with a countable k-network) (see Michael [1966]), and K-spaces (i.e., regular spaces with a cr-locally finite k-network) (see OMeara [1971]). A k-network B for a space Xis called closed if each member of 9 is closed in X .
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LaSnev [1966] characterized closed images of metric spaces as Frechet spaces with an almost refining sequence of hereditarily closure-preserving covers comprising a network for the space; for the details, see LaSnev's paper [1966]. Foged [1985] gave a new simple characterization for the closed images of metric spaces as Frechet spaces with a a-hereditarily closure-preserving closed k-network. We shall give the proof due to the Foged's paper.
3.2. Lemma. Let f : X -, Y be a closed map with X metric. Then every compact set of Y is the image of some compact set of X . Proof. Without loss of generality, we can assume Y is itself compact. Then Y contains no closed copy of S,*. Hence, every BF - I ( y ) is compact by Proposition 1.13. For each y E Y , take py E f - I ( y ) and put C, = Bf - I ( y ) if B f - ' ( y ) # 0,andC, = { p , } i f B f - ' ( y ) = 0 . L e t C = u { C , l y ~Y}.Then C is closed in X . Hence g = f I C is a closed map such that each g - ' ( y ) is compact and g ( C ) = Y. Then C = g - ' ( Y ) is compact and f ( C ) = Y. 0 3.3. Lemma. Let 9 be a hereditarily closure-preserving closed family in a Frdchet space X . Then the collection { nW*I W* is a.finite subfamily of .%} is also hereditarily closure-preserving. Proof. Assume the contrary. Then there exists a collection (9: Iu E A 1 consisting of finite subfamilies of 9 such that, for some G, c S = U{G, I u E a } is not closed in X . Then, there exists a sequence converging to a point not in S with x, E G,(,,).Since each G,(,, c ()We(,,),there exists {R, I m E N) c W with xn(,)E R,. Since { R, I m E N} is hereditarily closurepreserving, {x,(,) I m E N} is discrete in X. This is a contradiction. 0
n9:.
The following is easy, so we omit the proof. 3.4. Lemma. Let W be a hereditaril-v closure-preserving family in a space X . ZfL = { x, I n E N} is a sequence converging to x with L $ x, then there exists an m E N such that { x , I n > m} meets only jinitely many R E 9.
3.5. Lemma. Let X be a Frdchet space with a a-hereditarily closure-preserving k-network 9 = U{P,,ln E N} with 9,c 9'n+l. For x E X, let L be a sequence converging to x with L $ x, and let U be a neighborhood of x. Then there exists an n E N such that L is eventually in Int U { P E 9, 1P c U}.
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Proof. Assume the contrary. Let 9;= { P E 9,, 1 P c U } for each n E N. Then we can choose a subsequence {x,, I n E N} of L such that x, E U Int UP,,’. Since x, E U - UP,,’, there exists a sequence {x,,I m E N} in U - upn’converging to x,. Let S = {x,,,,,l m, n E N}. Since x E S and x 4 L , there exists a sequence T = {x,(,)~(,)I j E N} in S, with n ( j ) < n ( j + l), converging to x. Since 9 is a k-network for X, T is eventually in some UP;. But, for n ( j ) > m, x,,(,),,,(~)4 UP;.This is a contradiction. 0
3.6. Theorem (Foged [1985]). A space X is LaSnev if and only [f X is u Frtchet space having a a-hereditarily closure-preserving closed k-network. Proof. “Only-if”: Let f:M --* X be a closed map such that M is metric. Then Xis Frtchet by Proposition 1.3. Let W be a a-discrete base for M.Since f i s closed, 9 = { f(B)1 B E W }is a a-hereditarily closure-preserving closed cover of X. Moreover, by Lemma 3.2,B is a k-network for X. Hence X has a a-hereditarily closure preserving closed k-network 9. “If”: Let B = U { 9 , I n E N} be a a-hereditarily closure-preserving closed k-network for X. Here we can assume that 9,c 9,+, , and each 9,is closed under finite intersections by Lemma 3.3. For P E P,,, let R , ( P ) = P Int u ( Q E P,I P Q}, and 49, = { R , ( P ) 1 P E .9’,,). First we show that the following assertion (*) holds: (*) For x E X , let L = {x,,lnE N} be a sequence converging to x with L $ x and U be a neighborhood of x. If L is eventually in Int u{PE I P c U } , then L is eventually in Tnt UyP; and U.9: c IT, where 9,= { R E W,I R n L is infinite}. To show that (*) holds, let V = Int Up, - u{Q E 9, u W,,I Q n L IS finite}. Then, by Lemma 3.4, L is eventually in V. We show that V c UW;. Let v E V . Then j E Q E P,, implies Q n L is infinite. Thus, by Lemma 3.4, 9,, is point-finite at y , hence P( y) = E 9, I y E Q} E 9,. Moreover, y $ u{Q E 9,l P ( y ) c t Q } , then y E R,(P(y ) ) . Since y E V , R,,(P(y ) ) n L is infinite; thus R , ( P ( y ) ) c 9;. Then y E R , ( P ( y ) ) c UWA. Hence V c (J.%’i. Then L is eventually in Int UW;. Next, we show UW: c U.Note that, for each R,(P) c So;, there exists a Q E 9“such that R,,(P) c P c Q and Q c U . Indeed suppose not. Then Int U{QE PnlQ c U } c Int U{Q E P,, I P Q Q} c X - R,,(P).Thus L is eventually in X - R,(P). Thus L f\R,(P) is finite. This is in contradiction with L n R J P ) being infinite. Then U.9; c 0’. Hence assertion (*) holds. Now, for each n E h, let W,*= W,u {X - Int UW,,} and put a,*= { R, I a E Z,,}.We say that a collection N of subsets of X is a net at x E X if x E O N and every neighborhood of x contains a member of N.
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Let M = {a E ll Z,,l {R,(,,)InE N} is a net at some x E X } and M be a subspace of the product space of the discrete spaces I,. Then M is metrizable. We definef: M -, X by f(a) = x if and only if {Ru(n) In E N} is a net at x . We show that f is a closed continuous surjection. (i) f ( M ) = X: Let x E X . If x is isolated in X , { x }E 9,, for some n E N; hence, R,(x) = { x } .If x is not isolated in ,'A there exists a sequence L in Xconverging to x with L $ x . For each n E N, choose R,(,,) E W,* such that A,,,,,n L is infinite. If it is impossible, choose R,,(,,) E W,* such that Rut,,) 3 x . In any case x E I?,(,,). Then it follows that {Rut,,) I n E N} is a net at x by Lemma 3.5 and assertion (*). Hence, !'(a) = x. (ii) f i s continuous: Let U be open in X . Let a E ~ - ' ( Uandf(a) ) = x. Then there exists an m E N such that x E R,(,,,, c U. Thus a E ( T I t ( m ) = a(m)} c f - ' ( U ) . Thenf-'(U) is open in M. (iii) fis closed: Let F be closed in M. Suppose that x E f ( F ) - f ( F ) . Then there exists a sequence { x,, I n E N} inf(F) converging to x. For each n E N, choose a,,E F nf - ' ( x n j . By Lemma 3.4, there exists an m E N such that {R E 92: I R n {x,,I n 2 m} # 0} is finite. But for each n E N, Rum(,) E 9: contains x,,. Thus there exists an infinite subset N, of N such that, say a(l) = a,,(]) for all n E N,. By induction, we can choose a sequence { N l )i E N) of infinite subsets of N such that N 1 + , c N i , and o ( i ) = a,(i) for all n E N i . For each i E N, choose n ( i ) E Ni with n ( i ) -= n(i + 1). Then the sequence {a,,,,,l i E N} in F converges to a = (a(l), a(2), . . .), hence a E F. We show a ~ f - ' ( x ) . For each i E N and n E N i , x,, E RUci,= Thus x E for each i E N. Let U be a neighborhood of x in X. Since L = {x,,(~)I i E N} converges to x , by Lemma 3.5 and assertion (*), there exists a k E N such that L is eventually in U { R E WkI R n L is infinite} c U.But, c U. Then if i Z k, then n ( i ) E Ni c Nk. Thus Ru(k)= Run,,)(k) {R,,,,) I n E N} is a net at x ; hence,f(a) = x . Thus, x E ~ ( F This ) . is a contradiction. Thus, f ( F ) = f ( F ) . Then f is closed. This completes the proof. 0
By Theorems 3.6 and 1.15, we have the following theorem. ( l ) 4 2 ) is due to O'Meara [1970], and (1)-(3) is due to Guthrie's unpublished result.
3.7. Theorem. For a regular space X,the following are equivalent. (1) X is metrizable. (2) X is a first countable space with a a-locally finite k-network. (3) X is a first countable space with a a-hereditarily closure-preserving k-network.
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3.8. Theorem (Burke, Engelking and Lutzer [1975]). A regular space X is metrizable if and only if it has a a-hereditarily closure-preserving base. Proof. The “only-if” part follows from the well-known fact that every metric space has a a-locally finite base. “If”: From Theorem 3.7, it suffices to show that Xis first countable. Let = U{a,, I n E N} be a a-hereditarily closure-preserving base for X. Let x E X be a nonisolated point in X . We show that for each n E N, { E E a,,I x E a } is finite. Suppose that there exists an infinite subcollection {E,,In E N} of some a,, with X E E n . For each n E N, let G,, = X U { B I E E ~ , , , X $ B } . T ~=~ n~{{G~, ,}l n ~N} andeachG,,isopeninX. Let H,,= &-, n E,, n G,,for each n E N, where H, = X. Since x is not isolated in X , x E H I- { x } . But x 4 H,,- H,,,for any n E N. On the other c En for each hand, HI- { x } = U{H, - H,,+,lnE N} and H,,- H,,,, n E N. Thus x E H,,- H,,,, for some n E N. This is a contradiction. Hence, { B E Ix E B } is at most countable. This shows that X is first countable.
0 4. Spaces with certain point-countable covers Mainly we consider metrization of M-spaces with certain point-countable (not necessarily open or closed) covers. Here a cover is point-countable if every point is in at most countably many elements of it. 4.1. Lemma. Let S be a point-countable cover of a space X . If X is determined by { U 9 1 9 c 8 is finite}, then every countably compact K c X is covered by some finite 9 c 8.
Proof. Suppose not. For each x E X , let {P E 8 I x E P} = {P,,(x)I n E N}. We can choose a sequence A = { x,, I n E N} in K such that x,, # ? ( x i ) for i, j c n. Since K is countably compact, A has an accumulation point x . Since A - { x } is not closed in X , there exists a finite 9 t 9 such that A n ( U 9 ) is infinite. Then some F E 9 must contain infinitely many x,. Then F = ? ( x i ) for some i andj, and there exists an n > i,jsuch that x, E ? ( x i ) . This is a contradiction. In the following proposition, the result for case (1) (respectively (2)) is due to Foged [1985] (respectively Gruenhage, Michael and Tanaka [1984]). A map f:X -+ Y is an s-map if every f - I ( y ) is separable.
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4.2. Proposition.
Each of the following implies that X has a point-countable k-network. (1) X is a Lainev space. ( 2 ) X is the quotient s-image of a metric space. ( 3 ) X is dominated by metric subsets.
Proof. (1): By Theorem 3.6, X has a a-hereditarily closure-preservingclosed k-network U ( 9 , In E N}. Let D, = {x E X l 9 , is not locally finite at x } for each n E N. Then each D, is a closed discrete subset of X. Indeed, let ( x , 1 m E N} c 0,. Since F,is not point-finite at each x,, for each m E N there exists an F,,, E 9, - {FkI k < m } such that x, E F,. Then {x, 1 m E N} is closed and discrete in A’. But, since Xis Frechet, D, is closed and discrete ’{F- D,, IF E F,}u { { x }Ix E D,,}, and 9 = U{F,,’ In E N}. in X . Let 9,= Then 5 is a point-countable k-network for X. (2): Letf: M + X be a quotient s-map with M metic. Let @ be a a-locally finite base for M , and let 8 = {f(B)I B E B } .Then 9’is point-countable. To show that 8 is a k-network, let U be open and K be compact with K c U . We note that g = f l f - ’ ( U ) is quotient andf-’(U) is determined by its open cover % = { B E @ I B c f - ’ ( U ) } . Thus, by Proposition 2.3 (3), U is determined by a point-countable cover 9’= {f(B)I B E B } . Then, by Lemma 4.1, K c lJ9 c U for some finite 9 c 8‘.Hence, 8 is a point-countable k-network for X. (3): Let X be dominated by a closed cover { X u1 a } of metric subsets. For each a, let Ya = Xu - u { X , I < a}. Let B, be a a-locally finite base for F, and let 3: = B, n Y,. Then, B = U{B:I a } is point-countable. To show that B is a k-network, let U be open and K be compact with K c U.Then K meets only finitely many Y,. Indeed, suppose that there exists a D = {.u, I n E N} with x, E K n Yun,where a, < a,,+’. Then each D n Xu.is finite. Hence D is closed discrete in X . But, since K is compact, D has an accumulation point. This is a contradiction. Let { a ] Ya n K # 8) = ( a , , a,, . . . , a,}. For each a,, there exists a finite Fa,c a,, such that Y,, 17 K c c U. Hence,
u9,,
K c U{Fa,n Y,,li= 1 , 2 , .
,
.,n}
c U.
Then K c U5F c U for some finite 9 c B. Hence, B is a point-countable k-network for X . 0 The following example shows that it is impossible to add the prefix “closed” to “k-network” in the previous proposition.
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4.3. Example. A LaSnev, CW-complex which has no point-countable “closed” k-networks. Proof. Let X = {m} u u{LaIo!< w , } (La# co) be the quotient space obtained from the topological sum of w , many closed intervals [0, 11 by identifying all zero points with a single point co. Then X is a LaSnev, CW-complex. Suppose that Xhas a point-countable closed k-network 9. Let 8’= {FE 9IF 3 m}. Let 9, = ( F E 9’ 1 F meets infinitely many La}. Since 9, is countable, let 9, = {FnI n E N}. For each n E N, choose x, E F, such that x, # m and x, is not in any Lacontaining x, through x,-~. Let 9, = 9‘- 8,.Then the union of elements of F2meets at most countably ? the points many La.Choose Lawhich does not meet any element of 9-nor x,. Let K = L, u (00). Let G = X - { x , l n E N}. Then K i s compact and G is open with K c G. Since every element of the k-network 9is closed, no element of 9 both contains infinitely many points of K and is contained in G. Thus 9 is not a k-network. This is a contradiction. Hence, X has no point-countable closed k-networks. il 4.4. Remark. More generally, we have the following due to Tanaka [1980]: Let f:X + Y be a closed map with X metric. Then Y has a point-countable closed k-network if and only if every Bf -‘( y) is Lindelof. Indeed, for the “if” part, we can assume that every f - I ( y ) is Lindelof. Then this part is proved as in the “only-if’’ part of Theorem 3.6. For the “only-if” part, Y contains no closed copy of S,, in view of the proof of the previous example, where S,, is similarly defined as S,. Then, for each y E Y , any uncountable subset of Bf - I ( y ) has an accumulation point as in the proof ) metric, Bf - I ( y) is Lindelof. of (1)-+(2)in Proposition 1.13. Since Bf - - ‘ ( y is
4.5. Definition. For a space X , we consider the following conditions. (C,) X has a point-countable cover 9 such that if x E U with U open in X , then there exists a finite 9 c B such that x E Int U 9 c U 9 c U , and xE
09.
(C,) (C,) (C,) closed K t (C,)
Same as (Cl), but without requiring x E 0 8 . Same as (C,) with U = X - ( y } . X has a point-countable cover 9 such that if K c X - { y} with K countably compact, then there exists a finite 9 c 9 such that c X - {y}. Same as (C,) with K compact.
u9
4.6. Proposition. (C, ) +(C2) +(C3) +(C, ) +(Cs ). When X is a k-space, (C,)++(C,).
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Proof. We prove only (C3)+(C4) (respectively (C5)-4C4)for X being a k-space). Let B be a cover as in (C,) (respectively (C5)).For y E Y, let By = { U S I f c B is finite with U S j y } . Since X - { y } is determined by By,B satisfies (C,) (respectively (C,)) by Lemma 4.1. 0
A cover B of a space Xis called separating if, for any points x , y E X with x # y , there exists a P E B with x E P c X - { y } . A space X has a Gs-diagonal if its diagonal is a G,-set of X 2 ;equivalently, there exists a sequence {%,,In E N} of open covers of X such that for any points x , y with x # y , y 4 St(x, @,,) for some n E N (Ceder [1956]). We note that every metacompact space with G,-diagonal has a point-countable separating open cover. A space X is called subparacompact if every open cover of X has a a-locally finite closed refinement. Every a-space (i.e., space with a a-locally finite closed network), more generally, every semi-stratifiable space is a subparacompact space with a Gs-diagonal; see Creede [1970]. Let X be a space. For each x E X , let T, be a finite multiplicative family of subsets of X containing x . The collection U { T, I x E X} is called a weak base for X if F c X is closed in X if and only if, for each x 4 F,there exists a Q ( x ) E T, with Q ( x ) n F = 0;see Arhangel'skii [1966, p. 1291. 4.7. Proposition. (1) If X has a point-countable separating open cover, then X satisfies (C,). (2) (Burke and Michael [1976]). If X is a a-space (more generally, a subparacompact space with a G6-diagonal), then X satisfies (C,). (3) If X has a point-countable k-network, then X satisfies (C5). When X is a k-space, X satisfies (C,). (4) If X has a point-countable weak base, then X satisfies (C,). ( 5 ) If X is the quotient s-image of a metric space, then X satisfies (C,).
Proof. (1) is clear. (3) follows from Proposition 4.6. (5) follows from (3) and Proposition 4.2. So we prove (2) and (4). (2): Let d = U { d nI n E N} be a a-locally finite closed network for X , with each d,,3 X . For each n E N and W c d,,,let P,,(W) = nW - U ( d n - a), and let B,, = {P,,(W)IW c d,,}.Let B = U{Y,,In E N}. We show that B satisfies (C3). Since each 9"is disjoint, B is point-countable. Let x E U = X - { y } . Then there exists an n E N such that y E A but x 4 A for some A € & , , . For each Z E X , let W z= { A E ~ ~ ~ z ELet A } . f = {P,,(W)1 1c g,}. Then it is easy to see that S is finite, and U S c U . To show that x E Int(US), let V = X - U ( d n - a,). Then x E V and V is open in X . So we need only show that V = U S . Let z E U S . Then
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z E P,(W) for some W c B,, so z E P,(B) c X - U ( d , - W)c X U ( d n - W,)= V. Conversely, let z E V. Then Bz c a,,so Pn(Bz)E 9. Since z E P,,(Wz), z E U s . Hence V = UP. Then X satisfies (C3). For a more general case where X is a subparacompact space with a Gs-diagonal, X also satisfies (C,) in the same way as in the above. (4): It is sufficient to show that Xis a sequential space with a point-countable k-network. Let T, = U{T, I x E X }be a point-countable weak base. For x E X,let T, = {Q,,(x)I n E N}. To show Xis sequential, let F be not closed in X.Then there exists an x # Fsuch that Q,,(x) n F # 0 for each n E N. Let x, E Q,,(x) n Ffor each n E N. Then the sequence {x,, 1 n E N} in Fconverges to the point x not in F. Thus X i s sequential. Next, to show that T, is a k-network, let U be open in X. Since X is sequential, so is U . While any sequence A in X converging to y is eventually in any element of T,. Indeed, suppose that there exist Qn(y ) E T, and a subsequence B of A with Qn(y ) n B = 0.Then, for any p $ B, there exists a Q , ( p ) E Tpwith Q,(p) n B = 0.Thus B is closed in X.This is a contradiction. Then U is determined by a point-countable cover { T E Tcl T c V } . Thus, by Lemma 4.1, T, is a k-network for X. For the proof of the following lemma due to MiEenko [1962], see, for example, Nagata [1985, p. 4041. 4.8. Lemma. Let 8 be a point-countable collection of subsets of X . Then there exist at most countably many minimal finite covers of X by elements of 8.
Burke and Michael [ 19761 proved the following proposition for compact spaces. 4.9. Proposition (Balogh [19791). Every countably compact space X satisfy-
ing (C,) (equivalently, (C,)) is metrizable.
Proof. Let 8 be as in (C,).We assume that 8 is closed under finite intersections. Let 8’ = U{V c 8 I V is a minimal finite cover of X}.Then by Lemma 4.8, 8‘is countable. We show that 8’ also satisfies (C3). Let x E X - { y } . Then there exists a finite 9 c 8 such that
U 9 c X - {y}. We can assume that x $ Int U s ’ for any 9‘ 9. Then it suffices to show x E Int
U 9
that 9 c 8’.Since Int
c
UP
$
U { 9 - {F}} for each F E 9, there exists
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an xFE F n I n t ( u 9 ) such that xF4 - { F } } .Let A = { x F I F E 9). Since 9 satisfies (C3), for each z E X - A , there exists a finite szc 8 such that z E Int UFz c c X - A.
u9,
D e f i n e W c B b y W = 9 ~ ( ( U { 9 ~ l z E X - A } ) . L e t W * =( U 9 1 9 t W is finite}. Then {Int R I R E a*}is an open cover of X ; hence, Xis determined Since 8 is point-countable, by this open cover. Thus Xis determined by W*. by Lemma 4.1, W has a minimal finite subcover Y. Clearly, Y c 8’.If F E 9, then F is the only element of W containing xF, so F E 9.Hence 9 c Y c 9’. Let A’Z = { X - I n t ( U 9 ) l F c 9‘is finite}. Then Jl is a countable, closed cover of X such that { x } = n { M E A I x E M } for each x E X . F o r x E X , l e t { M E A ’ Z I x E M } = {M,,lnEN}andL, = n { M i l i < n } for each n E N. Since X is countably compact, for each neighborhood U of x , x E Lj c U for some i E N. This implies that the collection of all finite intersections of elements from A? is a countable network for X . Hence, X is a compact space with a countable network. Then, as is well-known, X is metrizable. 4.10. Lemma. Let X be a c-space, and A c X . I f 8 is a point-countable collection of subsets of X , then there exists at most countably many minimal finite subcollections 9 c 9 such that A t Int(U9), where minimal means that A $ I n t ( u F ’ ) if9’ 5 9.
Proof. Let 0 be the collection of all finite subcollection of 9.For each 9 E @, let A(9)= { B c X I B c I n t ( u F ) , and if 9’ s 9,then B 4 Int(U9’)). Suppose that, for uncountably many 9 E 0,A E A(9). Since@ = U{@,InEN}where@, = {9~@119 =1 n},thereexistannEN and an uncountable Y c @, such that A E A(9)for each 9 E Y . Now, let 9be a maximal subcollection of 9 such that W c 9for uncountably many SEY.Obviously,O < 1 9 1 < n.LetY* = { F E Y I W c F } . L e t F E Y * . Then A E d ( 9 ) and 9 5 9; hence, A c t Int(UW). Let x E A - Int(UW), and E = X - UW.Then x E E. Since Xis a c-space, there exists a countable L t E with x E t.While, x E Int(U9). Then L meets some P E 9. But, 9 is point-countable, and Y* is uncountable. Then L meets some PoE 9which is a member of uncountably many 9 E Y*. But Po n L # 8 and L n UW = 8,so Po4 9. Then W 5 W u {Po}. Moreover, 43 u {Po}c 9 for uncountably many 9 E Y . This is in contradiction with the maximality of W.Hence, for at most countably many 9 E 0,A E A’Z(9). Thus the lemma holds. 0
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4.11. Theorem (Burke and Michael [1972, 19761). Let X be a regular space. Then the following are equivalent. (( l)tr(2) holds without the regularity of X.) (1) X has a point-countable base. (2) X satisfies (Cl). (3) X is a k-space satisfying (Cz). (4) X is a c-space satisfying (Cz).
Proof. (1)+(2) is clear. (2)+(3): Since X satisfies (Cl), X is first countable. Thus (3) holds. (3)+(4): By Propositions 4.6 (1) and 4.9, every compact set of Xis metrizable. Then a k-space Xis sequential. Thus, by Proposition 1.8, Xis a c-space. (2)+(l): Let B be a cover of Xas in (Cl). Let @ be the collection of all finite subcollections of 9’. For each 9 E 0,let A($) = { A c X I A c Int US, and if 9’ 9, then A
s
V ( 9 ) = Int
cl
Int
US’},
U ( d ( 9 ) n 9).
Let Y = { V ( 9 ) I 9 E @}. Then we show that Y is point-countable. If E V ( 9 ) , then there exists an A E A(9)n 9 with x E A. Since 9 is point-countable, it is sufficient to show that, for each A E d ( F ) , {FE @ I A E A(9)}is at most countable. But, since X satisfies (Cl), X is E @ I A E A(9)) is at most countfirst countable. Then, by Lemma 4.10, (9 able. Next we show that Y is a base for X. Let x E Xand W be a neighborhood of x in X. By (C,), there exists an 9 E @ such that x E Int (J9 c W. Here we can assume that if 9‘5 9, then x $ Int Since x E Int by (Cl), there exists a W E @ such that x E Int UW, UW c Int US, and x E nW. Obviously, if C E W, then C E A(9).Then C E A ( 9 )n 9, thus Int UW c V ( 9 ) .Hence, x E V ( 9 ) c W. Hence, Y is a point-countable base for X. (4)+(2): Let 9be a cover of Xas in (Cz).Let @ be the collection of all finite subcollections of 9. For each 9 E 0,let x
u9’.
A(9) = {x
E
u9,
XI x E Int U9, and if F’ 9, then x $ Int(UF’)).
For each P E 9, let P’ = P u ( U { d ( 9 ) 1 9E 0,P E F}.Then P‘ c P, because if x E A(9)for some 9 E @ with P E 9, then x E Int(U9), but x $ Int(U(9 - {P})); hence, x E I‘. Let 8‘ = {P’l P E B}. Since X is a c-space and 9 is point-countable, by Lemma 4.10, if x E A(9),then {FE @ l x E d ( 9 ) is ) at most countable. This implies that 8’is pointcountable. Thus it is sufficient to show that 8’ satisfies (Cl). Let x E X
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and W be a neighborhood of x . Choose an open set U in X with E U c 0 c W. By (C2), there exists an 9 E 0 such that x E Int U 9 and U 9 c U . Here we can assume that x E A(9).Let 9'= {P'l P E 9}. Then x E Int U 9 ' ) . Since x E A(9),x E Moreover, if P E 9, then P' c P c c 0 c W; hence, US' c W. Then 8' satisfies (Cl). This 0 completes the proof. x
n9'.
The following theorem is due to Guenhage, Michael and Tanaka [1984]. The equivalence of (1) and (2) for paracompact M-spaces was obtained by Burke and Michael [1976]. Recall that a space is an M-space if and only if it admits a quasi-perfect map onto a metric space. Concerning M-spaces, see Morita [1964, 19721, or Nagata [1985, VI.8, V11.2 and 31. 4.12. Theorem.
The following are equivalent. (1) X is metrizable. (2) X is an M-space satisfying (C3). (3) X is an M-space satisfyng (C.,).
Proof. (1)+(2) is obvious, and (2)+(3) follows from Proposition 4.6. (3)+(2): Since Xis an M-space, there exists a quasi-perfect mapf: X + M with M a metric space. Since eachf-'( y) satisfies (C4),f-'( y) is compact and metrizable by Proposition 4.9. Thenfis perfect. So Xis paracompact. Hence X is regular. First we show that X is first countable. Let x E X and f ( x ) = y. Let { U, I n E N} be a neighborhood base of y in Y with U,+, c U,. Sincef-'( y ) is first countable, there exists a sequence { V ,1 n E N} of open sets in X such that { V , n f-'(y ) I n E N} is a neighborhood base of x inf-'( y). Since Xis regular, there exists a sequence { W,I n E N} of neighborhoods of x in X such that @"+, c W, c f - ' ( U , ) n V,. Let X,E W, for each n E N. Sincefis closed, the sequence { x, I n E N} has an accumulation point x' in X, hence x' = x . Thus { W , 1 n E N} is a neighborhood base of x . Thus, X is first countable. Now, we show that every cover 8 as in (C,) satisfies (C3).Let x E X - { y} for some y E X. Let Py = { P E 81P $ y } . To show that x E Int U 9 for some finite 9c Py, suppose not. For each countable C c X , let { P E P ~ I CP #~
S}
= {P,(C)lnEN}.
Since X is first countable, we can choose a sequence {C, I n E N}, C , = { x } , such that x E c,,and C, n p.(q) = 8 for i,j .c n. The last condition implies that no P E Pymeets infinitely many C, . Let A, = U{C, I k 2 n } for each n E N. Since x E 2,for each n E N, there exists a convergent sequence L in
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X - { y} meeting every A,, hence meeting infinitely many C,. But L is covered by some finite 9 c By,so some P E Bymeets infinitely many C,,. This is a contradiction. Hence, B satisfies (C3). (2)41): Since Xis an M-space, there exists a quasi-perfect mapf: X + M with M a metric space. Since eachf-'( y ) satisfies (C3), it is compact. Thus fis perfect. Hence Xis a paracompact M-space. Then, if Xhas a G,-diagonal, Xis metrizable (see Okuyama [1964], Borges [1966], or Nagata [1985, VII.31). But, sincef' is perfect, X 2 is a paracompact M-space. Also X 2 satisfies (C3). Then it is sufficient to show that every closed set of X is a Gs. Now we show that every closed subset A of X is a Gsin X: Let B be a point-countable cover of X as in (C3). We note that X is first countable in view of the proof of (3)+(2). Then, by Lemma 4.10, for each U c X there exist at most countably many minimal finite 9 c B such that ( U n A) c Int US. Label these collections {S(U,n) I n E N}. For U c X and k E N, let U(k) = n{Int U S ( U , j ) l j < k } n U . Let { V , l n E N} be a sequence of a locally finite open covers of M such that, for each y E M, {St(y , V,,)1 n E N} is a neighborhood baseofy in M. For eachn E N, let a,, = { f - ' ( V ) I V E For each n, k E N, let W,k = U ( k )1 U E a,,}.Then, obviously, each W,,k is an open subset of Xcontaining A. We show that A = W,,I n, k E N}. Let x E X - A, and S, = f-'(x). Since S, n A is compact and 9 satisfies (C3), there exists a finite f c B such that
c}.
u{
(S,n A)
c Int
n{
US c U S c X
- {x}.
Let G = (X - A) u (Int US). Then G is an open set containing S,. Since f is closed, St(x, 4,,)c G for some n E N. Let { U E a,,I U 3 x} = {Ul, U2, . . . , Us}. Then, for any i < s,
U, n A c St(x, 4,) n A so we can choose a minimal
c
G n A c Int
US,
Slc S such that V , n A c Int UR. Then = max{k, I i < s}. We note that if
8 = S(U,, k,) for some k, E N. Let k i
< s, then
V,(k) c USG(V,,k,) = U% c US. Let U E 4,. Since x 4 U S , if x E U , x 4 U,(k).If x 4 U,x 4 U(k). Then x 4 U ( k ) for any U E 4,,.Thus x 4 Wnk.This completes the proof. 4.13. Remark. We shall give some results by means of condition (C5). A space Xis called 0-rejinable(or submetacompact) if for every open cover 4 of X , there exists a sequence {4,, I n E N} of open refinements of 4 such that
each point of X is in at most finite number of elements of some 4,,.Such a
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sequence of open covers is called a &refinement of 4. Metacompact spaces, subparacompact spaces are O-refinable, and every countably compact 6-refinable space is compact (see Wicke and Worell [1965] or Nagata [1985, V.41). A space X is called a wA-space if there exists a sequence {a,, I n E N} of open covers of X such that if x E X and x, E St(x, %,) for each n E N, then the sequence { x,, I n E N} has an accumulation point y in X. When y = x (equivalently, {St(x, @,) I n E N} is a neighborhood base of x), such a space is called developable. Developable spaces and M-spaces are wA-spaces. Hodel [ 19711(respectively Shiraki [1971]) proved that every regular, O-refinable wA-space with a point-countable separating open cover (respectively point-countable weak base) is a Moore space (i.e., regular, developable space). By means of condition (C,), we can generalize these results as follows (see Tanaka [198.]).
Proposition 1. Every regular, 6-refinable wA-space X satisfying (C,) is a Moore space. Indeed, since X is a regular, O-refinable wA-space, there exists a sequence {@,,In E NjofopencoversofXsuchthatforeachx E X , C , = r){St(x, %“)I n E N} is compact, and {St(x, 42,) In E N} is a neighborhood base for C, (Burke [1970]; for the proof see Gruenhage [1984, p. 4321, for example). For each i , j E N, let qij= 4fi x q.Forp E X 2 ,let C, = n{St(p, ai,)Ii , j E N}. Then C, is compact, and {St(p, 4!lij) I i, j E N} is a neighborhood base for C,. For each i E N, let {4/I k E N} be a &refinement of 9Zi. For each i, j , k, 1 E N, let = %/ x q‘. Now we show that each closed set of X 2 is a Gd. Each C, is metrizable by Proposition 4.9. Hence, Xis first countable. Let 8 be a cover of X as in ((2,). We assume X E 8.Then 8 satisfies (C,) in view of the proof of (3)+(2) in Theorem 4.12. Thus X 2is a first countable space with a point-countable cover 9 x 9 satisfying (C,). Let A be a closed set in X 2 . Then, by Lemma 4.10, for each U c X 2 , there exist at most countably many minimal finite 9 c 9 x 8 such that U n A c Int UP.Label these collections { P ( U ,n)l n E N}, and let U(m) = n{Int U 9 ( U , n ) ( n < m} n U . For each i , j , k, I, m E N, let W ( i , j ,k, I, rn) = U{U(m)I U E a:}. Then we can show that A = W(i, j , k, 1, m) I i, j , k, I, m E N} by the same way as in the proof of (2)+(l) in Theorem 4.12. Hence, A is a G,-set of X 2 . Thus X has a G,-diagonal. But, X is a regular, &refinable wA-space. Hence X is a Moore space by Hodel [1971],Shiraki [1971], or Chapter 9 by Nagata, Theorem 2.16.
4!lt
n{
As other results by means of condition (C,), the following propositions are shown in Gruenhage, Michael and Tanaka [1984].
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Proposition 2. Every countably bi-k-space X (in the sense of Michael 119721) satisfying (C,) satisfies (C,). When X i s a regular space with apoint-countable k-network, X has a point-countable base. Proposition 3. Every regular, strong C-space (in the sense of Nagami [1969]) satisfying (C,) is a 0-space. Combining Theorem 4.12 with Proposition 4.7, we have the following metrization theorem on M-spaces. The result for (1) is well-known. When X is paracompact, the result for (2) and ( 5 ) was obtained by Nagata [1969] and Filippov [1969] respectively. For the result for (2) and (3), see Shiraki [1971]; for (4) and ( 5 ) see Gruenhage, Michael and Tanaka [1984]. 4.14. Theorem. Let X be an M-space. Then each of thefollowing implies that X is metrizable. ( I ) X is a o-space. (2) X has a point-countable separating open cover. (3) X has a point-countable weak base. (4) X is a k-space having a point-countable k-network. ( 5 ) X is the quotient s-image of a metric space. 4.15. Remark. For case (I), we have more generally, that if X has a G,-diagonal, then X is metrizable (see, for example, Nagata [1985, VII.31). Indeed, Chaber [I9761 showed that every countable compact space with a G,-diagonal is metrizable (hence, compact). Thus, an M-space Xis paracompact. Hence case (2) holds. Then X is metrizable. For case (4), the k-ness of X is essential. Indeed, Frolik [1960] showed that there exists a countably compact subspace Xof p ( N ) such that every compact set of Xis finite (hence, Xis an M-space having a point-countable k-network), but X i s not metrizable.
5. Quotient s-images of locally separable metric spaces We consider metrization of quotient s-images of locally separable metric spaces. In this section, we assume that all spaces are “regular”.
5.1. Proposition. Let f : X --* Y be a quotient s-map such that X is metric, and Y is strongly Frkchet. Then Y has a point-countable base. When X is moreover locally separable, Y is metrizable.
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Proof. Let 3 be a o-locally finite base for X, and let 9’ = f ( B ) .Since each f - ’ ( y ) is separable, 9’ is a point-countable cover of Y. We show that 9’ satisfies (C,). Let y E Y and U be a neighborhood of y in Y. Let g = f l f - ’ ( U ) , and let A?’ = {B E B I B c g - ’ ( U ) } .Then g is a quotient map and g - ’ ( U ) is determined by the open cover 4’. Then, by Proposition 2.3 (3), U is determined by a point-countable cover 4 = g ( 3 ’ ) . To show that 9’ satisfies (Cl), it is sufficient to show that y E Int U{C I C E 4’}and y E 04’ for some finite 9‘ c 4. Let {U,ln E N} be an enumeration of all finite unions of elements V E 4 with y E V. Let V , = U{Q I i 6 n} for each n E N. Suppose that y $ Int V , for any n E N, hence y E Y - V , for each n E N. Since Y is strongly FrCchet, there exists a sequence { y. I n E N} in Uconverging to y with y , $ V,. Since { y , I n E N f is not closed in U , there exists a V E 4 such that V n { y, I n E N} is not closed in V. Then y E V, hence V c V , for some n E N. Since V , meets only finitely many y,, so does V. This is a contradiction. Hence y E Int V , for some n E N. Thus 9’ satisfies (Cl).Then, by Theorem 4.11, Y has a point-countable base. When X is moreover locally separable, we can assume that each member of 9 is separable. In view of the above proof, Y is locally separable, hence Y has a point-countable base V = {G, I a E A} of separable subsets. For a, a’ E A, let a a’ if there exists a sequence {G,, I i = 1, 2, . . . , n> of elements of V such that G, = G,,, G,. = Gun,and G , n G,,+, # 8 for each i < n. Then “ ” is an equivalence relation. Hence, the index set A can be decomposed as U(A, I p E B} for example. Note that each member of V meets only countably many others, so each A, is at most countable. For each /3 E B, let Y, = U{GyI y E A,}. Since each Y, is open and closed in Y , Y is the topological sum of Y,’s. But each Y, has a countable base (GylyE A,}, so Y, is separable metrizable. Hence Y is metrizable.
-
-
5.2. Corollary (Stone [1956]).
Let f :X -P Y be an open, s-map. If X is locally separable metric, then Y is metrizable.
The following example shows that the local separability of X in the previous corollary is essential even if j-is an open, finite-to-one map. A nonmetrizable space which is the open, finite-to-one image of a metric space.
5.3. Example.
Proof. Let Y be the upper half plane. For each real number r and n E N, let ((x, y) 1 y = 1 x - r 1 < l/n} be a basic neighborhood of (r, 0), and let the other points be isolated. Let Y = {Grlr } , where G, = { ( x , y ) I y = I x - r I}.
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Then Y is a point-finite open cover of Y by metric subsets. Let X be the topological sum of G,’s, and f be the obvious map from X onto Y. Then X is metrizable, and f is an open, finite-to-one map. But Y is not normal, thus not metrizable. 0 As for the metrizability of the open compact image of a metric space, we have the following. A mapf: X + Y is compact if every f -’( y ) is compact.
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5.4. Theorem (Hanai [1961]). Let f:X Y be an open, compact map with X metric. r f Y is collectionwise normal, then Y is metrizable. Proof. For each n E M, let 9Yn be a locally finite open cover of X such that the diameter of each element of 9Yn is less than l / n , and let an= f(g,,).Then each %fn is a point-finite open cover of Y. Since Y is collectionwise normal, each %fn has a a-locally finite open refinement Yn(for the proof, see for example, Engelking [1977, p. 4001).Then Y = UYnis a a-locally finite open cover of Y. To show that Y is a base for Y, let y E Y and U be a neighborhood of Y in Y. Since f -’( y ) is compact, there exists an n E N such that e(f- I ( y), X - f - ’ ( U ) ) > l/n, where e is the metric on X . Then St(y , an)c U. Then Y is a base for Y. Thus Y has a a-locally finite base Y . Thus Y is 0 metrizable. 5.5. Theorem (Tanaka [I 9831). Let f : X + Y be a quotient s-map such that X i s locally separable metric. Then Y is metrizable if and only if Y contains no (closed) copy of S, and S,.
Proof. We only prove the “only-if‘’ part. Since Xis locally separable metric, Xis determined by a locally finite closed cover 9of separable metric subsets. Since f is a quotient s-map, Y is determined by a point-countable cover 8 = f ( S ) . To show Y is Frtchet, suppose Y is not Frtchet. Since Y is sequential, by Proposition 1.18, X contains a countable subspace M which, with the sequential closure topology, is a copy of S,. Let S = U{P E 8 1 P n M # 0). Then S is a countable union of elements of 8. Since each element of 9 is hereditarily Lindelof, so is S. Hence, each point of S is C, in S. Then, in view of the proof of Proposition 1.20, M can be assumed to be closed in S. To show M is closed in Y, let 9‘ = {PE 8 I P n M = S}. Since Y is determined by 8,by Proposition 2.3 ( l ) , Y is determined by 8’u IS}. Since M n P = 8 for each P E 8’and M n S is closed in S, M is closed in Y. Since Y is sequential, so is the closed set M. Hence, M is a copy of S, . Then Y contains a closed copy of S,. This is a contradiction. Hence, Y is Frtchet.
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Since Y contains no closed copy of S,, by Proposition 1.1 1, Y is strongly Frechet. Thus Y is metrizable by Proposition 5.1. 0 5.6. Corollary. Let X be a space determined by a point-countable cover of separable metric subsets. Then X is locally separable metrizable if and only if it contains no closed copy of S, and no S,. Proof. The “only-if” part is obvious. To show that the “if” part holds, let X be determined by a point-countable cover % of separable metric subsets. Let X * be the topological sum of W, and let f :X * + X be the obvious map. Then X * is locally separable metric, and f is a quotient s-map by Proposition 2.3 (4). Then the “if” part follows from Theorem 5.5. 0 A space X is of pointwise-countable type if each point of X is contained in a compact set having a countable neighborhood base in X . Locally compact spaces, and first countable spaces are of pointwise-countable type. We note that any space of the pointwise-countable type contains no closed copy of S, and no S,.
5.7. Corollary (Filippov [1969]). Let f :X + Y be a quotient s-map such that X is locally separable metric. If X is a space of pointwise-countable type, then X is metrizable. In the following, the result for case (1) was obtained by Nogura [1983]. 5.8. Theorem. Suppose that X i s embedded in a countably compact, c-space. Then each of the following implies that X is metrizable. ( I ) X is a Lainev space. (2) X is dominated by metric subsets. ( 3 ) X is the quotient s-image of a locally separable metric space. Proof. Since Xis countably compact, it contains no closed copy of S, and S,. Hence, by Proposition 1.9, X contains no copy of S, and S,. Thus X contains no copy of S, and no S,. Thus X is metrizable by Theorems 1.15, 2.10, and 5.5. 0 6. K-Metrizable spaces and S-metrizable spaces As classes of generalized metric spaces, ic-metrizable spaces, and b-metrizable spaces can be defined by means of annihilators. We consider metrization of these spaces.
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6.1. Definition. Let X be a space and 9 be a collection of closed sets of X. Let cp: X x 9 -+ Iw be a nonnegative, real valued function. cp is an annihilator for 9 if it satisfies p(x, F) = 0 if and only if x E F. p is monotone if F, c F2 implies cp(x, F , ) 2 p(x, F2) for every x E X. cp is continuous if cp(x, F ) is continuous inx for every F E 9.cp is additive if, for every 9’ c 9, U 9 ’ E 9 and cp(x, US’) = inf(cp(x, F)I F E 9’}. cp is linearly additive if, for every 9‘c 9 with linearly ordered by inclusion, E 9 and cp(x, U 9 ’ ) = inf{cp(x, F) 1 F E 9’}.
u9‘
For a space X , let F ( X ) , R ( X ) and Z ( X ) be the collection of all closed sets, regular closed sets and zero-sets in Xrespectively. - Here a set A in Xis regular closed (= canonically closed) if A = Int A. Let X be a metric space. For x E X and I; c X , let d ( x , F) be the usual distance function from x to F. Then d : X x F ( X ) + R is a monotone, continuous, and additive annihilator for F ( X ) . A completely regular space Xis called rc-metrizable if there exists a monotone, continuous, and linearly additive annihilator for R ( X ) .Also, Xis called additively rc-metrizable if we strengthen “linearly additive” to “additive” in the above. The notion of rc-metrizable spaces was introduced by SEepin [1976, 19801 as a generalization of metric spaces and locally compact groups. SCepin [1976] showed that any product of rc-metrizable spaces is rc-metrizable. A completely regular space X is called S-metrizable if there exists a monotone, continuous annihilator for Z ( X ) . By virtue of Borges [1966, Theorem 5.21, a space is stratifiable if and only if there exists a monotone, continuous annihilator for F ( X ) . Thus, every 6-metrizable space is stratifiable if and only if it is perfectly normal. A space X is called continuously perjectly normal if there exists a bicontinuous annihilator cp for F ( X ) ; that is, cp(x, F) is an annihilator which is ,jointly continuous when the topology on F ( X ) is induced by the Vietories topology. As for the metrizability of continuously perfectly normal spaces, see Gruenhage [1976], and Zenor [1976]. An annihilator cp : X x f -+R has the semi-closure condition if, for each x E X and 9’ c .?Fwith x E US’, inf{cp(x, F) I F E S’} = 0. A collection 9 of closed sets of X is a base for closed sets if every closed set of X is an intersection of members of 9. For an annihilator cp:X x 9 Iw, define z ( x , y ) = sup(cp(x, F)I y E F E 9 )for each x, y E X . Let &(x) = { y E X l z ( x , y ) < l/n}. -+
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6.2. Proposition. Let 4 be a base for closed sets of X . Then an annihilator cp :X x % + R satisfies the semi-closure condition if and only if { Int S,,(x) I n E N} is a neighborhood base of x in X for each x E X . Proof. The “if” part is obvious, so we prove the “only-if’’ part. Let x E X . Then, for each neighborhood U of x, there exists an F E % such that F $ x and F I> X - U . Then, for some n E N, cp(x, F ) > I/n, so S,,(x)-c U . We show that x E Int S,,(x). Suppose that x 4 Int S,,(x), and let A = X - S,,(x). For each y E A, there exists an Fy E % such that y E Fy and cp(x, F y ) > 1/(2n). Thus, inf{cp(x, Fy)I y E A} # 0. But, since X E A, x E U{Fyly E A}. Thus, inf{cp(x, F,) I y E A} = 0. This is a contradiction. Hence, x E Int S,(x). Therefore, x E Int S,(x) c U . This completes the proof. 6.3. Definition. A space (x, z) is called a B-space if there exists a function g : X x N -, z such that, for all X E X and n E N, x ~ g ( x n), , and if x E g(x,,, n ) for each n E N, then the sequence { x,,I n E N} has an accumulation point. wA-spaces and semi-stratifiablespaces are /3-spaces (see Hodel [1971], or Gruenhage [1984, p. 4761, for example). A space ( X , z) is called a y-space if there exists a function g : X x N + z such that for all x E X and n E N, x E g(x, n), and if y,, E g(x, n) and x,, E g( y,,, n) for each n E N, then the sequence {xnln E N} converges to x. Such a function is called a y-function for X.
The following is due to Hodel [1972].For the proof, see, for example, Gruenhage [1984, p. 4921. 6.4. Proposition. Every
p-, and y-space is developable.
6.5. Proposition (Suzuki, Tamano and Tanaka [1987]). Let 4 be a basefor closed sets of a space ( X , z). Suppose that there exists a continuous annihilator cp :X x 4 + R with the semi-closure condition. Then X is a y-space. Proof. Define g : X x N + z by g(x, n) = Int S,,(x). Then x E g(x, n) by Proposition 6.2. For x E X and n E N, let y,, E g(x, n) and x,, E g(y,,, n). Suppose that {x,,ln E N} does not converge to x. Hence, { x , l m E M} $ x for some infinite subset M of N. Then there exists an F E 4 such that F $ x a n d F 3 { x m \ mE M}.Sincez(y,, x,) < l/m,cp(y,, F ) < l/m.But,since { y, I m E N} converges to x, {cp( y,, F ) I m E M } converges to cp(x, F).Then q ( x , F) = 0, hence x E F. This is a contradiction. Hence, {x,,I n E N} converges to x. Then g is a y-function. Hence, X is a y-space. 0
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6.6. Corollary. Every additively u-metrizable space is a y-space. 6.7. Theorem (Isiwata [ 19871). Every additively u-metrizable 8-space is developable. In particular, every additively u-metrizable, stratiJiable space is metrizable.
Proof. The first half follows from Proposition 6.4 and Corollary 6.6. The latter part follows from the facts that every stratifiable space is a paracompact 8-space, and every paracompact developable space is metrizable. 0 The following example, due to Suzuki, Tamano and Tanaka [1987], shows that not every additively u-metrizable space is metrizable.
6.8. Example. The Sorgenfrey line is an additively u-metrizable space, but is not metrizable.
Roof. The Sorgenfrey line X is the set of all real numbers R with the base consisting of all intervals [x,y). Let d(x, y ) = min{ 1, Ix - y I} for each x, y E R. Define an annihilator cp :'A x R(X)+ R by cp(x,F ) = d(x, F n [x, 00)) for each x E Xand each F E R(X).We show that cp is a monotone, continuous and additive annihilator. But, it is shown that cp is a monotone and additive annihilator, so we prove the continuity of cp. Let x E X and F E R(X). Case (i): F n [x, co) = 8. Then U = [x, co) is a neighborhood of x and, for each y E U , q ( y , F) = 1 since F n [ y , co) = 8. r7ase (ii): 1; n ix, co) # 8 and x 4 F. Define z = inf(F n [x, 00)). Then x < z, rp(x, F ) = d ( x , z), and, for each E > 0 with x < x + E < z, U = [x,x + E ) is a neighborhood of x satisfying that if y E U, then cp(y,F) = d ( y , F n [ Y , 00)) = d ( y , F n [x,co)) = d ( y , z ) . Since d(x, z ) - E < d( y , z ) < d ( x , z), we have I cp( y , F ) - cp(x, F ) I < E . Case (iii) x E F. Suppose that E > 0 is given. Since Fis regular closed, there is a point z E F with x < z < x E . Let U = [ x ,z). If y E U, then
+
CAY,F )
= d(y, F
n [ Y , 00))
< d(y, 4
Since cp(x,F ) = 0,I q( y , F ) - q(x, F ) I < at x.
E.
<
6.
Hence, cp(x, F) is continuous
0
6.9. Definition. A space X is called monotonically normal, if, to each pair ( H , K) of disjoint closed sets of X,one can assign an open set D(H, K) such that H c D(H, K ) c C1 D ( H , K) c X - K , and if H c H' and K I> K', then D(H, K) c D(H', K ) . Such a function D is called a monotone normality operator for X.
Y. 7unaku
308
For the proof of the following, tor example. see Gruenhage [1984, p. 4601. 6.10. Proposition. Every stratiJiable space is monotonically normal. In particular, every LaSnev space, or every space dominated by metric subsets is
monotonically normal. Let X be a monotonically normal space. Then X contains no copy of S, and no S, if there exists a monotone annihilator cp : X x R(X) -+ R satisfying (*) below. (*) For each increasing countable S c R ( X ) , if x E C1 US. then inf{cp(x, F ) I F E S } = 0. 6.11. Lemma.
Proof. For the case of S,, the proof is similar, so we prove that X contains no copy of S,. Suppose that X contains S, = {m} u (a,,1 n E N} u (N Y N)” Let D be a monotone normality operator for X . Define Bflm= C1 D((n, rn), (n). Then {Bflm I n, m E N) is a family of regular closed sets of X satisfying that (i) for each function f :N + N, 03 $ Cl(U(B,,, I n, rn E N,m < f ( n ) } ) . Indeed, 00 4 Cl{(n, m) I n, m E N, rn < f ( m ) ) . While, if rn < f ( n ) , then
D((n, 4,
c
D(Cl{n, m) In, m E N, m
< f ( n ) ) ,~ 0 ) .
Then u { B f l m l nm,
E
N, m
< f(n)}
E
m
C1 D(Cl((n, m ) l n , m
< f ( n ) > ,03)
c X -
E
N,
(003.
Hence, co 4 Cl(U(B,,,In, m E N, m < f ( n ) } ) . On the other hand, there exists a sequence {%?n I n E N} of finite subfamilies o f {Brim 1 n, rn E N} satisfying that (ii) cp(00, UW,,) < I/n, and (iii) if B,, E W,, then i 2 n. Indeed, 00 E Cl(U(B,,I i , j E N, i 2 n}). Then, since cp satisfies (*), for each n E N, there exists a finite Snc {B,,I i,jE N, i 2 n } such that cp(col ugn) < Iln. In E N}. By (ii) and the monotonicity of cp, cp(co, C1 UU) = 0. Let %? = U{%?,, Hence, 00 E C1 U%?.But, by (iii), for each n, % n {B,,,lm E N} is finite. Hence, by (i), 00 4 C1 U%. This is a contradiction. Hence, Xcontains no copy of S,. This completes the proof. 0 The following theorem is due to Suzuki, Tamano and Tanaka [1987]. The result for a LaSnev space is due to Tamano [1985].
Melrization 11
309
6.12. Theorem. Let X be a u-metrizable space. If X is a Lakev space or a space dominated by metric subsets, then X is metrizable.
Proof. By Proposition 6.10, X i s monotonically normal. Then, by Lemma 6.1 1, X contains no closed copy of S, and no S, . Then X is metrizable by 0 Theorems 1.15 and 2.10. 6.13. Corollary. Let X be the quotient image of a locally compact, and
separable metric space. If X is a u-metrizable space, then X is metrizable.
Proof. Letf: M + X be a quotient map such that M is locally compact and separable metric. Then M is determined by a countable cover { C, I n E N} of compact metric subsets. Then, by Proposition 2.3 (3), X i s determined by a cover { f (C,)I n E N} of compact metric subsets. Then, by Proposition 2.4 (l), X is dominated by metric subsets. Hence, X is metrizable by Theorem 6.12. 0 Stratifiable spaces are b-metrizable. But the converse does not hold. Indeed, every P-space (i.e., every zero-set is open) is obviously b-metrizable. But not every point of a P-space is a Gs (see Gillman and Jerison [1960, p. 641). Hence not every P-space is stratifiable. DraniSnikov [1978] proved that a 6-metrizable space X i s stratifiable if X is a c-space, a countably compact space, a locally compact space, or X satisfies the Suslin condition; that is, every pairwise disjoint collection of open sets is at most countable (hence, among b-metrizable spaces, the separability and the Suslin condition are equivalent by Ceder [1961, Theorem 2.53). We shall give the proof of Dranihikov’s result above-mentioned. 6.14. Lemma. Let X be a b-metrizable space. Then every countably compact
subset of X is jirst countable.
Proof. Let C be a countably compact subset of X . First we show that C satisfies the Suslin condition. Suppose that C does not satisfy the Suslin I a < w , } of open sets of X such condition. Then there exists a collection (0, that 0, n C # 8 for each a, and 0, n 0, n C = 8 if ct # /?. For each ct choose x, E 0, n C and a cozero-set U, such that x, E U, c 0,. Let e : X x Z ( X ) + R be a b-metric on X . Then there exist an n E N and {tl, I i E N} such that @(xu,, X - U,,) > l/n for all i E N. Since C is countably compact, the sequence {x~, I i E N} has an accumulation point x E C . Let
Y . Tanaka
310
F =X
-
U{U,,I i E N}. Since x E F, e(x, F)
=
0. But,
1 n
e(x,,, F ) 2 @(xu,,X - U,,) > - for all i E N,
hence, e(x, F) 2 l / n . This is a contradiction. Thus C satisfies the Suslin condition. Now, we show that C is first countable. Since C is regular and countably compact, it is sufficient to show that each point of C is a Gsin C. Let p E C. Defineg: C -+ R byg(x) = sup{&, Z ) l p E Z E Z ( X ) } .Since { p } = g-’(0), it is sufficient to show that g is continuous. Since g is, obviously, lower semi-continuous,we show that g is upper semi-continuous. For each a > 0, let Fa = { x E Clg(x) 2 a } . Then Fa is closed in C. Indeed, let y E Fa n C and E > 0. For each x E Fa, since g(x) 2 a, there exists a Z , E Z ( X ) such that p E Z , and e(x, Z , ) > a - E . Since e is continuous, there exists a neighborhood V,of x in X such that e(x’, Z,) > a - E for each x’ E V,. Let V = { V, n C I x E Fa}.Then y E UV.Since C satisfies the Suslin condition, there exists a countable A c Fa such that y E U{ V, n C I x E a } . Indeed, let V = { V , l a < y} and, for each a < y, let
uu
=
U{V,lS\< a1 - U{V,lB < 4.
Since { U, I a < y} is a pairwise disjoint collection of open sets of C , there exists a countable A c [0, y) such that U, # 8,if a E Aand U, = 8 if a $ A. Then, for each a < y , V , c u { V , l u E A}. Thus, UV = U{ K l a E A}; hence, y E V ,I a E A}. Now, let Z = n { Z , I x E A}. Then p E Z E Z ( X ) ,and e(x’, 2) > a - Eforeachx’ E u{V,n Clx E A}. Since y E u{V,n Clx E A } , e( y , Z ) 2 a - E . Then g( y) 2 a; hence, y E Fa.Therefore, g is continuous. This completes the proof. 0
u{
rc-metrizability is productive (SEepin [1976]), but unlike this, as for d-metrizability, we have the following corollary. 6.15. Corollary. Let X be the product of spaces X,. I f X is d-metrizable, then
all but countably many spaces Xu must be a single point.
Proof. Suppose not. Then X contains a compact‘set K, which is the product of uncountably many copies of two points. Since K is not first countable, in view of Lemma 6.13, this is a contradiction. 0 A space is a quasi-k-space if it is determined by a cover of countably compact subsets. k-spaces and locally countably compact spaces are quasi-kspaces. Nagata [1969] showed that a regular space is a quasi-k-space if and only if it is the quotient image of a regular M-space.
31 1
Melrizarion 11
6.16. Proposition. Let X be a b-metrizable space. If X is a quasi-k-space, a c-space or X satisjies the Suslin condition, then X is stratifiable. Proof. Let X be a quasi-k-space. Since every countably compact set of X is first countable by Lemma 6.14, X is sequential. Then X is a c-space by Proposition 1.8. Let F be a closed set of X. Define f:X -+ Iw by f ( x ) = sup{g(x, Z) I F c Z E Z ( X ) } ,where e is a b-metric on X.Then F = f-' ( O). Moreover, since X is a c-space, f is continuous by the same argument as in the proof of Lemma 6.14. Then F is a zero-set in X. Thus every closed set of Xis a zero-set in X. Since Xis b-metrizable, Xadmits a monotone, continuous annihilator for F ( X ) . Then X is stratifiable. In case where X satisfies the Suslin condition, every closed set of X is a zero-set in view of the proof of Lemma 6.14. Hence, X is also stratifiable. 6.17. Theorem. (1) Every additively 6-metrizable space X is metrizable. (2) Every 6-metrizable wA-space X is metrizable.
Proof. (1): X is a stratifiable space by Propositions 6.2 and 6.16. Then each regular closed set of X is a zero-set. Thus a 6-metric space Xis ic-metrizable. Hence X is metrizable by Theorem 6.7. (2): Let x E X. Since X is a regular wA-space, - there exists a sequence { U, I n E N} of neighborhoods of x such that U,, c U,, and if x, E U,, then U,I n E N}. the sequence { x, I n E N} has an accumulation point. Let K = Then K is countably compact. Since Xis b-metrizable, the point x is a Gsin K by Lemma6.14. Then there exists a sequence { V ,I n E N} of open sets of X such that V,+, c V , c U,, and { x } = n{V,ln E N}. Then it follows that { V ,I n E N} is a neighborhood base of x in X. Thus Xis first countable. Hence X is stratifiable by Proposition 6.16. Then, since X is a wA-space, X is metrizable (cf. Creede [1970], or see Gruenhage [1984, p. 4591, for example).
,
n{
0 References Arhangel'skii, A. V. [I9661 Mappings and spaces, Russian Math. Surveys 21, 115-162. Balogh, Z. [I9791 On relative countably compactness, Uspekhi Math. Nauk 34, 139-143. Borges, C. R. [I9661 On stratifiable spaces, PaciJic J . Math. 17, 1-16. Burke, D. [I9701 On p-spaces and wA-spaces, PaciJic J. Math. 35, 285-296.
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Burke, D. and E. Michael [I9721 On a theorem of V. V. Filippov, Israel J. Math 11, 394-397. [1976] On certain point-countable covers, Pacific J. Math. 64, 79-92. Burke, D., R. Engelking and D. Lutzer [ 19751 Hereditarily closure-preserving collections and metrization. Proc. A M S 51, 483-488. Ceder, J. G. [1961] Some generalizations of metric spaces, Pacific J . Math. 4, 105-125. Chaber, J. [1976] Conditions which imply compactness in countably compact spaces, Bull. Acad. Polon. Sci. Ser. Math. 24, 993-998. Cohen, D. E. [I9541 Spaces with weak topology, Quart. J. Math., Oxford Ser. (2) 5, 77-80. Creede, G. D. [I9701 Concerning semi-stratifiable spaces, Pacific J. Math. 32, 47-54. DraniSnikov, A. N. [I9781 Simultaneous annihilation of families of closed sets, K-metrizable and stratifiable spaces, SOV.Math. Dokl. 19, 1466-1469. Engelking, R. [1977] General Topology (Polish Scientific Publishers, Warszawa). Filippov, V. V. [ 19691 Quotient spaces and multiplicity of a base, Math. Sbornik 80,521-532; English translation in: Math. USSR Sb. 9, 487496. Foged, L. [I9851 A characterization of closed images of metric spaces, Proc. AMS 95, 487-490. Franklin, S. P. [I9651 Spaces in which sequences suffice, Fund. Math. 57, 107-1 15. [1967] Spaces in which sequences suffice 11, Fund. Math. 61, 51-56. Franklin, S. P. and B. V. S. Thomas [1977] On the metrizability of k,-spaces, Pacific J . Mafh. 72, 399-402. Gillman, L. and M. Jerison [1960] Rings of Continuous Functions (Van Nostrand, Princeton, NJ). Gruenhage, G. [I9761 Continuously perfectly normal spaces and some generations, Trans. AMS 224,323-338. [1980] k-spaces and products of closed images of metric spaces, Proc. AMS 80, 478-482. [I9841 Generalized metric spaces, in: K. Kunen and L. E. Vaughan, eds., Handbook of SetTheoretic Topology (North-Holland, Amsterdam) 423-501. Gruenhage, G., E. Michael and Y. Tanaka [1984] Spaces determined by point-countable covers, Pacific J. Math. 113, 303-332. Hanai, S. [I9611 On open mappings 11, Proc. Japan Acad. 37, 233-238. Harley, P. [I9721 Metrization of closed images of metric spaces, in: Lecture Notes in Maihemaiics 378 (Springer, Berlin) 188-191.
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Heath, R. W., D. J. Lutzer and P. L. Zenor [1973] Monotonically normal spaces, Trans. A M S 178, 481-493 Hodel, R. E. [1971] Moore spaces and wA-spaces, Pacific J. Math. 38, 641-652. [I9721 Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J . 39,253-263. Isiwata, T. [I9871 Metrization of additive K--metricspaces, Proc. A M S 100, 164-168. Kannan, V. [I9801 Every compact T , sequential space is Frkchet, Fund. Marh. 57, 85-90. LaSnev, N. [1966] Closed images of metric spaces, Sov. Math. Dokl. 7, 1219-1221. Michael, E. [I9561 Continuous selections I, Ann. Marh. 63,361-382. [I9661 &,-spaces, J. Math. Mech. 15, 983-1002. [I9721 A quintuple quotient quest, General Topology Appl. 2,91-138. MiSEenko, A. S. [I9621 Spaces with a pointwise denumerable base, Sov. Math. Dokl. 3, 855-858. Morita, K. [1953] On spaces having the weak topology with respect to closed coverings, Proc. Japan Acad. 29, 537-543. [I9641 Products of normal spaces with metric spaces, Marh. Ann. 154, 365-382. [1972] Some results on M-spaces, in: Topics in Topology, Colloquia Mathematica Societas Jdnos Bolyai 8 (North-Holland, Amsterdam). Morita, K. and S. Hanai [I9561 Closed mappings and metric spaces, Proc. Japan Acad. 32, 10-14. Nagami, K. [I9691 Z-spaces, Fund. Marh. 65, 169-192 Nagata, J. [I9501 On a necessary and sufficient condition of metrizability, J. Inst. Polytech. Osaka City Univ. I, 93-100. [I9691 Quotient and bi-quotient space of M-spaces, Proc. Japan Acad. 45, 25-29. [I9691 A note on Filipov’s theorem, Proc. Japan Acad. 45, 30-33. [I9851 Modern General Topology (2nd revised edition) (North-Holland, Amsterdam). Nogura, T. [I9831 Countably compact extensions of topological spaces, Topology Appl. 15, 65-69. Nogura, T. and Y. Tanaka [1988] Spaces which contain a copy of S,,, or S,, and their applications, Topology Appl. 30, 51-62. Okuyama, A. [I9641 On metrizability of M-spaces, Proc. Japan Acad. 40, 176-179. Olson, R. C. [I9741 Bi-quotient maps, countably bi-sequential spaces, and related topics, General Topology Appl. 4, 1-28.
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OMeara, P. [1970] A metrization theorem, Math. Nach. 45, 69-72. [I9711 On paracompactness in function spaces with the compact-open topology, Proc. AMS 29, 183-189. Qepin, E. V. [I9761 Topology of limit spaces of uncountable inverse spectra, Russian Math. Surveys 31, 155-191. [1980] On K-metrizable spaces, Marh. USSR lzvesrija 15, 407440. Shiraki, T. [I97 I] M-spaces, their generalizations and metrization theorems, Sci. Rep. of Tokyo Kyoiku Daigaku 11, 57-67. Siwiec, F. [1971] Sequencecovering and countably bi-quotient mappings, General Topology Appl. 1, 143-1 54. [I9751 Generalizations of first axiom of countability, Rocky Mountain J. Math. 5, 1-60. Smirnov, Yu. M. [I9511 A necessary and sufficient condition for metrizability of topological spaces, Dokl. SSSR 77, 197-200. Stone, A. H. [I9561 Metrizability and decomposition spaces, Proc. AMS 7, 690-700. Suzuki, J., K. Tamano and Y. Tanaka [ 19871 K-metrizabte spaces and stratifiable spaces, Questions Answers General Topology 5, 167-171. [Proc. AMS., 198., to appear]. Tamano, K. [I9851 Closed images of metric spaces and metrization, Topology Proc. 10, 177-186. Tanaka, Y. [I9791 A characterization for the product of closed images of metric spaces to be a k-space, Proc. AMS 74, 166-170. [I9801 Closed maps on metric spaces, Topology Appl. 11, 87-92. [I9831 Metrizability of certain quotient spaces, Fund. Marh. 119, 157-168. [ 198.1 Point-countable covers and k-networks, Topology Proc., to appear. Tanaka, Y. and Zhou Hao-xuan [I9841 Spaces dominated by metric subsets, Topology Proc. 9, 149-163. Whitehead, J. H. C. [I9491 Combinatorial homotopy I, Bull. AMS 55, 213-245. Wicke, H. and J. Worell [1949] Characterizations of developable spaces, Canad. J . Math. 17, 820-830, Zenor, P. [I9761 Some continuous separation axioms, Fund. Marh. 90, 143-158.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 9
GENERALIZED METRIC SPACES I
Jun-iti NAGATA Symposium of General Topology, Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka, 543 Japan
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LaSnev space and K-space . . . . . . . . . . . . . . . . . . . . . . . . . Developable space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . M-space and related topics. . . . . . . . . . . . . . . . . . . . . . . . . Universal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 I5 3 I5 326 337 351 365
Introduction The primary purpose of this chapter is to supplement Sections VI.8 and VII.2 of Nagata [1985]. That is, we are going to discuss here some of the significant results of generalized metric spaces that were not treated in detail there. Doubtless M I - and M3-(= stratifiable) spaces are among the most important generalized metric spaces. But discussions on those spaces and especially on the “ M I = M3?” question will be left to Chapter 10. Note that all spaces are at least T I throughout this chapter, and that X denotes such a space. Also note that z ( X ) denotes the topology of X and N the set of all natural numbers. Also note that x, + x means that the point sequence { x nI n E N} converges to x. Nagata’s book [1985] will be referred to frequently with the abbreviation Na. 1. LaSnev space and K-space
LaSnev space ( = the image of a metric space by a closed continuous map) and K-space (defined in the following) are deeply related with each other
J. Nagata
316
despite the fact that they are defined differently.’ They both (K-space with Frechet condition) formulate important classes which are located between the MI-spaces and the metric spaces. 1.1. Definition. A collection 4 of subsets of X is called a k-network if for any compact set C and its open nbd U , there is a finite subcollection 4*of 9such that C c U 4 * c U . 4 is called a wcs-network if for any x E X , its nbd U and a sequence x, converging to x, there is a finite subcollection 9* of 4 such that { x, x,, x,+ I , . . .} c U 4 * c U for some n E N. B is called a cs-network if “a finite subcollection” in the above definition is replaced with “a single element”. 1.2. Definition.’ A regular space X is called an K-space if it has a a-locally finite k-network.’ It is obvious that every metric space is K. Also note that every K-space has a a-locally finite closed k-network.
1.3. Lemma. Let { G , I a E A } be a discrete collection in X with a a-closureThen for each G, we can select a preserving closed wcs-network u,“=’F,. sequential nbd W, such that W, n W,. = 8 whenever a # a’. Here W, is called a sequential nbd of G, if each sequence {x,} converging to x E Fa is eventually in W,. Proof. Assume 4, c 4,+ I without loss of generality. For each a E A define that m
-
U ( F E 4 , I F n G,
=
@}I.
It is obvious that W, n W,. = 8 if a # a‘. To see that W, is a sequential nbd of G,, let x, -+ x E G,. Then there is a finite subcollection 4*of some 4, such that U S * c X - U { G , I /l# R} and {x,} is eventually in UB*.Thus (x,} is eventually in
U {F E 4,I F and accordingly in W, .
n G, =
8 for /? #
a} -
U{ F E 4,I F n G,
=
8) 0
‘K-space was defined by OMeara [I9711and studied by OMeara, Guthrie, Foged, Gao and others. The LaHnev space was first studied deeply by Lainev [ 19661and later by many other people. *A regular space is called an KO-space if it has a countable k-network. KO-space is an interesting generalization of separable metric space. See Michael [I9661 for KO-spaces.
317
Generalized Metric Spaces I
1.4. Theorem (Foged [1984], Gao and Hattori [1986/87]).' The folfowing are equivalentfor a regular space X . (i) X is an K-space, (ii) X has a a-locallyfinite wcs-network, (iii) X has a a-discrete k-network, (iv) X has a a-discrete wcs-network, (v) X has a point-countable a-closure-preserving closed k-network, (vi) X has a point-countable a-closure-preserving closed wcs-network.
Proof. (i)+(ii) is obvious. To prove the converse, let U,"=I 9, be a a-locally c 9,+, . Observe that finite wcs-network of X , where we may assume that 9, each point of X is Gs and thus every compact set of X is first countable. Let C be a compact set and U its open nbd. To each x E C assign i(x) = min{iE N I t h e r e i s F ~ e . s u c h t h a t x ~cF U } . I f s u p { i ( x ) ( x ~ X }= 00, then select x,, n E N such that i(x,) > n. Since C is first countable, {x,} contains a subsequence converging to x E C. Assume, for brevity, that x, + x. Then there is a finite subcollection F*of some Fnsuch that c U . There is F E9* in which {x,} is cofinal. {x, x,, x , + ~ ,. . .} c Select m > n such that x, E F. Then, since F E9,i(x,) , < n, which is impossible. Now, let k = sup{i(x)(x E X}; then for each x E C there is F(x) E Fksuch that x E F(x) c U . Since Fk is locally finite and C compact, there are at most finitely many elements of Tkthat meet C. Thus U= :, 9, is a a-locally finite k-network of X , i.e. Xis an K-space. Similarly we can prove that (iii) = (iv) and (v) = (vi). (iii)+(i)+(v) is also obvious. So it suffices to prove (v)+(iv). Suppose I9 , is a point-countable a-closure-preserving closed k-network such that 9, c 9,+, . Further we may assume that each 9, is closed to finite intersections. For each n select an open cover V, each of whose elements meet at most countably many members of %,. Then select a a-discrete closed cover U,"= of X that refines V,. This is possible because X is obviously subparacompact. (See [Na, Theorem V.71.) By use of Lemma 1.3 we assign to each G E % ,,, its sequential nbd W ( G )such that { W ( G )I G E g,,} is a disjoint collection. For each pair (G, F) E 4, x 9, and I E N, we define that
u9*
u,"=
H(G, F, 1)
=
u { F nF'IF'E
&, F'
c W(G)}.
(1.1)
Then H ( G , F, I) is obviously a closed set contained in W ( C ) .Put #(n, m, f) = { H ( G , F, I)IG E g,,, F E 9", F n G # @}. 'Actually, Foged proved (i) = (ii) = (iii) = (iv) for cs-networks instead of wcs-network. Gao and Hattori proved the equivalence of (v). (vi) for cs-networks with the other conditions. Guthrie [I9731 initiated characterizations of KO-and K-spaces by use of cs-networks.
318
J . Nagata
Then, since { W(G)J G E grim} is disjoint and each G E 4, meets at most countably many members of S,, %(n, m,I ) is star-countable. Thus it is a sum of countably many disjoint closed collections. Moreover %(n, m, I) is closure-preserving, because each of its members is a sum of members of Sl n F,, which is closure-preserving. Thus X ( n , m, 1 ) is o-discrete. Now we claim that X = U= /,: %(n, m, I ) is a wcs-network of X . To prove it, let U be an open nbd of x E X and {x,} a sequence converging to x. Then there is a finite subcollection { F , , . . . , F k } of some 9, such that
,
i,I;;cU,
(1.2)
r=l
and
k
{ x , } is eventually in
UE. I=
(1 -3)
I
We may assume x E I I;;. There is G E g , for some m such that x E G. Then {x,} is eventually in W ( G ) . Now we can prove that for some finite subcollection 9*of U 9 * c W ( G ) and {x,] is eventually in UP*.Because, otherwise we let 9'= {F,', F;, . . .} = {US* IF*is a finite subcollection of U,"l S,such that { x,} is eventually in US*, and such that each member of S*contains x}. (Recall that U,"lPn is point-countable.) Then F,' n (X - W ( G ) )# 0 for allj E N. Since 9, is closed to finite intersections, so is F'. Hence putting F," = F; n . . . n F,', we can select
u;=,S,
yJ E F," n ( X - W ( G ) ) , j E N. (1 *4) Then { y,} cannot converge to x because W ( G )is a sequential nbd of x. Thus there is an open nbd V of x and a subsequence { y,, I i E N} of { y, l j E N} such that V n { y,,} = 0.There i s j for which F,' c V. Select i such that j , 2 j . Then by (1.4) yJ,E c V, which is a contradiction. Thus for some I E N there is a finite subcollection F*of S, such that { x , } is eventually in UP* and such that US* c W ( G ) .Now it follows from (1.1) and (1.3) that { x , } is eventually in H(G, F , , I) u . * . u H(G, Fk, I ) while by (1.1) and (1.2) the last set is contained in U:= I 4 and accordingly in U . This proves that % is a wcs-network of X.Therefore (iv) is established. 0 1.5. Corollary. Let X,, n E N be K-spaces. Then the product space X = I l ~ = , Xis, an K-space.
Proof.
Let 9, be a o-locally finite wcs-network of X,. Then put F,
X
. . . X Fk
X
n
n#n),
X,I&EF,,, ?flk
i = 1, . . . , k ; n l < . * * < n k ; k € N
Generalized Metric Spaces I
319
It is obvious that 9 is a-locally finite. It is also easy to see that 9 is a 0 wcs-network of X . Hence by Theorem 1.4 X is an K-space. 1.6. Example. Let S(N) be a (nonmetric) hedgehog with countably many spines. (See [Nayp. 2451.) S(N) is a Lainev space because it is the image of a discrete sum of segments by a closed continuous map. Then S(N) x S(N) is not Frkchet. Because, if it is Frkchet, then it is metrizable by Corollary 1.16 of Chapter 8, which is not true. Thus S(M) x S(N) is not LaSnev, since every LaSnev space is Frkchet. On the other hand S(N) is an K-space. Because S ( A ) = U,"= I I,, , where I,, = [0, 11, and for a-locally finite bases 9, of,I,, n E N, U,"19,, is a a-locally finite k-network of S(N). Hence S(N) x S(N) is an K-space that is neither LaSnev nor Frkchet.4 We recall here Foged's theorem on Lainev space (Theorem 3.6 of Chapter 8). 1.7. Theorem. A regular space X is Lainev 18it is FrPchet and has a a-hereditarily closure-preserving' closed k-network.'
1.8. Corollary. Every Frdchet K-space is a LaSnev space. Now, let us turn to characterizations in terms of the g-function (Definition 1.1 of Chapter 7). 1.9. Theorem (Nagata [ 1986/87]). A regular space X is a Lainev space if X
is FrPchet and has a g-jiunction g satisfying (i) ifx,, + p E X and ifx,, E g(n, yn),n E N, then yn + p, (ii) i f y E g(n, 4 , then g(n, y ) = g(n, 4, (iii) if a sequence {xi}satisfies for some n, xi 4 g(n, xi) or xj 4 g(n, xi) whenever i < j , then { x i } is discrete as a set in X . (Namely { x i } has no cluster point.) Let us begin with some lemmas that will be used later to prove the theorem. 1.10. Lemma. Under the FrPchet condition, (iii) of Theorem 1.9 is equivalent to (iii)' if a sequence { x i } satisfies for some n xi4 g(x, xi) or xi 4 g(n, x i ) whenever i < j , then { x i } contains a subsequence which is discrete as a set. 4See Tamano 11982, 19851 for subspaces of products of LaSnev spaces. 'See Definition 1.10 of Chapter 7.
6The following fact is of some interest in comparison with Theorem 1.7. A regular Frkhet space X is stratifiable iff it has a o-closure-preserving k-network (Kanatani, Sasaki and Nagata [1985]).
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320
Proof. To prove (iii)'+(iii), let {x,} satisfy the said condition. Assume that {x,} is not discrete and x is a cluster point of {x,}. Then, since X i s FrCchet, { x,} contains a subsequence { x,,} that converges to x. Then by (iii)' { x,,} must contain a discrete subsequence, but this is impossible. This means {x,} is discrete, and (iii) is proved. 0 1.11. Lemma. A collection {FaI a E A } of subsets of a regular Frtchet space X i s hereditarily closure-preserving ifffor any choice of xi E Fa(i),i E N, where a ( i ) # a ( j ) for i # j , { x i } is discrete as a set in X (or equivalently { x i } contains a subsequence that is discrete as a set in X . )
Proof. It suffices to show the sufficiency of the condition. Suppose {FaI a E A } is not hereditarily closure-preserving. Then we can choose G, c Fa for each a E A such that {G, I a E A } is not closure-preserving. There exists A' c A and x E Xsuch that x E UaEA,Ga Since X is Frechet, there is a sequence {x,,} c U a E A I Gsuch a that x, + x . Since x q! G, for each a E A', {x,} is eventually in X - G,. Thus we can select a subsequence { x , } of {x,} and a ( i ) E A' such that x,, E G,(,) c Fa(,,and a ( i ) # a ( j ) whenever i # j . Since x,,, + x, and x # x,, for i E N, every subsequence of {x,,} is not discrete. So, the said condition is not satisfied. 0
UaEA,ea.
1.12. Lemma. If 9 is a o-hereditarily closure-preserving k-network of a regular Frtchet space X,then so is 9 = {PI F E 9}.
Proof. 9 is a k-network because X is regular. Let us show that if Tois hereditarily closure-preserving, then so is go. Let x, E 6 ,F, E Fofor i E N. Suppose {x,} is not discrete in X.Then there is a subsequence { x,,} of { x,} and x E X such that x,, -+ x and x,, # x, j E N. For brevity, suppose x, + x and x # x,. Moreover, we may assume x, q! 6 . For each i we select a sequence S, c F, converging to x, such that x $ S , . Then x E Up"=, S, - Up"= and hence { S ,I i E N} is not closure-preserving, which contradicts that Fois hereditarily closure-preserving. Hence { x, } must be discrete, and thus by Lemma 1 . 1 1 Pois hereditarily closure-preserving. This proves the lemma. 0
,s,
1.13. Lemma. Every o-hereditarily closure-preserving k-network of a T,-space X is a a-hereditarily closure-preserving wcs-network, and vice versa.
Proof. The proof is similar to the proof (ii)+(i) of Theorem 1.4, and is therefore left to the reader.
Generalized Metric Spaces I
Proof of Theorem 1.9. Suficiency. For each x
E
32 1
X and n E N we define that
Hn(x) = [n{g(n, Y ) I x E g(n9y)}l n [f7{gc(n,Y ) I x E gc(n, ~ ) } l .
where gc(n, y ) = X - g(n, Y ) ,
u W
X,, =
{H,(X)IXE
X},
&?' =
It=
X,,.
I
Then we can prove that X is a a-hereditarily closure-preserving wcs-network of X. To prove that X, is hereditarily closure-preserving, let x i E H n ( y i ) , i~ N,
where H,(y,) # H,,( y,) if i # j. By virtue of Lemma 1.11 it suffices to show that {x,} is discrete. Since H , ( y i ) # H , ( y , ) if i # j , eithery, E g(n, z) andy, E gc(n, z) for some z E X , or elseyi E gc(n, z) and y, E g(n, z) for some z E X. In the first case xi E g(n, z) and xi E gc(n, z) follow from the definition of H,,(x). Hence by (ii) we have g(n, x i ) c g(n, z) $ xi.In the second case we have xi $ g(n, x,) in a similar manner. Thus by (iii) { x i } is discrete. That is, X, is hereditarily closurepreserving. To prove that 2 is a wcs-network, let xi + x E U , where U is an open set. By (i) there are io and n such that x, $ g(n, y) for all y E X - U and i 2 i,. Hence x i E H,,(x,) c U for i 2 io. If there are infinitely many distinct H,(xi)%,then { x i }contains a discrete subsequence, because X, is hereditarily closure-preserving. But this is impossible, because xi -,x and we may assume xi # x. Hence there is {i,, . . . , ik} c {i E N I i 2 i,} such that { x i }is even. &?' is a wcs-network, and hence by tually in H,,(x,,)u . * . u H n ( x i k ) Thus Lemma 1.13 it is a a-hereditarily closure-preserving k-network of X. Therefore by Theorem 1.7 X is a LaSnev space. Necessity. Let X be a LaSnev space. Then by Theorem 1.7, X has a Fn, where each F,, is a-hereditarily closure-preserving k-network F = hereditarily closure-preserving. By Lemma 1.12 we may assume that each member of F is a closed set and that F,, c F,+ I . Define g : N x X --* T ( X ) by g(n, X) = X - ~ { F F En I x $ F } .
ur=,
Then it is obvious that g is a g-function of X and satisfies (ii). To prove (i) assume x, + p E X and x, E g(n, y,), n E N. Suppose that U is an open nbd of p. Then { x i I i 2 i,} c U for some io. There are n E N and F , , . . . , Fk E F,, such that {Xili
2 io}
C
FI U
* * *
U
Fk
C
u.
322
J. Nagata
Let m = max(i,,, n); then { x i I i 2 m} c Fl v for each i 2 m. Hence Fk E { y i l i 2 m} c Fl
V
* *
V
Fk c
*
*
v Fk c
U,and F l , . . . ,
u
follows from xi E g(i, y i ) and the definition of g. Thus yi + p proving (i). To prove (iii), assume that { x i } is a sequence satisfying xi $ g(n, xi) or xj $ g(n, x i ) whenever i < j . By Lemma 1.10 it suffices to show that { x i } contains a discrete subsequence. Put
9 ( x i ) = {FE 9, I xi E F } . Then 9 ( x i ) # 9 ( x j ) whenever i < j . Thus we can select a subsequence {xi,} of { x i } and Fk E 9, such , that xik E Fk, k E N and Fk # Fh whenever k # h. Since 9, is , hereditarily closure-preserving, { xikI k E N} is discrete. Hence (iii) is proved. Since every Lainev space is Frtchet, necessity of the condition is proved. 0 It is not known whether the condition (ii) can be dropped from Theorem 1.9. But we can drop (ii) if (iii) is replaced with a stronger condition as in the following.
1.14. Theorem. A regular space X is Lainev iff X is Frtchet and has a g-function g such that (i) if x,, + p E X and ifx,, E g(n, y,,), n E N,then y,, + p , (ii) fi sequences { x i } and { y i } satisfy either (a) xi E g(n, y i )f o r each i E N,and xi E g"(n, y i ) whenever j > i, or (b) xi E gc(n,y i ) f o r each i E N, and xi E g(n, y i ) whenever j > i, then { x i } is discrete as a set in X . Proof. This theorem can be proved by an argument somewhat similar to but more complicated than the one of Theorem 1.9. See Nagata [1987] for the detailed proof, which is omitted here. 0 We can characterize Frkhet K-spaces in terms of the g-function as follows. 1.15. Theorem. A regular Frtchet space X i s an K-space iff it has a g-function g such that (i) ifx, + p E X and ifx,, E g(n, y,,), n E N, then y, + p , (ii) ifu E g(n, 4, then g(n, y ) = g(n, XI, (iii) {g(n, y ) I y E g(n, x)} is Jinite f o r each x E X ,
Generalized Metric Spaces I
323
(iv) i f a sequence { x i } satisfies for some n, xi $ g(n, xi) or xj $ g(n, x i ) whenever i c j , then { x i } is discrete as a set in X .
Proof. Suficiency. Construct H,,(x), &',, and &' as in the proof of Theorem 1.9. Then &' is at least a o-hereditarily closure-preservingwcs-network of X . Now, let x E X and n E N be fixed. Then there are at most finitely many distinct members among {g(n, x ) n g(n, y ) I y E X } . Because by (ii) g(n, x ) n g(n, y ) = U { g ( n , z ) I z E g(n, x ) n g(n, y ) } and by (iii) there are at most finitely many distinct g(n, z) for z E g(n, x). Thus {g(n, x ) n H,,( y ) I y E Y } is also a finite collection. However g(n, x ) n H,,( y ) # 8 implies H,,( y ) c g(n, x ) because of the definition of H,,( y). Therefore g(n, x ) intersects at most finitely many members of &',. This means that &' is a a-locally finite wcs-network of X , and hence Xis an K-space. Necessity. Let 'U : I 9,, be a o-locally finite closed k-network of X . Put g(n,x) = X - U { F E P , , I X $ F } , ~ E N , x E X .
Then it is easy to see that g is a g-function satisfying (i)-(iv).
0
1.16. Definition. A collection 9 of subsets of X is called a cs*-network of X if for each x E X , its open nbd U and a sequence x,, -B x , there is F E 6 such that F c U and { x , , } is cofinal in F. 1.17. Lemma. Every point-countable cs*-network 9 is a wcs-network.
Proof. Let U be an open nbd of x and x, + x . Let 6'= ( F E ~ I X Ec FV } = { F l , F 2 , . . .}. Assume { x . } is eventually in none of Fl v * * v Fk, k E N. Then select xnkE X - F, u . . . v Fk in such a way that n, c n2 < . . . . Then, since xnk + x , there is h and a subsequence { x,,!/} of { xnk} such that {xnk,}c Fh. For k, > h, we have xnS $ F h , a contradiction. Thus { x , } is eventually in Fl u . . . u Fkfor some k , which means that 9is a wcs-network.
0 1.18. Lemma. Let f be a closed continuous map from an K-space X onto Y . ZfBrf-l( y ) is Lindelof for each y E Y, then Y has a a-closure-preserving and point-countable closed cs*-network.
324
J . Nagata
Proof. Select x( y ) from each f X'
=
-I(
y ) with Brf-'( y ) =
0} u
{ x ( y ) l y E Y, Brf-'(y) =
0 to put
[U{Brf-'(y)ly
E
Y}].
Then X' is K and the restriction o f f to X' is a closed continuous map from X' onto Y with Lindelof pre-image of each y E Y. Thus we may assume without loss of generality that f - I ( y ) is Lindelof for each y E Y. Suppose Uf= F,is,a 0-locally finite closed wcs-network of X . Then put
,
3 n
= f(9n)
= {f(F)IFEsn}.
Sincef - I ( y ) is Lindelof for each y E Y and 9, is locally finite, f - I ( y ) meets at most countably many members of F,, Thus . 3, is point-countable and I 3,, is a cs*-network of Y. closure-preserving.Now, let us prove that 3 = Suppose y , -+ y E V in Y, where V is an open set and y , # y . Put
u,"=
H,
=
U { f - ' ( y , , ) I n 2 m } , m E N.
Then for any choice of x , E H,, m E N , the sequence { x , } has a cluster point $ y . On inf - I ( y ) , because otherwise (x,> n f - I ( y ) = 0,and hencef the other hand, f ( { x , , } contains ) an infinite subsequence of { y,,} that converges to y , which contradicts that f ( { x , } ) is closed in Y. Now, fix a sequence { x , } such that x,,, E H, and a cluster point x E f - I ( y ) of { x , }. Since { x } is G, and Xis regular, we can put { x } = G, for closed nbds G,, of x such that G, XI G,,,. Since x E R,,, n E N, we can select p,, E G,, n H,,, n E N. Now we can show that x is the only cluster point of { p , } . Because, suppose x # x'; then x' 4 G,, for some n. Hence X - G,, is an open nbd of x', and { p,,} is eventually in G,. Thus x' is no cluster point of { p , , } . On the other hand, as observed before, { p , } has a cluster point in f - I ( y ) , which must be x. We claim that p , -+ x . Because, otherwise there is an open nbd W of x and a subsequence { p , , } of { p , } such that { p , , } c X - W . Since p,, E H,,, c Hi, { p , , } has a cluster point p E X - W . Since p is a cluster point of { p , } as well, p = x , which is impossible. Now there are F,, . . . , Fk E U,"=,F, such that
(m)
{ p , , l n 3 no} c Fl
U
* ' '
U
Fk
C
f-'(v).
Then f ( p , , ) E f ( H , , ) = { y , l m 2 n } , i.e. f ( p , ) = y,, for some n' 2 n. Hence { y,.In 2 no} c f ( F , ) u . . . u f ( F k ) c V . Thus { y,,} is cofinal in I 3,. Therefore 3, is a cs*-network of Y. some off (F, ), . . . ,f (Fk) E
=:u
u,"=,
0 1.19. Theorem (Gao [1987a]). Let f be a closed continuous map from an K-space Xonto Y . ZfBrf-'( y ) is Lindeloffor each y E Y, then Y is an K-space.
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325
Proof. By Lemma 1.17 and Lemma 1.18, Y has a a-closure-preserving and point-countable closed wcs-network. Thus by Theorem 1.4 Y is an K-space.
17 1.20. Corollary. Ifthere is a perfect map from an K-space X onto Y, then Y is an #-space. The following theorem is interesting in comparison with the theorem of Stone, Morita and Hanai "a, Theorem VI.141. 1.21. Theorem (Gao [1987b]). Let f be a closed continuous map from a metric space X onto Y. Then Y is an K-space 1fBrf-I ( y ) is Lindelof f o r each y E Y.
Proof. Sufficiency follows directly from Theorem 1.19. Necessity follows from Remark 4.4 of Chapter 8. While LaSnev spaces are the closed continuous images of metric spaces, it is natural to try to characterize FrCchet K-spaces as images of metric spaces. 1.22. Theorem (Gao and Hattori [1986/87]). A regular space Y is a FrPchet H-space fi it is the image of a metric space by a closed continuous s-map.7
Proof. Necessity. Since Y is Lagnev by Corollary 1.8, there is a metric space X and a closed continuous map f from X onto Y. By Theorem 1.21, Brf-l( y ) is separable for each y E Y. Define X' as in the proof of Lemma 1.18. Then the restriction g off to X' is a closed continuous s-map from the metric space X ' onto Y. Suficiency. This follows from Theorem 1.21. 0 Finally let us prove the following (partial) converse of Corollary 1.18. 1.23. Theorem (Gao [1987b]). Assume the Continuum Hypothesis. Then a Lainev space X is a FrPchet K-space i f x ( X ) < o1.'
Proof. Necessity. Let X be a FrCchet K-space. Then by Theorem 1.22 there is a closed continuous s-mapffrom a metric space A4 onto X . Let x E A '; then 7The inverse image of each point of Y is separable. * x ( X ) denotes the character of X,and x ( X , x ) the character at x .
326
J. Nagata
I d ( f - ' ( x ) )I Q If-'(x) 1 < w"
Thus by "a, =
VIII.2.B)I and the Continuum Hypothesis,
wI.Now let % ( p ) be a countable open nbd base of p and
Put 9 =
U1@(P)IP Ef-'(X)}.
Let V be a given nbd of x in X. Thenf-'(V) is a nbd of Then 1421 Q 0'. f-'(x) in M.Sincef-'(x) is Lindelof, there is a countable subcollection 9* of 42 such that f - ' ( x ) c u9*c f - ' ( V ) . Now X - f(M is an open nbd of x contained in V. There are at most o1number of countable subcollections of 42. Hence x(X, x) Q ol, proving x ( X ) < 0'. Suficiency. Let X be a LaSnev space with x ( X ) Q 0'. In view of the proof of Theorem 1.9 we know that X has a a-hereditarily closure-preserving wcs-network H = U,"=I H, , where we can see from the construction of H, that each H,, is disjoint. Let 2,= {HI H E H,}. Then we can prove that 2,is point-countable. Assume the contrary; then for some x E X , (HI x E R E $,} is uncountable. Thus we may suppose x E fi,, CJ < wI,where H , E X, . Since X, is disjoint, we may assume x # HE for all a < ol. Since Xis Frkhet, we can select sequences {xi"I i E N} for all a < w1 such that H,3 xi" + x # xi". Since x ( X ) Q wl, there is a nbd base {U,(x)Ia < w l } of x . Select x i E U,(x) n ($1 i E N} for every a < q. Thenx E { x ; l a < wI}-- {x;la < ol}whilex; E H, E #,,a < w,.This contradicts that &,' is hereditarily closure-preserving. Thus 2,is pointcountable. Namely ' 2,is a a-closure-preserving and point-countable closed wcs-network of X.Hence X is an K-space by Theorem 1.4. 0
u%*)
u,"=
1.24. Example. In view of the proof of necessity of Theorem 1.23, we see, without assuming the Contuum Hypothesis, that x ( X ) < 2" holds for every Frkchet K-space X. Consider a nonmetrizable hedgehog S ( A ) with I A I 2 2". Then, since x ( S ( A ) , 0) > 2", S ( A ) is no K-space though it is a LaSnev space. 2. Developable space A motivation of the theory of generalized metric spaces is to factor metrizability into simpler conditions. In that way we can understand metrizability better than ever before by studying the simple factors separately. By Bing's Theorem m a , Corollary (ii) of Theorem VI.41 metrizability is factored to collectionwise normality plus developability. In the present section we shall show that developability can be further factored to several subconditions. 'ddenotes density. We denote by wand w , the countable and the least uncountable cardinals, respectively. Later we shall use w , to mean the first uncountable ordinal.
Generalized Metric Spaces I
321
Let us begin with Worrell and Wicke's Theorem, which is interesting because - besides from the above mentioned point of view - it was the first to characterize developable spaces in terms of base. Bases and sequences of open covers are frequently used in metrization theory, where we recognize that they are considerably different in their natures, and sometimes the former are easier to handle. 2.1. Definition (Arhangel'skii [1963]). A base W of X is called of countable order (BCO) if for each x E X and any BiE W, i E N satisfying x E I Bi and Bi+,s Bi,{Bi( i E N} is a nbd base of x.
nz
2.2. Proposition. Every developable space X has a BCO.
Proof. Let {4?JnI n E N} be a development of X such that > 42* > . . . . Generally for each open cover 4 we can construct an open cover 42' such that 9'3 9 and Q' is closed under the sum of chains." To do so, let e0= 9, and define 42u by transfinite induction on the ordinal a as follows. Suppose 4YU,tl < /3 have been defined. Then is Uu
u,<,@,.
n,?,
v,
v
2.3. Lemma. Let V be a BCO of X . Then there are subcovers 6 of V ,i E N such that = { Ku10 < o! < z i } (well-ordered) and (i) KI - Uu<sKI.# 0 f o r each /3 < zi, (ii) put a(i, x ) = min{a I x E Ku}, x E X , i E N; then V,,(,,,, c Ku(n,x) whenever n < m, and moreover Vmu(m,x) s V&,) f V.,(,,,) # {x}.
<
Proof. Let % = {V,,Ia < z l } c V be an open cover such that V,@- U,<s&u# 8 for each /3 < zI. Suppose %, . . . , K-l have been 'Owe mean by a chain a collection of sets which is totally ordered by inclusion.
J . Nagaia
328
defined. Then for each x E X with a(n - 1 , x) = 0 we select V ( x )E V such that V ( x ) c V,-,, and such that V ( x ) s V,-lo unless V,-,, = {x}. Then we well-order all such V(x)’s as { V,10 < y < yl} = V(O),where we assume V , - (Jocy,
u
=
V ( a ) = {Gagla’< T,},
U
where the order of Vnis the one derived from the order of Vn-I and the orders of V(a), a < T , - ~ . It is easy to see that {Vn1nE N} satisfies the desired 0 conditions. 2.4. Lemma. Let V,, n E N be the open covers obtained in Lemma 2.3 and put w, = “Y-, v Vn+l v * . Then each W, is a base of X satigying the following condition. If x E n Z l W;, Y E“w; and W;+l c W;, i E N, then { W;l i E N} is a nbd base of x.
-
Proof. Let x be a given point of X . Then by Lemma 2.3 Vna(n,x) 3 V n + l m ( n + l , x ) 2 V n + Z a ( n + Z , x )
3
.
* *
3 x.
If equality holds somewhere in the sequence, then by (ii) of Lemma 2.3 the sequence contains { x}. Otherwise, since V is a BCO, { Vmu(m,x) I m 2 n } is a nbd base of X. (Recall that Vnc “Y-, n E N.) Namely Wnis a base of X . Now assume x E W;, W; E “w; and W;+l t W;, i E N. Note that for each n, some member of V, contains W,, because W, E V k for some k 2 n and V k < Vn.Thus we denote:
V,,(,,
is the first member of Vnthat contains some of J+(, i E N. (2.1) Suppose Ka(,,)3 W;, V,+la(n+l)3 W; and i > n 1. Further assume E V h (h 2 i ) . Let y be a point of X such that = (There is such y because of (i) of Lemma 2.3.) Then V(n)
=
+
w
Y E W c Vna(n) n K + l a ( n + l ) and a(n, Y ) 1 , Y ) = a(n 1). a(n
+
+
w.
=
a(n),
(2.2) Because if e.g. a(n, y ) < a(n), then y E Thus by (ii) of Lemma 2.3 F = &a(h,y) c Vna(n,v) contradicting the definition (2.1) of V,, . Thus (2.2) is proved.
Generalized Metric Spaces I
329
Hence by (ii) of Lemma 2.3 and (2.2), either Kol(,)= { x } or V,,(,, 3 K+l.(n+Z) holds. If the latter is the case for each n E N, then, since *Ir is a BCO, (V(n)InE N} is a nbd base of X . Hence { Kli E N} is a nbd base ofx (recall (2.1)). If the former is true for some n E N, then { Y} contains the singleton { x } , and thus it is a nbd base of x . 0 2.5. Lemma. If X is 0-rejinable, then for every open cover 9 of X there is a countable closed cover {Fl, F2, . . .} such that for each i there is an open cover %f such that qi < 9 and ord.r9i < co for all x E 4..
u:,
Proof. k t q.be a &rejinement of %('I) and put Fnk = {x E X 1 ordX9, < k } . Then {&In, k E N} is the desired countable closed cover. 0 2.6. Theorem (Worrell and Wicke [1965]). A T,-space X i s developable zflit is 6-rejnable and has a BCO.
Proof. Necessity of the condition follows from Proposition 2.2 and the fact that every developable space is 0-refinable. (See [Na, Proof of Corollary (ii) of Theorem VI.4 and V.4.D)].) To prove the converse, assume X is a O-refinable space, and W,,,n E N are bases of X obtained in Lemma 2.4. Now from { WnI n E N} we construct open covers A?,,, n E N with A?,, c W,, open covers A?,,l, n , i E N and countable closed covers 9,n, E N as follows. = K . Then by use of Lemma 2.5 we construct open To begin with, put refinements (covers) g I f ,i e N of Bl and a countable closed cover 9, = { F l fI i E N} such that ordrA?lf< co for each x E F l f .( K lmay coincide for distinct i.) Assume that open covers all, A?,f, i E N and closed covers %/ = {F,I i E N} have been defined for all 1 < n - 1 in such a way that ord,Blf < 00 for x E Then to each x E X we assign W ( x ) E W, such that W(x) n (U{4,li,j < n, x
4 4,})
=
8,
(2.3)
and W ( x ) = U { B I B E U { . % I ~<, ~n , x E F , } , x E B } .
(2.4)
Then we put
a, = { W ( X ) I X E X } . Note that a, c W,.
(2.5)
Further, by use of Lemma 2.5, we construct open refinements (covers) Bnf, i E N of a, and a countable closed cover 9, = {F,,,I i E N } such that ord,A?,,, < 00 for each x E F,,. "Each
$2,
is an open cover with %, <
$2
and for each x E X there is such that ord, $2, < a.
J. Nagata
330
Now we claim that {a,I n E N} is a development of X.Assume the contrary. Then for some nbd U of a point x of X St(x, a,,)cjz U for all n E N. Now define a sequence n, = 1, n2, n,, . . . of natural numbers such that x
for each i 2 2.
E Fnl+...+ni-,.”,
(2.6)
# 8 follows. Thus, Then if x E W ( x )E Bnl+ ... + n i , x E W ( x ’ ) n F,,+ ... . . + n,-,andn, * * ni > n,fori 2 2, sincen, * ni > n, by (2.3) we have x’ E F,,+ .. ,,;. Hence by (2.4) W(x’) c B for every with B 3 x’. What we have observed is that every member B E a,,,+ .. of Bfl1 +... +, that contains x is contained in some member of anl+ ,.. Since ordxanl+ . _ . < CO follows from (2.6),
+
+
+
+
+
,
,
%
= {BE
a n l +...+n,-l.n,
I X E B C LU }
is finite and nonempty by the above observation. To see B E Pi we assign such that B c W ( B ) .Then x E W ( B ) U . Put W ( B ) E BnI+ .._
e
9i = { W ( B ) I BE Pi}, i
,
> 2.
If W E li+ c afll + ,. + , , then by the above observation W c B for some B E Pi. Hence 9,+< di follows. Thus we have a decreasing sequence g2 > 9,> . . . of open collections such that Si c aflI + ._. ( i 2 2), each is finite, and x E W U for each w E Uim_,L&. (2.7) Now, it is easy to select E g i , i > 2 such that W, =I W, =I * . . Since ‘ x E nim,2cI/; and K EK - , ,by Lemma 2.4 {Kli 2 2} is a nbd base of x. Note that K E follows from
,
,
+
K ~ g ci
c
anl+...+ni-l
wnl+...+ni-l
c
K-1,
which follows from (3), n, + . * + nip,2 i - 1 and “w, 2 wk+l.(Recall Lemma 2.4 for the last relation.) This contradicts (2.7), and hence {a,,I n E N} is a development of X. Therefore X is developable.” 0 9
2.7. Corollary (Arhangel’skii [19631). A paracompact T2-spaceXis metrizable if it has a BCO.
Now, let us turn to Brandenburg’s characterization of developable spaces in terms of base, which may be compared with Nagata-Smirnov’s theorem or Bing’s theorem in metrization theory. ‘*Worrelland Wicke [I9651gave another characterizationof developable spaces in terms of base as follows. A base I is called a &base if I = U:=lIfl and for every x E X and its nbd U, there is n such that ordxIm< a, and x E B c U for some B E I”. Then Xis developable iff it has a 0-base and every open set of X is an F,-set.
Generalized Metric Spaces I
33 1
2.8. Definition (Brandenburg [ 19801). An open collection 4 = { U, Ia E A} is called dissectable if there are closed sets F(a,n), a E A , n E N satisfying the following conditions: (0 ua = Un"IF(a, n), (ii) U{F(a, n) I a E A} is a closed set for each n E N, (iii) for each fixed n E N and for each x E U{F(a,n) I a E A } , U, I a E A , x E F(a, n)} is a nbd of x. Then {F(a,n)la E A , n E N} is called a dissection of 4.
n{
2.9. Proposition. Every open cover of a developable space is dissectable.
Proof. Let 4 = {UaIaE A} be a given open cover and 4, > 42> . . * a development of X. Well-order each aito denote it by ai = { V, 10 < /3 < t i } . Define closed sets eBn, i, n E N, fl < ti by =
&fin
{ x ~ X I s t ( x4,) , c uip} - UluiyIr <
B).
Then it is almost obvious that St(CB,, 4,) c QS, {eB, I B < t i }is a discrete closed collection for each i and n, and U{CsnI /3 < ti,n E N} = X. Now put
F(a,i, n)
= U{F,BnIB <
ti,
St(epn, 4,) c Val,
a E A , i , n N. ~
Then we claim that {F(a, i, n)la E A ; i, n E N} is a dissection of 4. It is obvious that (ii) and (iii) of Definition 2.8 are satisfied. To prove (i), let x E U,. Then St(x, ai)c U, for some i. There are fi c t i and n E N for which x E KBn.Then St(J&, a n ) c
v;, E fai.
Thus St(CB,, 4,) c St(x, % i ) c
u,.
Hence x
E
qBnc F(a, i, n).
Therefore U, =
U$=, F(a, i, n) proving (i).
2.10. Theorem (Brandenburg [1979]). A topological space X is developable i@ itj" has a a-dissectable base 4 , i.e. 4 = U;=,4,, and each 4, is dissectable.
Proof. Necessity follows directly from Proposition 2.9. To prove sufficiency, let 4Vn = {Vala E A,} and let {F,(a,i ) I a E A , , n E N} be a dissection of 4,. Fix n, i E N. For each A' c A, we define Ki(A') =
(n{u,laE A'})O n (n{x- F,(a, i ) ( aE A ,
- A'})
332
J . Nagata
Then by (ii) V,,(A’)is open. Further, let Vni= { & ( A ’ ) I A ’ c A , } .
Then “y,, is an open cover of X by (ii) and (iii). Now we claim that { V,,I n, i E N} is a development of X. To see this, let W be a nbd of X. Select n E N and a E A, such that x E U, c W. By (i) there is i for which x E F,(a,i ) c U,. Let x E V,,(A’)E V,,.Then a E A’ and hence V,,(A’) c CJ, c W. Hence St(x, V,,)c U, c W proving our claim.
0 Now, let us turn to a characterization in terms of annihilators. 2.11. Lemma (Alexandroff and Niemytzki). Let X be a semi-metrizable space with a semi-metric Q. I ~ satisfies Q the following condition, then X is , -, 0, then e(x,,, y,) -, 0. developable: Ife(x,,, x) -, 0 and ~ ( y , x)
Proof.
Let %,
=
{ U I U is an open set of X with 6 ( U ) < l / n } ,
where 6 ( U ) = sup{~(x,y) 1 x, y E U } . Then %, covers X. Because if x E X , by the given condition there is E > 0 such that 6(S,(x)) < l / n , where S,(x) = ( y E Xle(x, y) < E } . Thus x E (S,(x))o E %,. Now we can show that {%,In E N} is a development of X. Suppose W is a nbd of x; then S,,,(x) c W for some n E N. Let x E U E %,; then, since e(x, y) < l / n for ally E U , we have x E U c S,,,(x) c W. Namely St(x, %,,) c W. Thus X is developable. 0 2.12. Theorem (Brandenburg [1988]). A TI-spaceX is developable i f X has a monotone annihilator cp (for the closed sets) which is upper-semi-continuous on X x F ( X ) , where the space 9 ( X ) of closed sets of X has the $finite topology. l3
-
Proof. Suppose 92, > %* > . . is a development of X. Then we define map cp:X x 9 ( X ) -+ [0, 11 by if x
0
44x9 F )
=
1 min{n E N I St(x, %,) n F =
I3SeeChapter 7, Definition 1.5 and Definition 4.1
0}
E
F,
if x $ F.
Generalized Metric Spaces I
333
cp is obviously a monotone annihilator of X . Let us show that cp is upper semi-continuous.Assume q(x, F ) < a and x 4 F. Then by the definition of cp, l/n, < a for no = min{n E N 1 St(x, n F = S}. Since St(x, n F # 8, there is U E such that x E U 4 X - F. Let y E U and G E ( U , X ) ; then G n U # 8. Hence St(y, n G # 8. Thus min{n E N I St(y, %,) n G = 8} 2 no, i.e. q ( y , G ) < l/no < a. If x E F, then for each a > 0 = cp(y, F ) we can select n E N such that I/n < a. Suppose x E U E %,,. Then for each ( y , G ) E U x ( U , X ) , G n U # 8. Thus St(y, @,) n G # 8, i.e. cp(y, G ) < l/n < a. Hence cp is upper semicontinuous on X x 9 ( X ) . Conversely, assume that cp is an annihilator of X satisfying the said condition. Then define e : X x X + [0, 00) by
a,,)
e(x3 Y ) = c p k { Y } ) +
cp(Y7
bl).
Since e is obviously a semi-metric, let us show that e is compatible with T ( X ) , i.e. {S,(x) I E > 0 } is a nbd base of each x E X . Let W be an open nbd of x. Then cp(x, W c ) > E > 0 for some E . For each y E W‘, cp(x, { y } ) > E because cp is monotone. Thus e(x, y ) > F . This proves that S,(x)c W . To show that S,(x) is a nbd of x for each E > 0, we assume the contrary. Then x E X - S,(x). Since cp(x, {x}) = 0 < + E and cp is upper semicontinuous, there is a nbd of x such that cp( y , { z } ) < + E whenever y , z E U . Select y E U n ( X - S,(x)). Then e(x9 v) = cp(x, { Y ) ) + cp(Y, {XI, <
E,
which contradicts y S,(x). Thus S,(x) is a nbd of x, that is, e is compatible with t(X). Finally we can prove that e satisfies the condition of Lemma 2.1 1 . Suppose e(x, x,) + 0 and e(x, y,) -+ 0. Thenx, -+ x and y, + x. Since q(x, {x}) = 0 and cp is upper semi-continuous, for each E > 0 there is a nbd U of x such that cp(y, { z } ) < E whenever y , z E U. Thus for sufficiently large n we have e(xn, ~
That is, e(x,, y,)
n )=
+
(~(xn,{ y n > > + CP(Y,,{ x n } ) < 2 ~ .
0. Therefore by Lemma 2.11 X is deve10pable.I~ 0
Now, let us try again to factor developability into subconditions, but this time by use of different types of conditions. “As is easily seen from the proof, it is sufficient that q ( x , { y } ) is upper semi-continuous on X x X . Compare this theorem with Corollary I .9of Chapter 7.
J. Nagata
334
2.13. Definition. Xis said to have a G$-diagonal if there are open covers Vl, V2,. . . such that for any distinct points x, y of X and for some n E N,
Y
4 St(x, KIT,).
2.14. Proposition. Every regular O-refinable space X with a G,-diagonal has a Gd-diagonal.
Proof. Let a1, a2,. . . be a sequence of open covers of X such that for all distinct points x, y and some n E N, y # St(x, %,,). Let Vnbe an open cover such that 6 c anand Uy=I Vni a &refinement of V,,.Then {Vni I n, i E N} has the property in Definition 2.13. 0 2.15. Proposition (Hodel [1971]). Every wA-space with a G$-diagonal is
developable. Proof. Let {ai I i E N} be a sequence of open covers satisfying wA-condition, * be a sequence of open such that 42, > a2> . . . . Let Vl > V2> = aiA K , i E N. To covers with the property of Definition 2.13. Put prove that (W,,I n E N} is a development of X , we assume the contrary that St(x, W,,)Q W, n E N for an open nbd W of X E X . Then select x, E St(x, Wn)- W , n E N. By wA-condition of ai, {x,,}has a cluster point x,. Thus 1
m
xo
m
n w x , “w;,) = n wx,“y;,)
n=l
-
n=
which is a contradiction, because x,, developable.
=
I
4 W , and accordingly x, # x. Thus Xis 0
2.16. Theorem (Creede [1970], Hodel [1971]). For a regular space X the following are equivalent. (i) X is developable, (ii) X is semi-stratiJiable and wA, (iii) X is a 8-rejinable wA-space with a G,-diagonal, (iv) X is a wA-space with a Gd-diagonal.
Proof. See [Na, VI.8.B)I for (i)+) and [Na, VI.8.D) and V.4.D)] for (ii)+(iii). Note that (iii)+(iv) follows from Proposition 2.14 and (iv)+(i) 0 from Proposition 2.15. 2.17. Theorem (Burke [1970]). A completely regular 8-rejinable space Xis a p-space iflit is a wA-space.
Generalized Metric Spaces I
335
Proof. Suppose Xis a wA-space with a sequence {aiI i E N} of open covers satisfying wb-condition. Select open covers i E N such that f < $!li.
<, Assume x E 6 E x , i E N. Then, since n,?,c c nZ=,St(x, q.), and the last set is countably compact because of wA-condition, n ,; < is countably compact. Since X is O-refinable, so is n,?,c. Thus by m a , V.4.F)] n,?,c is n;=,c,
n E N. Then, since x, E St(x, $!l,,), {x,} compact. Now, suppose x,, E has a cluster point. Hence by Burke's Theorem [Na, Theorem VI.291 X is a p-space. Conversely, assume that Xis a p-space with a sequence V, > V, > * * of open covers satisfying the condition of "a, Theorem VI.291. Let U { K I i E N} be a @-refinementof 6. Let U{KIi2 I i2E N} be a &refinement of K, A V,. Generally we denote by U{KI. . . i k I i, E N} a &refinement of A *v,. Then we claim that I i,, . . . , ik E N; k E N} satisfies wA-condition. Suppose xi,,.._,ik E St(x, We select a sequence {n,, n,, . . .} such that ord,K, < co, ordxWnln2 < co, . . . . Then Wnl > W n I n 2 > * * * . Since S = { ~ , , ~ . . .E~N} , l kc St(x, W,,,),
there is W, E W,,,such that x E W, and {xnI,,,,,~ k E N} is cofinal in W,. Let W, n S = S , . Since S, is eventually in St(x, W,,,,Jn W, there is W, E WnIn2 such that x E W, and S , is cofinal in W, n W,. Let W,n W,n S , = S,. Continue this process to define W,, W,, . . . and S 3 S, 3 S, 3 * ,where x E w&E Wnl,,.nk and s&c W, n . * n wk. Thus we have a subsequence {xi, xi, . . .} of {~,,,...,~l k E N} such that x; E W, n * . . n W,. Since W,,,,,,,< V&, there are V,, k E N such that w k c 5. Then xi E V, n . n V,. Hence by the property of { V,}, {x;} and accordingly {x,,,...~,I k E N} has a cluster point. Thus the original sequence { x ~ ~ I. i,, , , ~. ,. . , ik E N; k E N}, too, has a cluster point, and our claim, namely X is a wA-space, is proved.
-
+
The following theorem follows from Theorems 2.16 and 2.17. 2.18. Theorem (Kullman [1971]). A completely regular space X is developable if it is a 8-rejinable p-space with a Gd-diagonal.
In this respect Hodel [I97 11 proved that every regular O-refinable wA-space with a point-countable p-base is developable. 2.19. Theorem (Heath [1965]). A semi-metricspace X with apoint-countable base $!l is developable.
J . Nagata
336
Proof. Let us denote by e the semi-metric of X and let S,,(x) = { y E XI e(x, y ) < I/n}'. We also denote by U,,(x), n E N the members of Q that contain x. Then we define h(n, x)
=
S,,(x) n Un(x), n E N, x
E
X.
Now, we well-order all points of X and denote by p(n, x) (n E N, x E X) the first point p such that h(n, p) 3 x. Then we define a g-function g : IV x X -+ r ( X ) by
< n}) n (n{q(p(i, x ) ) l j G n, i < n, x E q ( p ( i , x))}).
g(n, x> = &(x) n (n{h(i, p(i, 4 ) I i
(2.8) Now we claim that '9" = { g ( n , x) I x E X } , n E N form a development of X. Suppose not; then for some x E X and some nbd W of x, St(x, gi)r$ W , i E N. Hence for each i there is xi E X such that x E g(i, xi) r$ W. Note that by (2.8) x E &(xi) and hence xi -+ x. Select I , m E N such that S,(x) c U,(X) c
w.
(2.9)
Then p(I, x) E Um(x), because for any point p E X - Um(x) h(I, p) S,(x) n U,(x) $ x holds. Thus
Um(x)= U,(p(I, x)) for some k
E N.
=
(2.10)
Note that U,(p(l, x)) n h(I, p(I, x)) is an open nbd of x. Hence there is io E N such that xi E U,(p(l, x)) n h(I, p(I, x)) Thus p(I, x i )
< p(I, x)
for all i 2 io.
(2.1 1) (2.12)
for i 2 io,
where < denotes the well-order of X. On the other hand i 2 I implies x E g(i, x i ) c h(1, p(I, xi)) because of (2.8). Thusp(I, x) < p(I, x i ) for i 2 I . Combine this with (2.12) to obtain p(1, x i ) = p(1, x) for i 2 max(io, I),
(2.13)
xi E U,(p(I, x i ) ) for i 2 max(io, I )
(2.14)
Thus follows from (2.1 1). Hence i 2 max(i,,, I, k ) implies , = U ~ ( P ( XI) I, g(i, xi) c U ~ ( P ( Ixi))
=
um(x) c W
because of (2.Q (2.1 I), (2.13), (2.10) and (2.9). This contradicts g(i, xi) Q W. Hence {g,,In E N} is a development of X , and X is developable. 0
Generalized Metric Spaces I
337
2.20. Example. The butterfly space (X in [Na, Example VI.31) is a semimetric space with a point-countablep-base, but it is not developable. Because it is MI and accordingly paracompact and T, but nonmetrizable.
3. M-space and related topics The primary purpose of this section is to supplement [Na, VII.2,3] to give some examples concerning M-spaces and discuss generalizations of M-spaces. Throughout the section all spaces are at least TI. We know that the countable product of paracompact M-spaces is paracompact M "a, Corollary 2 of Theorem VII.31. But, if the paracompactness condition is dropped, the product theorem does not hold.
3.1. Lemma. Let F be an injinite closed subset of p(N), where N is the discrete space of all natural numbers. Then I F 1 = 2,". Proof. Note I B(N) I = 2'" (B. PospiSil's theorem m a , Exercise VIII.41). We can easily select y,, y,, . . . E F and mutually disjoint open nbds V,, G , . . . of y l , y,, . . . , respectively in p(N). Let Y = ( y l , y,, . . .}; then we claim that 7 = p( Y). Suppose f is a real-valued bounded continuous function on Y. Then define g : N --* R by f(yn) i f x N ~ n V,,
g(x)
=
{
a3
.O
ifxEN -
U V,. n= I
Then, since g is bounded and continuous, we can extend it to a continuous function 2: B(N) -+ R.Now it is easy to see that the restriction of 2 to F is a continuous extension off. This proves that 7 = fi(Y). Hence IF1 2 0 1 = I p ( Y )I = 2,", proving the lemma. 3.2. Lemma (Novak [1953]). Let N be the discrete space of all natural numbers. Then there are nonempty subsets P, Q of fi(N) - N such that P n Q = 0 and such that P u N and Q u N are countably compact. Proof. Let So,SI,. . . , S,, . . . , t < 2 be the collection of all countable subsets of p(N). Then we define disjoint subsets P,,Q, of p(N) - N for t < 2,'" such that (PC}and {Qt} are increasing sequences satisfying IP,I < 2,"', IQ,I < 2,'".
338
J. Nagata
The construction will be done by use of induction on g as follows. Let = ( y o } ,where x,, yo E B(N) - N and xo # y o . Assume P, and Q, have been defined for all l < a. Since I U { P t u Q , 15 < a } I < 2'" and I S, I = 2'" by Lemma 3.1, we have
Po = { x o } , Q,
IS, - (N u s a ) -
u (P,u QOl >
W.
,
Thus we can select distinct points xa,ya E 3, - (N u S,) - U,<.(Pt u Q,). Put Pa =
(,U P t ) u { x u } ,
Qa
(U
=
Qt) u
( ~ a } .
,
Then
P, u Q,
c B(N)
IPaI < 2'",
- N,
leal
P a n Q,
=
8,
< 2'"s
while P, c Pa and Q, c Q, for ( < a. Now, put P = U { P , l ( < 22w},Q = u { Q , l < c 22"}. Then P u Q c b(N) - N, P n Q = 8. Let us prove that P u N and Q u N are countably compact. Let S be a countable subset of P u N. Then S = Sa for some a < 2'". Now, since xu E S, n P - S,, x, is a cluster point of S, in P.Hence P u N is countably compact. Similarly, Q u N is also countably compact.
0 3.3. Example (Isiwata [1969]). Let x E b(N) - y(N), where y(N) denotes the real compactification of N. Put A = N u P u { x } , B = N u Q u { x } , where P and Q are the subsets of b(N) - N obtained in Lemma 3.2. Then A x B is no M-space. Proof. First observe that b(N) - y(N) is nonempty because N is not pseudo-compact. (See m a , IV.61.) Since x E B(N) - y(N), there is a continuous functionf: b(N) + [0, co) such thatf(x) = 0, andf(x') > 0 for all x' E N m a , IV.6.E)I. Put Z, = { x ' E p(N)If(x') 2 l/n}, n E N. Then b(N) - Z , is an open nbd of x in B(N). Now, in order to prove that A x B is not M , we assume the contrary. Then A x B has a sequence {%,In E N} of open covers satisfying wA-condition. Put 2 = (x, x); then 2 E A x B. Since St(2, 42,) is an open nbd of x in A x B, there is an open nbd U, of x in b(N) such that ( A x B ) n (U, x U,) c St(2, 42,)
and
u, = B ( W
-
z,,
on+,= u,
(3.1) in b
N.
(3.2)
Generalized Metric Spaces I
By [Na, IV. 1.A)], { x } is no G,-set in 8(N), and hence there is y E y # x . Since
339
n=:
I
U,, with
we have y E @(N)- N. Let V be an open nbd of y such that x#
v.
(3.3)
Then, since V n V , is an open nbd of y and accordingly V n U,,n N # we have ( V x V ) n (U,, x U,,) n A # 0,
0,
where A = { ( x ’ , x ’ ) I x’ E N}. Pick f,,E ( V x V ) n (U,, x U,,) n A.
(3.4)
Then by (3.1), f,,E St(f, %,,). However, we can show that {f,, I n E N} has no cluster point in A x B. Let i = (zl, z 2 ) E A x B. If zI # z 2 , then there are nbds Wl and W, of zI and 2, in j(N), respectively, such that W, n W, = 8. Then (W, x W,) n A = 0. Hence by (3.4) Z cannot be a cluster point of { x k } .Thus let Z = (z, z). If z E N, then z E 2, for some n. Then by (3.2) i 4 U,,,, x U,,,.Hence i is no Ifz 4 N, then, since A n B = N u { x } ,it follows from cluster point of {f,,}. (z, z ) E A x B that z = x . Since x 4 by (3), I = ( x , x ) cannot be a cluster (See (3.4).) Thus {f,,} has no cluster point, proving that A x B point of {f,,}. is not M. 0 3.4. Remark. Actually we have shown that A x B is not wA. Since every countably compact space is M and every M-space is wA, Example 3.3 shows that the product of two M-spaces (countably compact spaces, wA-spaces) need not be M (countably compact, wA). It is known that the image of a normal M-space by a perfect map (more generally by a quasi-perfect map) is M (see [Na, VII.2.F)]), but this is not the case if the normality condition is dropped.
3.5. Lemma. Let W[wl] be the space of all ordinals not greater than ol,the first uncountable ordinal, with the order topology. Put
s
=
W[Oll x W b l l - {<% Ol)}.
Suppose f : S -+ R is a real-valued bounded continuous function. Then for some a < 8, f is constant on {(a’, ol) E Sla’ > a } u {(q, 8’) E SIB‘ > a } .
J . Nagata
340
Proof. It is easy to see that for every P < wI there is a(P) < o1such that f i s constant on {(a', B ) E S I a' > a(B)} and that for every a < o1there is B(a) c o1such that f is constant on {(a, p') E SIP' > P(a)}. Thus we assume that
8') E S with a' > a, p' > P(a'), f ( x ) = b for all x = (a', P') E S with /?' > a, a' > a(P'). Then we can select a < aI < a, < . . . < o1and a < PI < /I2 < . . . < w , f ( x ) = a for all x = (a',
such that BI
> B(al), a2 > a(BI),
B2 >
B(a,), a3
=.
a(B2),
. .. .
Now, put p = {(a19
PI),
Q
PI),
=
{(a29
(a,,
P,), Pz), . . .I. * *
.I1
Then f ( P ) = a, f ( Q ) = b. Let a. = supi ai, = supi pi; then (ao,Po) E P n 0. Hencef((a,, Po)) = a = 6 . This proves the lemma.
3.6. Corollary. Let S be the space in Lemma 3.5 and f : S + I" a continuous mapfrom S into the Hilbert cube I". Then the conclusion of Lemma 3.5 remains true.
0
Proof. Obvious.
3.7. Example (Morita [1967]). Let S be the space in Lemma 3.5. Put L = { ( a , W I ) E S l a < ol},
M = {(W l , P ) E S I P < w11.
For each n E N, let S,,be a copy of S and (P, a homeomorphism from S onto S,,. Let X be the discrete sum of S,, n E N. Identify the point ( P ~ , , - ~ ( Xwith ) q2,,(x)for each x E L and the point cp,,,(x) with (P~,,+~(x) for each x E M. By doing so, we obtain the quotient space Y of X. We denote by f the quotient map from X onto Y. Note that Xis a Tychonoff M-space because it is locally compact, T,and a discrete sum of countably compact spaces. It is also easy to check that f is a perfect map. However, we can prove that Y is no M-space. Assume the contrary; then by Morita's Theorem [Na, Theorem VII.31 there is a quasi-perfect map g from Y onto a metric space 2. Consider the composite map $,, = g 0f 0 (P, : S + Z. Then $ , , ( S )is countably compact and metrizable, and thus it is separable and metrizable. Hence $,,(S) can be regarded as a subset of the Hilbert cube I". Hence by Corollary 3.6 there is
Generalized Metric Spaces I
34 1
a < oIsuch that
$,,(x) = c,, (constant) for x = (a', ol)E L with a' > u, and for x = (ol, a') E M with a' > a. Since i,bZn-'(x)= $2n(x) for x E L , and &,,(x) = $ z n - I ( ~for ) x E M, all n E N follows. This means that
c,, = cI for
g-l(Cl)
nf(Sn)
z 0,
n E N.
Selecty,, E g - ' ( c , ) nf (S,,),n E N. Then { y,} has no cluster point, and hence g-'(cl) is not countably compact contradicting that g is a quasi-perfect map. Thus Y is no M-space. Next we are going to characterize the perfect images of M-spaces. 3.8. Definition (Ishii [1967]). A space X is called an M*-space if there is a sequence {giI i E N} of locally finite closed covers that satisfies wA-condition. 3.9. Proposition. Every M-space is M *
Proof. Use Morita's Theorem [Na, Theorem VI1.31. The details are left to 0 the reader. 3.10. Definition. Let f be a multi-valued map from X to Y such that f(x) # 0 for every x E X and f - I ( y ) = {x E XI f(x) 3 y} # 8 for every y E Y. Such a map f will be called an m-map. Then for each C c X and D c Y
f ( C ) = U{f(x)lx E CI,
f -IP)= Uif - ' ( y ) l y E
01.
An m-mapfis called perfect (quasi-perfect) iff - ' ( G ) is closed in X for every closed set G of Y ,f (F) is closed in Y for every closed set F of X and f - I ( y ) is compact (countably compact) for every y E Y. Iff ( x ) is countably compact for each x E X , then f is called Y-countably compact. 3.11. Lemma. Let f be a quasi-perfect, Y-countably compact m-map from X to Y. If X is an M*-space, then so is Y. > . . . of locally finite closed covers Proof. X has a sequence '3, > satisfying wA-condition. Then it is easy to see thatf(9,) = { f (G) I G E q}, i E N are closure-preserving, point-finite and accordingly locally finite closed
342
J. Nagata
covers of Y . Let GI =I G, =I . . . be a sequence of non-empty closed sets of Y such that GI c St( y o , f ( g l ) ) i, E N. Then put
H, = f-IW n St(f-'CYo), 4).
(3.5)
Now { H II i E N} is a decreasing sequence of nonempty closed sets of X . Assume I HI = 8 to prove the contrary. Then for each x E f-'(y o ) ,there is i ( x ) E N such that
np"=
w x , gl(xJn
H I ( X )
=
8.
(3.6)
Because, otherwise from wA-condition it would follow that Now, put
V,(X)
=
X
-
U { G E g I l x$ G } ,
HI #
8.
(3.7)
K = U { V , ( x ) I x ~ f - l ( yand ~ ) i ( x ) = i } , i E N. Then each K is open and Up"=I 3 f-'(y o ) . Since f-'(y o ) is countably compact, UfI=,K I> f-'( y o ) for some k . Let x' € f - ' ( y o ) ;then x' E for
some i < k, i.e. x' E v ( x ) for some x ~ f - l ( y o ) with i ( x ) = i. By (3.7) this means that x' E G E gl implies x E G. Thus St(x', 4 ) c St(x, 4). Since i < k, this implies that St(x', gk) c St(x, g I )c X - HI c X - H k . (Recall (3.6).) Thus S t ( f - ' ( y o ) , gk) n Hk = Hk =
8
follows from (3.5). Since this contradicts that Hk# 8, we have proved nEl H, # 8. Select x E n E lH I ; then f ( x ) n GI # 8, i E N. Since f ( x ) is countably compact, GI # 8. This proves that { f ( 4 )I i E N} satisfies wA-condition. Hence Y is M*. 0 3.12. Lemma.
Y is an M*-space fi there is a metric space X and a perfect, Y-countably compact m-map from X to Y.
Proof. Sufficiency of the condition follows from Lemma 3.1 1. To prove necessity, assume Y is an M*-space with a sequence 3, > g2> . . of locally finite closed covers satisfying wA-condition. Let gi= {G, I a E Ai} and A = U E I A i .Denote by N ( A ) Baire's 0-dimensional space, i.e. the product of countably many copies of the discrete space A. Define a subspace X of N ( A ) and an m-mapf: X -P Y by m
(a1, a,,
. . .) E N ( A ) I aiE A ~ i ,E N; i= I m
f ( ( a l , az, . . .>I
= i= I
G,, for ( a l , a,, . . .) E X .
Generalized Metric Spaces I
343
Then f ( x ) is obviously countably compact for each x E X, because {9i} satisfies wA-condition. Namely f is Y-countably compact. Suppose H is a closed subset of Y and x = ( a l , a2, . . .) E X - f - ' ( H ) . Then
f(x) = nZ=,Gmi c Y
-
H.
Hence by wA-condition of {gi}
8
n Gai n H =
G,, n *
Then N ( a l y. . . , a,) = of x such that
for some i.
{(PlyB2, . . .) E XlBI = a,, . . . , Pi = a,} is a nbd
f ( N ( a , , . . . , a,)) n H =
8.
This implies that
N ( a I ,. . . , a,) nf-'(H) =
8
in X .
This means x $ f - ' ( H ) , proving thatf-'(H) is closed. Next, assume y E Y. Suppose x I ,x 2 , . . . ~f-'( y ) and x, = (a:, a;, . . .), n E N. Since y E GaiE 4,i E N, and since gl is locally finite, a( can take on at most finitely many distinct values. Thus aJi
I
- a:'
-
...
=
-
BI
for some infinite subsequence { j l , j 2 ., . .} of { 1, 2, . . .}. Similarly we can select an infinite subsequence { k l , k2, . . .} of { j l , j 2 ., . .} such that a;l
=
... -
=
P2.
. . .} of { k l ,k2,. . .} such that . . . = B3
Select a subsequence {Il, 12, a$ = a;I
=
*
Repeating this process we get a sequence B,, P2, . . . of B, E A, and a sequence N 3 S1 3 S, 3 * * of infinite subsequences of N such that a: = B, for all s E S,. Then put
-
x =
(B,, B 2 ,
*
- .).
Since y E G,, n G,, n * * # 8,x E X and moreover x E f-'(y). It is easy to see that x is a cluster point of { x , } . Hencef-'( y) is countably compact. Since X is metrizable,f - I ( y ) is compact. F,, where each F, is a Let F be a given closed set of X. Then F = union of closed sets of the form N(a,, . . . , a,). Now, let us prove thatf(F) is closed in Y. Suppose y $f(F); thenf-'( y) n F = 8 in X. Sincef-l( y) is compact,f-'(y) n Fl n . . . n F, = 8 for some n. Thus y $ f ( F l n . . . n F,).
J . Nagata
344
Since . . . , 9, are locally finite closed covers, f ( F l n . . . n F,) is a closed set. Hence Y - f ( F l n . . . n F,) is a nbd of y disjoint from f ( F ) . This proves that y 4 f ( F ) , and hence f ( F ) is closed in Y. Thus f is a perfect 0 map. 3.13. Theorem (Nagata [1972]). a perfect map f from X onto Y.
Y is an M*-space ifthere is an M-space and
Proof. Sufficiency of the condition follows from Proposition 3.9 and Lemma 3.1 1 because the perfect map is a quasi-perfect Y-countably compact m-map. Let us prove necessity. By Lemma 3.12 there is a metric space S and a perfect Y-countably compact m-map f from S to Y. Let i be the identity embedding of S into / ? ( S ) .Then put
X = { ( Y , s) E y x S l y E f ( 4 ) . We claim that X is closed in Y x / ? ( S ) .Let ( y , z ) E Y x p(S) - X . Then z 4 f - I ( y ) in / ? ( S ) .Select open sets U and V of / ? ( S )such that U =I f - I ( y). V 3 z, U n V = 0.This is possible because f - I ( y ) is compact and accordingly closed in / ? ( S ) .Put U' = Y - f ( S - U ) . Then, since f is perfect, U' is an open nbd of y in Y. U' x V is a nbd of ( y , z) in Y x / ? ( S ) .Let ( y', 2') E U' x V ; then y' 4 f ( z ' ) meaning that ( y', z') 4 X . Thus (U' x V ) n X = 8, proving that X i s closed in Y x / ? ( S ) . Let ns be the projection from Y x S onto S and cp the restriction of ns to X . Then we can prove that cp is a quasi-perfect map. It is obvious that cp is continuous. For each s E S , cp-'(s) = f ( s ) x {s} is countably compact. To see that cp is closed, let F be a closed set of X and s E S - cp(F). Then cp-'(s) n F = 8 in X . There is an open cover { W,, W,, . . .} off (s) in Y such that (W, x SI/,(s))n F = 0, n E N, where S,(s) denotes the c-nbd of s in S . Sincef (s) is countably compact, we can find n and an open nbd of f ( s ) in Y such that cp-'(s) c W x Sl/,(s) c YxS-F. Then W' = S - f - ' ( Y - W ) is an open nbd of s in S. Thus P = Sli,(s)n W' is an open nbd of s in S. Let us prove that P n cp(F) = 8. Assume q E cp(F); then q = cp(y, q ) for some y E Y satisfying ( y , q) E F. Since F n (W x S&)) = 8, this implies ( y , q) 4 W x S,,,(s). Namely either y 4 W or q 4 Sli,(s) holds. If q 4 Slln(s),then q Sll,(s)n W' = P. If y 4 W , then y E Y - W , and hence f-'(y) cf-'(Y - W ) c
s-
W'.
Generalized Metric Spaces I
On the other hand, since ( y, q) E X , q which implies q
4 S,,,(s)
n W' =
~ f - l (
345
y) follows. Hence q E S - W',
P. -
In either cases we have q 4 P,meaning that P n cp(F) = 8. Thus s 4 cp(F), proving that cp(F) is closed in S . Hence cp is a quasi-perfect map from Xonto S. Therefore X is an M-space by [Na, Theorem VII.31. Let n y denote the projection of X onto Y. Then it is easy to see, by use of an argument similar to the previous one, that n y is a perfect map. Thus the theorem is proved. 0 3.14. Remark. Example 3.7 shows that there is an M*-space which is not M. 3.15. Corollary. Every normal M*-space Y is an M-space. Proof. By Theorem 3.13 there is an M-space X and a perfect mapffrom X onto Y. Thus by [Na, VII.2.F)I Y is an M-space. (To be precise, the above quoted proposition was proved in case that X was normal. But by modifying the proof slightly, we can see that the proposition is also true in case that Y is normal.) 0
Now we are going to generalize M*-spaces further.
{e.
3.16. Definition (Nagami [1969]). Let I i E N} be a sequence of locally = n{FI x E F E x E X , i E N. If finite closed covers of X . Let C(x, Ti) xiE C(x, E.),i E N implies that the point sequence (xi}has a cluster point, is called a C-network and X a C-space. Furthermore, if then {Ti} C(x) = C(x, E.)is compact for each x E X , then is called a strong Z-network and X a strong Z-space.
np"=,
e.},
{e.}
It is obvious that for paracompact spaces the two concepts, C and strong C coincide.
3.17. Proposition. Every M*-space is a C-space, and every regular a-space is a strong &space. Proof. Obvious.
0
3.18. Example. Any normal a-space X which is not metrizable is a nonM*-strong C-space. Because if Xis M*, then by [Na, Corollary 2 to Theorem
346
J. Nagata
VII.51 it is metrizable contradicting the assumption. For example, a nonmetrizable hedgehog is such a space. On the other hand, any paracompact T,-M*-space which is not metrizable is a non-a-strong C-space. 3.19. Definition. Let X be a cover of X by compact sets and 9a collection of closed sets of X. If any K E X and any open nbd U of K there is F E 9 such that K c F c U, then 9 is called a X-network of X." 3.20. Proposition. X i s a strong C-space iy it has a cover X by compact sets where each Fi is locally finite. and a X-network
u:, e.,
Proof. Obvious. 3.21. Proposition (Nagami [1969]). X =,:II X, is strong X.
If X,, i E N are strong
Proof. Let be a cover of X, by compact sets and Up, %-network of X,.. Then put X
=
C-spaces, then
e,a a-locally finite
X , X X , X . -= . { K , x K , x ~ ~ ~ ~ K ~ E ~ , ~
-
m
ej,x - . . x R,i x fl
P ( j l , .. . , j i ) =
(Xk),
k=i+l
j , , . . . , j i e N,
i E N.
Then each 9 (jl,. . . ,j i ) is a locally finite closed cover of X. Let U be an open nbd of K , x K, x * E X . Then, since K I x K, x . . . is compact, there are i E N and open sets UI, . . . , V, in XI, . . . , X,, respectively, such that
-
n 3~ U. m
K , x K ~ x * * *Uc, X . * . X Q X
j=i+ I
Since U I ,. . . , V, are open nbds of K l , . , . ,K., respectively, there arej i , . . . , E,i such that
j i E N and F, E FIj,, . . . ,6 E
K , c F, c U I , .. . , Ki c 4. c U,. Then
n m
K , x K ~ x * * * c F , x . * - x F , xX , C
U,
j=i+l
and
n m
F, x
* * *
x
6x
3 ~ 9 ( j .~. , ,j i .) .
j=i+l
"9 was called a (mud k)-neiwurk by Michael 119701, to whom we owe this concept.
Generalized Metric Spaces I
347
Thus U { F ( j l., . . ,j i ) l j l ,. . . , j i E N; i E N} is a X-network of X. Hence by Proposition 3.20 X is strong X. 0 The following theorem generalizes ma, Theorem VI.271 and m a , Theorem VII.3-Corollary] at the same time.
3.22. Theorem (Nagami [19691). Let X,, i E N be paracompact T,-X-spaces. Then X = :I I X, is a paracompact T2-%space.
Proof. Since each X, is strong X, X is also strong X by Proposition 3.21. Thus all we have to prove is that X is paracompact. Let U{%,lj E N} be a strong X-network of X,. Then, since X, is paracompact, by [Na, V.3.D)], there are locally finite open covers aij, j E N such that Fijis a shrinking of aij, i.e. = {Fala E A } , qij= {Uala E A} and Fa c U,, a E A. Now F,jlx . . . x q7, x IIkmi+,{ X k }is a shrinking of the locally finite open cover 421j,x * . x aij,x I I ~ = i {Xk}. + l For brevity we assume that {% Ii E N} is a strong X-network of X while each 8.is a shrinking of a locally finite open cover qi. Moreover, we assume that % is closed to intersections. Suppose Y is a given open cover of X. To each F E Fi which is covered by finitely many members of Y we assign a finite open cover W ( F )as follows. Fix a finite subcollection {V,, . . . , V,} of Y such that F c V, u * * * u V,. Denote by U ( F ) the member of q.such that F shrinks U(F). Then we put
ej
W(F)
=
{V, n U ( F ) , . . . , V, n U ( F ) } .
Let = U { W ( F )I F E
6 and F is covered
by finitely many members of Y } . Then % is a locally finite open collection such that < V . Now, let us prove that Uzl"w; is a cover of X. Suppose x E X. Then since C ( x ) is compact, C(x) c V, u . * u 4 for some V,, . . . , V, E Y .There is F E Fi for some i such that C ( x ) c F c 6 u * . . u 4. Thus x E U W ( F ) c U-W;, proving our claim. Hence by Michael's Theorem "a, Theorem V.11 X is paracompact. Among other properties of X-spaces, Nagami [1969] proved that every X-space is a Morita's P-space and that iffis a quasi-perfect map from X onto Y, then X is X iff Y is X. Let us discuss metrization of %spaces and their relations with a-spaces.
J. Nagata
348
3.23. Proposition (Ishii and Shiraki [ 19711). Every point-countable p-base 42 of a countably compact space X is at most countable. Proof. By MiEenko's lemma [Na, VII.3.B)I there are at most countably many minimal (finite) covers by members of 4. Thus it suffices to show that for each member U, of 4 there is a minimal finite cover by members of 9 that contains U,. To do so, assume U, # 0 and U, # X. Pick x, E U,. Select xIE X - U, and put = { U € % ~ X , EU
$ x,].
If X - U, is covered by then since X is countably compact, { U o }u has a minimal finite subcover to which U, belongs. Otherwise select x2 E X - U, - U q I and put %2
= {UE%'IX2E
u j x,}.
If X - U, - U%, is covered by %2, then { U , } u u %2 has a minimal finite subcover to which U, belongs. If we can continue this process indefinitely to get an infinite point sequence xI,x2,. . . and a sequence %2, . . . of open collections, then the point sequence has a cluster point x E X - U, u U [=: l(U42n)].Select U E Q such that x E U $ x,. Then x I ,x2,. . . 4 U, which contradicts that x is a cluster point of {xi]. Thus the process must end after a finite number of steps. Then 17, belongs to a mimal finite cover by members of 4. Hence 1921 < o. 0
3.24. Proposition (Shiraki [ 19711). Every C-space with a point-countable p-base is a a-space.
{e
Proof. Let I i E N} be a C-network of Xand % a point-countablep-base of X. We assume that 9, and 42 are closed to finite intersections. To each x E X and i E N we assign C,(x) = C(x, and C(x) = C,(x). Then C(x) is countably compact. By MiSEenko's lemma there are at most countably many minimal finite covers of C,(x) by members of %, which we denote by %(x, i , j ) , j E N. We denote by V ( x , i, j ) the collection of all finite sums of members of @(x, i, j ) . Then
e)
np"=,
%(x, i , j ) = {C,(x) - V ! V E V ( x , i , j ) }
is a finite closed collection. Since { C,(x) I x E X} is a locally finite closed cover of X , %(i, j ) = b,(%(x,i, j ) I x E X} is a locally finite closed coliection of X. Thus Y = (Up41Fl)u ( U z = l % ( i , j ) )is a a-locally finite closed cover of X.
Generalized Metric Spaces I
349
Now we claim that 9 is a network of X . Let x E X and let W be an open nbd of x. If C(x) c W, then there is F E Up"=I%. c 9 such that x E C ( x ) c F c W. If C(x) Q W, then to each y E C(x) - W we assign U( y) E 9 such that y E U ( y )j x . Note that C(x) - W is countably compact. By Proposition 3.23 { U( y ) I y E C ( x ) - W } u { W} is at most countable. Thus it has a minimal finite subcover, say { U( y l ) , . . . , U( y k ) , W} covering C(x). Then we can select ZiE
C(X) - U ( y 1 ) LJ
U(yi+I)
* '
*
*
..u
U(yi-1)
U(yk)u
for i = 1 , . . . , k. Since 9 is closed to finite intersections, for each p E C ( x ) - U ( y , ) u . * * u U ( y k ) we can select U ' ( p ) E 9 such that p E U ' ( p ) j z I , . . . , Z k . By Proposition 3.23 {UYI),
* *
.
9
W Y k ) , U'(P)lP E C(X) - W Y I )
u*
* *
u U(Yk)}
is at most countable. Hence it has a minimal finite subcover, say 9'= { U( y , ) , . . . , U( y k ) , U ' ( p l ) ,. . . , U'(p,)}covering C(x). Since U9' is an open nbd of C ( x ) , for some i Ci(x) c
(U9')
n (U(yt) u
*
. . u U( yk) u W ) .
Then 9'is a minimal finite cover of C,(x) by members of 9.Thus 9'= 9 ( x , i , j ) for some j , and U( y I ) u * . . u U ( y k )E V ( x , i , j ) . Hence D = Ci(x) - U ( y l )u * . . u U ( y k )E %(x, i , j ) .
Observe that x E D c Wand D E 9.Therefore 9 is a o-locally finite closed network of X , proving that X is a o-space. 0 3.25. Corollary. Every T,-X-space X with a point-countable base is developable. Proof. By Proposition 2.24, X has a 0-locally finite closed network and thus it is semi-stratifiablep a , VI.8.B)I. Since Xis first countable, by p a , Corollary to Theorem VI.251, X is semi-metrizable. Hence, by Theorem 2.19, X is developable. 0 3.26. Theorem (Shiraki [1971]). X is metrizable normal X-space with a point-countable base.
#i
it is a collectionwise
J. Nagata
350
Proof. This theorem follows from Corollary 3.25 and Bing's Theorem ma, Corollary to Theorem VI.41.
3.27. Lemma. Let X be a T,-strong Z-space with a Gs-diagonal. Then there is a a-locally finite closed collection Y such that for each x E X, (-){YIXEFEY} = {x}.
Proof. Let U : I 45;. be a strong Z-network, where we assume that each Siis closed to intersections. Since X has a G,-diagonal, it has a sequence {%,,In E N} of open covers such that n?=_,St(x,%,) = {x} for each x E X. ma, VI. 1.B)]. Assume that F E @. is covered by finitely many members of %j, say U , , . . . , uk; then we put
T'(F)=
{F - U , , .. . F 3
- Uk}.
(To be precise, we fix such a cover to define 3 . ( F ) . ) Put R, = U{?.(F)I F E 45;..}; then 45;., is a locally finite closed collection in X . Now we can prove that Y = (UE145;..)u (U&,45;.,) satisfies the desired condition. Suppose x # y and y E C(x). Select j such that x $ W Y , %I.
(3.8)
Since C(x) is compact, C(x) c U I u . . u U, for some U, E aj,h = 1, . . . ,k. There are i E N and F E 45;. for which C(x) c F c UI u . u uk. Hence y E F - G for some G E T.(F)c Y. Since F - G c U for some U E 3, it follows from (3.8) that 9
XE
F - (F - G ) = G j y .
Suppose y 4 C(x); then select i and F E 45;.. such that X E
C(X) c
F jy.
Note F E 3. Thus Y satisfies the desired condition.
0
3.28. Lemma. r f X is a &pace with a a-locallyfinite closed collection UE Yi satisfying the condition in Lemma 3.27, then X is a a-space. Proof. Denote by UE145;.a X-network of X . Assume that Fic %+,, 3 c Yi+l and both %. and Yi are closed to finite intersections. Put Si= e. A Yi. Then UgI Zi is a Z-network satisfying the same condition as Uz I q.Let x E X and U be an open nbd of x. If C(x) c U, where C ( x ) = (-)El Ci(x, &.),
Generalized Metric Spaces I
35 1
then for some H E Up"-,.)E4:,x 6 C(x) c H c U.If C(x) U,then, since C(x) - U is countably compact, there is H E Uz I 3.such that x E H c X (C(x) - U). Then C(x) c U u (X - H). Select H' E Uzl$. such that C(x) c H c U u ( X - H ) . T h e n x ~ H n Hc Uand HnHEU:,S,. Hence Up"-,4. is a docally finite closed network of X. 0 3.29. Theorem (Shiraki [1971]). A regular space E-space with a Gs-diagonal.
X is a a-space ifl it is a
Proof. Necessity of the condition is obvious. is a Z-network Assume that X is a E-space with a Gs-diagonal and Uz I Fi of X.Then for each x E X, C(x) = C(x, Fi)is countably compact T2 and has a G,-diagonal. Thus by m a , Theorem VII.51, C(x) is metrizable and accordingly compact. Namely X is strong Z. Therefore by Lemma 3.27 and Lemma 3.28 X is a a-space. 0 4. Universal spaces The concept of universal space is quite important in general topology. We can visualize abstract spaces by embedding them as subspaces of a concrete (universal) space. The product I' of closed intervals and generalized Hilbert space H(A) are examples of universal spaces for the Tychonoff spaces and metric spaces, respectively. (See p a ] . ) However there are not so many universal spaces known for generalized metric spaces. On the contrary, sometimes we can prove that there is no universal space. To begin with, let us give a precise definition to the concept of universal space. 4.1. Definition. There are several definitions of universal space that are, slightly different from each other. Let X be a class of topological spaces. (a) If X, E X and if every X E X is homeomorphic to a subspace of X,, then X, is called a universal space for X . (b) If X , E X and if X E X holds iff X is homeomorphic to a subspace of X,, then X, is called a universal space for X . (c) If X, E X and if X E X holds iff Xis homeomorphic to a closed subset of X,,then X , is called a universal space for X . In the following discussions we shall specify in which sense we are talking about universal space. Throughout this section all spaces which are often denoted by X are at least TI-spaces.
352
J . Nagaia
First let us show that the metacompact developable spaces have a universal space.
4.2. Proposition. X is metacompact and developable fi it has a development {an I n E N} consisting of point-jinite open covers. Proof. Necessity of the condition is obvious. Conversely, suppose X has a development %, > %, > . . . ,where each %, is point-finite. Let % be a given open cover; then there is an open cover V = U;=,VnsuchthatV < %andV, < % , , , n ~ i Y . L e t V=~ { K I ~ E A , } , F:, = { X EXISt(x, %,) c K}. Then {F:i I o! E A,} is a locally finite closed collection for each i, n E N, because V, is point-finite. Besides up"=,F:,= holds since {%,I i E N} is a development. Put
w,,=
61,
tl E
K,
K
( U { F L b E 4)) u Cu{F:fl-,b
=
-
A,, E
A,})
u .. . u ( U { F ~ ~ ' I a ~ A n - l } ) ~An. Then W = { W,, I a E A , } u { W,, I a E A , } u . * . is a point-finite open cover of Xsuch that W < Y < %. Thus Xis metacompact. (The details are 0 left to the reader.) 4.3. Corollary. If X has a countable base { U ,, U,, . . .} such that each U, is an F,-set, then X is metacompact and developable.
u,Zlcj
ej,
ej},
Proof. Let U, = for closed sets j E N. Then %ii = { U,, X i , j E N obviously form a development of x. Thus the corollary follows from 0 Proposition 4.2.
4.4. Corollary. The product of countably many metacompact developable spaces is metacompact and developable. Proof. Obvious.
0
4.5. Corollary. Any subspace of a metacompact developable space is metacompact and developable. Proof. Obvious.
0
Generalized Metric Spaces I
353
4.6. Proposition (Chaber [ 19841). Let D = llr=I N, where each N, is the set of all nonnegative integers. For each d E D , d,, denotes the nth coordinate of d. Dejine B,(i) = { d E Did, 2 i } , n, i E N, B,,(i,j)
=
( D - B,,(i)) u Bn+l(j), n, i , j E N.
Then we define the topology of D by the subbase
W
= {B,(i)ln, i E N}
u {B,(i,j)ln, i, j E N} u { D } .
Then the space D is T I ,metacornpact and developable. Proof. It is obvious that D is T , . To see that D is metacompact and developable, note that I $ I i6 f w and each member of a is F,. Thus D has a countable base consisting of F,-sets. Hence, by Corollary 4.3, D is metacompact and developable. 0 4.7. Proposition. Let t be an (infinite) cardinal numberI6 and D' = n,,,D, the Cartesian product o f t copies of D. Then define a subset S(t) of D' by S(z) = { s E D' I the coordinates of s are 0 except at mostfinitely many}, where we denote by 0 the point of D whose coordinates are all zero. We introduce a topology into S(t) as follows. Let s = {s(a)I a < t} E S(t) satisfv s(a) = 0 for all u # u I , . . . , a k . Then we define that the sets of the form U,, x . * * x Uakx Ila+a,...akBal' form a nbd base of s in D, where U,, , . . . , Uakare open nbds of s ( a l ) ,. . . ,s(uk) in D,,, . . . ,Dak,respectively, andfor all a # u l , . . . ,uk, B, are copies of an open nbd B of 0 in D. Then S(z) is a metacornpact developable T,-space of weight T. Proof. It is obvious that S(t) is a TI-space of weight z. To prove the rest, Put D(U1, . ak) = {S E S(T)lS(U,), . . . s ( c ( k ) # 0, S(a) = O 9
3
for a # a I , . . . , a k } , u , , . . . , ak <
t.
Then, since D(u,, . . . , d k ) is a subspace of the finite product D,, x . * . x D,, of the copies of D , it is metacompact and developable. Suppose @,,(ul, . . . , a k ) , n E N is a development of D ( a l , . . . , a k ) such that @,(al, . . . , uk) > @,,+l(al,. . . , a k ) and such that each @,(ul, . . . , a k ) is point-finite. I6r also denotes the first ordinal with cardinality r. !'To be precise, we should denote it by (Ue, x . . x Urn,x sake of brevity, we omit nS(r) in the following.
n,,,, . . . m k BnJ S(r). But for the
354
J. Nagata
Let A?'
2
2
un
. * . be an open nbd base of 0 in D. Then put
n
=
a
una
c
s(t),
where U,,, = B,, for all a < t. Also put
Un(al,. . . , ak) =
n
n
U,,, c
,#al .. ' a k
D,, a,,. . . , ak <
t,
a#al ...ak
where U,,, = B,, for all a # a',. . . , ak. Furthermore, we define open collections
%(a1, . . . , ak) =
{ u x un(al,. . . ,ak)I u E %n(aI, . . . ,a,)}
and
%
=
n E
(U{K(aI,. . . , ak)laI, . . . , ak <
t ; k E N}) u {Un},
N.
While V,,is obviously an open cover of S(t), we can show that it is point. . . , a;)can finite. Because, if s E D(al, . . . , ak),then no member of V,,(ai, . . . , ctk}. Thus from the point-finitecontain s unless {a;,. . . , a;} c (a1, ness of V,,(a;, . . . , a;}it follows that there are at most finitely many members of V,,that contain s. It is obvious that U,, is the only member of V,,that contains (0). Thus V,,is point-finite. Suppose W = n,,, W,is a nbd of s E D(a,,. . . , ak),where W, is some open nbd of s(a)in D, for a = a,,. . . ,ak,and W, = B,, for all a # al,. . . ,ak. Select m n such that s(aI),. . . , s(ak)# B,,, in D and such that St(s, %,,(al, . . . , ak))c Wal x
*
-
*
x
Wak.
Then the only members of V,,,that may contain s are those in Vm(al , . . . , ak). Thus St(s, V,,,)c w. Since U,, is the only member of V,,that contains {0}, {St({o}, %)In E N} = {Kin E N} is a nbd base of (0). Thus {V,,I n E N} is a development of S ( t ) consisting of point-finite open covers. Hence by Proposition 4.2 S(z) is metacompact and developable.'* 0 4.8. Corollary.
The product space S(t)" of countably many copies of S ( t ) is a metacompact, developable, T,-space with weight t. 18S(r) may be regarded as a kind of generalization of hedgehog or star-space.
Generalized Metric Spaces I
355
4.9. Example. D gives an example of a metacompact developable T,-space which is not metrizable, because D is obviously not regular. Another example of such a space can be found in p a , Example V.71. 4.10. Proposition. Let { V ( i ) l iE N} be a sequence of open subsets of a developable space X such that V ( i ) 3 V ( i 1) and n Z l V ( i ) = 8. Then there is a continuous map f from X into D such that
+
f - I ( D - (0)) = V(l), f - ’ ( B 1 ( i ) ) = V ( i ) for i E N, and
X - f-’(B,,(i,j)) c V(1) - ~ ( j f)o r n , i , j N. ~
Proof. For each n E N we define a sequence { V , ( i )I i E N} of open sets satisfying (4.1)
K ( i ) = V(i),
V , ( i ) 3 V,(i
+ 1)
m
and
V , ( i ) = V,+l(i)c V(l),
n V,(i) = 8,
(4.2)
i e N,
(4.3)
i= I
V,(i,j ) = ( X - V , ( i ) ) u K + l ( j )is an open set for i, j
E
N.
(4.4)
Define K ( i ) , i E N by (4.1) and assume that v ( i ) , . . . , V,(i),i E N have been defined. Then we define K + l ( i ) ,i E N as follows. Since each closed set of X is G,,
n U,,(i,j) m
X - K(i) =
for some open sets U , , ( i , j ) , jE N,
j= I
where we assume U,,(i,j )
3
U,,(i,j
+
+ 1). Then we define that
+
(4.5)
To prove V , + l ( j )= V , + l ( j I), we use K ( j ) = V , ( j l), U,,(i,j) 2 U,,(i,j 1) and < ( j ) n U , , ( j , j 1) = V J j ) . To prove V,+l(i)= 8, let x E X . Then x 4 V , ( j )for somej. Assume x E V,(i)for i < j - 1. Then by (4.5) x 4 U,,(i,j’) for somej’ 2 j and for all i < j - 1. Now it is easy to see that x 4 V,+,( j ’ )by use of (4.6) and (4.2). Thus (4.2) is satisfied by V , + l ( i ) i, E N.
+
nzl
+
J . Nagata
356
It is obvious that K + , ( j ) satisfies (4.3). To prove (4.4), assume i 2 j . Then K ( i ) c K ( j ) c K + ' ( j )follows from (4.2) and (4.3). Thus by (4.4) K ( i , j ) = X . Assume i < j . Then W , j ) = ( X - K(iN u
K+lW =
Un(i,j) u
K+Im
follows from (4.5). To prove the opposite inclusion, assume x E U,,(i,j ) u K + , ( j )and especially x E U,,(i,j). If x 4 K ( i ) , then x E K ( i , j ) is obvious. If x E V,,(i), then X E K ( i ) n U,,(i,j) c K + i ( j )c K ( i , j ) follows from (4.6) because of i < j . Thus anyhow we can conclude Un(hj) u K + i ( j )
=
Kkjh
proving that K ( i , j ) = U n ( i , j )u K + , ( j ) is an open set. Put K(0) = X , n E N.
(4.7)
Now we definef: X + D by f(x) = { L W I n E N}, x E x, wheref,(x) = max{iIx E K ( i ) } . Then it is obvious that f-'(B,,C)) =
W ) , f-'@,,G,
j ) ) = K(i, j ) ,
because of the respective definitions of the sets concerned. Since B (defined in Proposition 4.6) is a subbase of D and K(i, j ) is open by (4.4), this proves that f is continuous. f - ' ( D - ( 0 ) ) = V(1)
follows from (4.1), (4.3) and (4.7) andf-'(B,(i)) = V ( i )follows from (4.1). On the other hand, X - f-'(B,,(i,j))
=
X - K(i,j)
=
X
-
= K(i)
(X -
KC))
" K+IW
- K + I W = V(1) - V ( j )
follows from the definition of K ( i , j ) combining with (4.3) and (4.1).
0
4.11. Proposition. Let { U a l u c z} be a point-$nite open collection in a developable space X . Then there is a continuous map f from X into S(z) such t h a t f - ' ( D ( u ) ) = U, for each u < z, where D(u) = { s E S(z) Is(u) # O } .
351
Generalized Metric Spaces I
Proof. Denote by { a f l / nE N} a development of X with Fun
= {X E
a,,> an+]. Put
XISt(x, a,,) < U}.
(4.8)
For a fixed a we put
(4.9) (4.10) Then { < ( i ) I i E N} is a decreasing sequence of open sets of X such that V , ( i ) = 0.Thus we can define a continuous mapf,: X + D satisfying the condition of Proposition 4.10 for { V , ( i ) l i E N}. Further we define f : X + D'by
np"=]
f ( x ) = {f,(X)b <
Sincef,-'(D - (0))
=
TI,
x
E
X.
U, and { U, I a < z} is point-finite,
f ( X ) c S ( r ) and f - ' ( D ( a ) ) = U,.
Now, let us prove that f is continuous. Let f ( x ) = s = {s(a)la < T}
E
S(T)
and let V be a nbd of s in S ( t ) . Then there are open nbds . . . , k and a nbd B of 0 in D such that i= I
n
a #ul
B, c V,I9
q of s(ai) for i
=
1,
(4.11)
. . . ak
where B, = B for a # a ] , . . . , c t k . Suppose B = B f l ( i , j ) Then . it follows from Proposition 4.10 and (3) that for each a # ul, . . . , ak x$X-f,-I(B,) = X-f,-'(B,,(i,j))
= K(1)- K ( j ) c Fajp1.
From (1) and point-finiteness of { U a l a < z} it follows that {Fan[a < z} is locally finite. Hence { X - f , - ' ( B f l ( i , j )I)a < z} is also locally finite. Thus U = n { f , - l ( B f l ( i , j ) ) l a# a ] , . . . , a k ) is a nbd of x such that f , ( U ) c Bn(i,j ) = B, for each a # a ] , . . . ,ak. Since eachf, is continuous, c q, i = 1, . . . , k . Then we can select a nbd U,' of xa, such that h,(U,') W = U, n * . . n u k n U is a nbd of x such that f ( W ) c V. (Recall (4.1l). This proves that f is continuous. 0
4.12. Theorem (Chaber [1984]). For each (infinite) cardinal z, S(z)O is a universal space for the metacompact developable T I-spaces of weight < z in the sense of (b) of Definition 4.1. I9We omit the symbol nS(r).
J . Nagata
358
Proof. Let X be a metacompact developable TI-space of weight < t. Then X has a development a, > 42, > * * consisting of point-finite open covers with I 4?!il < T . Let 4?!i = { Ui I a < z i } (zi < z). Then by Proposition 4.1 I, for each i there is a continuous mapf; from Xinto S(T)such thatJ-’(D(a)) = Ui for each a < t. Now, define f : X + S ( T ) by ~ f(x) = {J(x)liEN}, X E X .
Then it is almost obvious that f is a topological map from Xinto S(z)O. Thus the theorem is proved by virtue of Corollaries 4.5 and 4.8. Now, let us turn to some negative results, i.e. we will show that some generalized metric spaces have no universal space.
4.13. Definition (Mrbwka [1954]). Let N be the set of natural numbers and Y a collection of almost disjoint infinite subsets of N , where S , , S, E Y are called almost disjoint if S , n S, is finite. Then we introduce a topology into N v Y as follows. Every point of N is isolated, and each basic nbd of S E Y is of the form { S } v (S - a finite subset of S). 4.14. Proposition. The space N v Y defined in Definition 4.13 is completely regular developable, and N is dense in N v 9. Further, if Y is a maximal collection of almost disjoint infinite subsets of N, then every infinite subset of N has a cluster point in N v 9. (Such an Y is called simply a “maximal collection” in the following discussions.)
Proof. The easy proof is left to the reader.
0
4.15. Proposition (Mr6wka [19771). There are 22”pairwise nonhomeomorphic spaces of the form N v 9for maximal collections Y .
-
Proof. Define that S, S, for infinite subsets S , , S, of N if S , - S, and S, - S , are finite. We select a member from each of thus defined equivalence classes and denote by Yothe collection of the selected members. We regard Yofixed throughout the proof. Then we consider only maximal collections Y with Y c Yo. First, let us show that there are 2’” distinct maximal collections. Consider a fixed decomposition of N into two infinite disjoint sets N, and N,. Let .49
Generalized Metric Spaces I
359
and Y2be maximal collections of almost disjoint infinite subsets of N, and N,, respectively, such that I.4c;I = IY21 = 2". There are such collections. Because, e.g. consider a sequence of successive decompositions of N, into disjoint infinite subsets, {N,}, {Nlil I i , = 1, 2 } , {Nlili21i,, i2 = 1, 2 } , . . . , where N , = N , , u N12,NIjl = N l j l lu N , i , 2 , .. . . Foreach { i , , i2, . . .}we select a point sequence {xil i E N} such that x, E Nljl, x2 E Nljli2,. . . . Then we obtain 2" mutually almost disjoint infinite subsets of N,. Then a maximal containing these subsets satisfies the desired condition. Now, collection 9, denote by cp a one-to-one map from SP, onto 9,Then . for each S E 9,, we S u rp(S). Then Yp = {S'I S E Y ; } is obviously select S' E Y; such that S' a maximal collection in N. Since rp # rp' implies 9, # Yq., and there are 22" distinct cp's, there are 2,'" distinct maximal collections. Suppose h is a homeomorphism from N u Y onto N u 9",where Y and Y' are maximal collections in N (with Y t Yo,Y' c Yo). Then h(N) = N and the homeomorphism is determined by the values on N. Hence there are at most 2" spaces of the form N u Y' which are homeomorphic to N u 9'. Thus there are 22"spaces of the form N u Y which are mutually nonhomeo0 morphic.
-
4.16. Proposition. Let X be a T,-space. Then X has at most I XI" first countable subspaces Y such that Y has a countable dense subset of which each infinite subset has a cluster point in Y.
Proof. Let Y , and Y2 be first countable subspaces of X such that D c Y , n Y,, D is countable and dense both in Y , and Y 2 ,and each infinite subset of D has a cluster point both in Y, and Y,. Then for each y E Y,, there is a sequence { y , I n E N} c D such that y , -,y . On the other hand { y , } has a cluster point y' E Y,. Since X is T,,y = y'. Namely y E Y,, meaning Y, c Y,. Similarly we have Y2 c Y , , and accordingly Y , = Y,. Since there are 1x1" countable subsets of X, there are at most subspaces Y satisfying the said condition. 0
1x1"
4.17. Theorem (Van Douwen [1979]). There is no universal space for the completely regular (regular, T2-) separable developable spaces in the sense of Defnition 4.1(a). Proof. Combine Propositions 4.14, 4.15 and 4.16.
0
In the following arguments (through Remark 4.25) all spaces are at least regular.
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J. Nagata
4.18. Definition (Tsuda [1984]). Let t denote an infinite cardinal as well as the first ordinal with cardinality t.Let T be a set with I TI = t;then consider countably many (disjoint) copies K,,i E N of Tand put To = U ,: T,..Suppose 9 is a maximal filter of T ; then S ( 9 ) = To u (9) is introduced with a topology as follows. Each point of To is isolated in S ( 9 ) , and U c S ( 9 ) is U n 3 F f o r all a nbd of 9 iff 9 E U and for some n E N and F E 9, i 3 n. 4.19. Proposition. The space S ( 9 ) defined in Dejnition 4.18 is a a-discrete2' stratlfiable (= M 3 )space with weight < 2'.
0
Proof. The easy proof is left to the reader.
4.20. Proposition. There are at most 2' spaces of the form S ( 9 ' ) which are homeomorphic to a $xed S ( 9 ) . Proof. Let h, : S ( 9 ) S ( 9 , ) and h 2 : S ( 9 )+ S(9,) be homeomorphisms. Assume hl = h2on To;then we can prove that 9, = 9, To .prove it, assume the contrary, i.e., that FI # 9,. Thus we suppose, e.g. that there is F E 9, - 9'. Then put F, = UzlF,,where F, is a copy of F in Then U = {FI} u {F,} is a nbd of FIin S(gI).Thus h,h;'(U) is a nbd of 9 'in S(9,). Hence there are n E N and G E 9 'such that
r..
h,h;l(U) n
3
G for all i 3 n.
Since h,h;I is the identity map on T o , h,h;l(U) n T,
=
F,.
Thus F, 2 G E 9, for all i 2 n , which implies F E 9', a contradiction. Thus h is determined by the values on To. Since 1 T o != t, there are at most 2' spaces of the form S ( 9 ' ) which are homeomorphic to S ( 9 ) . 0 4.21. Proposition. T with I TI = t.
There are 2,' distinct maximaljlters of the discrete space
Proof. Let [0, t] be the compact space of all ordinals < t with the order topology. Then, by [Na, Theorem VIII.91, the product [0, TI,' of 2' copies of [0, z] is a compact T,-space with a dense subset D such that I D I = t. Letf be a one-to-one map from T onto D . Then, since f is continuous, there is a 20Wemean by a u-discrete space a space which is the sum of countably many discrete (in itself) subspaces.
Generalized Metric Spaces I
36 I
continuous extensionr: j ( T ) + [0, t]" off. (See m a , Corollary 1 to Theorem IV.21.) Now T((P(T)) =
mw))= S O
=
D
=
[O,TIZ0.
HenceTis an onto map. Therefore I j ( T ) I 2 I[O,
TI2'
I
= 22'.
Thus I j ( T ) l = 22'. Since j ( T ) consists of the maximal filters of T [Na, IV.2.F)], the proposition is proved. 0 4.22. Proposition. There are 2'' a-discrete stratijiable spaces with cardinality t and weight 6 2' which are mutually nonhomeomorphic.
0
Proof. Combine Propositions 4.19, 4.20 and 4.21.
4.23. Proposition. Let X be a a-space with weight 6 2'; then X has at most 2' subsets of cardinality 6 t. Proof. It is obvious that X has a a-locally finite network I %, I < 2'. Hence
1x1 < (2')"
urZlan such that
= 2'.
Thus the cardinality of the collection of all subsets Y of X with I Y I not exceed (Ty = 2'.
< t can 0
4.24. Theorem (Tsuda [1984]). For each (injinite) cardinal 't, there is no universal space in the sense of (a) for the classes of spaces with weight 6 2': (i) a-spaces, (ii) stratijiable spaces, (iii) a-discrete stratijiable spaces, (iv) stratijiable spaces with dim < 0." Proof. Note that every stratifiable space is a. (See [Na, VI.8.1) Also note that dim S ( 9 ) < 0 and then combine Propositions 4.22 and 4.23. 0 4.25. Remark. Actually Tsuda [1984] showed the following spaces of weight < 2' have no universal space either: M,-spaces, stratifiable p-spaces and some other spaces. (See Chapter 10 for the definitions of these spaces.) 2'See Nagata [1983].
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J. Nagata
Now, let us turn to another type of spaces, paracompact M-spaces. (See [Na, VII.21.) From now on all spaces are at least T2.
< t , and f a perfect map from X onto a metric space Y. Then X is homeomorphic to a closed subset of I' x Y , where I' denotes the product of t copies of the closed interval I = [0, I].
4.26. Proposition. Let X be a paracompact M-space with weight
Proof. L e t 9
=
{Ualct < t } b e a b a s e o f X w i t h I Q I
((Un,UB)ICG B < t; u a c us}. To each ( U a , U s ) E S we assign a continuous map gab:X
< r.Thenput
9 =
+
[0, I] = Iaasuch
that ga,(ua) = 0, Then define g =
ga,(X -
fl g,,: x +
a.B
by
up) =
n I,,
=
1.
I'
a.B
x E x* It is easy to see that g is a topological imbedding of X into I'. Now, define cp :X --t I' x Y by g ( x ) = {ga,(x) Ia,
cp(4
B<
= (g(x),f(x)), x
t},
E
x.
Then cp is a topological imbedding of X into I' x Y and cp(X) is closed. The detail is left to the reader. (See the proof of w a , Corollary 3 to Theorem VII.31.) 0 4.27. Corollary. I' x H ( T ) is a universal space for the paracompact M-spaces of weight < t in the sense of (a), where H ( T ) denotes the (generalized) Hilbert space with the index set T such that 1 TI = t.22
Proof. Let X be a paracompact M-space with weight < t. Suppose 9 is a base of X with 19I < T. Note that there is a perfect map f from X onto a metric space Y. (ma, Corollary 1 to Theorem VII.31.) Then it is easy to see that weight of Y < T. Because for the collection Y of all finite sums of members of 9, { Y - f ( X - V )I V E Y }is a base of Y. Then Y is homeomorphic to a subset of H ( T ) by p a , Theorem VI.101. Thus by Proposition 4.26 X is homeomorphic to a subset of I' x H ( T ) , proving the corollary.
0 " H ( T ) in this corollary can be replaced with S"(T), the countable product of copies of the metric hedgehog S ( T ) .
Generalized Metric Spaces I
363
However, I' x H ( T ) is not necessarily a universal space in the sense of (c) as will be seen in the following. 4.28. Lemma. For a metric space Z with weight < z the following are equivalent. (i) Every paracompact M-space X of weight < z is homeomorphic to a closed subset of I' x Z. (ii) Every metric space M with weight < z admits aperfect map onto a closed subset of Z.
Proof. (i)+(ii). By (i) we may regard M as a closed subset of I' x Z. Then, let n be the projection of I' x Z to Z. Suppose F is a given closed subset of Mand y E Z - n ( F ) . Then, since n-l( y) n F = 8, for each (x, y) E n-'( y) we can find a nbd U ( x ) x V ( x ,y ) such that U ( x ) x V ( x , y ) n F = 8, where U ( x ) and V ( x , y) are nbds of x and y in I' and Z, respectively. Cover the compact set n - ' ( y ) with {U(xi)x V ( x i ,y)l i= 1, . . . , k}. Then V(xi,y) is a nbd of y disjoint from n ( F ) . Thus y 4 n ( F ) , proving that n ( F ) is closed in Z. Hence n is a perfect map from M onto n ( M ) , which is closed in Z. (ii)+(i). By Proposition 4.26 we may regard X as a closed set of I' x M , where M is a metric space with weight < z, and there is a perfect map f from X onto M. By (ii) there is a perfect map g from M onto a closed set F of Z. Then by Proposition 4.26 we may regard M as a closed set of I' x F. Now I' x M is a closed subset of I' x I' x F. Since the last set is homeomorphic to I' x F, which is closed in I' x Z, we can regard I' x M as a closed subset of I' x 2. Hence X is homeomorphic to a closed subset of I' x Z. 0
4.29. Lemma. Let f be a perfect map from a subset S of X onto Y. Then G = {(x, f ( x ) ) I x E S } is closed in X x Y. Proof. Let (x, y) E X x Y - G; then x 4 f -'( y ) . Sincef - I ( y) is compact, there are disjoint open sets U and V of X such that U 3 x, V 3 f - I ( y). Thus y 4 f ( S - V ) . Sincef is closed, there is an open nbd W of y in Y such that W n f ( S - V ) = @.Thenf-'(W) n (S - V) = OinX.Hencef-'(W) n U = 8. Thus U x Wis a nbd of (x, y ) such that (U x W) n G = 8, proving that G is closed in X x Y . 0 4.30. Theorem (Kato [1983]).
Let z be an infinite cardinal; then thefollowing are equivalent. (i) There is a universal space for the paracompact M-spaces of weight < z in the sense of (c).
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(ii) There is a universal space for the metric spaces of weight of ( 4 .
< T in the sense
Proof. (i)+(ii). Let Xo be a universal space for the paracompact M-spaces of weight < z in the sense of (c). Then there is a perfect map from Xo onto a metric space Yo.By Proposition 4.26 X , may be regarded as a closed subset of I' x Yo.Then we claim that H ( T ) x Yosatisfies the condition of (ii), where I TI = 7 . Suppose M is a given metric space of weight < T. By Lemma 4.28, there is a perfect map f from M onto a closed set F of Yo. On the other hand, M can be regarded as a subset of H ( T ) . Thus by Lemma 4.29, G = { ( x ,f ( x ) ) I x E M } is a closed set of H ( T ) x F. Since M is homeomorphic to G, and H ( T ) x F is a closed set of H ( T ) x Yo,M is homeomorphic to the closed set G of H ( T ) x Yo.Thus our claim is proved. (ii)+(i). Let M , be a universal space for the metric spaces of weight < T in the sense of (c). Then I' x Mo satisfies the condition of (i). Because, if X is a paracompact M-space with weight < T, then there is a perfect map f from X onto a metric space M with weight < T. Then M is homeomorphic to a closed set F of Mo. Then X is homeomorphic to a closed set of I' x F by Proposition 4.26. Thus X is homeomorphic to a closed set of I' x M,, proving our claim. 4.31. Theorem (Kato [1983]). Let T be a cardinal satisfying w < z < 2". Then there is no universal space for the paracompact M-spaces of weight < z in the sense of (c). Proof. By Theorem 4.30 it suffices to show that there is no universal space for the metric spaces of weight < T in the sense of (c). Let X be an arbitrary metric space with weight < T. Let R be the real line and B the collection of all subsets of R.We also denote by 2 the collection of all subsets of R that are homeomorphic to a closed set of X. Then 1 9I = 2*". Suppose F is a closed set of X and 2(F)
=
{F' E 2 I F' is homeomorphic to F } #
8.
Let C(F) be the set of all continuous functions from F into R. Then, since F is separable, I C(F)l < (2")" = 2". Thus I2(F)I < IC(F)I
< 2".
On the other hand, each closed set F o f Xis determined by its countable dense subset. Thus there are at most 1 X 1" of them. Note that I X 1"' < (T")" < 2". S i n c e 9 = U{9(F)I F i s aclosed set ofX}, we have 1 9 1 < 2" x 2" = 2".
Generalized Metric Spaces I
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This proves that I 9 I > I % 1, and hence some subsets of Iw are homeomorphic to no closed subset of X. Hence Xis no universal space for the metric spaces of weight < T in the sense of (c). This proves the theorem. 0
References Arhangel’skii, A. V. [I9631 Certain metrization theorems, Uspehi Mat. Nauk 18, 5 (1 13) 139-145. Brandenburg, H. [I9801 Some characterizations of developable spaces, Proc. A M S 80, 157-161. [I9881 Upper semi-continuous characterizations of developable spaces, to appear. Burke, D. [I9701 On p-spaces and wA-spaces, Pacific J. 35, 285-296. Chaber, J. [I9831 Another universal metacornpact developable T,-spaces of weight m, Fund. Math. 122, 247-25 3. Creede, G . [ 19701 Concerning semi-stratifiable spaces, Pacific J. 32, 47-54. Douwen, E. K. van [I9791 There is no universal separable Moore space, Proc. A M S 76, 351-352. Foged, L. [I9841 Characterizations of K-spaces, Pacific J. 110, 59-63. Gao, Z. [I 987al N-space is invariant under perfect mappings, Questions Answers General Topology 5, 271-279. [l987b] The closed images of metric spaces and Frichet K-spaces, Questions Answers General Topology 5, 28 1-291. Gao, Z . and Y. Hattori [ 1986/87] A characterization of closed s-images of metric spaces, Questions Answers General Topology 4, 147-1 5 I . Guthrie, J. [I9731 Mapping spaces and cs-networks, Pacijic J. 47, 465471 Heath, R. W. [I9651 On spaces with point-countable bases, Bull. Acad. Polon. Math. Ser. 13, 393-395. Hodel, R. [1971] Moore spaces and wA-spaces, Pacijic J. 38, 641-652. Ishii, T. [I9671 On closed mappings and M-spaces I, Proc. Japan Acad. 43, 752-756. Ishii, T. and T. Shiraki [I9711 Some properties of wM-spaces, Proc. Japan Acad. 47, 167-172. Isiwata, T. [I9693 The product of M-spaces need not be an M-space, Proc. Japan Acad. 45, 154-156
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Kanatani, Y ., N. Sasaki and J. Nagata [ 19851 New characterizations of some generalized metric spaces, Murh. Juponicu 30,805-820. Kato, A. [ 19831 Non-existence of universal paracompact M-spaces, J. Ausrr. Muth. SOC.35, 281-286. Kullman, D. [I9711 A note on developable spaces and p-spaces, Proc. A M S 27, 154-160. LaSnev, N. S. [I9661 Closed mappings of metric spaces, Soviet Marh. 7 , 1219-1221. Michael, E. [1966] &,-spaces, J. Murh. Mech. 15, 983-1002. Morita, K. [ 19671 Some properties of M-spaces, Proc. Japan Acud. 43, 869-872. Mrowka, S . [I9541 On completely regular spaces, Fund. Murh. 41, 105-106. [I9771 Some set-theoretic constructions in topology, Fund. Muth. 94, 83-92. Nagami, K. [I9691 Z-spaces, Fund. Math. 65, 169-192. Nagata J. [ I9721 Some theorems on generalized metric spaces, Theory of Sets and Topology, A collection of papers in honour of F. Hausdorff, 377-390. (19831 Modern Dimension Theory, (Heldermann, Berlin rev. & ext. ed.). [ 19851 Modern General Topology (North-Holland, Amsterdam, 2nd. rev. ed.). [1986/87] Characterizations of metrizable and Lahev spaces in terms of g-function, Questions Answers General Topology 4, 129-139. [I9871 A note on LaSnev space, Questions Answers General Topology 5, 203-207. Novak, J. [I9531 On the Cartesian product of two compact spaces, Fund. Murh. 40,106112. OMeara, P. [I9711 On paracompactness in function spaces with the compact open topology, Proc. A M S 29, 181-189. Shiraki, T . [ 19711 M-spaces, their generalizations and metrization theorems, Sci. Rep. Tokyo Kyoiku Duiguku, Ser. A 11, 57-67. Tamano, K. [I9821 On a local property of products of LaSnev spaces, Topology Appl. 14, 105-1 10. [1985] Closed images of metric spaces and metrization, Topology Proc. 10, 177-186. Tsuda, K. [I9841 Non-existence of universal spaces for some stratifiable spaces, Topology Proc. 9, 165171. Worrell, J. M. and H. H. Wicke [ 19651 Characterizations of developable topological spaces, Cunud. J. Muth. 17, 820-830.
K.Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V.(1989)
CHAPTER 10
GENERALIZED METRIC SPACES I1
Ken-ichi TAMANO* Faculty of Liberal Arts, Shizuoka University, Ohya, Shizuoka 422, Japan
Contents
I. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of basic results. . . . . . . . . . . . . . . . . . . . . . . . . . . Closure-preservingcollections and definitions of various stratifiable spaces . . . . Expandability, extension property, and sup-characterization of stratifiable spaces . Classes of M,-spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closed maps and perfect maps . . . . . . . . . . . . . . . . . . . . . . . Related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367 368 371 382 388 396 400 405 406 407
Introduction The purpose of this chapter is to give an introduction to the recent development of the theory of stratifiable spaces. The Nagata-Smirnov metrization theorem says that metrizable spaces are precisely those spaces which have a a-locally finite base. Replacing the locally finite condition by closure-preserving type conditions, Ceder [ 19611 defined Mi-spaces, i = 1, 2, 3, as generalizations of metrizable spaces and proved MI + M2+ M 3 . Borges [1966] renamed M,-spaces “stratifiable spaces” and proved several important results concerning these spaces. Gruenhage [ 19761 and Junnila [19781 independently proved that M 2 = M 3 . Ceder’s classic problem whether M3 implies M I still remains open. Not only the *Present affiliation: Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240, Japan.
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K. Tamano
problem but also various problems concerning stratifiable spaces are unsolved. In this chapter, our main concerns are classes of spaces between the class of stratifiable spaces and the class of stratifiable p-spaces. Stratifiable p-spaces are stratifiable spaces which can be embedded in a product of countably many paracompact F,-metrizable spaces, and are known to be hereditarily MI.The question whether these classes are equal remains unsolved. The author ventures to have a fervent hope that some reader will prove that they are equivalent classes. There are plenty of fruitful properties for those spaces proved by Ceder, Borges, Gruenhage, Junnila, Heath, It6, Mizokami, Oka and others. Stratifiable spaces possess nice preservation properties under various topological operations. MI-spaces have a simple and natural definition. Stratifiable p-spaces have various dimension theoretical properties. If they are unified, then stratifiable spaces will have more validity and utility. All the requisite background in general topology and stratifiable spaces can be found in Nagata [ 19851. For the survey of generalized metric spaces, we recommend Burke and Lutzer [1976], Gruenhage [ 19841and Chapter 9 by Nagata. Throughout this chapter, all spaces are assumed to be regular TI.A map is a continuous surjection unless stated otherwise. The letter N denotes the set of positive integers and i, j , k, m and n are always used to denote members of N. The author wishes to thank J. Nagata and H. Ohta for their many helpful comments and suggestions.
1. Review of basic results
This section contains a review of basic results concerning stratifiable spaces. We state them without proofs. The proofs can be found in Nagata [1985] or Gruenhage [1984]. Two lemmas on monotonically normal spaces are proved. 1.1. Definition. A collection d of subsets of a space Xis closure-preserving if Cl(Ud’) = U{Cl A : A E d ’ }for each d’ c d . d is closure-preserving at x E X if x E Cl(Ud’) implies that x E U{Cl A : A E d’}for each d’ c d. 1.2. Definition. Let X be a space and A9 a collection of subsets of X . A9 is a quasi-base for X if for each open set U of X and a point x E U , there exists B E L24 such that x E Int B c B c U . A quasi-base consisting of open sets of X is a base for X.
Generalized Meiric Spaces I1
369
1.3. Definition. Let X be a space and A a subset of X . A collection $2 of subsets of X is an outer quasi-base of A in X if every member of $2 is a neighborhood of A, and for each neighborhood V of A, there is U E $2 such that A c U c V . An outer quasi-base consisting of open sets is an outer base. 1.4. Definition. An M,-space is a space with a a-closure-preserving base. An M,-space is a space with a a-closure-preserving quasi-base. An M,-space is a space with a a-cushioned pair-base. (We do not state the definition of a a-cushioned pair-base.: 1.5. Definition. A space Xis stratijiable (resp., semi-stratijiable) if there is a function G which assigns to each n E Nand a closed set H of X , an open set G(n, H) containing H satisfying (i) and (ii) (resp. (i’) and (ii)) of the following: (i) H = C1 G(n, H); (i’) H = nnG(n,H ) ; (ii) G(n, H) c G(n, F ) whenever H c F.
n,,
The function G is called a stratijication (resp., a semi-stratijication). Replacing G(n, H ) by G’(n, N ) = nk,,G(k, H), we always assume that G(n + 1, H ) c G(n, H) for each n E N.
1.6. Definition. A collection d of subsets of a space Xis called a network for X if every open set of X is a union of members of d .A space is a a-space if it has a a-locally finite network. It is known that a space is a cr-space if and only if it has a a-discrete closed network. 1.7. Definition. A space Xis monotonically normal if to each pair (H, K ) of disjoint closed subsets of X, one can assign an open set D(H, K) such that (i) H c D(H, K ) c C1 D ( H , K) c X - K; (ii) if H c H‘ and K =I K’, then D ( H , K) c D ( H ’ , K ’ ) . The function D is called a monotone normality operator for X . Recall that two subsets S, T of X are separated if S n (CI T ) = (CI S ) n T = 8. 1.8. Lemma (Borges [1973], Heath, Lutzer and Zenor [1973], K. Masuda [1972]). Any monotonically normal space X has a function D which assigns to each pair ( S , T ) of separated subsets of X an open set D(S, T ) satisfying (i) S c D(S, T ) c C1 D(S, T ) c X - T ; (ii) i f S c S’ and T =I T‘, then D ( S , T ) c D(S‘, T ‘ ) ;and (iii) D(S, T ) n D ( T , S ) = 8.
370
K. Tamano
Proof. Let Gobe a monotone normality operator for X . Define G(H, K) = Go(H,K ) - C1 Go(K,H). Then G is also a monotone normality operator and satisfies that G ( H , K) n G(K, H ) = 8. For each pair (S, T ) of separated subsets of X , define D(S, T ) = U{G({x}, C1 T ) : xE S}. Then C1 D(S, T) c X - T. Indeed if y E T, G({y}, C1 S) is an open neighborhood of y which does not meet D(S, T), because G ( { x } ,C1 T ) n G({y } , C1 S ) c G({x},{ y } ) n G({y } , {x}) = 8 for any x E S.Then it is easy to check that D satisfies (i), (ii) and (iii). The following lemma is essentially due to Gruenhage [1980] and is described in Mizokami [1983]. 1.9. Lemma. Let A be a closed set of a monotonically normal space X . Then to each open set U of the subspace A , one can assign an open set E ( U ) of Xsuch that (i) E ( U ) n A = U ; (ii) for any open collection 4 of the subspace A , (Cl(U{E(U): U E 4})) n A = Cl(U4); (iii) i f 4 is a closure-preserving open collection of the subspace A , then the collection Y = U {Yu: U E @), where 9;= ( S :S is a subset of X with U c S c E ( U ) } , is closure-preserving at every point of A.
Proof. Since Xis monotonically normal, there is an operator D satisfying the properties of Lemma 1.8. For each open set U of A, U and A - C1 U are separated. Define E ( U ) = D(U, A - C1 U ) - (A - U ) . Clearly (i) holds. To show (ii), assume that x E A - Cl(U4). Then D ( { x } , Cl(U4)) is an open neighborhood of x in X which does not meet Cl(U{E(U): U E 4}). Indeed, for each U E 4 , E ( U ) n D ( { x } , Cl(U4)) c D ( U , A - C1 U ) n D ( { x } , Cl(U4)) c D(Cl(U4), {x}) n D ( { x } , Cl(U4)) = 8. The property (iii) easily follows from (ii). 0 1.10. Theorem. The following conditions are equivalent for a space ( X , z): (i) X is stratijiable. (ii) X is M3. (iii) X is M 2 . (iv) There is a function g : N x X + T such that (a) {XI = n,g(n, and (b) if y 4 H, where H is closed, then y $ Cl(U{g(n, x): x IZH ) for some n E N.
Generalized Metric Spaces II
37 I
Recall that a space is called perfect if every closed set is a Gs. 1.11. Theorem. The following implications hold: (i) metrizable * MI * M2 = M , = stratijiable normal, monotonically normal a. (ii) a =s semi-stratijiable * perfect.
* paracompact, perfectly
Generally, let X be a class of topological spaces. Consider various conditions to be satisfied by X , as follows: (a,) (resp., (a2);(a3))If X’ is a subset (resp., a closed subset; an open subset) of X E X , then X’ E X . (b,) If X, E X for each i E N , then lIiENX,.E X . ( b , ) i f X ~ X a n dY E X , t h e n X x Y E X . (ci) ( i = 1,2, . . . , 5 ) If Y is the image of X E X under a continuous map f satisfying the condition (i) of the following, then Y E X . (1) a closed map; (2) a perfect map; (3) a closed irreducible map; (4) a perfect irreducible map; ( 5 ) a finite to one closed map. (do) If X = UieNX,for closed subsets X, E X , i E N , of X , then X E X . (d,) (resp. (d2))If Xis a stratifiable space satisfying X = UieNX,for closed subsets X, E X , i E N , of X (resp. X = XI u X2 for closed subsets X , and X 2 ) , then X E X . (e) If X is dominated by a closed cover { X , : a E A} such that X , E X for all a, then X E X , where the closed cover { X , : a E A} is said to dominate X if the following holds: A subset F of X is closed if there is a subcollection { X a : aE A’} of the cover such that F c U { X a :a E A’} and X , n F i s closed for every tl E A’. 1.12. Theorem. Stratijiable spaces satisfy all above conditions except (d,,).
2. Closure-preservingcollections and definitions of various stratifiable spaces The purpose of this section is to give lemmas for closure-preserving collections and mosaical collections which will be used in later sections. Some of them are easy to prove, so the proofs are left to the reader. After giving them, we define various classes of stratifiable spaces, and show a diagram and a table concerning these classes. 2.1. Lemma. Let F be a closure-preserving closed collection of a space X and A a subset of X . Then B I A = { F n A :F E F}is a closure-preserving closed
K . Tamano
372
collection of A. Furthermore, i f A is closed in X , then 9I A is a closure-preserving closed collection of X .
2.2. Lemma. Let 42 be a closure-preserving open (resp. regular closed) collection of a space X . Suppose G is a clopen set in X . Then 42 1 G = { U n G : U E 42} is a closure-preserving open (resp. regular closed) collection of X . Closure-preserving regular closed collections play an important role in the theory of MI-spaces. Indeed we have the following theorem. 2.3. Theorem. A space X is an M I-space i f and only i f X has a a-closurepreserving quasi-base consisting of regular closed sets.
Proof. If X is an MI-space, then there is a a-closure-preserving base g. Then %? = (C1 B : B E ?? is aIclosure-preserving } quasi-base consisting of regular closed sets. Conversely, if X has a a-closure-preserving quasi-base 59 consisting of regular closed sets, then = {Int C: C E %?}is a a-closurepreserving base. 0 2.4. Lemma. Let F be a closed (resp., regular closed) set of a space X . Suppose 42 is a closure-preserving closed (resp., regular closed) collection of the subspace F. Then 42 is also a closure-preserving closed (resp., regular closed) collection of X .
2.5. Lemma. Let 42 = U {a,,:n E N } be a a-closure-preserving outer quasibase of a closed set F of a space X consisting of closed (resp. regular closed) sets of X . Suppose { H,, :n E N } is a decreasing sequence of closed (resp. clopen) neighborhoods of X with F = n { H , , : n E N } . Then u{%,,lH,,:n E N } is a closure-preserving outer quasi-base of F consisting of closed (resp. regular closed) sets of X . 2.6. Lemma. Let 42 = (J{a,, : n E N } be a a-closure-preserving outer base of a closed set F of a space X . Suppose {G,,:n E N } is a decreasing sequence of clopen sets o f X with F = G,, : n E N }. Then (J{@,,I G,, :n E N } is a closurepreserving outer base of F.
n{
2.7. Lemma. Let 42 be a closure-preserving open coilection of X . I f 0 is a dense subset of X , then the collection 42 I D = { U n D : U E 42} is a closurepreserving open collection of the subspace Q.
Generalized Merric Spaces 11
373
Proof. Since D is dense in X , Cl,(U n D) = (C1,U) n D for any open set U . Since 42 is a closure-preserving collection of X , {CI,U: U E %} is a closure-preserving closed collection of X . Thus by Lemma 2.1, { (Cl, U ) n D : U E a} is a closure-preserving collection of D. Hence { U n D : U E %} is a closure-preserving collection of D. 0 2.8. Theorem. If D is a dense subspace of an M,-space X , then D is an M I-space. Proof. Let A? be a a-closure-preserving base for A’. Then by using the above lemma, it is easy to see that B I D = { B n D : B E B } is a a-closure-preserving base for D. 0 2.9. Lemma. Let dj,be a closure-preserving collection of a space X , and A;, = udj.for each 3, E A. If the collection { Aj.:Iz E A} is locally finite at a point x E X . then uj.dj.is closure-preserving at x. 2.10. Lemma. Let di be a closure-preserving collection of subsets of a space X, for each i = 1,2, . . . ,n. Then theproduct ll:=, di= {n,”, , A i :Ai E difor i = 1,2, . . . ,n} is a closure-preserving collection of the product space l l y = I X;. Now let us define a mosaical collection which will play an important role throughout this chapter. Although the term was defined in Tamano [1985], such a collection has been used by many topologists. 2.11. Definition. Let d be a collection of subsets of a space X . A a-discrete closed cover 4 of X is a mosaic for d if for each A E d there exists a subcollection FAof 4 such that A = U F A = X - sA). In other words, 4 is a mosaic for d if 4 is a a-discrete closed cover of X such that for each F E 4 and A E d,F n A = 8 or F c A . A mosaical collection d is a collection which has a mosaic 4 for d.The partition induced by d is the where P ( d ’ ) = - U { d - d’). disjoint cover { P ( d ’ ) : d ’ c d}, Note that P(0) = X -
u(4
ud.
nd’
It is easy to see that a collection d has a mosaic 4 if and only if the partition induced by d has a a-discrete closed refinement 9 as a cover. The following theorem is essentially due to Siwiec-Nagata [1968]. They proved for a-spaces. The theorem for semi-stratifiable spaces is in Junnila [1978].
K. Tamano
314
2.12. Theorem. For any semi-stratifiable space X , every closure-preserving closed collection 9 of X is mosaical. Proof. Let G(n, H ) be a semi-stratification for X . For each subcollection 9‘of 9, define P(F’)
=
OF’ - U(F
P,,(9’)
=
n9’
-
9’1, P(8)
=
X - U 9 , and
- G(n, U ( 9 - F’)), Pn(8) = X
-
C(n,
u9).
Then P,(S’) is a closed set and P ( 9 ’ ) = U,P,(F’). So { P n ( 9 ’ ):9’ c 9, n E N } is a closed cover of X which refines the cover ( P ( 9 ’ ):9‘c 9} of the partition. It remains to show that { P n ( 9 ’ ): 9‘c 9} is a discrete collection for each n. To see this, let x E X . Define X = {FE % : x E F} and U = C(n, {x}) - U(S - X ) . Then U is an open neighborhood of x. Suppose F‘ c 9 and 9’ # X . If there exists F E 9‘- 2,then P,(9’) c F and U c X - U ( 9 - 2)c X - F. So P n ( 9 ’ )n U = 8. On the other hand, if there exists F E X - F’, then P,(9’) c X C(n, U(S - 9’)) c X - G(n, F ) c X - C(n, {x}). Thus P,(S’) n U = 8.
0 2.13. Lemma. (i) Every locally finite open (closed) collection of a perfect space is mosaical. (ii) (Worrell and Wicke [1965]) Every point-finite open cover of a perfect space is mosaical. 2.14. Lemma. Let d and W be mosaical collections of a space X and V c d.Let 9 be the smallest collection of subsets of X satidying that d c 9 and 9 is closed under arbitrary unions, intersections, and complements. Then the following collections are mosaical: (a) W,
(b) 9,
(c) d u 93,
(d) {A n B : A
E
d ,B
E
a},
(e) ( A U B : A E ~ , B E W ) , (f) {A - B : A E ~ , B E W } . (g) iff: Y + X i s a continuous onto map, thenf is a mosaical collection of Y.
-I(&)
= { f - ‘ ( A ): A
E
d}
Proof. (d), (e), (f) follows from (a), (b), (c). Let 9 and X be a mosaic for d and respectively. Then 9 is also a mosaic for %? and 9.Define X = ( F n H : F E 9, H E X } .Then it is easy to see that .T is a o-discrete closed cover and a mosaic for d u 93. To see (g), note that f -‘(9) is a 0 mosaic for f -I(&).
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2.15. Definition. We call a collection d of subsets of a space X strongly point-finite if there exists an open collection { U ( A ) :A E d } such that A c U ( A ) for each A E d ,and { A E d :x E U ( A ) } is a finite collection for each x E X . A collection d is star-finite if for each A E d,the collection { A ’ E a2 : A n A’ # 8) is a finite collection. 2.16. Lemma. Every mosaical collection d of a collectionwise normal perfect space X has a star-Jinite and strongly point-finite mosaic. Proof. Let 9 = U n P nbe a mosaic for d ,where each Fnis a discrete closed collection. Define F, = U(ukG,,Pk). It follows from the perfect normality of X that there are closed sets H,,,, such that X - Ffl = U,H,,, and the collection {Hn,,:m E N } is star-finite for each n. Define X,,, = Pn+, 1 Hn,,, = { F n H,,, :F E Sn+, }. Then it is easy to see that 9, u {Un.,Xn,,) is a starfinite mosaic for d . To complete the proof, it suffices to show that every a-discrete and star-finite closed cover H of a collectionwise normal space is strongly point-finite. Let Hn, n E N , be discrete closed collections with H = u n H f For l . each H E Hn,take an open neighborhood U ( H )of H such that U ( H ) c X - U { H ’ E UkGn3EPk : H n H’ = 8} and { U ( H ) :H E H , } is a discrete collection. Now it is easy to check that the collection { U ( H ): H E H } guarantees the strong point-finiteness of 2. 0 Let { A , : 1E A} be a collection of subsets of a space X indexed by A. We say { Aj.: 1 E A} is point-finite if for each point x E X , (1 E A :x E A , } is a finite set.
2.17. Lemma. Let { A , : 1 E A} be apoint-finite mosaical collection of a space X . If di is a mosaical collection of the subspace Aj, for each 1 E A, then d = is a mosaical collection of X .
u,d,
Proof. Let 9 be a mosaic for the collection ( A , :1 E A} in X . Since { A, : A E A} is point-finite, for each F E 9, there is a finite subset AFof A such that F c A, for each 1 E A F , and F n A , = 8 for each 1 E A - AF. For each A E AF, since d,is mosaical in A , , d,I F = { A n F : A E dj.}is a mosaical collection of F. Thus by Lemma 2.14(c), d I F = I F : 1E AF} is a mosaical collection of F. Let 2;. be a mosaic for d I F in F. Then the a-discrete closed collection H = UlHi of X is a mosaic for d . 0
u{d;,
2.18. Lemma. A strongly point-finite collection d = { A j , :1 E A). of a perfect space X is mosaical if each A, is closed or open in X .
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Proof. It follows from the strong point finiteness of d that there is an open collection { U ,: I E A} such that for each 1E A, A .; c U , ,and for each x E X , { I E A :x E U , } is a finite set. Then, since { U, : I E A} u {X}is a point-finite open cover of X , the collection { U ,: 1 E A} is a point-finite mosaical collection of X by Lemma 2.13. On the other hand, the collection {A,} with only one element is mosaical in the perfect space U, for each I. Thus by the above lemma, d is a mosaical collection of X. 0
2.19. Lemma. Ifd is a mosaical collection of a a-space X , then there is a a-discrete dense subset D of X such that A n D is dense in A for every A E d. Proof. Let X be a a-discrete closed network for X and 9 a mosaic for d . For each H E X and F E 9 with H n F # 8, pick a point x ~E H , n ~ F. Define D = { x ~ H . ~E: X , F E 9, H nF # Then it is easy to check that D is the desired subset of X. 0
s}.
2.20. Lemma. Let ( X , t) be a paracompact a-space and 9 = U,,9,, a a-discrete open collection of X . Then there is a metrizable topology e on X with @c@cr.
Proof. Since Xis a paracompact a-space, there is a network 9 = U,%, such is , a discrete closed collection. By using the collectionwise normalthat each 9, ity and the perfect normality of X , there is a discrete open collection V,,,,= { V,(F) : F E F,, for }each n, m E N such that V,(F) = F for each F E 9. Since Xis perfectly normal, there are continuous functionsf,,,, g,,, n, m E N , from X into the unit interval [0, 11 such thatf,;(O) = X - (UV,,,,), g;'(O) = X - (U9,,). Define continuous functions p,,, and v, on X x X as follows:
om
i
- L,m(Y)l
If,.m(x)
Pn,m(X, Y ) =
f,,m
vn(x, Y ) =
(x)
+
Y)
if {x, Y } c Vm(F) for some F E T,,, otherwise;
Ign(x) - gn(Y)I if {x, y } c U for some U E 9,,, gn(X) gn(Y) otherwise.
Now define
It is not difficult to check that d is a metric function on X and the topology c t. 0
e induced by d satisfies that 9 c e
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2.21. Lemma (Bregman [1983], Oka [1983]). Let ( X , z) be a paracompact a-space and 9 = UnF,, a a-closure-preserving closed collection of X . Then there is a metrizable topology e c z on X such that each F,,is also a closurepreserving closed collection of ( X , @).Furthermore, $a a-discrete open collection 9 and a metrizable topology p c T are given, then e can be chosen to satisfy 9 v p c c z.
Proof. For each n E N , let 2, = U,X,,, be a mosaic for F,, where each Xn,m is a discrete closed collection of ( X , T). For each H E X,, define V ( H ) = X - U { F E F,,: F n H = S}. Then V ( H )is an open set. For each n, m E N , take a discrete open collection @, = { U ( H ):H E X,.,} such that H c U ( H ) c V ( H ) . By the above lemma, there is a metrizable topology e on Xsuch that U,,,9,,, c e c z. We show that each 9, is a closure-preserving closed collection of ( X , e). So suppose S’c 9,. We show that US’ is closed in ( X , e). To see this, let x E X - US’. Since X, is a mosaic for F,’, there are m E Nand H E X,,, such that x E H t X - UF’. By the definition of @”,,, we have H c U ( H ) c V ( H ) c X - ~ { F Sn: E F n H = 8) c X - US’. Hence U ( H ) is an open set of ( X , e ) with U ( H ) n (US’) = 8, which implies that US’ is closed. To complete the proof, suppose 42 and p are given. By the above Lemma, it is easy to choose e so that 9 c e c T. Now define e’ to be the topology generated by the subbase e v p. Then it is easy to check that e’ is the desired one. 0 2.22. Definition. Let d be a collection of subsets of a space X . d is almost locallyfinite at x E X (Itb-Tamano [ 19831) if there is a finite collection %? of subsets of Xsuch that for each A E d ,A = C n Vfor some C E %? and some (not necessarily open) neighborhood V of x. d is finitely closure-preserving at x E X (Ohta [198.]) if for any subcollection d‘of d,there are a neighborhood U of x and a finite subcollection %? of d‘such that U n C l ( U d ’ ) = U n CI(u%‘). d is almost locallyfinite in X (resp.jnitely closure-preserving in X ) if d is almost locally finite (resp. finitely closure-preserving) at every point of X . 2.23. Lemma. Every almost locally finite collection is finitely closurepreserving. Every finitely closure-preserving collection is closure-preserving. The following lemma was described in Junnila-Mizokami [ 19851. The almost local finiteness of such collections was essentially proved in Tamano [1983].
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2.24. Lemma. Let 4 be a mosaical open collection of a stratijiable space ( X , t). Then there are open collections 4' = { U ' : U E 4 } ,0 < r < 1, of (X, t) satisfying (i) each 4' is an almost locallyjinite and mosaical open collection of ( X , t); (ii) C1,Us c U' whenever r < s; (iii) U = U{U':O < r < l}, U' = U { U J : r< s .c l} for each U E Q and0 < r .c 1; (iv) there are a metrizable topology e on X with e c t and a a-discrete closed collection .@ of ( X , e) such that H - U and H - u'for each H E .@, U E 4 , 0 .c r < 1, are closed sets of ( X , e).
Proof. Let 9 = U , 9 , be a mosaic for 4, where each 9, is a discrete closed collection. By Lemma 2.16, we may assume that 9 is strongly point-finite. So there is an open collection { V ( F ):F E 9} satisfying (1) if F E 9,, then F c V ( F ) c G(n, F ) , where G(n, F ) is the nth stratification of F; (2) { V ( F ):F E 9,} is discrete for each n E N; ( 3 ) { F E 9 : x E V ( F ) } is a finite collection fo each x E X . For each F E 9, let f F :X + [0, 11 be a continuous function from X into the unit interval satisfying ( 4 ) f F W = F; ( 5 ) C1(fL1([O, 1))) = W ) . For each F E 9 and 0 < r < 1, define K,(F) = f;I([O, r]). Note that (6) F c Kr(F) c Int K s ( F ) c V ( F ) whenever 0 < r < s < 1. For each U E 4 and 0 < r < 1, define U' = X - ( U ( K , ( F ) : F E9, F n U = @}).Let%'= { U ' : U E ~ } . We show that W , 0 < r < 1, is the desired collection. First we show that each %' is an open collection. To see this, note thatX- U = U ( F : F E ~ , F ~ U = @ ) ~ U { K , ( F ) : F E ~ , F ~ U = @ for each U E 4 because 9 is a mosaic for 4 . So it suffices to show that (7) { V ( F ):F E 9, F n U = @}is locally finite at every point of the open set U . Indeed if (7) is satisfied, then, since K r ( F ) c V ( F ) by (5), { K , ( F ) : F E 9, F n U = @}is also locally finite at every point of U . Thus U' is an open set of X . To show (7), suppose x E U . By the property of a stratification, there exists m E N such that x 4 C1 G(m, X - U ) . Then for each n > m and F E 9, with F n U = 8, by (I), we have V ( F ) c G(n, F ) c G(m, X - U ) . Thus x $ Cl({ V ( F ): F E Fn,n > m, F n U = S}). On the other hand, by (2), the collection { V ( F ) :F E 9,n,< m, F n U = S}, which is the finite union of discrete collections, is locally finite in X.Thus (7) is satisfied.
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Next we show (ii). By the definition of U s ,the set X - u{Int K,(F) :F E 9, F n U = S } is a closed set of Xcontaining Us.So if r < s, then it follows F n U = S} c X from (6) that C1 U s c X - U{Int K , ( F ) : F E9, U ( K , ( F ) : F E f , F n U = S } = U'. To show (iii), note that F = n{K,(F):O< r < l}, K,(F) = n { K s ( F ) : r < s < l} for each F E 9 and 0 < r < 1, by the definition of K,(F) and (4). Using the local finiteness of { V ( F ):I; E 9, F n U = S} at every point of U, and using the fact that K,(F) c V(F), one can easily show (iii). Now we show (i). It follows from (3), (6), and Lemma 2.18 that the is a mosaical collection for each r with 0 < r < 1. collection { K J F ) : F E 9} Thus by Lemma 2.14, W is also a mosaical collection. Let x E X, and we show that %'is almost locally finite at x. Put X = {K,(F):F E 9, x E K,(F)}.Then by (3), X is a finite collection. Define W = {X - UX' : X' c X } u {X}. We claim that V is a finite collection of subsets of X which guarantees the almost local finiteness of 4' at x. We must show that for every U' E W , U' = C n V for some C E W and some neighborhood V of x. We distinguish three cases. Case 1. x $ U . Take an element F, E 9 with x E F,. Since 9 is a mosaic for 9, we have F, c X - U. Note that x E F, c Int K,(F,) c K,(F,) c X - U', and Kr(F,) E X . Define V = U' u K,(F,). Then Vis a neighborhood of x , X - K,(F,) E %, and U' = (X - K,(F,)) n V. Case 2. x E U'. Then U'is a neighborhood of x, X E W,and U' = X n U'. Case 3. x E U and x 4 U'. By (7), { K , ( F ) :F E 9, F n U = S } is locally F n U = 0,x E K,(F)} c X , and finite at x . Define X' = { K , ( F ) :F E 9, X" = { K , ( F ) : F E9, F n U = 8, x 4 K J F ) } . Let C = X - U X ' E %, and V = U - UX".Then Vis an open neighborhood of x and U' = C n V. Finally we show (iv). It follows from (l), (3), (9,Lemma 2.18, and Lemma 2.14(c) that there is a mosaic S f o r the collection 9 u {f;'([O, 1)) : F E S}. Let Q be the set of rationals. Then 3 = (f;'((q, 1)):F E 9, q E (0, 1) n Q} is a a-discrete open collection, so by Lemma 2.21, there is a metrizable topology e on Xwith 3 c e c z such that S is a a-discrete closed collection of (X, e). Observe that (8)f;'((r, 1)) is an open set of (X, e) for any real number r with 0 < r < 1, because f;'((r, 1)) = U(f;'((q, 1 ) ) : q E (r, 1) n Q). It remains to show t h a t H - U , H - U r f o r e a c h H E S , U E 9 , 0 < r < 1,areclosedsetsof (X, e).Suppose H E S and U E 9.Since %' is a mosaic for 9, it follows that S is a mosaic for 9. Thus we have H n U = 8 or H c U . Hence H - U is a closed set of (X, e). Next, suppose H E S,U E 9 and 0 < r < 1. If H n U = 8, then H - U' = H. If H c U , then by the fact f;'([O, 1)) c V ( F )and the fact that S is a mosaic for {f;'([O, 1)): F E 9},
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and by (7), the collection { F E 9 : H nf;'([O, 1)) # 0,F n U = S} = { F E 9 : H c f;'([O, l)), F n U = 0) is a finite collection. Hence by the H n K , ( F ) # 0, fact that K,(F) c f;I([O, l)), the collection 9 = { F E 9: F n U = fl} is finite. Note that H - U' = H n ( U { K , ( F ) :F E U } ) .So it suffices to show that if H n K,(F) # 0 for F E 9, then H n K , ( F ) is a closed set of ( X , e). Since H c f;I([O, 1)) and K,(F) = f;I([O, r]) = X f;l((r, l]), it follows from (8) that H n K,(F) = H n (X - f;'((r, 1))) is a closed set of ( X , e). 0 Now let us turn to the definitions of various classes of stratifiable spaces.
2.25. Definition. Let 9' be the class of stratifiable spaces whose every closed set has a closure-preservingopen outer base. Let A? be the class of hereditarily MI-spaces, i.e., M,-spaces every subset of which is also an MI-space. 2.26. Definition. A space is an Fa-metrizablespace if it is a countable union of closed metrizable subspaces. A space is a p-space (Nagami [1970]) if it can be embedded in the product of countably many paracompact Fa-metrizable spaces. 2.27. Definition (Tamano [ 19851). A space X is a mosaic space if X has a a-mosaical base, i.e., a base W = UnWnwhere each W nis a mosaical open be the class of stratifiable mosaic spaces. collection of X. Let As generalizations of the class of stratifiable p-spaces, Mizokami [ 19841 defined the class of stratifiable spaces with an M-structure, and Tamano [ 19831defined the class of regularly stratifiable spaces. Junnila and Mizokami [1985] proved that these three classes are equivalent. The notion of mosaic spaces simplifies and unifies these methods and results. Thus some theorems and lemmas on mosaical collections or mosaic spaces and their proofs in this chapter are essentially due to some of these three persons even if unreferenced.
2.28. Definition (Oka [1983]). A closed collection 9 of a space X is an encircling net if the collection {X - U 9 ' :9'c 9, UP' is a closed set of X} forms a base for the topology of X.A space X is an EM,-space if X is a stratifiable space with a a-discrete encircling net.
2.29. Lemma. (i) Every mosaic space is a a-space, hence every normal mosaic space is perfectly normal. (ii) Every stratiJiable mosaic space is an EM,-space.
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Proof. We only show (ii). Let 8 = U,,@,, be a a-mosaical base for X, and Fna mosaic for 8,,for each n E N. Then it is easy to see that 9 = lJ,,Tn is 0 a a-discrete encircling net. We conclude this section with a diagram and a table. The diagram in Fig. 1 indicates implications holding among classes of stratifiable spaces. It should be noted that it is not known that any of these implications can be reversed. Also no other implications are known. Table 1 shows what classes of stratifiable spaces can be preserved under respective topological operations. The definitions of properties from (a,) to (e) can be found in the last paragraph of Section 1. stratifiable pspaces = stratifiable mosaic spaces ( A )-
.1
spaces with a o-almost locally finite base (~499)
1
spaces with a a-finitely closure-preserving base ( 9 W 9 )
1 EM3-spaces ( 1 stratifiable spaces in the class 9 (9) 1 hereditarily M,-spaces (X')
M I-spaces (Al)
stratifiable spaces
1
=
M3-spaces = M,-spaces (A3)Fig. 1.
2.30. Remark. (i) The conditions (a,) and (az)are equivalent for Aland 9. For Al, see Theorem 4.1. (ii) Let X be a class of stratifiable spaces. Suppose that a collection %? c X dominates a space X. Since a space dominated by a collection of stratifiable subspaces is stratifiable (Borges [1966]), X is stratifiable. By the definition of a domination, the collection % is a closure-preserving closed collection. Then by Theorem 2.12, %? has a mosaic 9. Thus if the class X is hereditary for closed sets, X is closed under discrete unions, and X is closed under unions of countably many closed sets, then X E X . The condition (e) for X', A, and &A3 can be proved by this observation. The condition (e) for A?was proved by Mizokami [1984]. (iii) Let { X i :1E A} be spaces, and letp E X = Ole,,Xi,where the symbol 0 denotes the box product. Ep = { x E X : x ( 1 ) = p ( A ) for all but finitely
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TABLE 1 Properties
Classes
(a, ) hereditariness (a2) hereditariness for closed sets (a,) hereditariness for open sets (b,) countable productivity (b2) finite productivity (c,) closed images (c2) perfect images (c3) closed irreducible images (c4) perfect irreducible images (c5) finite to one closed images (d, ) countable unions (d,) finite unions (e) domination (f) 8-products
O
? ?
? ?
O O
O O
0 0
0
0
0
0
0
0
0
0
o
0
0
0
0
0
0
O
?
0
0
0
0
0
o
0
0
0
0
? ? O
? O ?
? ? ?
? ? ?
O O
0
? ? ?
? O O O
0
O O
0
0
0
0
0
0
?
?
0
O
?
?
O
O
0
?
0
0
? ? ? ?
O
O
?
0
0
0
? O
O ?
?
? 0 ?
0
0
0
0
0 0 0
0 0 0
O O O
many I E A} is called a =-product. Borges [1978] and San-ou [1977] independently proved that if each X,is stratifiable, then so is E p . It is well known that if Epis perfectly normal, there is a o-discrete closed cover 9of Ep such that each member F E 9is homeomorphicto a closed subspace of the product of a finite subcollection of {X,: I E A}. Hence if a class X of stratifiable spaces satisfies the condition (a2),(b2),(d,), and the condition that it is closed under discrete unions, then it is closed under E-products. The condition (f) for A, and &A3 follows from this fact. The condition (f) for dL?S is due to It6 [1982].
3. Expandability, extension property, and supcharacterization of stratifiable spaces
'
Stratifiable spaces, stratifiable spaces in the class 8,and normal mosaic spaces have respective kinds of expandability. 3.1. Definition (Borges [1983]). A space X is CP-expandable (resp. WCPexpandable) if for each collection { U, :I E A} of open sets of X and closurepreserving collection {I;n:I E A} of closed sets of X with F, c U,, there
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exists a closure-preserving collection { W, : 1E A} of open neighborhoods (resp. (not necessarily open) neighborhoods) of F, such that CI W, c U,. As an immediate consequence of the definition, we have a lemma.
3.2. Lemma. A normal space X is CP-expandable (resp. WCP-expandable) if and only if for each closure-preserving closed collection 9 of X,there is a closure-preserving collection 4 of open sets (resp. subsets) of X such that 4 contains an outer neighborhood base of each F E 9 as a subcollection.
Proof. Sufficiency is obvious by the normality of X . To show the necessity, let 9 = {F,:I E A} be a closure-preservingclosed collection of X.For each 1E A, let { c,y : y E I-,} be the collection of all open neighborhoods of F,, and E r,, 1E A} is also a closurelet Fi.y = F, for all y E r,. Note that {F,,y:y preserving closed collection of X . Hence there is a closure-preservingcollection 4 = { Wi,7:y E r,,1 E A} of neighborhoods of F,,y satisfying the definition of CP-expandability (resp. WCP-expandability). It is easy to check that 4 is the desired collection. 0 3.3. Theorem (Mizokami [1984];Yamada [1984]). A space X i s a stratijiable space in the class 9 if and only if X is a CP-expandable a-space.
Proof. Suppose that Xis a CP-expandable a-space, and let &' = U,Mnbe a a-discrete network for X . Then by the above lemma, for each n, there is a closure-preserving open collection 4,,of X such that %, contains an outer base of each H E X;, as a subcollection. It is easy to check that 4 = U,,%, is a a-closure-preservingbase for X.Hence X is M ,. By the same argument, we can show that every closed set F of X has a closure-preserving outer base because the collection { F} with only one element is closure-preserving.Thus Xis a stratifiable space in the class 9. To show the converse, suppose that Xis in the class 8.Let { U , : I E A} be an open collection and 9 = { F, :Iz E A} a closure-preservingclosed collection of X with F, c U, for each I E A. By Lemma 2.12, there is a mosaic &' = UnMn for 9, where each JEP, is a discrete closed collection of X . For each n E N, take a discrete open collection { V ( H ): H E X,,}satisfying that H c V ( H ) c G(n, H ) , where G(n, H ) is the nth stratification of H . Since Xis in the class 9,each H E &' has a closure-preservingouter base BIH in X such that each member of which is contained in V ( H ) . Now define Xi = { H E X : H c F,} and Xi,,,= Xi n X,,.For each I E A and H E Xi, take an open set B,,H E BIH with B,,H c U,. And define W, = {B,,H: H E X i } ,
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x,,,,},
ux,
Wi,, = {B,,,:H E Wi = uWi, y.,,= UWi,,.Obviously Fj. = c Wj, c U , .Note that the normality of X implies that we need only show that c U , instead of CI Wj, c U , . It remains to show that { Wj,: I E A} is closure-preserving. Suppose that A' c A and x E Cl(ui,,,, Wj,).We show that x E Ur,,C1 W;,. Without loss of generality, we may assume that x 4 Uj,,/\.y,, which implies that x 4 Uj,,,,,Fj..Since { F j , : IE A} is a closurepreserving closed collection of X , Uie,,,Fl is a closed set of X. Then by the definition of a stratification, there is k E N with x 4 CI G(k, Ur,,,,Fj.). Observe that U;,,. = U(lJ{W;,.,:I E A', n E N}); and for each I E A', if n 2 k and BL,"E Wi,,, then BA,, c V ( H ) c G(n, H) c G(k, H) c G ( k , FA)c G(k, Ure/\.Fj,).Thus U(U{%,.,,:I E A', n 2 k}) c G(k, ui,,,,Fj,). Now define N = X - C1 G ( k , Uj,,,,,Fj,). Then N is an open neighborhood of x which c Cl(U((J{Wi.,:I E A', n < k})). Since satisfies that N n CI(U,,,,. UW, c V ( H )for each H E X,, and { V ( H ):H E A?,,} is a discrete collection H E X,,,, n < k} is a for each n E N , it follows from Lemma 2.9 that closure-preserving collection. Thus u{Wj,,, : I E A', n < k}, which is a subcollection of u{WH: H E X,,, n < k}, is a closure-preserving collection. This implies that there exist p E A' and m < k with x E Cl(UWp.,). Consequently, x E CI W,. The proof is completed. 0
w,
w,
w,)
u(9,:
In the proof of the above theorem, we have shown the following theorem essentially due to Borges and Lutzer [1973]. 3.4. Theorem. Every stratijiable space in the class S is an M,-space. 3.5. Lemma (Ceder [1961]). Every closed set of a stratzjiable space has a closure-preserving outer quasi-base consisting of closed sets. Proof. Let F be a closed set of a stratifiable space X , W = U{W,:n E N} a a-closure-preserving quasi-base consisting of closed sets of X . By the perfect normality of X , there is a decreasing sequence {H,,: n E N} of closed neighborhoodsofFsuchthatF = n{H,:nEN}.DefineQ, = W,IH,, = { B n H,,: B E W,,}. Then by Lemma 2.1 and Lemma 2.9,Q = U{Qn :n E N} is closurepreserving at every point of X - F because { H , : n E N} is locally finite at every point of X - F. Therefore the family V of unions of subcollections of Q is also closure-preserving at every point of X - F. Now define W = { V E V : F c Int V } . Then W is closure-preserving in X . We show that W is an outer quasi-base of F. Suppose G is a neighborhood of F. Since W is a quasi-base for X , for each point x E F, there are n(x) E Nand B, E Wn(r)with x E Int B,r c B,r c G. Define V = U { B , n H,(x):x E F}. Then V E V ,and F c Int V c V c G, which implies V E W . 0
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By the above lemma and an argument similar to the proof of Theorem 3.3, we have the following theorem.
3.6. Theorem (Yamada [1984], Gruenhage). A space Xis stratiJiable ifand only if X is a WCP-expandable o-space.
3.7. Lemma. Let F be a closed set of a normal mosaic space X . Then there is a mosaical outer base of F in X . Proof. Let 9 = U,,gnbe a a-mosaical base for X . We may assume that 9,c Bn+,for each n E N. Since X is perfectly normal, there is a sequence {C,}, of open sets of X such that n,G, = F and Cl C,, c G, for each n E N. Observe that by the perfect normality of X , {G, - F } , u { F } is a mosaical point-finite collection of X . Define Y = U,(B,,I (G, - F ) ) u { F } . Then by Lemma 2.17, “Y- is a mosaical collection of X . Now define 4 = { U V ’ : V’ c “Y-, UV’ is an open neighborhood of F in X}. Then by Lemma 2.14(b), 4 is a mosaical collection of X. It remains to show that 4 is an outer base of F. Let W be an arbitrary neighborhood of F in X . Then for each x E F, there are n(x) E Nand B ( x ) E such that x E B ( x ) c W . Define Y’ = { B ( x ) n GflC,,- F : x E F } u { F } . Then Y’ c Y and it is easy to see that U V ’ is an open neighborhood of F contained in W . 0 The following theorem is a generalization of Lemma 3.6 of Mizokami [1984]. Furthermore if X is a stratifiable mosaic space, then we can show that any closure-preserving closed collection of X can be expandable to an almost locally finite and mosaical open collection (Tamano [I 9891). 3.8. Theorem. A paracompact o-space X is a mosaic space fi and only if for each mosaical collection d of X , there is a mosaical open collection 4 of X such that for any A E d and a neighborhood W of A , there is U E 42 with A c U c W .
Proof. The sufficiency is easy. Indeed, since X is perfectly normal, every discrete closed collection of X is mosaical. Thus by an argument similar to the proof of Theorem 3.3, we can see that X is a mosaic space. To show the necessity, let 9 be a mosaic for a mosaical collection d of a paracompact mosaic space X . By Lemma 2.16, we may assume that % is a strongly point-finite collection. Hence there exists an open collection { V ( F ): F E 9} such that F c V ( F ) , and the collection {FE F :x E V ( F ) }is a finite collection for each x E X . By Lemma 3.7, each F E 9 has a mosaical outer base
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YFin V(F).ThenbyLemma2.17and2.18,Y = U{YF:F~9}isarnosaical collection of X . Now define Q = {UY’:Y’ c Y } .Then it is easy to see that Q is the desired collection. 0 Now let us turn to extension properties due to Mizokami [1984].
3.9. Theorem. Let A be a closed subset of a stratiJiable space X . Then for each open set U of the subspace A, there is a collection Yuof (not necessarily open) subsets of X such that (i) V n A = (Int V) n A = U, (C1 V) n A = Cl U, and V - A is closed in X - A for each V E Yu; (ii) for each open set W of X with W n A = U, there is V E Yuwith
Vc
w;
(iii) Y = U{Yu: U is open in A } is closure-preserving at every point of X - A ; and (iv) $42 is a closure-preserving open collection of the subspace A , then the collection U { Yu: U E Q} is closure-preserving in X .
Proof. By the perfect normality of X , there is a decreasing sequence of open neighborhoods of A in X with nnCl G,, = A. Since A is a a-space, there is a a-discrete network 9 = U n 9 , for A, where each Snis a discrete closed collection. By Lemma 3.5,for each F E 9, there is a closure-preserving outer quasi-base WFof F in Xconsisting of closed sets of X . Since each Pnis discrete and U S n c A c G,,,we may assume that each Wn= U{WF:F E Fn}is closure-preservingand UWnc G,, .Note that by Lemma 2.9 and the fact that {G,,},, is locally finite in X - A, W = UnWnis closure-preserving at every point of X - A. Now for each open set U of A, let E ( U ) be an open set in X of Lemma 1.9, and define Yu= {UW’:W‘ c W , UW‘ c E(U), W‘ n WF # 8 for each F E 9 with F c U } . Then it is easy to check that Y = U{Yu: U is open in A} has the desired properties. 0 An extension property similar to the condition of the following theorem is called the property (ECP) by Nagata [1973].Mizokami [1983]proved that a stratifiable space is in the class 9’ if and only if it has the property (ECP).
3.10. Theorem. Let A be a closed set of a stratiJiable space X. If X - A is in the class 8,then for each open set U of the subspace A, there is an open collection Yuof X satisfying: (i) V n A = U , (C1 V) n A = C1 U for each V E Yu;
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(ii) for each open set W of X with W n A = U, there is V E Yuwith
V c
w;
(iii) Y = U{Yu: U is open in A } is closure-preserving at every point of X - A; (iv) if9 is a closure-preserving open collection of the subspace A , then the collection U{Yu: U E a} is closure-preserving in X.
Proof. Since Xis stratifiable, for each open set U of A, there is a collection Yuof subsets of X satisfying the properties of Lemma 3.9. For each open set U of A, let E( U )be the open set of Lemma 1.9. Define Y = U {Yu: U is open in A}. Since Y I(X - A ) = { V - A : V E Y } is closure-preserving in X - A and X - A is in the class 9,by Theorem 3.2 and 3.3, for each V E Y , there is an open collection Wvof X - A such that Wvis an outer base of V - A in X - A, and W = U {Wv:V E Y }is closure-preserving in X - A. Now define Yi = {U v W :W EWvfor some V E Yu,and W c E(U)}. Then it is not difficult to see that Y' = {Yi : U is open in A} has the desired properties. As an application of WCP-expandability we have the following theorem.
3.11. Theorem (Borges and Gruenhage [1983]). A space ( X , t ) is stratijiable if and only if for each U E t , one can find a continuous function fu: X --* Z = [0, 11 such that fi'(0) = X - U and, for each 4 c t , sup,,,f, is continuous.
Proof. Let ( X , t) be a stratifiable space with a a-closure-preserving quasibase L3 = U,,L3,,consisting of closed sets of X. Denote by Qo the set of rational numbers in (0, 11. We show that for each q E Qo, one can define a closure-preserving closed collection a4= { U q: U E t} of X such that ( 1 ) if q < p , then Up c Int Uq,and (2) U = U{U4:qE Qo}. First we define @I/", n E N by induction on n. Define 9' = { U': U E T}, where U' = U { B E :B c U}.Assume that %I/" has been already defined. Define V(n, U ) = Ul/"v ( U { B E L3,,+':B c U } ) . Then {V(n, V ) :U E t} is a closure-preservingclosed collection and for each U E t, V(n, U ) c U. By Theorem 3.6, there is a closure-preservingclosed collection = { U'/"+l: U E t } with V(n, U ) c Int U'/"+l c U'l''+l c U.Note that U = U { U " " : n E N } for each U E t,because U { B E g,,: B c U } c U1/" for each n E N . Now enumerate Qo - { l / n :n E N } = {q,,:n E N } . Then by a fashion similar to the above, by induction on n, we can easily define closure-preserving
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collections 4'",n E N such that Up c Int U q c U q c Int U' whenever r c q c p. It is easy to check that 4'7,q E Qosatisfy the desired properties. Now for each U E T, define f , : X + Z by
if x 4 U , f U ( 4
=
sup{q E Q o : xE U q } if x E U.
Clearly f L 1 ( 0 )= X - U by (2). First we show that each f , is continuous. Supposef,,(x) > r. Then there are q l , q2 E Qowith q1 > q2 > rand x E U". Then by (l), x E Uq' c Int Uq2c U q 2 .So Uq2is a neighborhood of x with f U ( U q 2c ) [q2,11 c (r, I ] . On the other hand, suppose f , ( x ) < r. Pick q E Qo with f , ( x ) < q < r . Then by the definition of f , , x 4 U 4 . Then X - U q is a neighborhood of x withf,(X - U q ) c [0, q] c [0, r), which completes the proof of the continuity of f , . Next let 4 c T, and we show that f = sup,,,f, is continuous. Since each fu is continuous, it suffices to show that f is upper semicontinuous. So suppose f ( x ) c r. Pick q E Qo with f ( x ) c q < r. Then by the definition of f , , x 4 U 4 for each U E 4 . Since 4Yq is a closure-preserving closed collection, V = X - U { U 4 :U E 4 } is an open neighborhood of x such that fu( V) c [0, q] c [0, r ) for each U E 4 .Thusf(x) < q < r for each x E V. To show the converse, assume that f , , U E t, are functions satisfying the properties of the theorem. For each U E 4 , define g , = sup{f,,: V c U, V E T}. For each closed set Fof Xand n E N , let G(n, F ) = X - g;!,([l/n, I]). It follows from our assumption that each g, is continuous. So G(n, F ) is an open set containing F. It is easy to check that C(n, F ) satisfy the properties of a stratification. Hence X is stratifiable. 0 4. Classes of M,-spaces Our concern in this section is to prove some implications in Fig. 1 and preservation properties under topological operations in Table 1 . The following theorems (4.1 to 4.5) are due to It6 [1984a, 19851. 4.1. Theorem. A space X is hereditarily M I if and only if each closed set of X is MI. Proof. The necessity is obvious. To show the sufficiency, suppose that each closed set of X is M I . Let A be an arbitrary subset of X. Then by our assumption, C1 A is M I .Since A is dense in C1 A, by Theorem 2.8, A is also MI. 0
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4.2. Theorem. The following conditions for a stratiJiable space X are equivalent: (i) Every closed set of X has a closure-preserving outer base, i.e., X is in the class 9. (ii) Every point of X has a closure-preserving open neighborhood base.
Proof. (i)*(ii): Obvious. (ii)*(i): Since Xis stratifiable, by Lemma 3.5, F has a closure-preserving outer quasi-base 9 consisting of closed sets of X . On the other hand, by Lemmas 2.12 and 2.19, there is a subset D = U,,D,, of X such that each D,, is a discrete closed subset of X and U n D is dense in U for each U E 9.For each n E N , there is a discrete open collection { G, : x E D,,} such that each G,. contains x. By our assumption, for each point x E D, there is a closurepreserving open neighborhood base V,.of x. We may assume that if x E D,,, then UV,,c G(n, {x}) n G,., where G(n, H) is the nth stratification of H. We say a function cp is a choice function on U n D if c p : U n D + ( J { V , . : x E D} is a function such that cp(x) E V,.for each x E U n D. For each U E 4 and a choice function cp on U n D, define B(U, cp) = CI (U{cp(x) : x E U n D}). Now we show that the collection 93 = { B ( U , cp) : U E 4, cp is a choice function on U n D} is a closure-preservingouter quasi-base of F consisting of regular closed sets, which implies that the collection Int 93 = {Int B: B E 93} is a closure-preserving open outer base of F. Since each cp(x) is an open set, B(U, cp) is a regular closed set. Since U n D is dense in U E 9, we have F c Int U c U = CI ( U n D) c B(U, cp) for each B(U, cp) E g. Thus each B( U, cp) is a neighborhood of F. To show that 93 is an outer quasi-base, let W be an open set containing F. Take an open set W' with F c W' c CI W c W. Since 9 is an outer quasi-base, there exists U E 9 such that F c Int U c U c W'. For each x E U n D,there exists $(x) E V,.such that x E $(x) c W', since V,.is an open neighborhood base at x. Then B(U, $) c CI W' c W. Now it remains to show that 93 is closure-preserving. Let 5%' be a subcollection of 93.To show that Ua' is closed, let p E X - Ug'. Define 9' to be the set { U E 9 : B(U, cp) E 3'for some choice function cp on U n D}. Since U c B(U, cp) for each U E 4, p # US'. By the closure-preservingproperty of 4, u4' is closed. So by the definition of a stratification, there is a k E N such that p $ C1 G(k, u4').Define N = X - CI G(k, u4').Then N is an open neighborhood ofp and for each U E Q', n 2 k, and x E U n Dk,we have cp(x) c u"y; c G(n, (x}) c G(k, {x}) c G(k, u4') t X - N. Thus for each B(U, cp) E 93', B(U, cp) n N = (Cl(U{cp(x):x E U n (U,,
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Since {G, :x E ~ , c , D , }is locally finite, each Y, is closure-preserving, and UY, c G,, it follows that Y = u { . Y ; : x E U n < k D nis) closure-preserving. Hence the subcollection {cp(x):x E U n (unckDn), B(U, cp) E W.} of Y is closure-preserving, which implies that W' is closure-preserving at p. Thus
P # c1 (UW. It is easily seen that a countable decreasing neighborhood base of a point is closure-preserving. Thus by the above theorem and Theorem 3.4, we have another theorem. 4.3. Theorem. Everyfirst countable stratijiable space (= Nagata space) is an MI-space.
Recall that a space is zero-dimensionalif it has a base consisting of clopen sets. 4.4. Theorem. Every zero-dimensional MI-space X is in the class 8.
Proof. Let x E X. Then since X is a zero-dimensional perfectly normal space, there is a sequence {G,}, of clopen sets of X such that {x} = n,G,. On the other hand, since X is an MI-space, x has a a-closure-preserving neighborhood base. Thus it follows from Lemma 2.6 that x has a closurepreserving neighborhood base. Hence by Theorem 4.2, X is in the class 8.
4.5. Theorem. Every hereditarily M,-space is in the class 8,i.e. %' c 8.
Proof. Let X be a hereditarily MI-space. By Theorem 4.2, we need only show that every point x of X has a closure-preserving open neighborhood base. First we show that there are two regular closed sets HIand H , of Xsuch that (i)xEX=HluH2; (ii) for each i = 1, 2, if x E Hi, then there is a sequence {Hi,,:n E N} of clopen sets of the subspace Hisatisfying { x} = n{Hi,, :n E N}. To show this, using the perfect normality of X, take a sequence {G, :n E N} of regular open sets of X such that X = GI 3 C1 G2 3 G2 3 C1 G3 3 . . . , and {x} = n{G,:nE N}. Define HI = Cl(U{Cl(G2,-1 - G , , ) : ~ E N } ) and H2 = Cl{~{Cl(Gz,- Gz,+l):n EN}). Note that C1(Gk - G,,,) = Cl(Gk - C1 Gk+I )for each k E N, because each Gkis regular open. Hence the
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set Cl{Gk - G,,,) is a regular closed set of X . Define HI,,,= HI n C1 G2n-I= HI n G2n-2,H2,,,= H2 n C1 Gzn = H2 n G2,- I . It is easy to check that HI and H, are regular closed sets satisfying (i) and (ii). Next, to complete the proof of our theorem, for each i = 1, 2, if x E Hi, then take a closure-preserving outer quasi-base W iof Hiconsisting of regular closed sets of Hi. This can be done by the above property (ii), Lemma 2.6, and the fact that each Hi is an MI-space. Note that by Lemma 2.4, each giis also a closure-preserving regular closed collection in X . If x 4 Hi, then define Ui = 8.Now define W = {C, v C,:Cj E W j , i = 1, 2). Observe that by (i), each member of V is a neighborhood of x in X . Thus U is a closure-preserving outer quasi-base of x in X consisting of regular closed sets of X . Hence the collection { Int C : C E U } is a closure-preserving open neighborhood base of x in X. 0 4.6. Theorem. Every stratijiable mosaic space has a a-almost locally finite base, hence is hereditarily M I .
Proof. First we show that every stratifiable mosaic space has a a-almost locally finite base. Let 42 = U,,42,, be a a-mosaical base for X. Using Lemma 2.24, we can obtain almost locally finite open collections %,!Irn = {Ulim:U E %$},mE N,ofXforeachn E Nsuch that U = U{U1/"':mE N}. Then Un,rn42i'm is a a-almost locally finite base for X. It is easy to see that every subspace of a space with a a-almost locally finite base also has a o-almost locally finite base. Thus by Lemma 2.23, every space with a a-almost locally finite base is hereditarily M , . Now let us turn to preservation under unions. 4.7. Theorem (Mizokami [ 19831). Suppose that A is a closed set of a strat$able space X and X - A is in the class 8. (i) I f A is M I , then so is X . (ii) I f A is in the class 8, then so is X .
Proof. (i): Let $3 = U,,91nbe a a-closure-preserving base for the MI-space A. By Lemma 3.10, for each n E N , there is a closure-preserving open collection Vnof X such that for each B E a,,and an open set W of X with W n A = B, there exists V E Vnwith V n A = W n A and V c W. Since X - A is in the class 8, hence is M I , there is a a-closure-preserving base W = U,Wnfor X - A. We may assume that each Wnis closure-preserving
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in X . Define Y = (U,Y,) u W . Then Y is a a-closure-preserving base for X , which implies that X is M , . (ii): By Theorem 4.2, we need only show that every point of X has a closure-preserving open neighborhood base. The proof is similar to (i), hence omitted. 0 4.8. Lemma. Every open subset of a stratijiable space in the class 8 is also a stratijiable space in the class 9.
Proof. Assume that U is an open set of a stratifiable space Xin the class 8. By Theorem 1.12, U is a stratifiable space. Thus by Theorem 4.2, we need only show that every point x E U has a closure-preserving open neighborhood base Y in U . By our assumption, there is a closure-preserving open neighborhood base W of x in X . Define Y = { W E W : W c U } . Then it is easy to check that V is the desired one. 0 Compare the following theorem with (ii) of Theorem 4.7. 4.9. Theorem (Mizokami [1984]). I f a stratijiable space X is the union of countably many closed subspaces { X, : n E N } such that each X, is in the class 8, then X is in the class 8.
Proof. It follows from Theorem 4.7 and Lemma 4.8 that a stratifiable space which is the union of finitely many closed subspaces in the class 9 is also in the class 8. So we may assume that X, c X,,, for each n E N . Let n E N . Then by Lemma 1.9, for each open set U of the subspace X,, one can assign an open set E,(U) of X such that ( I ) E , ( U ) n X, = U ; (2) if 42 is a closure-preserving open collection of the subspace X,, then the collection 9’= U{Yu:U E a } is closure-preserving at every point of X,, where Yu = { S :S is a subset of X with U c S c E , ( U ) } . On the other hand, by Lemma 3.9, for each open subset U of the subspace X,, there is a collection V,.uof subsets of X such that (3) V n X, = (Int V ) n X , = U for each V E Y,,u; (4) for each open set W of X with W n X, = U , there is V E V,,uwith V c w; ( 5 ) let Y, = U{Y,,,: U is open in X,}. Then Y , I ( X - X,) = { V - X,: V E Y,}is a closure-preserving closed collection of X - X , . For each n 2 2, define Yn’to be the collection of all unions of finite subcollections of ( U k < , V kI )( X , - X,-,). Then Y,’ is closure-preserving in
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,
X , - X , - ,. By Lemma 4.8, we have X, - X,- E 8,so by Theorem 3.2 and 3.3, there is a closure-preserving open collection W, of X, - X,-, for each n 2 2 satisfying (6) for each V’ E V,’ and an open neighborhood G’ of V’ in X, - X , - , , there is W E W, with V‘ c W c G’. To complete the proof, by Theorem 4.2, we need only show that every point of X has a closure-preserving open neighborhood base. To see this, let x E X. Without loss of generality, we may assume that x E XI.Since XI E 8, there is a closure-preserving neighborhood base “w; of x in XI. Now define W to be the collection of all W = U, W, such that W, E W, and there are V , E V,, n E N satisfying that for each n E N, (7) W, u W, u . * u W, is open in X,; and if n >, 2, (8) K-I E *v;.,wl”w,u...ut+~l~ (9) w, u K-, = n k < n w vu, w, u . . u 4 ) ; (10) w, = (Uk<,Vk) n (X, - Xfl-1). We show that W is a closure-preserving open neighborhood base of x in X. First we show that each member of W is open in X. Let W E W and { W,},, { V,}, be as the above. Clearly W n X, = W, u W, u . . u W,. On the other hand, by (3), (7), (8), and (lo), we have W 3 V,-, 3 Int V,-, 3 W, u W, u . . . u W,-,. Hence W = U,, W, = U,V, = U, Int K , which implies that W is open in X. Secondly we show that W is a neighborhood base is a of x in X. Suppose G is an open neighborhood of x in X. Since neighborhood base of x in XI, there is W, E “w; with x E W, c G. By (4), there is V, E “t;.wlwith W, c V, c E l (W,) n G. Inductively define sequences { W,},, and { V,}, as follows. Assume that { W,},,, and { &},<,, are defined so as to satisfy the above conditions from (7) to (10) and (UksnW,) u (U,<, 4 ) c G. Note that it follows from (9) and our assumptions that
-
w , u w , u * . . u w, c (n,<,E,(w,
u
w,u
. . . u w,))
n G.
Hence by (4) and (7), there is V , E Vn,.wluwzu . uw, such that V , c n , < , + , E , (W, u W, u * * * u W,) n G. Then by the definition of W,, there is W,,, E W,+,satisfying (Uk
n (Xn+, - Xn)
K+,c (n,,,+l4(W,u W2 u . . . u 4 ) ) n G . Then by an argument similar to the above, W, u W, u . . . u W,,, is open in A’,,+,, which completes our induction. Now define W = U,, W,. Then c
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W E W and x E W c G. Finally we show that W is closure-preservingin X . Observethat W, u W, u * u W, c W c E,(W, u W, u * * u W,).By (2), (S), (9) and the fact that each W, is a closure-preservingopen collection of X, - X n - , , by induction n, it is easy to see that the collections { w u W , u - . - v w , ( = w n X , ) : w = U,W,EW, W,EW,)
and W are closure-preserving at every point of X,. Thus W is closure0 preserving in X. Combining the above theorem and Theorem 4.5, we have as a corollary. 4.10. Corollary. I f a stratijiable space X is the union of countably many' closed subspaces { X , :n E N } such that each X, is hereditarily M , , then X is
hereditarily MI. 4.11. Lemma. Every finite union of closed mosaic subspaces is a mosaic
space.
Proof. It suffices to show that the union X of two closed mosaic subspaces
u
A, and A , is a mosaic space. Let gi = {aij : j E N } be a a-mosaical base for Ai for each i = 1,2. We may assume that each gijcontains 8 as an element. Define @mn = { X - [ ( A , - B , ) LJ (A2 - B2)I:BI E g l m , B2 E % n } .
u{9,,,,
Then :m, n E N } is a base for X. Note that since Xis the union of two closed a-spaces, X is also a a-space, and hence is perfect. So the collection { A , , A , } is mosaical. Thus by Lemmas 2.14 and 2.17, each %m is mosaical in X , which completes the proof. 0 The following theorem is essentially due to Mizokami [1984]. 4.12. Theorem. Ifa stratijiable space X i s the union of countably many closed subspaces { X , :n E N } such that each X, is a mosaic space, then X is a mosaic space.
Proof. By Lemma 4.1 1, we may assume that X , c X , , , for each n E N.Let 9,be a a-discrete closed network for X, for each n E N, and define 9 = u,,9,. Then f is a a-discrete network for X.By an argument similar to the proof of Theorem 3.3, we need only show that each F E 9 has a mosaical outer base in X . Without loss of generality we may assume that F E9,.
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By Lemma 3.9, for each open set U of the subspace X,, there is a collection "v;,,, of subsets of X satisfying (3), (4), and (5) of the proof of Theorem 4.9. For each n > 2, define V,' to be the same collection as the proof of Theorem 4.9. It is easy to see that every subspace of a mosaic space is also a mosaic space. Hence the space X, - A',,-, is a mosaic space, and by Theorem 3.8, there is a mosaical collection W, of X, - X,-, satisfying (6) of the proof of Theorem 4.9. Let 7v: be a mosaical outer base of F i n X , . Now define W to be the collection of all W = U,, W, such that W, E W, and there are V , E Y,,n E N satisfying that for each n E N , (7), (8), and (10) of the proof of Theorem 4.9 are satisfied. Since each W, is a mosaical collection of X, - Xn-l,where X,, = 8,and since {X, - X , - , } , is a pointfinite mosaical collection of X , it follows from Lemma 2.17 that the collection U,Wnis a mosaical collection. Thus by Lemma 2.14(b), the collection W , each member of which is the union of a subcollection of U,W,, is also a mosaical collection. By an argument similar to the proof of Theorem 4.9, it is easy to see that W is an outer base of F in X. By Corollary 4.10, or by combining Theorem 4.6 and 4.12, we have the following corollary. This also follows from the combination of Theorem 5.3 and 4.6.
4.13. Corollary (Gruenhage [1980]). r f a stratijiable space X i s the union of countably many closed metrizable subspaces {X, :n E N }, then X is hereditarily MI *
For the productivity, we have the following theorem. 4.14. Theorem (It6 [1985]). The class 9 is countably productive.
Proof. Let X,, n E N, be spaces in the class 9. We show that the product X = l l n E N Xis, also in the class 9. By Theorem 1.12, X is stratifiable. Thus by Theorem 4.2, it remains only to show that each point x = ( x i ) E X has a closure-preserving open neighborhood base. Let 4Yi be a closure-preserving open neighborhood base of xi in X,, and { G , , :k E N } a decreasing sequence of open neighborhoods of x i in X, such that CI Gi,,:k E N } = { x , } . Define 42i,, = { U E 4?Zi: U c Gi,,). Now define Y,,= {(llis,,u) x (ll,,,,X,.): U,E 4Yi,, for each i = 1, 2, . . . , n}. Then it follows from Lemma 2.10 that Y, is a closure-preservingopen collection of X for each n E N. Define G,, = (lliGnGi,,,) x ( l l i , , , X , ) . Note that UY,, c G,, and {G, :n E N } is locally finite at every point of X - { x } . Then by Lemma 2.9, 9" = unEN^1/;1 is closure-preserving. Obviously Y is a base of x in X. The proof is completed. 0
n{
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5. Embeddings 5.1. Theorem (Heath and Junnila [1981]). Every stratijiable space is aperfect retraction of an MI-space in the class 8.
Proof. Let X be a stratifiable space. Denote by S the subspace ( 0 ) u { I/n :n E N } of the real line. Let Z be the set X x S topologized by isolating points of U { X x { l / n } :n E N } and leaving points o f X x ( 0 )with their usual neighborhoods of the product topology. For each closed set A of X, define A[n] = A x ( ( 0 ) u { I/k:k 2 n } ) and A [O] = A x (0). Since A is closed, A [n] is the closure of the open set A [n] - A [O] of Z. Thus A [n] is a regular closed set of Z. We show that Z is a stratifiable space in the class 8. Let 93 = U,,A?" be a a-closure-preserving quasi-base consisting of closed sets of X. Define qn= { B [ m ] : BE A?,,, m E N } and Vn= { { ( x , l / n ) } : xE X } . Then each %, is a closure-preserving regular closed collection, and each Vn is a discrete clopen collection. It is easy to see that (U,,@,,) u (U,,*V;.) is a quasi-base for Z. Thus by Theorem 2.3, Z is stratifiable. To see that Z is in the class 8,by Theorem 4.2 and by the fact that Z - X[O]is discrete, we need only show that for each x E X , the point ( x , 0) has a closure-preservingouter base in Z. By Theorem 3.5, x has a closure-preserving outer quasi-base V in X consisting of closed sets. Then it is easy to see that the collection { C[n]: C E W, n E N } is a closure-preserving quasi-base of ( x , 0) in Z consisting of regular closed sets, so the collection {Int,C[n] : C E V, n E N } is a closure-preserving open neighborhood base of ( x , 0) in Z . Thus it follows from Theorem 3.4 that every stratifiable space X is homeomorphic to the subspace X[O] of an MI-space Z in the class 8. Although the projection Z + X is a retraction when we identify X with X[O],to obtain a perfect retraction, we must construct a subspace Y of Z. By Theorem 2.12 and Lemma 2.19, there is a a-discrete dense set D = UnDnof X such that B n D is dense for every B E A?. We may assume D, c Dn+I for each n E N. Now define Y = X[O] u (U{D,,x { l / n } : n E N } ) . By the property of D, for each n E N , B[n] n Y is the closure of the open set (B[n] - B[O])n Y in Y, hence is regular closed in Y. So restricting the above argument to Y, we can easily see that Y is an M,-space in the class 8.Note that the fact that the above spaces Z and Yare in the class 9' was observed by It6 [1985]. It remains to show that the restriction f of the projection Z -+ X to Y is a perfect map. Clearly each f - ' ( x ) , x E X is compact. It remains to show that f is a closed map. Suppose F is a closed set of Z and x E X - f ( F ) . Since ( x , 0) 4 F, there is a neighborhood V of x in X and m E N with v[m] n F = 8. For each n < m, since x $ f ( F n ( X x { I/n})) c D,,
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and 0, is a discrete closed set of X , there is an open neighborhood V , of x in Xwith V , nf ( F n ( X x { l / n } ) ) = 8. Define U = V n (fin<,V,). Then U is an open neighborhood of x disjoint from f (F), which completes the proof.
0 Now let us characterize the class of stratifiable p-spaces. The following Lemma 5.2 and Theorem 5.3 and their proofs are combinations of results essentially due to Mizokami [1984],Junnila and Mizokami [1985],and Tamano [1983, 19851.
5.2. Lemma. Let X be a stratijiable space. Y a paracompact F,-metrizable space, and Z a space with X c Y x Z . Then there is a a-mosaical open collection 9 in X such that for each open set G of Y, n-'(G) is the union of a subcollection of 9, where n:X + Y is the restriction of the projection Y x Z - YtoX.
Proof. Express Y = U, Y, where each Y,, is a metrizable closed subspace of Y. It is well-known that the union of finitely many closed metrizable subspaces is metrizable. So we may assume that Y, c Y,,, for n E N. Define & = 8, and X, = n-'(Y,,). Let i4l = Ufli4l,,be a a-closure preserving quasibase for the stratifiable space X , where each 93, is a closure-preserving closed collection of X. We may assume that &I c,,93,,+, for each n E N. By Lemma 2.1 and Theorem 2.12, each i41n I (X, - X,- I ) is a mosaical collection of X, - X , - , . On the other hand, the disjoint cover {X, - X,- :n E N } by F,-sets of X is a mosaical collection of X. Hence by Lemma 2.17, the collection W = U{an I (X, - X,- ) :n E N } is a mosaical collection of X. By Lemma 2.16, there is a strongly point-finite mosaic 9 = U,F, for W, where each 9,is a discrete closed collection of X. Define F,, =,{F ,E 9,:, F c X, - Xn-l}.It should be noted that U((J,9,.,) = X, - X,.. I Now fix k E N. For each F E 9,. ,let ,3fk(F)be a locally finite closed cover of Fsuch that for each H E Hk(F),diam,(n(H)) < l / ( m k), where diam, is the diameter in the metric space (Y,, d,,).This can be done by using the paracompactness of Y, and the continuity of K. Define 3f:, = U ( 3 f k ( F ) : F E 9,, }, , 3fL = Urn&':, and 3fk = un3fL.By Lemma 2.13, each 3 f k ( F ) is a mosaical collection of F. It follows from Lemma 2.18 that 9 is a point-finite mosaical collection. Thus by Lemma 2.17, we can see that Z kis a mosaical collection of X. Now define 9k= {X - US':.%' c 3fk, and U3f' is closed in X}. Finally define 9 = Uk@. We show that 9 has the desired property. By Lemma 2.14(b), 9kis a mosaical open collection of X.
,
,
+
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LetGbeanopensetofY.DefineX&= { H E H [ , : H - n-'(G)# 0 } , X k = U,,,,X&, and Gk = X - UX'. To complete the proof, we show that each Gk is a member of Qk and n - ' ( G ) = UkGk.First we show that Gk E Q k , i.e., U S k is closed in X. Since every union of a locally finite closed collection is closed and X - n - ' ( G ) is a closed set of X contained in UXk, we need only show that X kis locally finite in the open set n-'(G). To show this, suppose x E n-'(G). Then since A? is a quasi-base, there exist no E Nand BE with x E Int,B c B c n-'(G). Observe that if n 2 no and m E N, then H n B = 0 for each H E X:m, because H: is a mosaic for c a,,and H - B # 0.Thus 9Y, I (X, - X,-,), (1)
B n U(U{X$,:n 2 no, m E N } ) =
0.
On the other hand, for each n E N,if x E X,, pick m(n) E N with l/m(n) < d,(n(x), Y, - G ) . If x X,, let m(n) = 1 . Define m, = max{m(n):n < 4). Note that if H E X&, for m 2 m,, then diam,,(n(H)) < l / ( m k) < l/mo. Since H - n - ' ( G ) # 0 for each H E X:,,, it is easy to see that
+
(2)
x
+
4 cl(U{UX&:n < no, m 2 m,)}.
Now it follows from (I), (2) and the local finiteness of each Xl,,, the collection X k = U,,,,X:,, is locally finite at x E n-'(G ), which completes the proof of the fact that Gk E Q k . To this end, we show that G = UkGk. Suppose x E G. Let mo E N be the same one as the above argument. Then it is easy to check that x E G"O. 0 5.3. Theorem. The following are equivalent for a stratijiable space X : (i) X is a p-space. (ii) X is a mosaic space. (iii) X can be embedded in the product of countably many stratijiable F, -metrizable spaces.
Proof. (iii)*(i) Obvious. (i)*(ii) Suppose that X is a stratifiable p-space and X c nieN where each Y, is a F,-metrizable space. It is well known that every finite product of paracompact F, -metrizable spaces is paracompact F, -metriable. Hence by Lemma 5.2, for each n E N , there is a a-mosaical open collection 9,,in X such
x,
that for each open set of n,,, U,, n;'(G) is the union of a subcollection of%,,, where n, : X -,n,,, Y, is the restriction of the projection nisN r;. + nidn r;. to X. Now define 9 = U,Q,,. Then 9 is a a-mosaical base for X. (ii)*(iii) Suppose that ( X , z) is a stratifiablemosaic space with a a-mosaical base 43 = U,Q,. Denote by Q the set of rational numbers. Define S = (0, 1) n Q. By Lemma 2.24, for each n E N, there are a countable family
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{%: :q E S } of open collections; a metrizable topology e,, on X with en c t; and a a-discrete closed collection S,,= Urn#,,,, of (X,en) such that (1) %: = { U Q :U E %,,} is a closure-preserving open collection of ( X , t) for each q E S; , Q = u { u p : p E s , p > q} = (2) u = U { U Q : q E S }= u { q u Q : q € S } U U{Cl,Up:p E S , p > q}; and (3) for each H E S,,,U E a,,, q E S , the sets H - U , H - U qare closed in ( X , Q n ) . By (1) and Lemma 2.21, we may assume that (4) Clz%j = {Cl, Uq: U E %,,} is a closure-preserving closed collection of (X, en) for each q E S ; Now define the topology p,, on X which has the collection 9 = e,, u (U{%::q E S } ) as a subbase. We show that each (X, p,,) is a stratifiable F,-metrizable space. First we show that (X, p,,)is a regular space. To see this and x E V, there is W E Y with we need only show that for each V E 9, x E W c ClpmW c V. But this follows easily from (2), (4) and the fact that each en is regular. Next we show that ( X , p,,) is stratifiable. Let & be ifa a-locally finite quasi-base for the metrizable space (X, en)consisting of closed sets. Define %? to be the collection of intersections of finite subcollections of W u (U{Cl, %: :q E S}).Then by (2) and (4), it is easy to see that %? is a a-closure-preserving quasi-base for (X, p,,) consisting of closed sets. Hence by Theorem 1.10, ( X , p,,) is stratifiable. Finally, we show that ( X , p,,) is F,-metrizable. It follows from (3) that (4)
p,,I H = enI H for each H
E
X,,.
Define X,,,, = US,,,. Since X,,,, is a discrete closed collection of ( X , en),it follows from (4) that A',,,,, is a closed subspace of ( X , p,,)with the topology which is the same as the subspace topology of ( X , en).Hence ( X , p,,)is the union of closed metrizable subspaces X,,,,, m E N . Now it can easily be seen that ( X , t) is homeomorphic to the diagonal of the product n,(X, p,,). 0 5.4. Definition. A space is an M,,-space if it has a o-closure-preserving base consisting of clopen sets.
Denote by dim X = 0 that the covering dimension of a space X is zero. It is well known that for a normal space X, dim X = 0 if and only if Ind X = 0, i.e., every closed set of X has an outer base consisting of clopen sets of X.
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5.5. Theorem. The following are equivalent for a stratijiable space X : (i) X is a stratijiable p-space with dim X = 0. (ii) X is an M,-space.
Proof. (i)=(ii) (Mizokami [1984]) By the above theorem, X has a a-mosaical base 42 = U,,42,,. Apply the construction of Lemma 2.24 for each %, . Since of Lemma 2.24 can be chosen such that dim X = 0, the functionsf F ,F E 9, K,,,(F) = f;I([O, l/m]) is a clopen set for each m E N . Then the resulting collection %,!I"' is a closure-preserving clopen collection. Hence by an argument similar to the proof of Theorem 4.6, we can show that Xis an &-space. (ii)*(i) (It6 [1984b], Junnila [1984]) Suppose X is an M,-space with a a-closure-preserving base 9 = U,,42,, consisting of clopen sets of X . By Theorem 2.12, each a,, is mosaical, so it follows from Theorem 5.3 that Xis a p-space. To show that dim X = 0, let F be a closed set of X. Define H,, = X - U { U E UkBn%k: U n F = O } . Then {H,,},, is a decreasing sequence of clopen sets of X with F = n,,H,,.By an argument similar to the proof of Theorem 3.5, we can show that every closed set of X has a closurepreserving outer base consisting of clopen sets. Hence dim X = 0. 0 Besides p-spaces, Nagami defined the class of free L-spaces as generalized metric spaces which can be characterized by a certain embedding theorem. His idea and technique much influenced the recent development of the study of stratifiable spaces by Japanese topologists. For this direction, see Nagami [1980], and Oka [1981]. 6. Closed maps and perfect maps 6.1. Definition. Let f : X + Y be a map, i.e., a continuous surjection. f is irreducible if no proper closed subset of X maps onto Y . For A c X , define f#(A) = { y € Y : f - ' ( y ) c A } = Y - f ( X - A ) . 6.2. Lemma. Suppose f : X + Y is a closed irreducible map. Then (i) for each open set U of X , f " ( U ) is an open set of Y with CNf " ( W )= f(C1 (ii) (Borges and Lutzer [1973]) $42 is a closure-preserving open collection of X , then f # (9)= { f # ( U ): U E 9 } is a closure-preserving open collection of Y .
w;
Proof. (i) is well known. For example, see Pears [1975]. To show (ii), let 42' c 42. Since 42' is a closure-preserving collection, it follows that
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U{Cl U : U E W }is a closed set of X. Hence, by (i) and the closedness off, ) f(U(C1 U : U E % ' } ) thesetU(C1 f # ( U ) : U E W } = U{f(Cl U ) : U E @ ' = is a closed set of Y, which implies that f # (a)is closure-preserving. 0
6.3. Theorem (Borges and Lutzer [1973]). Suppose f :X + Y is a closed irreducible map. (i) Iff is perfect and X is M I ,then so is Y. (ii) I f X is in the class 8,then so is Y. Proof. (i): Let B be a a-closure-preservingbase for X. We may assume that B is closed under finite unions. Then, using Lemma 6.2, we can see that f # (53) = { f # ( B ): B E B } is a a-closure-preserving base for Y. (ii): Since X is stratifiable, by Theorem 1.12, the closed image Y is also stratifiable. Let H be any closed set of Y. Then since X is in the class 9,the closed set f - ' ( H ) has a closure-preserving outer base 4 in X. Then by Lemma 6.2, f # ( 4 is ) a closure-preserving outer base of H in Y. 0 For a subset A of C,Bd A denotes the boundary of A in X.
6.4. Lemma (Okuyama [1968]). Let f:X + Y be a closed map of a normal a-space X onto a space Y. Then the set { y :Bd f - I ( y ) is not countably compact} is a a-discrete set of Y, i.e., the union of countably many closed discrete sets of Y.
6.5. Lemma (Gruenhage [ 19801). Let f : X --* Y be a closed map of a normal a-space X onto a space Y. Then there is a closed set A of X such that f I A : A + f ( A ) is a closed irreducible map and Y - f( A ) is open and a-discrete in Y.
Proof. By Corollary 2 in Section VII.3 of Nagata [1985] a subspace C of X is compact if and only if C is countably compact. So it follows from the above lemma that there is a subset B of Y such that Y - B is a-discrete and for each y E B, Bd f - I ( y ) is compact. For each y E Y such that Bd f - l ( y ) is empty, pick a point x( y ) in f - I ( y). Define a subset X' of X by X' = X (U{Intf-'(y):yE Y, Bdf-'(y)isanonemptycompactset}) - ( u { f - ' ( y ) { x( y)} : Bd f - I ( y) is empty}). Then X' is a closed set of X which maps onto Y. Replacing X by X', we may assume that f - I ( y ) is compact for each y E B. Define d = {F c X: Fis closed in Xand f(F) 3 B}. Partially order d by inclusion. Since f - I ( y) is compact for each y E B, for every chain %' in 8, 0% E 8.Thus by the Kuratowski-Zorn lemma, d has the minimal element
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A. Since A is closed and f ( A ) 3 B, it follows that Y - f ( A ) is open and o-discrete in Y. The minimality of A implies that there is no proper closed subset A’ of A such that f (A’) = f ( A ) 3 B. Hence f I A is irreducible. 0
6.6. Theorem. Let V be a class of stratijiable spaces. Suppose V has the following properties: (i) Any closed subset A of a space X E V is also in the class V; (ii) if a stratijiable space X has a closed subset A such that A E V and X - A is a-discrete, then X E V; (iii) iff :X + Y is a closed irreducible map and X E V, then Y E V. Then the class V is closed under closed images, i.e., iff: X + Y is a closed map and X E V, then Y E V . Proof. Let f:X + Y be a closed map and X E V. Note that Y is stratifiable by Theorem 1.12. By Lemma 6.5, there is a closed set A of X such that f I A is closed irreducible and Y - f ( A ) is open and o-discrete. By (i), A E V. Using (iii), we have f ( A ) E V. Finally by (ii), Y E‘3. 0 6.7. Theorem (It6 [1984a]). The image of a hereditarily MI-space under a closed continuous mapping is also hereditarily M I .
Proof. Let f:X + Y be a closed map of a hereditarily MI-space onto a space Y. It is enough to show that Y is MI. The class 2 of hereditarily M,-spaces obviously satisfies the properties (i) of the above theorem. It follows from Corollary 4.7 that 9 satisfies (ii). Combining Theorem 6.3 and Theorem 4.5, we have: (iii)’ iff: X’ + Y’ is a closed irreducible map and X’ E 2, then Y E 8. Thus by an argument similar to the proof of the above theorem, we can show that Y E8,which implies that Y is an MI-space. 0 6.8. Corollary (Slaughter [1973]). The closed image of a metrizable space is MI. Now let us characterize the perfect (closed) images of zero-dimensional stratifiable spaces.
6.9. Theorem (Oka [1983]). The following are equivalent for a space X : (i) X is an EM,-space. (ii) X is the image of a stratijiable space Y with ind Y = 0 under a perfect map, where ind Y = 0 means that Y is zero-dimensional.
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(iii) X i s the image of a stratijiable space Y with ind Y = 0 under a closed map. (iv) X i s the image of a stratijiable space Y with dim Y = 0 under a closed (perfect) map.
Proof. We give here only the proof of the equivalence of (i) and (ii). We refer the reader to Oka [1983] for the complete proof. (ii)*(i): Letf: Y -+ X be a perfect map and suppose ind Y = 0. Since Y is a a-space, there is a network X = U,Xnof Y such that each X, is a discrete closed collection of Y. It follows from the perfectness o f f that f(Sn) = { f ( H ): H E X , } is locally finite in X for each n. Thus by Lemma 2.13, there is a mosaic 9, forf(Xn). Define a a-discrete closed collection 9 of X by 9 = UnPn.We show that 9is an encircling net. To show this, let U be an open set of X and suppose x E U . Since f-'(x) is compact and ind Y = 0, there is a clopen set Vof Y withf-'(x) c V c f - ' ( U ) . Define X' = {HE X : H c Y - V}. Note that by the definition of 9, eachf(H), H E X , is the union of a subcollection of 9. Hence there is 9'c 9 with f(Y - V ) = f(UX') = Uf(X') = Then U F ' is a closed set satisfying that x E X - U 9 ' = X - f(Y - V ) c f(V) c U, which implies that 9 is an encircling net for X. (i)*(ii): Our proof is rather more geometrical than Oka's one. Let X be an EM,-space with an encircling net 9 = U n9 ,, where each snis a discrete closed collection of X. Take open collections { V ( F ):F E F}, and Yn,,, = { V',(F) :F E 9,}, n, m E N satisfying (1) V ( F ) and V,(F) are open neighborhoods of F; (2) { V ( F ):F E Pn}is a discrete collection for each n; (3) V,+,(F) c V,(F) c V ( F ) n G(m, F), where G(m, F) is the mth stratification of F. In general, for each discrete open collection Y'- of a space X , define a space Z ( Y ) = ((U(C1 V : V E Y } )x ( 0 ) ) u ( ( X - U Y ) x { l}) as a subspace of the product space X x D, where D is the set (0, l} with the discrete topology. Note that the projection p : Z ( Y ) -+ X is a perfect onto map, and, by Theorem 1.12, if X is stratifiable, then Z ( Y ) is also stratifiable. Define 2' = Z i ( Y ) = Z ( Y ) n (Xx { i } ) for each i = 0, 1. It is easy to see that (4) {Zi n p - ' ( U ) : U is open in X , i = 0, l} is a base for Z ( Y ) . Now define Z,,,, = Z(Y,,,). Let pn,,:Z,,,, + X be the projection, and Z;,,, = Z(Yn,,,) n ( X x { i } ) . Define a subspace Y of the product space IIn,',Z,,,,by Y = {(z,,,) E IIn,,Zn,,,:there is x E X such that pn,,(zn,,,)= x for all n, m E N } . Then it follows from Theorem 1.12 that Y is stratifiable. Let n,,, : Y -+ Z,,,, be the projection and f : Y -+ X be the map defined by
up'.
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f(y )
= pn,m(nn,m( y)) for some (any) n, m E N. Then the mapfis well-defined because of the definition of Y and it is easy to see thatfis a perfect map. It remains to show that ind Y = 0. To show this, we need only show that Y has a subbase consisting of clopen sets. It follows from (4) that the open collection
{n;k(Z:,, n p;k(U)) : U is an open set of X , n, m E N, i
= 0, I }
is a subbase for Y. Suppose that U is an open set of X and y E n;l(ZA,,n p ; l ( U ) ) = n$,(Z~,,) n f - ’ ( U ) . To complete the proof, we need only to show that there is a clopen set G of Y with y E G c n;k(Z;.,) n f - l ( U ) . Since n;!,(ZA,,) is clopen, it suffices to show that ( 5 ) there is a clopen set G of Y with y E G c f - ’ ( U ) . By the definition of an encircling net, there is 9‘= unFn’c S satisfying that Sn’c Sn, and US’ is a closed set of Xwith x = f(y ) E X - US’ c U. Since Xis stratifiable, there is k E N with x 4 C1 G(k, U P ) , where G(k, H) is the kth stratification of H. For each n E Nand F E Sn, define a clopen set H ( F ) of Zn,n+k and a clopen set K ( F ) of Y by H ( F ) = Cl(K+k(F))x {0}, K ( F ) = n;;+k(H(F)). Now define G = Y - (JEEP K(F). We show that G is a clopen set satisfying the condition (5). Since H ( F ) c P $ + ~ ( C K ~ + k ( F ) ) ,we get (6) K ( F ) c n;;+k(Pn,;+k(Cl K+k(FN) = f - ’ ( c W + k ( F ” By (3), if F E S‘, then x E X - CI G(k, US’) c
x - CI G(k, I;) t X
-
c1 V , ( F ) c
x - CI K + k ( F ) .
Thus y E G. On the other hand,
f(Y -
= Pn,n+k(an.n+k(Y- n ; ; + k W ( m
- PnJI+k(Zv+k -
H(F)) =
x - Yl+k(Fb
Thus we have
uFE/.w))= n F E S m- ~ ( n ) = n F E f a- ~ + m
S(G) = f ( y -
=
x - UFEF’K+k(F)c x - US’ c u.
Hence (7) G c f - ’ ( X - US’) c f - ’ ( U ) . Since each K ( F ) is clopen in Y, G is a closed set of Y. So it remains to show is locally that G is open in Y. By (7), we need only show that { K ( F ):F E 9’}
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finite at every point of the open set f -‘(X- US’). By an argument similar to the proof of Lemma 2.24, it follows from (2) and (3) that (C1 K + k ( F ) : F E S,‘, n E N } is locally finite in X - UF’. Thus by (6), { K ( F ):F E S’} is locally finite at every point off - ‘ ( X - US’), which completes the proof of (5). 0
6.10. Theorem (Mizokami [1984]). Every stratijiable p-space is the image of an Mo-space under a perfect map.
Proof (Outline). Let X be a stratifiable p-space. Then by Theorem 5.3 and Lemma 2.29, X is an EM, -space. Apply a construction similar to the above theorem. Then we can obtain a space Y and a perfect map f : Y -+ X. If we work much harder, the resulting space Y can satisfy dim Y = 0. Since Y is a subspace of a countable product of stratifiable p-spaces, it is easy to see that Y is also a stratifiable p-space. Then by Theorem 5.5, Y is an Mo-space.
Recently Ohta [198*] defined the class of M,-spaces with a a-finitely closure-preserving base. He proved that a space is a perfect image of an Mo-space if and only if it has a a-finitely closure-preserving fitting base.
7. Related topics We give here a list of references for some topics which we do not mention in the text. Characterizations of stratifiable spaces by neighbornets, g-functions, k-networks, point-networks: Junnila [19781, Kanatani, S, saki and Nagata [1985], Balogh [1985]. Universal spaces for stratifiable spaces: Tsuda [19841. Quasi-open, countably bi-quotient closed images of M , -spaces: Kao [1983]. Stratifiable Baire spaces: van Douwen [ 19771, Tamano 19891. Theory of retract for stratifiable spaces and related spaces: Cauty [1973, 19741, MardeSiC and Shostak [1980], Tsuda [1985/86]. Monotonically normal spaces: Heath, Lutzer and Zenor [ 19731, Borges [1973], San-ou [1974]. a-spaces: Okuyama [ 19711. Semi-stratifiable spaces: Creede [ 19701.
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8. Problems
The M3*MI
question:
- Is every M3-spacean M,-space? (Ceder [1961]). Problems which are equivalent to the M3=>M1question: Does any point in a stratifiable space have a a-closure-preserving base? Is every (closed) subspace of an MI-space an M,-space? (Ceder [1961]). - Is the closed image of an MI-space an MI-space? (Ceder [1961]). - Is the perfect image of an Ml-space an MI-space? -
Problems whose positive answers give the positive answer to the M3*Ml question : - Can every M,-space be represented as the image of an Mo-space under a perfect mapping? (Heath and Junnila [19811). - Is every M3-space (M,-space) a p-space? (Mizokami [1984], Tamano [1983], Junnila and Mizokami [1985]). Problems whose negative answers give the negative answer to the M3*Ml question: - Does any point in an M,-space have a closure-preserving base? (Ceder ~96111, Or equivalently, - is any MI-space in the class 8?
-
-
-
Related Problems: Is the countable product of hereditarily MI-spaces hereditarily MI? (It6 [19841). Is the product of two hereditarily M,-spaces hereditarily MI? (It6 [1984]). Is the closed irreducible image of an MI-space an M I-space? Is every perfect image of a p-space a p-space? (Nagami [1970]). Is the perfect image of an Mo-space a p-space? (Mizokami [1984]). Characterize the closed images of Mo-spaces. Find a class of MI-spaces which contains the F,-metrizable spaces, and which is closed under closed subspaces, closed images, and countable products (Gruenhage [1980]). Is the union of two closed M,-spaces an M,-space? (Ceder [1961]). If a stratifiable space X is a countable union of closed MI subspaces, is X an MI-space? (Gruenhage [19801). Is every stratifiable space the countable union of closed MI subspaces? (Gruenhage [ 19801).
Generalized Metric Spaces I1
407
< n (resp. ind X < n) if and only if there exists a a-closure-preserving base 9Y for X such that dim Bd B < n - 1 (resp. ind Bd B < n - 1) for every B E 9Y? (Nagata [1973]). In particular, - is every strongly zero-dimensional MI-space an M,-space? - Is a space an M,-space if it has a stratification consisting of clopen sets? - Is every stratifiable space a perfect image of a stratifiable space with the covering dimension O? (Nagami [1970]). Or equivalently, - is every stratifiable space an EM,-space? (Oka [1983]). - Is every EM,-space an MI-space? (Oka [1983]). - Let X be an MI-space. Is it true that dim X
References Balogh, Z. [I9851 Topological spaces with point-networks, Proc. Amer. Math. SOC.94, 497-501. Borges, C. J. R. [I9661 On stratifiable spaces, Pacifc J . Marh. 17, 1-16. [I9731 A study of monotonically normal spaces, Proc. Amer. Math. SOC.38,21 1-214. [I9781 Direct sums of stratifiable spaces, Fund. Math. 100, 97-99. [19831 Expansion of closure-preserving collections and metrizability, Math. Japonica 28, 67-7 I. Borges, C. J. R. and G. Gruenhage [19831 Supcharacterization of stratifiable spaces, Pacifc J. Marh. 105, 279-284. Borges, C. J. R. and D. J. Lutzer [I9731 Characterizations and mappings of Mi-spaces, Topology Conference VPI,Lecture Notes Math. 375, (Springer, Berlin) 34-40. Bregman, Ju. H. (19831 A note about M,-spaces and stratifiable spaces, Comment. Math. Univ. Carol. 24,23-30. Burke, D. K. and D. J. Lutzer [I9761 Recent advances in the theory of generalized metric spaces, in: Topology: Proc. Memphis Stare University Conference, Lecture Notes in Pure and Applied Mathematics (Marcel Dekker, New York) 1-70. Cauty, R. [I9731 Convexit6 topologique et prolongement des fonctions continues, Compositio Marhematica 27, 233-271. [I9741 Retractions dans les espaces stratifiables, Bull. Soc. Math. France 102, 129-149. Ceder, J. G. [I9611 Some generalizations of metric spaces, Pacific J. Marh. 11, 105-125.
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Creede, G. D. [19701 Concerning semi-stratifiable spaces, Pacific J. Math. 32, 47-54. van Douwen, E. K. [I9771 An unBaireable stratifiable space, Proc. Amer. Math. SOC.67, 324-326. Gruenhage, G. [I9761 Stratifiable spaces are M 2 , Topology Proc. 1, 221-226. 119781 Stratifiable a-discrete spaces are MI, Proc. Amer. Math. SOC.72, 189-190. [I9801 On the M,*M, question, Topology Proc. 5, 77-104. [I9841 Generalized metric spaces, in: K. Kunen and J. E. Vaughan, Eds., Handbook of SetTheoretic Topology (North-Holland, Amsterdam) 423-501. Heath, R. W. and H. J. K. Junnila [I98 I] Stratifiable spaces as subspaces and continuous images of MI-spaces, Proc. Amer. Math. SOC.83, 146-148. Heath, R. W., D. J. Lutzer and P. L. &nor [I9731 Monotonically normal spaces, Trans. Amer. Math. SOC.178, 481-493. It6, M. [I9821 On 8-product of spaces which have a a-almost locally finite base, Proc. Japan Acad. 58, 250-252. [1984a] The closed image of a hereditary M,-space is MI, Pacific. J. Math. 113, 85-91. [l984b] Ma-spaces are p-spaces, Tsukuba J. Math. 8, 77-80. [I9851 M,-spaces whose every point has a closure preserving outer base are M , , Topology Appl. 19, 65-69. It6, M. and K. Tamano [I9831 Spaces whose closed images are MI, Proc. Amer. Math. SOC.87, 159-163. Junnila, H. J. K. [I9781 Neighbornets, Pacific J . Math. 76, 83-108. [I9841 A characterization of Ma-spaces, Proc. Amer. Math. SOC.91, 481484. Junnila, H. J. K. and T. Mizokami [ 19851 Characterizations of stratifiable p-spaces, Topology Appl. 21, 51-58. Kanatani, Y.,N. Sasaki and J. Nagata [I9851 New characterizations of some generalized metric spaces, Math. Japonica 30,805-820. Kao Kuo-Shin [I9831 A note on M,-spaces, Pacifc J. Math. 108, 121-128. MardeSiC, S. and A. Shostak 119801 On perfect inverse images of lacy (=stratifiable) spaces, Russian Math. Surveys 35, 99-108. Masuda, K. [I9721 On monotonically normal spaces, Sci. Rep. Tokyo Kyoiku Daigaku 11, 259-260. Mizokami, T. [I9811 On the dimension of p-spaces, Proc. Amer. Math. SOC.83, 195-200. [I9831 On a certain class of M,-spaces, Proc. Amer. Math. SOC.87, 357-362. [I9841 On M-structures, Topology Appl. 17, 63-89. [I9851 On M-structures and strongly regularly stratifiable spaces, Pacific J . Math., 131-141. [I9861 On functions and stratifiable p-spaces, Pacific J. Math. 125, 177-185.
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Miwa, T. [I9811 Retraction of MI-spaces, Topology Proc. 6, 351-361. Nagami, K. [I9701 Normality of products, Actes Congris Intern. Marh. 2, 33-37. [I9801 Dimension of free L-spaces, Fund. Math. 108, 21 1-224. Nagata, J. [I9731 On Hyman’s M-space, Topology Conference VPI, Lecture Notes Math. 375, (Springer, Berlin) 198-208. [I 9851 Modern General Topology (North-Holland, Amsterdam). Ohta, H. [198*] Well behaved subclasses of M,-spaces, Topology Appl., to appear. Oka, S. [I9811 A generalization of free L-spaces, Tsukuba J. Math. 5, 173-194. [I9831 Dimension of stratifiable spaces, Trans. Amer. Math. Soc. 275, 231-243. Okuyama, A. [I9681 a-spaces and closed mappings. 11, Proc. Japan Acad. 44, 478-481. [I9711 A survey of the theory of a-spaces, General Topology Appl. 1, 57-63. Pears, A. R. [I9751 Dimension Theory of General Spaces (Cambridge University Press, London). San-ou, S. [I9741 A note on monotonically normal spaces, Sci. Rep. Tokyo Kyoiku Daigaku 12,214-217. [I9771 A note on 3-product, J. Math. Soc. Japan 29, 281-285. Siwiec, F. and J. Nagata [I9681 A note on nets and metrization, Proc. Japan Acad. 44, 623-627. Slaughter, Jr. F. G. [I9731 The closed image of a metrizable space is MI, Proc. Amer. Math. Soc. 37, 309-314. Tamano, K. [I9831 Stratifiable spaces defined by pair collections, Topology Appl. 16, 287-301. [I9851 On characterizations of stratifiable p-spaces, Math. Japonica 30,743-752. [I9891 p-spaces, stratifiable spaces and mosaical collections, Math. Japon., to appear. Tsuda, K. [I9841 Non-existence of universal spaces for some stratifiable spaces, Topology Proc. 9, 165-171. [1985/86] ANR (paracompact M) versus ANR (stratifiable), Questions Answers General Topology 3, 87-94. Worrell, Jr. J . M. and H. H. Wicke [I9651 Characterizations of developable topological spaces, Canad. J. Marh. 17, 820-830. Yamada, K. [I9841 M,-spaces whose every point has a closure-preserving outer base are CP-expandable, Math. Japonica 29, 503-508.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 11
FUNCTION SPACES
Akihiro OKUYAMA Department of Mathematics, Kobe University, Nada, Kobe, 657 Japan
Toshiji TERADA Yokohama National University, Hodogaya, Yokohama, Japan
Contents I . Introduction and notation . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 3. Some properties of C,(X) . . . . . . . . . . . . . . . . 4. Some properties of C J X ) . . . . . . . . . . . . . . . . 5. Topological properties and linear topological properties . . 6. Topological properties and /-equivalence. . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . 2. Some properties of C,*(X)
. . . . . .
. . . . . .
. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 1 416 417 427 436
447 457
1. Introduction and notation Let X and Y be topological spaces and C(X, Y ) the set of all continuous mappings from Xinto Y . Then several topologies may be defined in C(X,Y ) according to the claim of researching of the topological structures in C(X, Y ) . Here, we are mainly concerned with C(X, Y )in the case of X being a Tychonoff space and Y the real line with the usual topology which will be denoted by C(X),and three types of topologies for C ( X ) : norm topology, compact-open topology and topology of pointwise convergence. In this section we state their definitions and some related facts which are either well known or fundamental and we introduce the topological situation of C ( X ) in view of our interest in Sections 2 , 3 and 4 according to the type of topology in the above order. Adding topological structure, C(X)has the linear structure; that is, C ( X ) is also treated as a topological linear space. In view of this point, we explain
412
A . Okuyama, T. Terada
the linear topological properties of C ( X ) in Section 5 and the influence of linear homeomorphisms to topological properties of underlying spaces. All spaces are assumed to be Tychonoff spaces; that is, completely regular T,-spaces.N denotes the set of all natural numbers and R denotes the real line with the usual topology. For a space X let C ( X ) be the set of all real-valued continuous functions on X and C * ( X )the subset of C ( X )consisting of all bounded functions. For a space X let Y ( X )be the set of all subsets of Xand X ( X ) the subset of 9(X) consisting of all compact subsets. In a metric space ( X , e), for any subset A of X and positive number E , put S ( A , E ) = {x E Xle(x, A) <
1.1. Definition. For a space X , let f
Ilf and for any E
IIA
= SUP{If(X)llX E
E
E}.
C ( X ) and A E 9 ( X ) . Then we put
4,
=- 0 we put
1.2. Lemma. Let d be a subset of Y(X)such that A v B B E d . Then the family
{Wf, A, & ) I f
E
C ( W ,A
E
E
d for any A,
d , E > 0)
forms a base of some topology on C ( X ) .
Using this lemma, we can define topologies on C ( X ) or C * ( X ) according to the choice of d . 1.3. Definition. (a) For d = {X}, let Yfl be the topology of C * ( X )induced by d .Then we call F,, the norm topology and denote the space ( C * ( X ) ,Ffl) by C,*(X). (b) For d = X ( X ) , let Y k be the topology of C ( X ) induced by d.Then we call Y k the compact-open topology and denote the space ( C ( X ) ,y k ) by ck(X).
(c) For d = { A E Y ( X )I A is a finite set}, let Ypbe the topology of C ( X ) induced by d . Then we call Ypthe topology of pointwise convergence and denote the space ( C ( X ) ,Fp)by C p ( X ) . (d) For a family 99 of compact subsets of OX (resp. P X ) , let be the the topology topology of C ( X ) (resp. C * ( X ) )induced by B. Then we call
413
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of 93 convergence and denote the space ( C ( X ) ,y # (resp. ) ( C * ( X ) , Fa))by C , ( X ) (resp. C ; ( X ) ) , where U Xdenotes the Hewitt realcompactification of X and PX denotes the Stone-cech compactification of X , further C ( X ) is identified with C ( u X ) and, of course, C * ( X ) with C*(j?X).
From the above definitions we can easily see the following relations between the above topologies. 1.4. Lemma. F,
=I
Fk holds on C * ( X ) and yk
=I
ypholds on C ( X ) .
The following examples show the differences between these topologies.
1.5. Example. defined by
Let I be the unit interval [0, I] and f, the function on Z
0
<x <
1/2"+',
Then the sequence {f,I n E N } converges to 0 in Fp,but does not converge to 0 in Fk= F,; that is, Fpis really weaker than F k on C(Z).
1.6. Example. Let X be the half-open interval [0, 1) andf, the function on X defined by 0 < x < 1 - 1/2" =
iO' + 2"x
1 - 2",
1 - 1/2"
<x
< 1.
Then the sequence {f,I n E N } converges to 0 in yk,but does not converge to 0 in y,; that is, y k is really weaker than F, on C ; ( X ) . We sometimes use other expressions for yk and Fp according to the following lemmas. 1.7. Lemma. For any K , , . . . , K, E X ( X ) and open sets V,, . . . , V , in R, Put U(K1, * * * Kn; V , . . . V,) = { f e C ( X ) I f ( K i ) c Y 9
for i = 1 , .
9
. .,n}.
Then the family { U ( K , , . . . , K,; V,,
. . . , V,)I K , , . . . , K , E X ( X ) ;V,, . . . , V , are open in R, n E N }
forms a base f o r some topology y on C ( X ) and .F coincides with
y k .
A . Okuyama. T. Terada
414
1.8. Lemma. For any points xI,. . . , x, in X and open sets V,, Put
U ( x l ,. . . , x,,;
v, . . . , V , )
=
{f
E
. . . , V , in R,
C ( X ) lf ( x i )E r/; for i = 1,
. . . , n}.
Then the family {U(X,,. . . , x,;
v, . . . , v.)I
XI,
. . . ,x,
E
X;
v, . . . , v,
are open in R, n E N } forms a base for some topology F on C ( X ) and F coincides with Fp. As general properties, the following are well known.
1.9. Theorem. Let X be a space. Then we have: (a) C p ( X )is a dense subspace of the product space RXand, therefore, C p ( X ) is a Tychonoflspace. (b) C , ( X ) is also a Tychonoflspace. (c) C,*(X) is a completely metrizable space. 1.10. Theorem. Any space X is embedded in Cp(Cp(X)).
Proof. For any point x E X , let cp(x) be a function on C p ( X )defined by cp(x)( f ) = f (x) for each f E C p ( X ) Then . cp(x) is continuous on Cp(X);that Hence, cp is a mapping from Xinto Cp(Cp(X)). Since X is, cp(x) E Cp(Cp(X)). is a Tychonoff space, it is easy to see that cp is one-to-one. To show that cp is continuous, let x be an arbitrary point of X and W(cp(x),{fi, . . . , h } ,E ) an arbitrary neighborhood of cp(x) in Cp(Cp(X)). Since eachf; is continuous at x, there exists a neighborhood U of x in X such thatf;(U) c S(f;.(x), E ) for each i. Then cp(U) c W(cp(x),{fi, . . . ,f,},E ) holds, which shows the continuity of cp. Finally, to show that cp is an open mapping from X onto cp(X), let G be an open subset of X and take a point cp(x) in cp(G). Since X is a Tychonoff space, there exists an f E C p ( X )such that f ( x ) = 1 and f ( y ) = 0 for y E X - G. Then we have W(cp(x),{f}, i)n q ( X ) c cp(G), which shows that cp(G) is open in cp(X). 0 To clear the situation of subsets of C ( X ) , the following concept is often used.
1.11. Definition. For a subset F of C ( X )and a point x of X , we say that F is equi-continuous at x, if for any E > 0 there exists a neighborhood U of x
Function Spaces
415
such that If (x) - f ( y ) I c E holds for each y E U and f E F. If F is equicontinuous at each point of X , we say that F is equi-continuous. 1.12. Lemma. Let F be a subset of C ( X ) .IfF is equi-continuous, then C1,F is also equi-continuous, where C1,F denotes the closure of F in C,(X). 1.13. Remark. CI,F and C1,F will be used to denote the closures of F in C,(X) and C,*(X),respectively. Since C1, F c Cl, F and C1, F c CI, F hold, in general, by Lemma 1.12 we can see that, if F is equi-continuous, then CI, F and C1, F are also equi-continuous. 1.14. Theorem. If F is an equi-continuous subset of C ( X ) , then the relative topology Yk I of T k to F coincides with the relative topology ypI of Y, to F. Proof. Since F , c y k holds, in general, it suffices to show that F , I 3 Y, I F . For this purpose, we show that, if a net {&I 3, E A} converges to f in (F, 6 I F ) , then it converges to f in (F, F k I F ) , too. Let W ( f ,K, E ) n F be an arbitrary neighborhood off in (F, 9, I F ) . For any point x of K, by the assumption for F, there exists a neighborhood U, of x such that If ( x ) - f ( y)l < + E for each y E U,and f E F. Since K is compact, a finite subcollection { U,, I i = 1, . . . , n } of { U, I x E K} covers K. Since {hI 1 E A} converges to f in (F, F , 1 F ) , there exists liin A such that h ( x i ) E W ( f , { x i } ,$ 8 ) for any 1 2 li.Take p in A with 2, < p for i = 1, . . . ,n. Then it is easily seen that, if 3, 2 p , h E W ( f ,K, E). This shows that the net {hI A E A} converges to f in (F, F ,IF). 0 1.15. Definition. For a subset F of C ( X ) , we say that F separates points of X if, for any distinct points x , y of X , there exists an f E F such that f ( 4 z f ( Y ) holds. As for the basic terminologies and knowledge, we refer the reader to books of Engelking [1977], Jameson [1974], Kelly [1955], Nagami and Kodama [1974] and Nagata [1985]. Especially Section 2 is largely based on the works mentioned. As for a characterization of F separating points of X , the following expression will be used. As its proof is very elementary, it is omitted.
1.16. Theorem. Let F be a subset of C ( X )and cp a mappingfrom X into C(F) defined by cp(x)(f ) = f (x) for each x E X and f E F. Then F separates points of X if and only if cp is a one-to-one mapping.
A . Okuyama, T. Terada
416
2. Some properties of C,*(X) First we will explain the following which are most fundamental relationships between C,*(X)and X in view of general topology. 2.1. Theorem (Jameson [1974]). Zf C,*(X) is separable, then X is a second countable space.
Proof. Let { f,ln E N} be a countable dense subset of C,*(X). For each n E N, put U, = {x E XllS.(x)I < 5). Then we show that the collection {U, I n E N} forms a base for X. Let x be an arbitrary point of X and U an open neighborhood of x, and let f be a continuous function from X into I = [0, I] such that f(x) = 0 and f 1 x - o = 1 . Since {fn} is dense in C,*(X), there exists an no E N such that 11 f - fn, 11 < $ holds. Then Ifn,(x) I < 3 holds, and if y is such that Ifno( y ) I < +,then f( y ) < 1 holds, which shows y E U ; that is, we have x E U,, c U . 0
2.2. Proposition (Jameson [ 19741). Let E be a linear subspace of C,*(X)such that for any disjoint closed subsets A , B of X there is a memberf of E that maps X into I and takes the value 0 on A and 1 on B. Then E is dense in C,*(X). Proof. Applying the hypothesis to the pair 0,X , we see that E contains the constant map 1 . Choose E > 0 and an arbitrary element g of C,*(X). We show that there is an element f of E such that IIg - f 11 < E . Let a = infg(X). There exists an integer n such that g(x) < a nE for all x E X. For 1 < r < n, define
+
A, =
{XE
Xlg(x)
B, = {x E Xlg(x)
< a + (r > a + rE}.
l)~},
Then A,, B, are disjoint closed subsets of X and so, by hypothesis, there is a memberf, of E that maps into I and takes the value 0 on A, and 1 on B,. Let
f
= a.1
+ & ( A +.**+fn).
Then f is certainly in E. Let x be an arbitrary point of X. There exists p such that 1 < p < n a n d a
+(p-
1)E
< g(x) < a
+ PE.
in B,, sof,(x) = 1. If r > p + < &(x) < 1, we have a + ( p - 1 ) ~< f(x) < a + P E . Hence, we have I g(x) - f(x) I < E , as required. If r
- 1, then x is
1, then x i s in A,, so
f,(x) = 0. Since 0
0
Function Spaces
417
2.3. Theorem. Let X be a compact space. Then C,*(X) is separable if and only if X is a second countable space.
Proof. The “only if” part is just 2.1. To prove the “if” part, let W be a countable base for Xwhich is closed under finite unions. For each pair ( U , V) of members of W with C1 U c V , letf;,,, be a continuous function from X into Z that takes the value 0 on U and 1 on X - V. Put E the linear subspace of C,*(X)generated by { l } u {hu,v, I U, V E 9, C1 U c V}. Then it is easy to see that E satisfies the assumption in Proposition 2.2. Hence E is dense in C,*(X). E is clearly separable and consequently, C,*(X) is separable. The following theorems are very useful when we discuss function spaces. As they are well known, we state them without proofs.
2.4. Theorem (Stone-Weierstrass Theorem). Let X be a compact space and P a subring of C ( X ) containing all constant functions on X . If P separates points of X , then P is dense in C,,(X). 2.5. Theorem (Ascoli Theorem). Let X be a compact space and F a subset of C , ( X ) such that C1 F [ x ] is compact for each x E X , where F [ x ] denotes the subset { f (x) If E F } of R . Then F is compact ifand only i f F is equi-continuous and closed in C , ( X ) . 3. Some properties of C,(X) To explain the topological properties of C , ( X ) , we use the following conceptions for X which are exclusive for general topology.
3.1. Definition. A space X is said to be a space of point-countable type if each point of X is contained in some compact subset of X which has a countable neighborhood base. 3.2. Definition. A space Xis called a q-space if, for each point x of A’, there exists a sequence { U,,I n E N } of neighborhoods of x such that each sequence {x, I n E N } with x,, E U,, for n E N has a cluster point in X. 3.3. Definition. A space Xis said to be hemi-compact if there is a countable subcollection {K, I n E N} of X ( X )such that for any K E X ( X )there exists m E N w i t h K c K,.
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A . Okuyama, T. Terada
As for the equivalence of topological properties of c k ( X ) and those of X , we can pick up the following theorems.
3.4. Theorem (McCoy and Ntantu [1986]). For a space X , thefollowing are equivalent : (1) C , ( X ) is a metrizable space. ( 2 ) ck(x)is afirst countable space. (3) ck(x)is a space of point-countable type. (4) C , ( X ) is a q-space. ( 5 ) X is a hemi-compact space. Proof. Implications (1)+(2)+(3)+(4) are clear from their definitions. (4)45): By the assumption for 0 E C,(X), there is a countable sequence { W(0, K,,, E,) In E N } of neighborhoods of 0 satisfying the condition of a q-space, where we can assume K,,c K,,,, for each n E Nand lim,,+mc,,= 0. Then we show that each K E X ( X ) is contained in some K,,. Suppose there exists K E X ( X )such that K is not contained in any K,,. Take a point x,, in K - K,, and let g,, E ck(x) such that g,,(x) = n and g,, = 0 on K,. Then g,, E W(0, K,,, E,) for each n E N . On the other hand, the sequence {g,,I n E N } has no cluster point. This is a contradiction. (5)+(1): Let {K,,I n E N } be a sequence satisfying the condition of the hemi-compactness of X and put S = 0 { K,, I n E N } (topological sum), and let $ be a natural mapping from S onto X. Then the induced mapping $* is an embedding of C,(X)into C,(S), where +* is defined as $*(f)= f o $ for eachfE C , ( X ) . It is easy to see that C , ( S ) is homeomorphic to the countable product n{C,(K,,)I n E N } , each factor of which is metrizable. Hence, ck(s) is metrizable and as its subspace C,(X) is also metrizable. 0 3.5. Theorem (McCoy and Ntantu [1986]). For a space X the following are equivalent : (1) ck(X)is a completely metrizable space. (2) ck(x)is a tech-complete space. (3) X is a hemi-compact k-space.
Proof. (1)+(2) is clear. (2)+(3): Since Cech-complete spaces are always spaces of point-countable type, (2) implies that X is a hemi-compact space by Theorem 3.4, and equivalently, C , ( X ) is a metrizable space. As a consequence, there exists a sequence {K,,In E N } satisfying the condition of the hemi-compactness of X with K,, t K,,,, for each n E N and C , ( X ) is a completely metrizable space.
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419
Now, we show that Xis a k-space. Suppose Xis not a k-space. Then there is a subset A of X such that A n K, is closed for each n E N, but A is not closed in X. Pick a point x E C1 A - A . Here we can assume x E Kl. Let f , E C ( X )withA(x) = 1 andf, = 0 on A n Kl andf, E C ( X )such thatf, I K2 is an extension ofA I K I over K2 such thatf, = 0 on A n K 2 . Continuing this process, definef, E C ( X )such thatf, I K n - , = f,- I I K,-l andf, = 0 on A n K,. Then ( f , l n E N} is a Cauchy sequence in C,(X)and, since C , ( X ) is completely metrizable, it converges to some g in C,(X). This g must be continuous; however, since g ( x ) = 1 and g = 0 on A , g is not continuous, which is a contradiction. This shows that X is a k-space. (3)+(1): Let {K,ln E N ) be a sequence in X ( X )satisfying the condition of hemi-compactness of X and S = 0 ( K , In E N } , and let $ be a natural mapping from S onto X. Remark that $ is a quotient mapping, because Xis a k-space. Then $* is an embedding of C , ( X ) into C , ( S ) as $*(C,(X)) being a closed subset of C,(S). Hence, by the same reasoning as the proof of Theorem 3.4, (5)4(1), C , ( X ) is homeomorphic to a closed subset of C,(S), which is completely metrizable. As a consequence, C , ( X ) must be completely metrizable. 0
3.6. Definition. A space X is called a submetrizable space if X is mapped onto a metrizable space by a one-to-one continuous mapping. Of such a situation it is sometimes said that X is condensed onto a metrizable space. 3.7. Definition. A space is said to be a-compact if it is a union of countably many compact subsets. As a weaker condition, a space is said to be almost * a-compact if it contains a dense, a-compact subspace.
3.8. Theorem (McCoy and Ntantu [1986]). For a space X , the folfowing are equivalent : (1) C,(X)is a submetrizable space. (2) Any compact subset of C , ( X ) is a G,-set. ( 3 ) Any element of C , ( X ) is a G,-set. (4) The diagonal of C,(X)x C , ( X ) is a G,-set. ( 5 ) X is an almost a-compact space. Proof. (1)+(2): Let $ be a one-to-one continuous mapping from C , ( X ) onto a metrizable space M. Then for any compact subset F of C , ( X ) , $ ( F ) is compact and, therefore, closed in M ; hence, $ ( F ) is a G,-set in M. As a consequence, F is a G,-set in C,(X). (2)+(3) is clear.
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n{
(3)+(5): By assumption, 0 must be a G,-set; that is, {0} = W(0,K,,, E,,) I n E N } for some K , , EX ( X ) and E,, > 0. It is enough to show that Y = U { K , I n E N} is dense in X. Suppose there is an x E X - C1 Y. Let f be a continuous function on X such that f ( x ) = I and f = 0 on CI Y. Then f does not belong to some W(0,K,,, E,,) and, on the other hand, f = 0 on K,,, which is a contradiction. (5)+(1): Let Y = U { K , l n E N } be a dense, a-compact subspace of X , where each K,, is compact. Put S = @{K,,In E N } and let $ be a natural mapping from S into X . Then the induced mapping $* : C , ( X ) + C , ( S ) is one-to-one and continuous. The space C , ( S ) is metrizable, because it is homeomorphic to the product space lI{C,(K,,)I n E N } , each factor of which is metrizable. As a consequence, C , ( X ) is condensed onto a subspace of C , ( S ) , which is metrizable. (1)+(4): Let $ : C , ( X ) + M be the mapping in the proof of (1)+(2). Then $ x $ is a one-to-one continuous mapping from C , ( X ) x C , ( X ) onto M x M. Since the diagonal of M x M is a G,-set, the diagonal of C , ( X ) x C , ( X ) is also a G,-set. (4)+(3) is clear. 0
3.9. Theorem (McCoy [1978]). I f X is a submetrizable space, then C , ( X ) is an almost a-compact space. Proof. Let $ be a one-to-one continuous mapping from Xonto a metrizable space M, and $ * : C , ( M ) + C , ( X ) the induced mapping by $. Then $*(C,(M)) is dense in C, (X). Because, for any f E C, (X) and any neighborhood W ( f , K , E ) off, by 2.4 there exists goE C ( $ ( K ) ) such that 11 f I goo$ I 11 < E . Since $ ( K ) is compact, there exists a continuous extension g E C ( M )ofg,. Then we have 11 $*(g) - f I1 .c E ; that is, $*(g) E W ( f ,K , E ) , which shows that $*(C,(M)) is dense in C , ( X ) . Hence, it is enough to show that C , ( M ) is an almost a-compact space. We will prove it dividing into several steps. Let U{.?d,,I n E N} be a o-locally finite base of R such that each .?dn is a locally finite open covering of R, whose mesh is less than lln and F,, a partition of unity subordinated to B,,, and put F,* = F,, u { l } , F,,‘ = { r f i . . . f k l k< n,Irl < I , f i , . . . , f k ~ F y u . . . u F , * } , F = ; {g, + . . . + g,lk < n, g,, . . . ,g, E F:} and F = U{F,,”lnE N } . ( I ) F is a subring of C , ( M ) . For anyf, g E F, it is easy to see thatf g and f g belong to F. Furthermore,foranyfE F a n d a E R , l e t n , ~ N s u c h a s I a I < n,.Thenn,fbelongs to F and af = a/no nof.
,
+
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(2) F separates points of M. For any two distinct points x, y in (M, d), pick n E N such as d(x, y) > 2/n. Then there exists an f E F, such that f(x) = 0 and f ( y ) # 0 hold. Since F, is contained in F, F separates x and y. (3) F: is equi-continuous. Let x be an arbitrary point of M and E a positive real number. For each i < n, since 9, is locally finite, there exists a neighborhood U, of x such that the subcollection {B E 9,l U, n B # @} is finite. It may be denoted by { B , , , . . . , BIk,>.As F, is subordinated to 9,,each B , is corresponded to some J;, E F, for j = 1, . . . , k,. By the continuity off;, at x, there is a positive number 6, such that S(x, 6,) c U, andf;,(S(x, 6,)) c S(f;,(x), ~/n’).Put 6 = min(6,l 1 < j < k,, 1 < i < n}. ForanygEF,”putg = g, + g,withg,EF,fori = 1 , . . . , k a n d k < n, and, for each i < k, set g, = rlgll . . . g,,,with g, E F: u . . v F,* f o r j = 1, . . . , I, and I, < n. From the fact (1) we can assume that each g,, is not constant on S(x, 6). Hence, each g, belongs to {Lq11 < q < k , , 1 < p < n} and therefore, we have g,(S(x, 6)) c S(g,(x), E/n’) for 1 < j < l,, 1 < i < k and by an easy calculation, we have g,(S(x, 6)) c S(g,(x), E/n) for 1 < i < k, and as a consequence, g(S(x, 6)) c S(g(x), E ) holds. (4) For any x E X and n E N , Cl,F,”[x] is compact. Iff is an arbitrary element of F,: then it is easily seen 1 f (x) 1 < n remembering that each F,,is a partition of unity. Hence, C1, F,”[x] is a closed subset of [-n, n], which is compact. As a consequence, using the facts (1) and (2) and Theorem 2.4 we can see that F is dense in ck(M)and using the facts (3) and (4), by Theorem 3.10 below, ClkF,” is compact for each n E N , where ClkF” , denotes the closure of F,” in ck(M). 0
+
-
The following is the Arzela-Ascoli’s Theorem in function spaces with the compact-open topology. 3.10. Theorem (Arzela-Ascoli, see Nagami and Kodama [1974]). subset of C ( X ) which satisfies the following two conditions: (1) F is equi-continuous. (2) Cl,F[x] is compactfor each x E X . Then ClkFis a compact subset of C , ( X ) .
If F is a
Proof. We sketch the proof. Put H = ClkF,then it suffices to show that any maximal filter in Hconverges. Let A?be an arbitrary maximal filter in H. For
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each x E X , H[x] is contained in ClRF[x] and, since Cl,F[x] is compact, the intersection n{clRM[x] I M E A } is not empty. Choose a point $ ( x ) in the intersection. Then $ is defined as a function on X . Now, we are going to show that $ is a continuous function in H and A! converges to $. By Remark 1.13, H is equi-continuous. Using this fact, we can see that $ is continuous. For each x E X , 6 > 0 and M E A, we have W ( $ , { x } , 6) n M # 8 and, by the maximality of A, we have W($, { x } , 6) n H E A!. Using this fact, $ belongs to C1,H. By Lemma 1.12 and Theorem 1.14 we have C1,H = ClkH = H and hence, $ E H. Finally, for each M E A we can see $ E C1,M which is equal to ClkM,and therefore, A converges to $ in H. This shows that H is a compact subset of C,(X).
0 For further topological properties of C,(X), we use the following notions. For a space X and an infinite cardinal t, we denote t ( X ) < 5 if, for any subset A of Xand a point x with x E C1 A, A contains a subset B such that the cardinality I B I of B is not greater than t and x E C1 B holds. The minimal cardinal of z which holds up t ( X ) < z is called the tightness of X. 3.11. Definition.
3.12. Definition. An open covering 4! of a space X is called a cover for
compact subsets of X if every compact subset of X is contained in some member of a. 3.13. Theorem (McCoy [1980bJ). For a space X , t(C,(X)) < Noholds ifand only if any cover for compact subsets of X contains a countable subcover for
compact subsets of X .
Proof. (Necessity):Let 4! be a cover for compact subsets of X . Then for any K E X ( X )there exists a U, E 4! with K c U,. For any such pair (K, U,), let fK,uK be a continuous function on X such that f K , U K = 1 on K and fK,,uK = 0 on X - UK.Then we can see 1 E Clk{fK,uKl K E X ( X ) } .By the assumption, there exists a countable subset { fK,,uK,1 i E N } with 1 E C1,{ fK,,UK, I i E N } . Now we show that { U,, I i E N} is a cover for compact subsets of X. Take an arbitrary K E X ( X ) . Then the neighborhood W(1, K, 1) of 1 contains somefKa,,., and hence fKn,., > 0 on K, which means K c UK,.This shows that { U,, 1 i E N } is a cover for compact subsets of X. (Suficiency):Let F be a subset of C , ( X ) and f E C1,F. For each K E X ( X ) and n E N there is an fK,,n in W(S, K, l/n) n F. Put UK,n= {x E XI If (x) fK,Jx) I < l / n } and a,, = { UK,n I K E .X(X)}. Then @, is a cover for compact
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subsets of X. By the assumption, q,,contains a countable subcollection "1T, = { UKi.,I i E N } which is a cover for compact subsets of X. Put Fo = {fK,,,,I i, n E N } . Then we can easily see f E ClkF,. This shows 0 t(ck(x)) < KO. In view of the point that topological properties of ck(X) may restrict those of X , we explain the following results.
3.14. Lemma. If X is an infinite compact space, then there is an infinite, discrete collection of closed subsets of Cp(X,I ) .
Proof. Let { x, 1 n E N } be a discrete subspace of Xand A = Cl,{x, I n 6 N } and A. = A - {x,l n E N } , and let, for each i E N, f; be a continuous function from A into I such that f ; ( x , ) = 1 for n < i and f ; ( x , ) = 0 for n > i and f;. = 0 on A o . Then { f i I i E N} is a discrete, closed subset of C,(A, I).Let cp* be the mapping from Cp(X,I) into C p ( A ,I) induced by the inclusion map cp : A + X , and put F, = c p * - ' ( { f ; } ) for each i E N. Since cp* is continuous and onto, {F,1 i E N } is the required collection. 0 3.15. Theorem (Pol [1974]). Let X be aparacompact k-space. I f C p ( X ,I ) or ck(x, I ) is normal, then the derived set Xdof X is a Lindelof subspace.
Proof. Suppose Xd is not a Lindelof subspace. Since X is paracompact and X d is a closed subset of X,there is a discrete collection { A j .11 E A} of closed subsets of Xsuch that I A I = K,and Int,A, n X d # 8 for each 1 E A. Put Fi = { f C(X, ~ I)[f = 0 on X - A,} for each L E A and F = {fc C ( X , I)]f = 0 on X - U { A i I 1 E A}}. Then F is closed in CJX, I) and hence, in ck(x, I), and it is homeomorphic to the product space H { F AI 1 E A}, because, as a mapping from n { F i 11 E A} to F, if we define it by V((A)AGA)(~) = x{h.(x)I A} for each (h.),sAE H { F , 11 E A) and x E X, then cp is homeomorphic with respect to any relative topology of y k and YP. Therefore H{F, I A E A} is normal in any case of the assumption. Since Xis a k-space and Int,A, n X d # 8, we can find an infinite compact subset K, of Int,A,. By Lemma 3.14 there exists a discrete collection { F , I n E N } of closed subsets of Cp(K,,I). Let $ be a mapping from Fi into Cp(K,,I) such that $(f) = f 1 and En = $-'(F,) for each n E N. Then $ is continuous with respect to the relative topology on FAof Fpand hence, the
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collection { E , I n E N } is discrete and each Enis closed in (Fi, F, I F i ) and also in (Fj.,FkI F i ) . This shows that in each Fi there exists a countable discrete collection of closed subsets. In this case, the product space 17{Fi 11 E A} is not normal (cf. Stone [1948]). This is a contradiction, which completes the 0 proof.
3.16. Lemma (McCoy [1980a]). r f X is a normal space and C,(X) is a Lindelof space, then any discrete collection of closed subsets of X is at most countable. Proof. Let {A;,I A E A} be any discrete collection of closed subsets of X . Put F = { f E C,(X) If ( A , ) E N for each A E A}. Then F is a closed subset of Cp(X).By the assumption for C,(X), F is a Lindelof space. We define a mapping cp from F to the product space N" by cp( f ) = (nj.)i.EA with n;. = f ( A j . )E N . Then cp is a continuous, onto mapping and therefore, N" is a Lindelof space. By Stone [1948], A must be at most countable, which completes the proof. 0
3.17. Theorem (Corson and Lindenstrauss [1966]). For a metrizable space X , the following are equivalent: (1) ck(X) is a Lindelof space. ( 2 ) C p ( X )is a Lindelof space. (3) X is a separable space. Proof. (1)+(2) is clear. (2)+(3) is also clear by Lemma 3.16. (3)+(1): Let d be a countable base for X and 547 a countable base for R . For any finite A , , . . . , A , E d and B , , . . . , B, E @ put
for i
=
1, . . . , n}.
Then the collection W = { W ( A , ,. . . , A,; B , , . . . , B,) I A , , . . . ,A , E d, B , , . . . , B, E 547, n E N } is countable and satisfies the condition: If f E U(K, V ) with K E X ( X ) and V open in R , then there exists a W E W such a s f E W c U ( K , V ) . Hence, if we set W* the collection of all intersections of finite members in W , then for any open covering 42 of C , ( X ) , W * contains a subcollection which covers C , ( X ) and refines 42. Since W * is also countable, it shows that ck(x)is a Lindelof space. 0
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We introduce the KO-space.It is a very useful concept which is closely related to function spaces. After a few preliminaries we also explain some of many related results.
3.18. Lemma (see Kelley [1955]). Let X and Y be spaces and K a compact subset of X. Then the mapping 5 from K x ck(X, Y ) into Y defined by {(x, f) = f ( x ) for each x E K and f E C,(X, Y ) is continuous. 3.19. Lemma (Michael [1966]). Let X be a k-space, Y a space and F a compact subset of ck(X, Y ) . Then the mapping $from X into -X( Y ) defined by $(x) = { f (x) If E F }for each x E X is upper semi-continuous; that is,for every open subset V of Y the set ( x E XI $(x) c V } is open in X .
Proof. Let V be an arbitrary open subset of Y and U = {x E XI +(x) c V}. Since X is a k-space, it is enough to show that U n K is open in K for each compact subset K of X. Let K E X ( X )and x E K n U . For each f E Fby Lemma 3.18, there exists a neighborhood G , of x in K and W, off in C,(X, Y ) such that g ( z ) E V whenever z E G, and g E W,. Since F is compact, there exists a finite subcollection { Ti1 i = 1, . . . , n} covering F. Then G = n{Gh I i = 1, . . . , n} is a neighborhood of x in K such that $ ( z ) c V holds for each z E G. This shows that G is contained in K n U.This completes the proof that K n U is open in K. 0 3.20. Definition. Let d and B be subsets of B ( X ) . Then B is called a network for d if for each A E d and open set V in X with A c V there exists a P E 9 such as A c P c V . Especially, if d is X ( X ) ,then a network for d is called a k-network, and if each member of d is a singleton, then a network for d is called a network. A space Xis called an KO-spaceif X has a countable k-network, and a space X is called a cosmic space if X has a countable network for X. 3.21. Theorem (Michael [1966]). an K,-space.
I f X a n d Yare &spaces, then ck(X, Y ) is
Proof. Suppose, first, that X is a k-space. Let d and W be countable k-networks for X and Y, respectively, such that both of them are closed under finite intersections. For each A E d and B E W,put W ( A , B) = { f E C,(X, Y )If(A) c B). Then the collection { W ( A , B) 1 A E d,B E W) will form a k-network for ck(X, Y).
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By Michael [1966], it suffices to show that for each compact subset F of C,(X, Y ) and open subset U ( K , V) of C,(X, Y) with F c U(K, V ) there exists W ( A , B) such as F c W ( A , 8 ) c U(K, Y ) , where K E X ( X ) and V is an open subset of Y. For such F and U ( K , V ) put G = {x E X l f ( x ) E V for eachfE F } . As F c W ( K , V ) , we have K c G. By Lemma 3.19 G is open in X . Let A , , A 2 , . . . be an enumeration of all A E d such that K c A c G , and let A; = A , n . . . n A,,for all n E N. Also, let B,, B 2 , . . . be all the members of a with Bi c V. We show that F c W ( A ; , B,,) t U(K, V) for some n E N. By the choice of A, and B,,, W(A,', B,,) is always contained in U(K, V). It suffices to show that F(A;) = {f(x)lx E A;, f~ F } c B,, for some n E N . Suppose there were no such n. Then, for each n E N , there is an x,, E A,' such that F(x,) = { f ( x , , ) I f F ~ } Q B,. Let E = K u {x,, I n E N}. Then it is easily seen that E is a compact subset of X and hence, F ( E ) = {f(x) I x E E, f E F } is a compact subset by Lemma 3.18. Since E c G, we have F ( E ) c Vand thus, F(E) c B,, c Vfor some n E N. In particular, F(x,) c B,,, which is a contradiction. If X is not a k-space, we consider the k-space k ( X ) , where k ( X ) denotes the space obtained by topologizing X so that a subset A of X is closed in k ( X ) if and only if A n K is closed in K for each K E X ( X ) . If X is an Ko-space, then k ( X ) is an KO-and k-space (cf. Michael [1966]). Hence, as was shown above, C,(k(X), Y) is an K,-space. Since C,(X, Y) is embedded in C,(k(X), Y) in a natural way, C,(X, Y) is also an K,-space.
3.22. Theorem (Michael [ 19661). For a space X thefollowing are equivalent: (1) X is an K,-space. (2) C,(X)is an KO-space. ( 3 ) C,(X)is a cosmic space.
Proof. By Theorem 3.21, (1)+(2) holds. (2)+(3) is clear by definitions. (3)+(1): Let 9 be a countable network for C , ( X ) and H = {SEC,(X)I f ( X ) c I } and, for each F E 9,let F* = { x E X l g ( x ) > 0 for each g E H n F}.Then it suffices to show that {F* I F E 9} forms a k-network for X . For this purpose, let K E X ( X )and V an open subset of X with K c V . Pickfin H such thatf = 1 on K andf = 0 on X - V . Here, we can assume f > 0 on V.Then U(K, (0, co)) is a neighborhood offin C,(X)and hence, there is an F E 9withf e F c U(K, (0, a)).For this F we can easily see that K c F* c V holds. 0
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4. Some properties of C'(X) In this section we explain the function space C J X ) with the topology of pointwise convergence and also the relationship between C , ( X ) and X. 4.1. Definition. A space X is called a Bake space if, for any sequence {G,, I n E N } of open dense subsets of X , the intersection n{G, I n E N } is also dense in X. The following theorem shows that the situation of C,(X) in R Xrestricts the space X.
4.2. Theorem (Dijkstra, Grilliot, Lutzer and van Mill [1985]). a G,-set in R X ,then X is a discrete space.
If C,(X) is
Proof. On the contrary, suppose Xis not a discrete space. Then there is an x, E X with x, E Cl(X - { x o } ) . Let g be a function on X defined by g ( x o ) = 1 and g = 0 on X - {xo}and Jl a mapping from R Xto R Xdefined by J l ( f ) = f g for e a c h f e R X .Then Jl is a homeomorphism such that Jl(C,(X)) c R X - C , ( X ) holds. Since by the assumption and Theorem 1.9 (a) C,(X) is a dense G,-set in R X ,Jl(C,(X))is also a dense G,-set in R X .Since R X is a Baire space, the intersection C , ( X ) n J / ( C , ( X ) )is still dense in R X , which contradicts to C,(X)n Jl(C,(X)) = 0. 0
+
4.3. Theorem (Dijkstra, Grilliot, Lutzer and van Mill [1985]). IfC,(X) is an F,-set in R X ,then X is a discrete space.
Proof. Let C,(X) = U{F,,lnE w } with each F,, a closed subset of R X and Fo = 0,where w denotes the set {0} u N.Now, suppose X is not a discrete space; that is, there is a nonisolated point xo in X. We choose a sequence {f,1 n E w } in I x and a sequence { U,,I n E w } of open neighborhoods of x, which satisfy the following conditions: ( I ) - & 6.6 G f i 6 * * * (2) u, 3 CI u, 3 UI 2 CI u, 3 u, 3 . . . , I (3) = 1, (4)f. = 1 - 2-" on U,, - { x , } , ( 5 ) f,,is continuous at each point x of X - { x , } , (6) f n + l = f, on X - un, (7) Iffis in R X such thatf = f, on (X - U,,) u { x o } ,thenfe F,. We choose these sequences by the induction for n E w. For n = 0, it is sufficient to put fo(xo) = 1 and fo(x) = 0 if x # xo, and put Uo = X. 9
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Suppose that for each i with 0 < i < n,f; and U, are well defined. Let g be a continuous function from X into [0, 1/27 such that g(x,) = 1/2"and g = 0 on X - U,,, and letf,+'(x,) = 1 andf,,'(x) = f,(x) min{ 1/2"+', g(x)} if x # x,. Then the set V = g-'((1/2"+', 1/2"-']) is a neighborhood of x, such that f,+l(x) = 1 - 1/2" + 1/2"+' = 1 - 1/2"+' for each x E V - {x,}. Since x, is a nonisolated point,f,+ I is not continuous at x, and, therefore,f,+ I does not belong to F,+ I . Since F,+ is closed in R X ,there exists a finite subset A of X such that, for each f E R X ,iff = f,,, on A, then f does not belong to F,,,,. Hence, put U,+, = ( V - A ) u {x,}. Then f,+' and U,,, are required ones. Continuing this process, { f,I n E o}and { U,,I n E o}will be found out. Put f = lim,,.+af,.Then, for each n E o,f = f, holds on (X - U,) u { x,} and by (7) f does not belong to U{F,,ln E a} = C,(X); that is, f is not continuous. On the other hand, f will be continuous. Because, for any point x E X , if x 4 n{U,l n E o},then x 4 C1 U,, for some n by (2). By (6) and (9,f is continuous at x. If x E U,,I n E o},then each U,, is a neighborhood of x such that f ( U , ) c (1 - 2-', l } by (3) and (4) and hence, we can see that f is continuous at x by (l), (9,(6) and (3). As a consequence,f is continuous, which is a contradiction. 0
+
,
n{
The following cardinal numbers will be related to C,(X). Especially, we can see that cardinal numbers for C p ( X )will be discussed concerning those of product spaces X" and X"'. 4.4. Definition. For a space X let c ( X ) be the cardinal number which is the minimum of T'S such that I 4 I < T for any disjoint collection 4 of nonempty open subsets of X. Then c ( X ) is called the cellularity or Souslin number of X .
4.5. Definition. For a space X let t ( X ) be the cardinal number which is the minimum of T'S such that any open covering of Xcontains a subcover whose cardinality does not exceed T. Then t ( X ) is called Lindelof number of X . 4.6. Lemma (Arhangel'skii [1985]). For a space X , c(C,(X)) <
KO holds.
Proof. It is not difficult to see c ( R X ) < KO.Since C,(X) is dense in R X by Theorem 1.9 (a), c(C,(X)) < KOalso holds. 0 4.7. Theorem (Arhangel'skii [1985]). For a space X , C p ( X )is paracompact ifand only if C p ( X )is a Lindelof space.
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Proof. We only have to prove necessity. Let y be any locally finite open covering of C , ( X ) and p the maximal disjoint family of open subsets of C,(X) such that for each H E p the collection { G E y I H n G # S} is a finite set. Then, by Lemma 4.6, I p I < No holds and as a consequence, I y I < No must be held. I7 4.8. Theorem (Arhangel'skii 119851).
only i f / ( X " ) < z for each n
E
For a space X , t(C,(X))
< z fi
and
N.
Proof. (Necessity, Pytkeev): Let n E N be an arbitrary fixed one and y an open covering of X". A finite family p of open subsets of Xis said to be y-small if, for any HI, . . . , H, E p, the product set HI x . . . x H,, is contained in some G of y. Let 6 be the set of all y-small families and for each p E 6 put A, = {fEC,(X)If = O o n X - U { H I H ~ p ) } , a n d p u t A= ( J { A , l p ~ 6 } . Then we show CI,A = C,(X). For this purpose, take an arbitraryfe C,(X), and a finite set K = {xI, . . . , x,} (n < rn) in X and E > 0. Let 0, be a family of open subsets of X such that for any (rl, . . . , r,,) E K" there exist V, , for i = I , . . . , n and . . . , V , in OK and G in y which satisfy ri E V , x . . . x V,c G . T h e n w e h a v e K c U { V ( V E ~ , K ) . F O r e a c h x E K p u t W, = n { V E O , I x E V } , and put A, = { W , I X E K } . Then we have K c W I W E A,} and we can easily see that A, is y-small. Let g be an elementofC,(X)suchthatg = f o n K a n d g = OonX - U ( W l W E & , } . Then g belongs to A , and therefore, to A and also, to W ( f , K , E ) ; that is, A n W ( f , K , E ) # 0.This shows f E C1,A. By the above fact, we have 1 E CI,A. Apply the assumption for 1 E CI,A. Then there is a subset B of A such that 1 B ( < z and 1 E C1,B hold. By the definition of A , there is a subset Soof 6 such that I goI < z and B is contained in the union U { A , I p E gob). For each p E goand each = (V,, . . . , V,) E p" pick a G , in y such as V, x . . . x V, c G,, andputy, = { G , 1 5 ~ p " and7 } = U { y , I p ~ 6 ~ bThen ). j j is a subcollection of y whose cardinality does not exceed z. Now, it suffices to show that j j covers X". For any point ( x l , . . . , x,,) of X " , since U = { f C~, ( X ) 1 f(xi) > 0 for i = 1, . . . , n } is an open neighborhood of 1 in C , ( X ) , there exists a po E So such as U n A , # 8. Pick g in U n A,. Then g ( x i ) > 0 for i = 1, . . . , n andg = OonX - U { H I H E po} and henceeachxibelongs to U{HI H E p o } . Choose Hi in po with xi E Hi for i = 1, . . . , n. Then the point (XI, . . . , x,) belongs to HI x . . . x H,, which is contained in some member of y,, which is also a member of j j . This shows that j covers X " .
u{
r
430
A . Okuyama, T . Terada
(Suficiency, Arhangel'skii): Let F be an arbitrary subset of C,(X) and f E C,(X) with f E C1,F. For each n E Nand point x = (x,, , . . ,x,,) of X", there is a g , E F such that I f ( x i ) - g,(xi)) < l / n holds for i = 1, . . . , n. Let, for each i, Ox, be an open neighborhood of xi in X such that I f ( y) - g,( y ) I < l/n for each y E Ox,and U, = Ox, x * . . x 0,"Then . the family 49" = { U, 1 x E x"} is an open covering of x".By the assumption, 42, contains a subcovering 4'2: of 49" whose cardinality does not exceed z. Put En = {g, I U, E 49:} for n E Nand E = U{E, I n E N}. Then E is a subset of F whose cardinality does not exceed z. It suffices to show f E C1,E. Let W ( f , { y,, . . . ,y,}, E ) be an arbitrary neighborhood off in C,(X). Here, we can assume that n is large enough so that l/n < E . Since y = ( y,, . . . , y,) is a point of x"and: % ! covers X " , there is a U, E 49: such as y E U,. By the definition of U, we have If ( yi) - g,( yi) I < l/n for i = 1, . . . , n and hence, I f ( y , ) - g,(y,)( < E for i = 1, . . . , n. This shows that g, belongs to the intersection W ( f , { y,, . . . , yn>,E ) n E, which proves f E C1,E.
o
4.9. Corollary. Zf X is a a-compact space; i.e. union of countably many compact subsets of X , then we have t(C,(X)) < KO. Proof. If Xis a a-compact space, then it is easily seen that, for each n E N , x" is a Lindelof space. Hence, by Theorem 4.7, t(C,(X)) < KOis clearly held.
0 4.10. Theorem (Asanov [1979]). For a space X , the inequality t ( X " ) t ( C , ( X ) ) holds for each n E N .
<
Proof. Let t ( C , ( X ) ) < z and n E N. We show t(X") < z. Let A be a subset of X" and x = (x,, . . . , x,) a point of X" with x E C1 A, and let U = U , x . . . x U,, be an open deighborhood of x in X" such that, if x, = x,, then U, = U, and, if x, # x,, then U, n V, = 8. We can assume x E CI(U n A) and therefore, that A is a subset of U. Put F = { f E C,(X)If(x,) = 1 for i = 1, . . . , n}. Then F is a closed subset of C,(X). As t ( C , ( X ) ) < z, we have t ( F ) < z. For any pointy = (y,, . . . , y,) of A, we put V, = {g E C,(X) lg( y,) > 0 for i = 1, . . . ,n}. The collection { V, I y E A} forms an open covering for F. By assumption, there exists a subset B of A such that I B I < z and U{V, I y E B} contains F. Hence, it suffices to show x E C1 B. On the contrary, suppose x 4 C1 B. Choose an open neighborhood U' = U [ x * * . x U,,' of x in X" such that V,' c U, for i = 1, . . . ,n and U' n B = 0,and pick an element
Function Spaces
43 I
f , i n P w i t h f , = O o n X - U{V,'li = 1 , . . . , n}. BythechoiceofB, there isayEBwith&EVy.Thenwehavef,(yj) > O f o r j = 1, . . . , nandhence, y j E U{V,'Ii = 1, . . . , n } f o r j = 1 , . . . , n. As y E B c A c U, we have y j E V, f o r j = 1, . . . , n and therefore, we have y j E V,' f o r j = 1, . . . , n. This means that y is in U' and B, as well. This is contradictory to U' n B = 8. Hence, we have x E C1 B, which completes the proof. 0
As for the following results, we restrict our attention to the case of a countable cardinal. The original results are obtained from more general cases. 4.11. Lemma (Zenor [1980]). Let X and Y be spaces a n d Z a second countable
space, and let f be a mapping from X x Y into Z such that (a) f ( x , ) is continuous on Y for each x E X and (b) X has the weakest topology for whichf ( , y ) is continuous on X for each y E Y. Then we have the following: (1) I f the countable product Y" of Y is a hereditarily Lindelof space, then X is a hereditarily separable space. (2) I f the countable product Y" of Y is a hereditarily separable space, then X is a hereditarily Lindelof space. Proof. For any x E X andJ E Y" withj = ( y J j E Oput , g(x, J ) = cf(x, yi))jEw. Then g is a mapping from X x Y" into Z" such that, by (a), g ( x , ) is continuous on Y for each x E .'A Let W be a countable base for 2". For each U E B and J E Y", put M ( J , U ) = { x E X l g ( x , J ) E U } . Then by (b) the collection { M ( j , U ) l J E Y", U E W } is a base for X . Using these notations we prove (1) and (2). (1): Let A be an arbitrary subspace of X . For any point x of A and any U E 9,put G(x, V ) = { J E Y " J g ( x ,J ) E U } . Then G ( x , U ) is an open subset of Y". By the assumption, there exists a countable subset A, of A such thatU{G(x, U ) ~ X E A=} U{G(x, U)Ix~A,}.Then,using(b),wecansee that the countable set U{A,I U E W}is dense in A; i.e. A is separable. (2): Since {M(J , U ) I j E Y", U E B } forms a base for X , it is sufficient to show that any subcollection d contains a countable subcollection d' of d such as ~ { G I G E=~ ~}{ G I G d'}. E For each U E W , put A, = { j E Y"I M ( J , U ) E d}.Then, by the assumption, there is a countable subset E, of A, such as Cl,E, 2 A,. Put d' = { M ( J , U ) l J E E,, U E B } . Then d'is a countable subcollection of d. Furthermore, if x is a point of U{G I G E d } ,then x belongs to some G = M ( j , V )in d.By the definition
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of M ( j , U ) we have g ( x , j ) E U . Since g ( x , ) is continuous on Y" and E, is dense in A,, there exists a j oE E, such that g ( x , lo) belongs to U . This shows x E M ( lo, U ) with j oE E and U E W and hence, x is in some member of d ' . In other words, it shows U { G ( GE d }= U{Gl G E a'},which means that X is a hereditarily Lindelof space. 0 4.12. Lemma. For a space X , the space Cp(X,R") is homeomorphic to the product space (Cp(X))". Proof. For each n E N , let n, be the projection from R" to R mapping the point of R" to its nth coordinate, and let II/ be a mapping from Cp(X,R") to (Cp(X))o defined by $(f ) = (II,f for each f E C,(X, R"). Then II/ is the required homeomorphism from C,(X, R") onto (Cp(X))". 0 0
4.13. Theorem (Zenor [1980]). For a space X , the following equivalences hold: ( I ) Cp(X,R") is a hereditarily Lindelof space i f and only if X" is a hereditarily separable space. (2) Cp(X,R") is a hereditarily separable space if and only if X" is a hereditarily Lindelof space. Proof. (Necessity): We use Lemma 4.1 1 for the triple ( X " , C p ( X ) ,R") instead of ( X , Y , Z ) and for the mapping II/ from X" x C p ( X )to R" defined by $(R, f ) = (f ( x n ) ) , e Nfor each R = ( x , ) , , ~E X" and f E C p ( X ) .Then they satisfy the conditions required in Lemma 4.1 1. Applying Lemma 4.12, we have the "necessity" of ( I ) and (2) by Lemma 4.1 1 (1) and (2), respectively. (Suficiency):Apply Lemma 4.11 to the triple (Cp(X,R"),X , R") and the mapping $ from C,(X, R") x X to R" defined by $(S, x ) = f ( x ) for each f~ C,(X, R") and x E X. Then we have "sufficiency" of (1) and (2) by Lemmas 4.1 1 and 4.12. As further relationship between C p ( X )and X we use the following concepts. 4.14. Definition. For a space X we define the cardinal numbers for X as follows: (1) d ( X ) is the minimum in the set {TI A is a dense subset of X with I A I = T} and d ( X ) is called the density of X. (2) $ ( X ) is the minimum in the set {z I any point of x is expressed as the intersection of z many open subsets of X} and $ ( X ) is called the pseudo-character of X.
433
Function Spaces
(3) nw(X) is the minimum in the set { IB I 1 B is a network for X } and nw(X) is called the net-weight of X . (4) iw(X) is the minimum in the set { w ( Y )I X is condensed onto Y } (see Definition 3.6) and iw(X) is called the i-weight of X . 4.15. Theorem (Arhangel'skii [1976]). wc, (XN.
For a space X we have nw(X) =
Proof. We first prove I~w(C,(X))d nw(X). Suppose nw(X) = t. Let 9 be 9 1 = t and ?d a countable base for R . Then the a network for X with 1 collection { U ( P , ,. . . ,P,,; . . . , V,)]P I , . . . ,P,,E 9, 6,. . . , V, E g,n E N } forms a network for C , ( X ) whose cardinality does not exceed t, where U ( P , , . . . , P,,;V,, . . . , V , ) = {YE C(X)If(<.) c Kfor i = 1, . . . , n}. This shows nw(C,(X)) < z. Next, we prove nw(X) d nw(C,(X)). By Theorem 1 .lo, Xis embedded in C,(C,(X)) and hence, we have nw(X) d nw(C,(C,(X))). By the inequality proved above we have nw(C,(C,(X))) < nw(C,(X)). As a consequence we have nw(X) d nw(C,(X)). This completes the proof.
v,
4.16. Theorem (Guthrie [1974]). For a space X iW(C,(X)) = W p ( W ) .
we have d ( X ) =
Proof. Since, for any space 2, we can easily see $ ( Z ) < iw(Z), in general, it suffices to show iw(C,(X)) < d ( X ) < +(C,(X)). To prove the first inequality, let d ( X ) = z and Y a dense subset of X with I YI = t. Then we have w(C,(Y)) < w ( R Y )by Theorem 1.9(a). Let cp be a mapping from C , ( X ) into C,( Y ) defined by c p ( f ) = f l ,,for eachfE C,(X). Then cp is a one-to-one, continuous mapping, since Y is dense in X . Put 2 = cp(C,(X)). Then we have w ( Z ) < t and cp is a condensation of C , ( X ) onto 2. This shows iw(C,(X)) < t. To prove the second inequality, let +(C,(X)) = z and y a family of basic open subsets of C,(X) with I y I < T and (0) = WI W E y}. For each W = W(0, {x,, . . . , x,,}, E ) E y , weputK(W) = ( 4 ,. . . , x,,}, and we set Y = U{K(W)I W E y } . Then we have I YI d t. Now, we show that Y is dense in X , which will complete the proof. Suppose there is an x E X - C1 Y. Let g be an element of C , ( X ) such that g ( x ) = 1 and g = 0 on C1 Y . Then g belongs to any W E y and hence, g must be 0. On the other hand, since 1 holds, g is not 0, which is a contradiction. g(x)
n{
4.17. Theorem (Noble [19741). For a space X we have iw(X) = d(C,(X)).
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A . Okuyama, T . Terada
Proof. By Theorems 1.10 and 4.16, we have iw(X) < iw(C,(C,(X))) = d(C’(X)). Now, we show another direction; that is, d(C,(X)) < iw(X). For this purpose, let iw(X) = z and cp a condensation of X onto a space Y with w ( Y) < z. Since nw( Y) < w( Y) holds, in general, by Theorem 4.15 we have nw(C,( Y)) = nw( Y) < w ( Y) < t. Since rp is a condensation, the induced mapping q* is a homeomorphism from C,(Y) into C,(X) and cp*(C,( Y)) is dense in C’(X). Hence, we have d(C,(X)) < d(cp*(C,( Y))) < nw(cp*(C,( Y))) = nw(C,(Y)) < t, which shows d(C,(X)) < t. This completes the proof of d(C,(X)) < iw(X). 0 4.18. Definition. A space X is said to be monolithic if nw(C1 A ) d I A 1 holds for each subspace A of X . 4.19. Theorem (Arhangel’skii [1976]). Z f X is a compact space, then C , ( X ) is monolithic.
Proof. Let F be any subspace of C,(X) and cp a mapping from X into the product space R F defined by rp(x)( f ) = f ( x ) for each x E X and f E F and put Y = cp(X). Then we have w( Y) < 1 F I * Noand the induced mapping cp* is a homeomorphism from C,(Y) into C , ( X ) . Since X is compact, cp is a perfect mapping and hence, it is a quotient mapping, cp*(C,( Y)) is a closed subset of C,(X). Since cp* is a homeomorphism, we have nw(C,(Y)) = nw(cp*(C,(Y))) and also, by Theorem 4.15, we have nw(rp*(C,(Y))) = nw(Y) < IFI.N,,. Now, it suffices to show that F is embeddable into cp*(C,(Y)), since cp*(C,( Y)) is closed in C,(X).For eachfe F, let n,%e the projection from RF tof th coordinate. Then anyfE F is just n p rp; that is,f = rp*(n, 1 y ) for each YE F, which showsf€ cp*(C,( Y)). Hence we have nw(C1,F) < nw(cp*(C,( Y)) < I F I * KO. The following concept seems to be valuable and interesting when we analyze the structure of C , ( X ) . 4.20. Definition. A space is said to be functionally complete if there exists a compact subset of C , ( X ) separating points of X . A compact space which is functionally complete is called an Eberlein compact space. 4.21. Lemma. Let X be a functionally complete space. Then any subspace Y is also functionally complete.
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Proof. Let cp be an inclusion of Y into X and cp* the induced mapping from C,(X) into C,(Y). Since cp* is continuous, cp*(F) is compact for any compact subset F of C,(X). Therefore, if X is functionally complete, then Y is also functionally complete. 0
4.22. Theorem (Arhangel’skii [ 19761). Let X , be a functionally complete space for n E N . Then the product X = II{ X , I n E N } is also functionally complete. Proof. Since R is homeomorphic to an open interval (- I/n, l/n), we identify C,(X,) to C,(X,, ( - l / n , l/n)). By the assumption, there exists a compact subset F, of C,(X,, ( - l / n , l/n)) separating points of X,. Let n, be the projection from X to X, and put F = (0) u U{a,*(F,) I n E N } , where a,*is the induced mapping by n, . Then we can see that F is the required compact 0 subset of C,(X). 4.23. Theorem (Arhangel’skii [ 19761). If X is a metrizable space, then X is functionally complete. Proof. Since Xis metrizable, Xis embedded into the product n { Z , I n E N} of countably many hedgehogs {&In E N} (cf. Nagata [1985, p. 2711). Each hedgehog is functionally complete. Because, if a hedgehog Z is 0 {IaI a E A } / {0}, then the subset F = {0} u { f , l a E A } of C,(Z) is the required one, wheref, denotes the function such thatf, is linear on Z, withf,(O) = 0 and L(l=)= 1 and such thatf, = 0 on any I, with fl # a. Hence, by Theorem 4.22, the product n { Z , I n E N } is functionally complete and by Lemma 4.21 X is functionally complete. 0
4.24. Theorem (Arhangel’skii [1976]). A space X is functionally complete if and only fi there exist a compact space Y and a one-to-one continuous mapping $from X into C,(Y). Proof. (Necessity):Let F b e a compact subset of C,(X)separating points of Xand cp the mapping from Xinto C,(F) defined by cp(x)(f ) = f ( x ) for each x E X and f E F. Then cp is a one-to-one, continuous mapping. Hence, put Y = F and = cp. This completes the proof. (Suficiency): Let cp be the embedding of Y into C,(C,(Y)) by Theorem 1.10, and put F = $* ocp(Y), where t+b* is the induced mapping by $. Then F is clearly a compact subset of C , ( X ) . It suffices to show that F separates points of X. For any two distinct points x , x’ of X , by the assumption, $ ( x )
+
A . Okuyama, T. Terada
436
and $(x’) are distinct points of C,( Y) and hence, *(x)( y ) # $(x’)(y ) for some y E Y. We show that ** o cp( y ) separates x , x’. Since $* 0 cp( y)(x) = cp( y)($(x)) = $(x)( Y ) and $* O cp( Y ) ( X ’ ) = $(x’)(Y ) , as well, ** cp( Y ) ( X ) and $* o cp( y)(x‘) are distinct, which completes the proof. 0 O
4.25. Corollary (Arhangel’skii [1976]). Z f X is a compact space, then C,(X)
is functionally complete. 4.26. Theorem (Arhangel’skii and Tkachuk [ 19851). For a compact space X
the following are equivalent: (1) X is functionally complete; i.e., X is an Eberlein compact. (2) There exists a compact space Z such that C,(Z) contains a compact set Y which separates points of Z and is homeomorphic to X . (3) There exists a compact space Z such that C , ( Z ) contains a cornpact set Y which is homeomorphic to X .
Proof. (1)+(2): By assumption, there is a compact subset F of C,(X) separating points of X . Define cp as a mapping from X into C,(F) by cp(x)(f ) = f ( x )for each x E X andf E F. Then cp is a homeomorphism, since Xis compact. Hence, it suffices to put Z = F and Y = cp(X). (2)+(3) is clear. (3)+(1): Let cp be the mapping from Z into C,( Y) defined by p(z)( y ) = y(z) for each z E Z and y E Y. Then cp is continuous and hence, q ( Z ) is a compact subset of C,(Y). By the assumption, there is a homeomorphism $ from X onto Y. The induced mapping $* is also a homeomorphism from C,(Y) onto C,(X). Put F = $*(cp(Z)).Then F satisfies all conditions for X to be functionally complete. 0 For further details and proceeding interests in the last three sections, we would like to recommend the Lecture Note of McCoy and Ntantu [1988]. 5. Topological properties and linear topological properties
We have learned that some topological properties of a space X are characterized by suitable topological properties of C J X ) or C,(X). We present here the attempt to translate purely topological properties of a space X into linear topological properties of C , ( X ) and C , ( X ) . First, let us recall the uniformity of a topological linear space. For a topological linear space E, let Y be the collection of all neighborhoods of zero of E. For each V E Y ,define a covering @ ( V ) of E by @(V) = {V
+ PIPEE)
Function Spaces
where V
+p
= {x
431
+ p I x E V}. Then we can easily show that
{WV) I V E V } is a uniformity on E compatible with the topology of E. This uniformity is called the uniformit-vof the topological linear space E. 5.1. Definition. Let E be a topological linear space. If every Cauchy net under the uniformity of E converges, then E is said to be complete. Likewise, if every Cauchy sequence converges, then we say that E is sequentially complete.
Completeness and sequentially completeness of a topological linear space are not topological properties, but they are linear topological properties.
5.2. Theorem (Buchwalter and Schmets [1973]). A space X is discrete ifand only i f C p ( X )is complete. Proof. Since the product of complete uniform spaces is always complete, if X is discrete, then C,(X)= R X is obviously complete. Conversely, let C p ( X )be complete. To show that X is discrete, it suffices to see that, for each point x of X, the characteristic function xX of { x} is continuous. For each finite subset F of X , there is an f F E C p ( X )such that fFlF
=
XxlF.
As the collection of all finite subsets of X with the order of inclusion, the set { f F } is considered to be a net. Further, we can easily show that I f F }is a Cauchy net and converges to xx. Then, since C p ( X )is complete, xx must be a member of C ( X ) . 0 A topological space Xis called a P-space iff-'(0) is open for a n y f e C ( X ) (see Gillman and Jerison [1960]).
5.3. Theorem (Buchwalter and Schmets [1973]). A space X is a P-space tf and only i f C p ( X )is sequentiully complete. Proof. Let X be a P-space and {fn) a Cauchy sequence in C p ( X ) Since . RX is complete, the Cauchy sequence has a limitfin R X .Hence it suffices to show thatfbelongs to C ( X ) .For this, fix x, E X and E > 0. Since X i s a P-space, the set V = { x E XI Ih(x) - f,(xo)l < + E for all n )
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A . Okuyama, T. Terada
is a neighborhood of xo. On the other hand, for each x n such that ifnix) - f(x)l G
I f ( 4 - f(xo>l G
E
X , there exists some
If,(xo) - f ( x d I G
$8,
are satisfied. Hence, for each x
E
5~
V,
E
holds. This shows that f is continuous. Conversely, let C,(X) be sequentially complete, and let f be an arbitrary member of C ( X ) .We will show thatf-'(O) is open. Iff-'(0) = 8,then it is obvious. We can assume that 0 G f < 1. For each n, there exists an f , E C ( X ) such that f,-'(O) = f-'(O), fn-'(l) = f-'([l/n, I]). It is obvious where x , - ! ( ~ )is that (fn} is a Cauchy sequence and converges to 1 the characteristic function off-'(O). It follows that 1 - x,-,(~)is continuous. Hence f --'(O) is open. 0 5.4. Definition. A real-valued functionfof a space Xis called k-continuous iff 1 is continuous for each compact subset K of X . A space X is called a k,-space if every k-continuous function on X is continuous.
5.5. Theorem (Warner [1958]). A space Xis a k,-space ifand only ifck(X) is complete. Proof. Let X be a k,-space, and {f,}a Cauchy net in ck(X). Then, for each point x E X , (f,(x)} is a Cauchy net in the real line R . Hence we can define the function f so thatf(x) is the limit of { f , ( x ) } for each x E X . Since, for every compact subset K of X , the net (f,I K } is a Cauchy net in the complete space C , ( K ) and { f ,I }. converges tofl K, the functionfmust be k-continuous. It follows that f is continuous. This shows that { A }converges in c k ( X ) . Conversely, let C, ( X ) be complete and f a k-continuous function on X . Then, for each compact subset K of X , there is an fK E C ( X ) such that j i I = f l K. We can consider the collection {fK} to be a Cauchy net which converges to f . It follows that f must be in C , ( X ) . 0 Let us recall some basic concepts of the theory of topological linear spaces. Let A be a subset of a linear space L. Then A is called convex if t A + (1 - t ) A c A for each t E [0, I]. If a A c A holds for each a such that I a 1 G 1, then A is called circled. If, for every x E L, there exists ex > 0 such that [0, ex]x c A , then A is said to be absorbent. Now, let A be a subset of a topological linear space E. Then A is called a barrel if it is absorbent,
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circled, convex and closed. A topological linear space is called locally convex if it has a neighborhood basis of zero consisting of convex subsets. These concepts are found in every standard textbook of topological linear spaces. Note that C J X ) and C , ( X ) are locally convex spaces. 5.6. Definition. Let E be a locally convex space. If every barrel is a neighborhood of zero in E, then E is called barrelled. The space E is called KO-barrelled(Husain [1966]) if every barrel which can be represented as the intersection of a sequence of closed, circled and convex neighborhoods of zero is itself a neighborhood of zero in E. From Baire category theorem, it follows obviously that every Banach space is barrelled. Hence, for each space X , the function space C,*(X) is barrelled.
5.7. Definition. Let X be a space. For a subset H of C ( X ) ,let K ( H ) be the set of all x E X which satisfies the following condition: (*)
for every neighborhood U of x there exists a n f e C ( X ) such that fix-,, = 0 a n d f # H.
A subset B of a space Xis called bounded if the restrictionfl for every f E C ( X ) .
is bounded
5.8. Proposition (Asanov and Shamgunov [1983]). I f H is a barrel in C, ( X ) , then K ( H ) is bounded.
Proof. Suppose that K ( H ) is not bounded. Then there exists an infinite discrete collection 4% of open subsets of Xsuch that U n K ( H ) # 8 for every U E 4%. Define inductively sequences of sets { U,,} in 42, of functions { A }in C ( X ) ,of compact sets { K , } in X , and of positive numbers { E , } in R such that (1)
(2) (3)
U,,, n U{Kili= 1, . . . , n } =
fn 4 H
8,
and f , ( x ) = 0 for every x
W A , K,,, 8 , )
nH =
E
X - U,,,
8.
In fact, let U , be an arbitrary member of %,A a function which satisfies the condition (*) of Definition 5.7 for U , . The compact set K , and the number E , are easily selected since H is closed. Then there is U2E 4% such that U2 n K , = 8, and so on.
440
A . Okuyama, T. Terada
Define c, = 1 and take c,+, inductively so that lCfl+l(.fdX) + (l/Cdfz(X) + . *
*
+ (1/cfl)AI(x))l <
&,+I
+
for every x E K,+] and 0 < c,+, < l/(n 1). It is possible since the function C { ( l / c i ) f ; I1 < i < n} is bounded on K , + , . Let g = Z{(l/ci)f;lie N}. Then the continuity of g follows from the choice of {A}. For each x E K,+, cn+,g(x) = cn+IZ((l/ci)f;(x)I 1 i G n} sincefk(x) = 0 for every k > n + 1. Hence
+ Sn+l(x),
lc,+,g(x) -Sn+,(x)I < & , + I bythedefinitionofc,+,.Thismeansthatc,+,gE W ( f . + , K,,+,, , and, by condition (3), cn+,g 4 H. So, c,+ ,g 4 H for all cn+I and c,+ I -, 0, thus H does 0 not absorb the function g, which contradicts the condition of H. 5.9. Theorem (Nachbin [ 19541, Shirota [1954]). For a space X , thefollowing are equivalent. ( I ) Every bounded closed subset of X is compact. ( 2 ) C k ( X )is barrelled.
Proof. (1)+(2): Let H be a barrel in C,(X).We will prove that there exists some Q > 0 such that W(0, K ( H ) , e) c H. Since K ( H ) is a bounded closed subset, K ( H ) is compact. It follows that H is a neighborhood of zero in C , ( X ) , which implies that C , ( X ) is barrelled. First of all, we shall show the following: Iff E C(X)andf(x) = 0 for every x E V , where V is some open neighborhood of K ( H ) , thenfE H. Suppose the contrary. Then we can find a compact subset K of X and a number E > 0 such that W(S, K, E ) n H = 8. Let F = K - V . For every x E F, there exists a neighborhood U, of x which satisfies the following condition: (*)
For every function g E C ( X ) , if g(X - 17,= ) {0}, then g
E
H.
The family { U, 1 x E F } covers F and hence, has a finite subcover { Ur,I i = 1, . . . , n } . Let {gili = 1, . . . , n} be a partition of unity subordinate to this finite subcover. Denoteg = Z { g , f l i = 1, . . . , n}. Theng E H sinceg,f E H by (*) for each i and the set H is convex. For each x E F, Idx) - fWl =
I (g,(x)f(x) +
= If(x)(g,(x) = 0.
+
. + s,(x)./’(x)) - f’(x)I . . . + g n w - 111 * ‘
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If x E K - F, thenf(x) = 0 and g(x) = 0. Consequently,f(x) = g(x) for each x E K and hence, g E W ( f ,K , E ) and g $ H, which is a contradiction. Recall that C,*(X) is barrelled. Since H n C * ( X ) is a barrel in C,*(X), there is r~ > 0 such thatf E H wheneverf E C * ( X )and I f(x) I < r~ for every x E X.Let e = $0.Then, i f f € W(0, K ( H ) , e), then there is a neighborhood V of the set K ( H ) such that If(x) I < e for every x E V. Let g(x) = max{f(x),
e } + min(f(x), - e } .
Then 2g(x) = 0 for every x E V. Hence 2g E H by the condition (*). Moreover, I2(f(x) - g(x)) I < 0 for each x E X. It follows that 2(f - g) E H. Sincef = 1/2 * (2g) + 1/2 (2(f - g)), f~ H. This completes the proof of (I )+). (2)+(1): Let A be a bounded closed subset of X. Then H = W(0, A, I) is obviously a barrel in C , ( X ) . From assumption (2), it follows that there are a compact subset K of X and E > 0 such that W(0, K , E ) c H. This implies that A c K. Hence A must be compact.
-
The above proof is due to Asanov and Shamgunov [1983]. 5.10. Theorem (Buchwalter and Schmets [1973]). For a space X , the following are equivalent. (1) Every bounded closed subset of X is finite. (2) C , ( X ) is barrelled. Proof. (1)+(2): Let H be a barrel in C , ( X ) . Then H is also a barrel in C , ( X ) . It follows that K ( H ) is bounded in X from Proposition 5.8. Hence, by the condition (l), the set K ( H ) must be finite. As the proof of Theorem 5.9, it is proved that W(0, K ( H ) , e ) c H for some e > 0. The proof of (2)+(1) is much the same as that of Theorem 5.9.
5.11. Definition. Let 9 be a locally convex topological property; that is, if E and Fare linearly homeomorphic locally convex spaces and E has 8,then F has also 8.Further, we assume that 9is stable under inductive limits and satisfied by any linear space endowed with its finest locally convex topology. For a linear space E with a locally convex topology F,the .P-spaceassociated to E is the linear space E with the coarsest locally convex topology Y which both satisfies 9 and is finer than Y, 5.12. Definition. For any space X , let X’ be the subspace of the Hewitt realcompactification UXof X defined by X’
=
!JiCIUxBIB
13
bounded in Y).
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A . Okuyama, T. Terada
Using this notation, we define the space Xufor each ordinal a by transfinite induction: if a = 0,
Xa
=
r‘
(Xu-1)’ U{X,l fi <
if a has the predecessor, a}
if a is a limit ordinal.
Then there exists the smallest ordinal A such that X , = X,+ this space Xi as p X .
,. We will write
The space p X is the minimum extension of X in O X such that every bounded closed subset of it is compact. Let us recall that C ( X ) = C ( u X ) as algebraic linear spaces. Since X c p X c uX, we can identify the space C ( p X ) with C(X). 5.13. Proposition (see Schmets [1976]).
The barrelled space associated to C,(X) coincides with C , ( p X ) f o r any space X .
Proof. As a result of Theorem 5.9, the space C&X) is barrelled. It is also obvious tha.t the topology of C , ( p X ) is finer than that of C,(X). Hence, it suffices to show that for every compact subset K of p X and each E > 0, W(0, K, E ) is a neighborhood of zero in the barrelled space associated to
C,(X). Claim 1 . The barrelled space associated to C,(X) is finer than C , ( p X ) . For this, it suffices to show that the barrelled space associated to C,(X) is finer than C,(Xa) for any a. This can be proved by transfinite induction. In fact, assume that the barrelled space associated to C , ( X ) is finer than C,(Xu). Then, for each bounded closed subset B of Xa,the set HE = { f e
C(X,)(If(x)l < E for every x
E
B}
is a barrel in C,(Xa) and hence in any locally convex space with a topology finer than that of C,(X,).It follows that HB is a neighborhood of zero in the barrelled space associated to C,(X). Since HB equals to the set HcI,,xB =
{ f C(Xa+l)I ~ If(x)I < E for every x E C1u,B},
every basic neighborhood of zero in C,(X,+ I ) must be a neighborhood of zero in the barrelled space associated to C,(X). Claim 2. For any compact subset K of p X and any E > 0, W(0, K,E ) is a neighborhood of zero in the barrelled space associated to C,(X).
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Since it is obvious that
HK
=
{ f E C ( p X ) I If(x)l
< E for every x E K }
is a barrel in C , ( p X ) , HK must be a barrel and hence a neighborhood of zero in the barrelled space associated to C,(X). 0 5.14. Corollary. A space X is bounded associated to C,(X) is normable.
if and only if the barrelled space
Proof. Let X be a bounded space. Then p X is the same as the Stone-Cech compactification PX. It follows that the barrelled space associated to Cp(X) is a Banach space. Conversely, assume that Xis not bounded. Then p X can not be compact. Hence, for any neighborhood V of zero in the barrelled space associated to C,(X), n { ( l / n )YI n E N } contains some nonzero function. This shows that the barrelled space associated to C p ( X )is not normable. 0 Since the topology of C,(X) is finer than that of C p ( X )and coarser than that of C , ( p X ) , the following are obvious.
5.15. Proposition. The barrelled space associated to C , ( X ) coincides with C , ( p X ) for any space X . 5.16. Corollary. A space X is bounded associated to C,(X)is normable.
if and only if the barrelled space
Next, we will study the KO-barrelledspace associated to C,*(X). For this, we introduce a linear topology on C * ( X ) studied in Definition 1.3 (d). Let &INobe the following collection of compact subsets of uX:
BN0= {Cl,,A I A is a countable subset of X}. Then the topology Tabon C * ( X ) is a linear topology on C * ( X ) . The topological linear space C * ( X ) with this topology is denoted by C& (X).
5.17. Proposition (see Schmets [ 19761). The K,-barrelZed space associated to C:(X) coincides with C&&X) for any space X . Proof. At first, we show that C&&X) is an KO-barrelledspace. Let H be a barrel in C&$X) and H = n{Fy,ln E N } , where W, is a closed, circled, convex neighborhood of zero for each n. Since H i s a barrel in the barrelled
A . Okuyama, T . Terada
444
space C:(X), there is a positive number a =- 0 such that f~ H whenever f c C * ( X ) and If(x) I < a for every x E X . On the other hand, for each n, there exists a neighborhood W(0,K,,, E,,) of zero such that W(0, K,,, E,,) c W,,, where K,, E guo. Let E = $a. Then (1)
W(0, Kn, 8 ) c Wn-
In fact, letfbe an arbitrary member of W(0, K,,, E ) . Then, since I 2f(x) I < a for each x E K,,, there is a g E C * ( X )such that g I Kn = 2fl Kn and Jg(x)I < a for any x E X. It follows that g E H c W,,. It is also obvious that 2f - g E W(0, K,,, E ) c W , . Hence f = ) g $(2f - g ) E W , since W , is convex. Now, let K = Cl,,U{K,, I n E N } . Then K is a member of au0. Further, it is obvious that W(0, K, E ) c W(0, K,,, E ) I n E N } c H. This shows that H is a neighborhood of zero in C&(X). Next, let A be an arbitrary countable subset of X . Then W(0,Cl,,A, E ) is a barrel in Cp*(X)which is represented as the intersection as follows:
+
n{
W(0, Cl,,A,
E)
=
n{W(0,F,
E)
I F is a finite subset of A}.
Hence, by the assumption, W(0, Cl,,A, E ) must be a neighborhood of zero in the KO-barrelledspace associated to C'(X), It follows that Csb(X) is the KO-barrelledspace associated to C,*(X).
5.18. Corollary. For a space X , the space C,*(X) is K,-barrelled f a n d only i f X is jinite. Similarly, we can prove the following. 5.19. Proposition (See Schmets [1976]). For a space X , let
W
= {Cl,,A
IA
is a a-compact subset of X I .
Then the topological linear space C,*(X) defined in Definition 1.3 (d) is the K,-barrelled space associated to C,*(X). 5.20. Corollary. For a space X , the space C,*(X) is K,-barrelled if and only if closure of a-compact subset of X is compact The essential of the following lemma is well known and found in Schmets [1976].
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5.21. Lemma. Let X be a space and g a collection of compact subsets of O X which defines a linear topology on C ( X ) as Definition 1.3 (d). Let u be a continuous linear functional of C,(X); that is, u : C , ( X ) + R. Then there exists unique compact subset of O X , denoted by supp(u), which satisfies the following conditions: (1) there is a member A of a such that supp(u) c A , (2) for any f E C ( X ) with f = 0 on supp(u), u ( f ) = 0 holds, ( 3 ) for any open subset U in O X such that U n supp(u) # 8, there exists some f E C ( X ) such that f = 0 on O X - U and u ( f ) = 1. In the next section, we will discuss about continuous linear functionals of C,(X).
5.22. Definition. Let F,,F2 be linear topologies on a linear space E such that F, > F2. Let u be a continuous linear functional on (E, Y,).Then u is called a special outer functional with respect to (E, Y2)if the following conditions are satisfied: (a) u is not continuous on (E, F2), (b) for each sequence { f , }of members in E, there exists a continuous linear functional u, on (E, Y2)such that u ( f , ) = u o ( f , ) for each n E N. 5.23. Proposition. For a space X , let @ be a collection of compact subsets in O X which defines a linear topology on C ( X ) . Further assume that X c U ( BI B E g}.Then a continuous linearfunctional u on C , ( X ) is a special outer functional with respect to C p ( X )ifand only ifthe compact set supp(u) is afinite subset of O X such that supp(u) n ( O X - X ) # 8. Proof. Let u be a special outer functional with respect to C p ( X ) .Assume that supp(u) is an infinite subset of O X . Then there is a sequence { Un} of disjoint open subsets in OXsuch that U, n supp(u)
z0
for each n. By Lemma 5.21, for each n, there exists a memberf, E C ( X )such that f,l ,,x-u, = 0 and u( f , ) = 1. Now, let uo be an arbitrary continuous linear functional on C p ( X ) .Then supp(u,) is a finite subset of X . Hence the set 8) { ~ I ~ u P P (n u ~un )
+
is finite. It follows that u o ( f , ) = 0 for a sufficiently large n. But, this is a contradiction since u satisfies u ( f , ) = 1 for all n. It is proved that supp(u)
A . Okuyama, T. Terada
446
must be a finite set. Further, since u is not a continuous function on C,(X), it is obvious that supp(u) n (OX- X ) # 0 (cf. Proposition 6.4). Conversely, let u be a continuous linear functional on C,(X) with a finite supp(u) such that supp(u) n (OX- X ) # 0. Let SUPP(U)=
{XI,
* * *
9
xk, Y I ,
* * *
9
yj},
where SUPP(U)n (OX- X ) = { y i ,
...
9
yj).
Then, by 6.4, u can be expressed as u = a16,,
+
*
*
+ akaxlr+ BlS,, + . . + BjS, *
where 6, is a linear functional defined by 6,( f ) = f ( z ) for each f~ C ( X ) . Now, let {fn} be an arbitrary sequence in C(X).Then, for each i = 1, . . . ,j, Zi = n { f n - l ( f . < y i ) ) I n= 1, 2, . . .} is a zero-set of OX (see Gillman and Jerison [1960]). Hence
zinX#O for each i = 1, . . . ,j . Let zi be a point in Zi n X for each i. Then
+ . .+
uo = alljx,
a
aka,
+ BIS,, + . + Bj", * *
is a continuous linear functional on C p ( X ) .Further, for each n,
+.
+ a k f , ( x k ) + B l f n ( z ~ )+ . . + aIf,(xl> + . . + akfn(xk) + + .. +
= alfn(xl> =
*
*
.
*
B l f n ( ~ I )
*
Bjfn(zj) Bjfn(Yj)
= u(fn).
It follows that u is a special outer functional with respect to C p ( X ) .
0
A space X is called KO-boundedif the closure of every countable subset of X is compact. 5.24. Theorem. For a space X , the following are equivalent. (1) X is KO-bounded. (2) The barrelled space associated to C p ( X )is normable and the KO-barrelled space associated to C , ( X ) has no special outerfunctional with respect to C,(X).
Proof. (1)+(2): Since every KO-boundedspace is obviously bounded, the first part of (2) is obvious by Corollary 5.14. Further, since every member of aN0 is a subset of X,for each continuous linear functional u on C a J X ) , the inclusion SUPP(4 = X
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441
holds by Proposition 5.17 and Lemma 5.21. Hence the &-barrelled space C,,(X) associated to C p ( X )has no special outer functional with respect to Cp(X). (2)+( 1): From the first part of (2) it follows that X is bounded. If X is not KO-bounded,then there is a point y in mX - X , where
mX
=
U{Cl,,A 1 A is a countable subset of X}.
Then the function 8, defined by S,( f ) = f(y ) is a continuous linear functional on C,,(X). From Proposition 5.23, it follows that 8, is a special outer functional with respect to C p ( X ) . 0 It is trivial that a space X is compact if and only if C,(X) is normable. Compactness is also characterized by a linear topological property on C,(X).
5.25. Theorem. For a space X , the following are equivalent. (1) X is compact. (2) The barrelled space associated to C p ( X )is normable and has no special outer functional with respect to C,(X). Proof. (1)+(2): If X is compact, then X = p X = fix hold. Hence the barrelled space associated to C , ( X ) is obviously normable and has no special outer functional with respect to C p ( X ) . (2)+(1): From the first part of (2), it follows that X is bounded. Hence fix = p X . On the other hand, by the second part of (2) it must be satisfied that p X = X. From these equalities it follows that X is compact. 0
6. Topological properties and !-equivalence 6.1. Definition (Arhangel’skii [1981]). Let X and Y be spaces. If the function spaces C p ( X )and Cp(Y) are linearly homeomorphic, then it is called that X and Y are !-equivalent.
If topological spaces X and Y are homeomorphic, then X and Y are obviously !-equivalent. But, the converse of this implication is not valid. 6.2. Example. Let X be the topological sum of two copies of the unit interval. We assume that
A . Okuyama. T. Terada
448
On the other hand, let Y be the topological sum of the closed interval [O, 21 and a one-point space { p}. Obviously X and Y are not homeomorphic. It is shown that X and Y are t-equivalent. In fact, define Q,: C ( X ) + C ( Y ) and Y : C ( Y ) + C ( X ) as follows: i f 0 < y < 1, f ( ( Y , 0)) Q,(f)(Y) =
1
f ( ( Y - 1, 1)) - ( f ( ( 0 , 1)) - f((L 0))) if 1 < Y G 2,
f((0, 1)) -
f((l9
ify = p ,
0))
where f E C ( X ) and y E Y. 'Y(g)((x,t ) ) =
+:{
if (x, t ) E [O, 11 x { O } , 1)
+ g(p)
if(x, t ) E
[o, 11 x
(11,
where g E C( Y) and (x, t) E X. Then the following are easy exercises: (1) Yo@ = id,,,,, Q,oY = id,,,,, (2) @ and Y are linear maps, (3) Q, and Y are continuous as maps between C , ( X ) and C,( Y). It follows that C,(X) and C,( Y) are linearly homeomorphic. This example shows that the classification of the class of topological spaces by [-equivalence is strictly coarser than that by topological equivalence. Hence, there arises the problem, for each topological property 8,whether 9 is invariant under [-equivalence. If a topological property 8 of a space X is characterized by some topological or linear topological property of C, (X), then 8 is invariant under [-equivalence. Hence, by results in the previous sections, we already know some topological properties which are invariant under /-equivalence. In this section we will study some other topological properties which are invariant under t-equivalence. At last, some examples of topological properties which are not invariant under [-equivalence will be given.
6.3. Definition. For a space X , let L ( X ) be the linear space of all continuous linear functionals on C , ( X ) . When we give the topology of pointwise convergence on L ( X ) ,it is denoted by L,(X). Obviously, L,(X) is considered as a subspace of C,(C,(X)). In Theorem 1.10, we showed that any space Xis embedded in C,(C,(X)); the embedding is the map cp : X + C,(C,(X))defined by cp(x)(f) = f(x) for each x E Xand f E C,(X). Since it is easily seen that cp(x) E L ( X ) for each x E X , we have the following.
449
Function Spaces
6.4. Proposition. The map cp :X
+
L,(X) is an embedding.
Let us recall the following fundamental lemma (see Kelly and Namioka [19631). 6.5. Lemma. Let u, uI, . . . , uk be linear functionals on a linear space L. If n{u;I(o)li = I ,
u-l(o),
. . . , k}
then u is a linear combination of uI, . . . , uk. 6.6. Proposition.
The image cp(X) is a Hamel basis of L(X).
Proof. Let u E L(X). Then there is a neighborhood U = W(0, {xi, . . . , x k } ,E ) of 0 in C,(X)such that u ( U ) c (- 1, I), since u is continuous. It is not so difficultto see that, if cp(xl)(f ) = . * = (p(xk)(f ) = 0 (i.e.f ( x l ) = . . * = ' f ( x k ) = O), then u( f ) = 0. It follows that u is a linear combination of cp(xi), . . . , cp(xk) from Proposition 6.4. On the other hand, it is obvious that {cp(x)1 x E X } is linearly independent. 0 From now on, the subset cp(X) of L J X ) will be identified with X for each space X. But, cp(x)is sometimes written as 6,. 6.7. Theorem. Let L,(X)* be the topological linear space of all continuous
linear functionals on L p ( X )with the topology of pointwise convergence. Then L,(X)* is linearly homeomorphic with C p ( X ) .
Proof. Define @: C p ( X )+ L,(X)* in the following way: @( f ) ( u ) = u( f ) for f
E
C,(X) and u E L p ( X ) .
Then it is obvious that @ is linear and one-to-one. Further, as the argument in the proof of Proposition 6.6, it is shown that @ is onto by using Lemma 6.5. Hence, we only show that @ is homeomorphism. Let W(0, { x } ,E ) be a subbasic neighborhood of zero in C,(X). Then @(W(O,{ X I ,
4)
= { @ ( f ) I IfW = { @ ( f ) II
I<
4
cp(x)(f1I <
4
(cp is that in Proposition 6.4.) =
{@(f)II@(f)(cp(x))I <4
= {P
E
L,(X)* I I A x ) I <
El.
A . Okuyama, T. Terada
450
It follows that @(W(O,{ x } ,E ) ) is a neighborhood of zero in L,(X)*. Conversely, let W(0, { u } , &) be a subbasic neighborhood of zero in L,(X)*, where u E L , ( X ) . Since u can be expressed as u = alxl
+
*
+ akxk
where al # 0, . . . , a/, # 0,
we can take a finite subset { x I ,. . . , x k } of X . Here, note that Xis identified with the subset of L,(X). Let U be the following neighborhood of zero in C,(X): U = n{W(O, {x,},&/(kla,l))li= I , . . . , k}. Then it is obvious that @ ( U ) c W(0, { u } , 6). This completes the proof that (9 is homeomorphism. This proposition also shows the following propositions. 6.8. Proposition. For any space X , every continuous function f : X be extended to a continuous linear functional @( f): L,(X) + R. 4.9. Proposition. Spaces X and Y are d-equivalent Lp(Y ) are linearly homeomorphic.
-+
R can
if and only if L,(X) and
By Proposition 6.6, every point u E L p ( X )can be uniquely expressed in the form u = alxl . . . + a,x,,
+
where x I , . . . , x, are distinct points of X , a l , . . . , a,, are nonzero real numbers. The number n is called the length of u with respect to the basis X, and we write d,(u) = n. We put
6.10. Proposition (Pavlovskii [1980]). (a) Let u = u , x , L',(u) = n. Then the sets of the form AIUI
+
* . *
+ . . . + a,,x, and
+ A,U,
form a neighborhood basis of u in L,(X), where each A, is a neighborhood of ai in R which does not contain 0, and U,, . . . , U, are pairwise disjoint neighborhoods of x l r . . . , x,, respectively. (b) L , ( X ) is closed in L,(X). Proof. (a): It suffices to show that the sets defined in (a) are neighborhoods of u in L , ( X ) . By Theorem 6.7, L,(X)* and C,(X)are identified. For each
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45 1
i = 1, . . . , n, there isf;' E C ( X ) such thatf;'(x,) = (af)-' andf;' = 0 on X - V,. There exist neighborhoods q' and q2of points 1 and (af)-' in R, respectively, such that (q2)-'c A,. Define a functionJ2: X + R such thatJ2(x,) = 1 andf;' = 0 outside (f;')-'(v*).Then
v'
u E n{(f;;')-'( 5 ' )n ( A 2 ) - ' ( R- (0)) 1 i = 1, . . . , n; n L,(X) c A,UI
+ . . + AnUn.
This completes the proof of (a). (b): By (a), every point of length n has a neighborhood in L p ( X )consisting of points of length at least n. It follows that L , - , ( X ) is closed in L p ( X ) .
n 6.11. Corollary. For each space X , X is a closed subset of L p ( X ) .
6.12. Theorem (Arhangel'skii [1985], Terada [1985]). Let 9'be a topological property which satisfies the following conditions: (1) 9' isfinitely productive. (2) 9 is preserved under the continuous images. (3) 9' is closed under the increasing closed sums; that is, if Fl c F7 c . ' is an increasing sequence of closed subsets of a space X such that each f has then the sum U{&l i E N } has the property 9. the property 9, (4) The closed unit interval [0, 11 has the property 9'. ( 5 ) S is closed-hereditary. Then a space X has the property 9 if and only if L J X ) has the property 9. Proof. From (3) and (4), it follows that the real line R has the property 9'. By (I), R" x X" has 9for each positive integer n. Then the continuous image L , ( X ) of R" x X" has 9 for each n. Since { L , ( X ) l n = 1, 2, . . .! is an increasing closed covering of L p ( X ) ,L p ( X )has 9' by (3). Conversely, since Xis closed in L p ( X ) ,if L p ( X )has 9, then X has 9 by (5). By this theorem, the following is obvious. 6.13. Theorem. Let 9'be a topologicalproperty which satisfies the conditions (1)-(5) in Theorem 6.12. Then 9 is invariant under L-equivalence.
For example, a-compactness is invariant under L-equivalence. 6.14. Definition (Pestov [1982]). If a space Xis embedded in the topological linear space E satisfying:
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A . Okuyama, T . Terada
X is a Hamel basis of E (and hence, L,(X)has also meaning in E), (2) the condition (a) of Proposition 6.10, then X is called a weak topological basis in E. (1)
6.15. Definition (Pestov [ 19821). Spaces X and Y are called weakly L‘-equiv-
dent if they are embedded in some topological linear space E as weak topological bases. 6.16. Definition.
Let 9be a topological property which is closed-hereditary and invariant under weakly [-equivalence. Then, for an infinite cardinal m, a space Xis called m-8-closed if, for any subset A of X with I A I < rn, CI,A has the property 9. 6.17. Theorem. m-9-closedness is invariant under [-equivalence.
Proof. Let Xand Y be /-equivalent and Xis m-@-closed. It suffices to show that Y is also m-%closed. We can assume that L p ( X ) = L p ( Y ) .Let B be a subset of Y such that I BI < m. Each point y E Y can be expressed uniquely by y = qxl
+ . + akxk, ‘ ’
a,, . . . , q # O ,
xi#xj ifi#j.
Let A,, be the set of all xiwhich appears in such an expression of some y E B. Since each point x E X can be expressed as a linear combination of members in Y , we can define the subset B, of Y as the set of all yi which appears in such an expression of some x E A,, . Continuing this process, we obtain sequences {A,} and { B,,} with the following properties: (1) B,, = B c B, c B, c - * - c Y, ( 2 ) A. C A , t A , c . . . c X , (3) Each point y E B,, is expressed as a linear combination of members in An. (4) Each point x E A, is expressed as a linear combination of members in
Bn+,-
( 5 ) IBI = ] A o ]= IB,] = J A , I = . ..holds. Let A’ = U{A,,ln E N } and B‘ = U{B,In E N}. Then it is obvious that I A’I = I B’I = I BI. The linear subspace L ( A ’ ) generated by A’ and the
linear subspace L(B’) generated by B‘ are clearly the same in L p ( X ) = Lp(Y). Further, it is not so difficult to see that the linear subspace L(C1,A’) generated by C1,A’ coincides with the linear subspace L(C1,B’) generated by
453
Function Spaces
C1,B'. It follows that CI,A' and CI,B' are weakly /-equivalent. Since CI,A' has the property 9, CI,B' must have the property 9. Hence CI,B has the property 9. 0 The following exciting result is obtained finally by Pestov [1982]. 6.18. Theorem. The covering dimension is invarianr under f-equivalence.
Since the perfect proof of this theorem requires more detailed discussion concerning linear topologies on linear spaces, we only explain the outline of the proof. Outline of proof. At first, Claim 1. Let X and Y be second countable spaces and weakly /-equivalent. Then dim X = dim Y holds. Suppose that X and Y are embedded in the topological linear space L as weak topological bases. For n E Nand 6 = ( m l , . . . , m,) E N" let L6 be the set of u E L such that / , ( u ) = n and u = a l x l + . . + a,x,, where a, E R - { 0 } ,x, E X, and /,(x,) = m, for i = 1, . . . , n. Then obviously, (1)
L = U ( L , l n EN,6 E N " } u (0).
Since nw(L) < KO,each L , ( X ) - L , - , ( X ) = { u E L l / , ( u ) set in L. Hence the following is also obvious. (2)
=
n} is an F,-
Each L6 is an F,-set in L.
Let Y6 = Y n La for each 6 E N" = u { N " I n E N } . Then (3)
for each y E Y, there exists an open neighborhood G of y that is expressed as a finite union of closed subsets of it which are homeomorphic to subspaces of X .
+
+
In fact, let y = a l x l . . z,x,, a, E R - { O } , x, E X and x, # x, for i # J . Then there is a disjoint family of open neighborhoods UI, . . . , U, of xI, . . . , x, in X respectively. Let U = R*UI . . . R*U,, where R* = R - (0). Then U IS a neighborhood of y in L , ( X ) - L n P l ( X )Let . p , : U + U, be the map defined by
+
PI(&
+
*
.
3
+ a:x:)
+
= x,'
for each i = 1, . . . ,n. Then p, is continuous. For 1 < i < n, x, is expressed and YI, # yik for j # k . as X I = flil yi, + ' ' ' + flim,yim,, fl, E R*, YI, Hence, there is a disjoint family of open neighborhoods KI, . . . , of
A . Okuyama, T. Teraab
454
+
+
= R*&, * * R*Cmj.Then we y i l , . . . , yi, in Y respectively. Let can define the continuous mapping qij: 4 + Cj for each j = 1, . . . ,mi in a way analogous to the definition o f p i .For 1 < i < n let %be an open subset of X such that c Q and n L,(Y) = n Ui.Let G = Y, n (R* W, . R* Wn).Then G is an open neighborhood of y in Y,. Let y’ be an arbitrary point of G. Then y’ = a;x; . * * @Ax;, x,! E U( c A’, and xl = /3i; yi; . * * flk,y;mi,yQ E Y for each i. It follows that y’ = yQ for some i, j. Hence, the set A , of all fixed points of the mapping qi,opil contains y’. It is obvious that
+ -+
+
G = g{Aijl1
+
+
+
< i < n, 1 < j < mi>.
Each A , is closed in G, and p i : A , + X is an embedding. This completes the proof of (3). Now, we can prove Claim 1. From (l), (2) and (3) it follows that Y is a countable union of closed subsets of it which are homeomorphic to subsets of X. Hence if dim X < k, then dim Y < k by the countable sum theorem in dimension theory. In general, for a space X let Fx be the family of all continuous mappings from X onto second countable spaces. Fx becomes a directed set by the relation < :f < g if there exists a continuous mapping CI$ :g ( X ) + f ( X ) such that f = u+ og. The following is an easy consequence of Pasynkov [1965]. (4)
For a space X,dim X < k if and only if the subset 9;o f f € Fx such that dimf(X) < k is cofinal in 4tx.
Claim 2. Let X and Y be /-equivalent and dim X < k. Then for anyf E .Fy there exist g E Fyand h E Fxsuch thatf < g, dim h ( X ) < k and g ( X ) and h ( X ) are weakly {-equivalent. Assuming this claim is true, if spaces X and Y are /-equivalent and dim X < k, it follows that the subset 9; of g E Fysuch that dim g ( X ) < k is cofinal in Fyby Claim 1 and hence, dim Y < k by (4). 0
At last, we study those topological properties which are not invariant under /-equivalence. For this purpose, we need some lemmas. For a nonTychonoff space X , let a X be the complete regularization of X (see Herrlich [1968]). Then the following is obvious. 6.19. Lemma. The topological linear spaces C,(X) and C,(aX) are linearly homeomorphic.
Function Spaces
455
Suppose Y be a subspace of a space X. We define the subspace Cp(X;Y ) of C,,(X) as follows: C,(X Y ) =
Cp(X)Ifl Y = 0).
6.20. Lemma (see Pavlovskii [1980]). Let Y be a subspace o f a space X . If there exists a continuous linear extension operator u : Cp(Y ) -+ C p ( X ) ,then C P ( X )N C P ( Y )x Cp(X;Y ) . Proof. Define 0:C p ( X )+ C p ( Y )x C,(X; Y ) as follows:
@(f)
=
(fl n f - 4fl Y ) )
f o r f c CP(X).
On the other hand, define Y : Cp(Y ) x C p ( X ;Y ) -+ C p ( X )as follows: W g , h)) = u ( g )
+h
for (g, h) E C p ( X )x Cp(X;Y ) .
Then obviously Y 00= idcp,,,, 00" = idCp(Y)x c p ( x : v ) . It is also obvious that @ and Y are linear homeomorphisms. 0 For a closed subset Y of a Tychonoff space X , let X / Y be the quotient topological space of X obtained by collapsing Y to one point *. Then the following lemma is trivial. 6.21. Lemma.
Cp(X;Y )
N
C,(X/Y;*).
6.22. Example. Metrizability, locally compactness and first countability are not invariant under /-equivalence. In fact, let R be the real line and 2 the subset of all integers in R. Then obviously R is locally compact and metrizable and R / Y is neither locally compact nor first countable. But, it will be shown that R and R/Z 0 Z are t-equivalent. We can assume that Z = { p } 0 Z where { p } is an isolated point outside 2. Cp(R)N- C,,(Z) x Cp(R;Z )
= C P ( Z )x 'v
N-
C p ( Z )x
C,(R/Z; *)
CP(P)x C p ( R / Z ;*)
C p ( Z )x C p ( R / Z )N Cp(Z0 R / Z ) .
6.23. Example. For each infinite cardinal K , there exist t-equivalent topological spaces X and Y which satisfy x ( X ) = Noand $ ( Y ) = IC.
A . Okuyama, T. Terada
456
+
Let V ( K + 1) be the space of all ordinals less than K I whose cofinality is less than w , with the usual topology. Let A ( K )be the one-point compactification of the discrete space of cardinality K . Let X = V ( K 1) and Y = A ( K ) 8 F(K l), where F(K + 1) is the subspace of V(K I ) consisting of all limit ordinals. Then
+ +
+
C,(X)
+
N
C,(V(ri
31
R x CP(V(x
1))
1:
R x C,(V(K + 1))
1:
+ 1); F(K + 1)) x ~,(F(K+ 1)) R x C,(V(K + ~ ) / F ( + K 1); *) x C,(F(K + 1)) C,(V(K -k l)/F(K + 1)) C,(F(K -k 1)) CP(ACK)8 F(K + I)) C P ( A ( K ) )x C,(F(K + 1))
1:
CJY).
N
1:
X
6.24. Example. Normality is not invariant under /-equivalence. For each ordinal a, let W(a) be the topological space of all ordinals less than a with the usual interval topology. Let us consider the product W ( w , )x W(w, + 1). Then it is well known that this space is not normal. On the other hand, it is also trivial that the complete regularization of the quotient space ( W ( 0 , )x W ( 0 ,
+ l ) ) / ( W w , x) f W I > )
is normal. Hence it follows that a ( ( W ( w , )x w o ,+ 1 ) ) / ( W w , )x fw,))) 8 W w , )
is normal. We will show that this space is t-equivalent to W ( o , )x W(w, + 1). Note that there is a continuous linear extension operator u : C , < W ( w , )x
{w))
-+
C , ( W ( w , )x W(w, + 1)).
Hence C,(W(w,)x W(O,
+ I))
N
R x C,(W(w,) x W(O,
+ 1);
N
R x C,(W(wi) x W(OI
1:
R x C,(a((W(w,)x w w , + 1 M W w l ) x
= c,
C,(a((W(w,) x
+ W(wl + Ww,
This completes the calculation.
W(wi) x
( ~ 1 ) )
+ I))
x C,(W(wi))
b,))*)); x C,(W(w,))
I ) ) / ( W ( w , x) { w , ) ) ) ) x C,<W<w,))
I ) ) / ( W w l )x
1)) 0 W W ) )
Function Spaces
457
References Arhangel’skii, A. V. [I9761 On some topological spaces that occur in functional analysis, Uspkhi Mat. Nauk 31(5) 17-32; English translation in: Russian Math. Surveys 31(5) 14-30. [ 198I] Classes of topological groups, Russian Math. Surveys 36, 15 1-1 74. [ 19851 Function spaces with pointwise convergence topology, Part 1; General Topology, Function Spaces and Dimension, 3-66. Arhangel’skii. A. V. and V. V. Tkachuk [ 19851 Function spaces and topological invariants, Publication of the Moscow University. Asanov, M. 0. [I9791 On cardinal invariants of the space of continuous functions (in Russian), Modern Topology and Set Theory, No. 2 (Idzevsk, Moscow). Asanov, M. 0. and N. K. Shamgunov [I9831 The topological proof of the Nachbin-Shirota theorem, Comment. Math. Univ. Carolinae 24,693-699. Buchwalter, H. and J. Schmets 119731 Sur quelques proprietes de l’espace C , ( T ) ,J. Math. Pures Appl. 52. 337-352 Corson, H. H. and J. Lindenstrauss [I9661 On function spaces which are Lindelof spaces. Trans. Amer. Math. Soc. 121, 476-491. Dijkstra, J., T. Grilliot, D. Lutzer and J. van Mill [I9851 Function spaces of low Bore1 complexity, Proc. Amer. Math. SOC.94, 703-710. Engelking, R. (1977) General Topo/ogy (PWN-Polish Scientific Publ., Warsaw) Gillman, L. and M. Jerison [ 19601 Rings of Continuous Functions (Princeton University Press, Princeton, NJ).
Guthrie, J. A. [I9741 Ascoli theorems and the pseudo-character of mapping spaces, Bull. Austral. Math. Soc. 10, 403-408. Herrlich, H. [ I9681 Topologische Retlexionen und Coreflexionen, Lecture Notes Math. 78 (Springer, Berlin). Husain, T. [I9661 Two new classes of locally convex spaces, Math. Ann. 166, 289-299. Jameson. G. J. 0 . [I9741 Topology and Normed Spaces, Chapman and Hall, Math. Ser. (Chapman and Hall, London). Kelley, J. L. [I9551 General Topology (New York). Kelly, J. L. and I. Namioka [I9631 Linear Topological Spaces (Princeton University Press, Princeton, NJ). McCoy, R. A. [I9781 Submetrizable spaces and almost u-compact function spaces. Proc. Amer. Math. Soc. 71. 138-142.
458
A . Okuyama. T. Teraah
[1980a] Necessary conditions for function spaces which are Lindelof spaces, Trans. Amer. Math. SOC.121, 476491. [1980b] Function spaces which are k-spaces, Topology Proc. 5, 139-146.
McCoy, R. A. and I. Ntantu [I9861 Completeness properties of function spaces, Topology Appl. 22, 191-206. [I9881 Topological properties of spaces of continuous functions, Lecture Notes Math. 1315 (Springer, Berlin). Michael, E. [I9661 K,-spaces, J . Math. Mech. 15, 983-1002. Nagami, K. and Y. Kodama [I9741 General Topology (in Japanese) (Iwanami-shoten, Tokyo). Nagata, J. (19851 Modern General Topology (North-Holland, Amsterdam, 2nd rev. ed.). Nachbin, L. (19541 Topological vector spaces of continuous functions, Proc. Nor. Acad. Sci. USA 40, 471-474. Noble, N. [I9741 The density character of the function spaces, Proc. Amer. Math. SOC.42, 228-233. Pasynkov, B. [I9651 On the spectral decomposition of topological spaces, Marh. Sb. 66, 35-79. Pavlovskii, D. S. [I9801 On spaces of continuous functions, Soviet Math. Dokl. 22, 34-37. Pestov, V. G. [I9821 The coincidence of the dimensions dim of /-equivalent topological spaces, Soviet Math. Dokl. 26, 380-383. Pol, R. [I9741 Normality in function spaces, Fund. Math. 134, 145-155. Schmets, J. [I9761 Espaces de Fonctions Continues, Lecture Notes Math. 519 (Springer, Berlin). Shirota, T. [I9541 On locally convex vector spaces of continuous functions, Proc. Japan Acad. 30,294-298. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. Amer. Math. SOC.54, 977-982. Terada, T. [ 19851 Linear homeomorphisms of function spaces and topological properties (in Japanese), General Topology Symposium Lecture Notes, 85-95. Warner, S. [ 19581 The topology of compact convergence on continuous function spaces, Duke Math. J. 25, 265-282. Zenor, P. [ 19801 Hereditary m-separability and the hereditary m-Lindelof property in product spaces and function spaces, Fund. Math. 151, 175-180.
K. Morita, J. Nagata, Eds., Topics in General Topology Q Elsevier Science Publishers B.V. (1989)
CHAPTER 12
N-COMPACTNESS AND ITS APPLICATIONS
Katsuya EDA Institute of Mathematics, University of Tsukuba, Tsukuba-Shi, Ibaraki, 305 Japan
Takemitsu KIYOSAWA and Haruto OHTA Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422 Japan
Contents I. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . . . N-compactness and N-compactifications . . . . . . . . . . . . . . . . . . . N-compactness vs. realcompactness . . . . . . . . . . . . . . . . . . . . . N-compactness of k , X . . . . . . . . . . . . . . . . . . . . . . . . . . . Rings and lattices of continuous functions . . . . . . . . . . . . . . . . . . Applications to abelian groups . . . . . . . . . . . . . . . . . . . . . . . Applications to non-ArchimedeanBanach spaces . . . . . . . . . . . . . . . Problems.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
459 460 461 467 474 479 483 497 516 5 18
Introduction This chapter is designed as a continuation of Chapter IV of Nagata's text, Modern General Topology [1985], which will henceforth be referred to by ma]. There we have learned realcompact spaces as well as compact spaces. A realcompact space is a topological space which can be embedded in the product of copies of the real-line R as a closed subspace. Likewise, a topological space is called N-compact if it is homeomorphic to a closed subspace of the product of copies of the countable discrete space N. In the first part, formed by Sections 2, 3 and 4, we give an introduction to N-compact spaces. In Section 2, as a unified viewpoint for realcompact spaces and N-compact spaces, we shall introduce the notion of E-compact spaces due to Mr6wka and investigate N-compact spaces under this perspective. By the definition,
460
K . Eda, T. Kiyosawa. H . Ohra
N-compact spaces can be regarded as a O-dimensional analogue of realcompact spaces. Strictly speaking however, the analogy is not complete; in fact, Nyikos proved that a O-dimensional realcompact space need not be N-compact. In Section 3, we consider the relationship between realcompact spaces and N-compact spaces. In Section 4, we give technical results and examples which are needed in the sequel. The second part, formed by Sections 5, 6 and 7, is devoted to applications. As was stated in “a], the algebraic structure of the ring C(X, R) of real-valued, continuous functions on a realcompact space X determines the topology of X , more precisely, if both X and Yare realcompact and C(X, R) and C( Y, R) are isomorphic as rings, then X and Y are homeomorphic. In the same way, we prove in Section 5 that the ring structure or the lattice structure of C(X, Z) determines the topology of an N-compact space X , where C ( X , Z) is the ring or the lattice of integer-valued, continuous functions on X. As various instances including this one show, many things go parallel to realcompact spaces. However, there are some aspects of, N-compact spaces which realcompact spaces do not have. In Section 6, we shall see such aspects in applications to abelian groups, where N-compactness will play an important role to reduce the reflexivity of abelian groups to the topological properties investigated in Section 4. Surprisingly, Banach spaces over certain non-Archimedean-valued fields have many features on the reflexivity that are similar to those of abelian groups. In Section 7, we apply N-compact spaces to such Banach spaces, with particular attention paid to their similarity. The final part, Section 8, lists open problems. At the end of each section, we note from which works we have obtained the results and examples. Since some of them are folklore, the authors in question were not always the first who proved or exhibited them. This chapter is written as self-contained as possible without repeating the proofs which can be found in “a].
1. Conventions and notation The symbols Z, Q and R denote respectively the set of integers, rational numbers and real numbers; in each case the usual algebraic properties are assumed. Let N and I denote the set of natural numbers and the closed unit interval, respectively, and let D = (0, l } . When viewed as topological spaces all of these have the usual topologies inherited from R. For topological spaces X i , I E A, the Cartesian product Hi,, X , possesses the product topology, unless stated otherwise. In case each Xi is identical to X , llj,s,,Xj,is written as X”. For I E A, the symbol el is reserved for the I t h
N-compactness and irs Applications
46 I
projection; however, given x E n,,,X,,we usually write xj, or x(A) for e;(x). For a subspace Y of nlehXlr the set of the form ( n , , , e ~ ’ ( U , ) )n Y, where M is a finite subset of A and each U , is an open set of X , , is called a basic open set in Y. The collection of all basic open sets in Y forms a base of Y. For a topological space X , CO(X) is the collection of all clopen ( = open and closed) sets in X , and w(X)is the minimal cardinality of a base of X. Except for Section 7, C ( X , Y) denotes the set of all continuous maps from X to a topological space Y. For a mapf: X + Y and S c X , f l denotes the restriction off to S. The identity is denoted by id. For a set A, I A I denotes the cardinality of A . As usual, an ordinal is equal to the set of smaller ordinals and, as a topological space, will have the order topology. Cardinal numbers are initial ordinals and w (resp. w , ) is used to denote the first infinite (resp. first uncountable) ordinal. Unless otherwise stated, topological spaces require no separation axiom; however, regular, completely regular, and compact spaces are assumed to be Hausdorff. All undefined terms can be found in “a].
2. N-compactness and N-compactifications We begin this section with a brief account of E-compact spaces which will enable us to introduce N-compactness simultaneously with compactness or realcompactness. Let E be an arbitrary fixed Hausdorff space. A space homeomorphic to a subspace (resp. closed subspace) of a product of copies of E is called an E-regular (resp. E-compact) space. For example, it follows from Tychonoff’s theorem (“a, Theorem 111. I , p. 801) that both I-regularity and R-regularity are complete regularities and I-compactness is just compactness. Moreover, by the definition, R-compactness coincides with realcompactness. Clearly, E-compact spaces are E-regular, and the class of E-regular (resp. E-compact) spaces is closed under taking arbitrary products and subspaces (resp. closed subspaces). Suppose we are given a topological space X, a collection { Y,},,, of topological spaces and a collection {fi.)j,Eh of maps, where : X + Y,. Then there exists a unique map f’ from X to nLEA Y , such thatfl, = @;.offoreach i E A. The mapfis called the diagonal and A is denoted by A;,s,&,. By the following lemma, if each f;. mup of { h } i E is continuous, then so is A;.,, f i , . 2.1. Lemma (“a, Section III.l.D, p. 821). Let f be a mup from u topologicul space X to a product space nLEA Y,. Then, f is continuous i f and only if e,. f is continuous.for each A E A. 0
462
K. Eda. T. Kiyosawa, H . Ohta
For a topological space X , let a, denote the diagonal map of C(X,E). Let Then a, is a continuous map from X to E", where C(X,E ) = {h}lE,,. a,X = a E ( X ) ,and define BEXto be the closure of a,Xin E". The spaces a,X and BEX are E-regular and E-compact, respectively. 2.2. Theorem. I f f is a continuous map from a topological space X to an E-regular space Y, then there exists a unique continuous map a, f :a E X + Y such that f = aEf 0 a,. Moreover, if Y is E-compact, then there exists a unique continuous map BEf :BEX+ Y such that f = BEfoaE.
Proof. Let C(X, E ) = {h}iEA. Since Y is E-regular, Y can be regarded as a subspace of EMfor some index set M . For each p E M,there is a A(p) E A such that h(,,)= e l o f . where e l is the pth projection of EM. Define F = ApEMel(,,), where el(plis the A(p)th projection of E". Then Fis a continuous map from E" to EM,and f = F o a,, because el o FOa , = el(,,)o a, = &,,, = el0f for each p E M . The following commutative diagram describes the situation:
Define a E f to be the restriction of F to a,X. Then f = a E f o a E and the uniqueness of a, f is clear. Moreover, assume that Y is E-compact, i.e., Y is closed in EM.Then, F(BEX) c Cl,,F(a,X) c Cl,, f ( X ) c Y, and hence the restriction BEf of F to BEX gives the required factorization off through BEX. The uniqueness of BEf follows from the fact that a,X is dense in BEX and Y is Hausdorff. U
2.3. Remark. In later sections, we consider various algebraic structures on C(X, E). For example, if E is a topological ring, then C(X, E ) turns out to be a ring under pointwise addition and multiplication. Theorem 2.2 shows that C(X, E ) is then isomorphic to C(a,X, E) and C(BEX,E ) . Hence, in those cases, we can assume that X is E-regular or even E-compact without loss of generality.
If X is an E-regular space, then a , is an embedding, i.e., a,X = X.Moreover, if X is E-compact, then BEX = X . 2.4. Proposition.
N-compactness and its Applications
463
Proof. Let C(X, E ) = Consider X to be a subspace of EMfor some index set M. Then, we may assume that for any distinct p, p' E M, there exists x E X such that xp # 5 . . Moreover, since e,,I E C(X, E ) for each p E M, we can assume that M c A and f , = e, I for each p E M. Define f = A M E M & and g = AiEA-ML. Then f is an embedding of X into EMand aE = fAg. Identifying x E X with f(x) E E M ,we have aEx = {(X, g(X)):X E X}
C
EMX EA-M= EA,
i.e., aEXis the graph of g. Since aE carries x to (x, g(x)), it is an embedding. Next, assume that X is E-compact, i.e., X is closed in EM.Then X x EAPM is closed in EA. Since EAPM is Hausdorff, the graph a,X is closed in X x E"-M.Therefore, a,X is closed in E", and hence flEX = a,X = X .
0 An E-compact space including X as a dense subspace is called an E-compactijkation of X. If X is an E-regular space, then by the above, flEX is an E-compactification of X to which every continuous map from X to an E-compact space admits a continuous extension. In this case, we call flEXthe maximal E-compactijication of X . 2.5. Theorem. Any E-compactiJication of an E-regular space X to which every f E C ( X , E ) admits a continuous extension is homeomorphic to PEX under a homeomorphism which leaves points of X fixed.
Proof. Let K be an E-compactification of X satisfying the stated extension property. Then the embedding aE can be extended to a continuous map g : K + BEX by the same way as in the proof of Theorem 2.2. On the other hand, the embedding of X into K has a continuous extension h to flEX by Theorem 2.2. Since (g 0 h) I is the identity of X , g 0 h is the identity of flEX and, similarly, h og is the identity of K. This means that g is a homeomorphism leaving points of X fixed. 0 By the preceding theorem, /?,X (resp. PRX)is the Cech-Stone compactification (resp. Hewitt realcompactification) of a completely regular space X . We now restrict our attention to the special case that E = D or N. A Hausdorff space is called 0-dimensional if it has a base consisting of clopen sets. Clearly, D and N are 0-dimensional, and the class of 0-dimensional spaces is closed under taking arbitrary products and subspaces. Moreover, it is easily checked that if X is a 0-dimensional space, then aDis an embedding. Hence, for any 0-dimensional space Z with I ZI 2 2, in particular Z = D
K. Eda, T. Kiyosawa. H . Ohra
464
or N , 2-regularity is identical to 0-dimensionality. It also follows that D-compactness is just 0-dimensional compactness. 2.6. Corollary. Let X be a 0-dimensionalspace. Then PDX is that 0-dimensional compactiJication of X in which every member of C O ( X ) has a clopen closure. Proof. The stated property implies that everyf E C ( X , D) has a continuous extension. Hence this follows from Theorem 2.5. 0
x
A subspace S of a topological space X is said to be G, -closed in X if for each X - S, there exists a G,-set G such that x E G and G n S = 8.
E
2.7. Lemma. A G,-closed subspace S of an N-compact space X i s N-compact. Proof. Let Y = X - S and y E Y. Since S is G,-closed in the 0-dimensional space X , there exists a decreasing sequence {U,,},,, c C O ( X ) such that y E n,,,,U, c Y. Put N, = X - n,,,,U,,. Then N,, is N-compact, because it - U,,) x { n } )of X x N, is homeomorphic to the closed subspace UnEN((U,,-, where U,, = X. If we select such the subspace N,, for each y E Y , then S = n , , E Y N which v, means that S is homeomorphic to the closed subspace {x:x,, = x , , . f o r e a c h y , y ’ Y ~ }
of
n,,
N, . Therefore, S is N-compact.
Since / I D ( P N X )= /IDXby Theorem 2.5, P,X lies between X and next theorem determines the position of P N X .
0
a, X. The
2.8. Theorem. Let X be a 0-dimensional space. Then b,X is the G,-closure of X in P, X , i.e., the intersection of all G,-closed subsets of PDX containing X . Proof. Let S be the G,-closure of X in D,X. By Lemma 2.7, S is N-compact. By Theorem 2.5, it suffices to show that each f E C ( X , N ) can be extended continuously to S. The map f has a continuous extension /IJ: b D X .+ fi, N . Since ( P , f ) - ‘(P,N - N ) is a G,-set disjoint from X , it does not meet S. Hence, (P, f ) I is the required extension off. 0 2.9. Corollary. .4 0-dimensional spucr X is N-compact if’ nnd only if Y is G,-closed in BDX.
Proof. ‘This follows from Proposition 2.4 and Theorem 2.8.
0
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Let d be a collection of closed sets of a topological space X such that A, B E d implies A n B E d .A subcollection p of d is called an d-filter if (i) 0 4 p , (ii) A E p and A c B E d imply B E p , and ( 5 ) for every A, B E p , A n B E p . An d-ultrafilter is a maximal d-filter. Like a usual filter, an d-filter p is an d-ultrafilter if and only if any member of d which meets every member of p is in p. An d-filter p is said to be countably complete (resp. to have the countable intersection property (= c.i.p.)) if for every c p, nneNAn E p (resp. n n e N A n# 0). An d-filter is called free or jixed according to the intersection of all its members being empty, resp. nonempty. We call a CO(X)-ultrafilter a clopen ultrafilter.
2.10. Theorem. Let X be a O-dimensional space. Then flDX is the space of all clopen ultrafilters on X topologized by letting sets of the form ( p : U E p } , U E C O ( X ) , be basic open sets. Proof. This immediately follows from “a, Theorem IV.2 and Section IV.2.E, pp. 138-1421, so we only state a track of the proof. Let K be the space of clopen ultrafilters on X described as above. First, prove that K is O-dimensional and compact. Second, observe that for each x E BDX, there exists unique px E K such that { x } = n { C l f l D x UU: E px}. Finally, prove that the map carrying x to px is a homeomorphism. 0 2.11. Corollary. Let X be a O-dimensional space. Then flNX is the subspace of BDX consisting of all clopen ultrafilters with c.i.p. Proof. Let p be a clopen ultrafilter on X , i.e., p E BDX. If there is {U,},,,, c p such that nns,Un = 0, then p E One,V. c flDX - X, where V, = (4:U,,E 4). Hence, it follows from Theorem 2.8 that p $ flNX. The converse is obtained by reversing the steps. 0 2.12. Corollary. A O-dimensional space X is N-compact if and only if every clopen ultrajilter on X with r.i.p. is jixed. Proof. This follows from Proposition 2.4 and Corollary 2.1 1.
0
We shall give some examples of N-compact spaces and maximal N-compac tifications.
2.13. Examples. (1) Let X = ol.Then, it follows from p a , Example IV.2, p. 1301 and Corollary 2.6 that BDX = o,+ I . Since wI + I is the Gbclosure of X,BNX = P,X by Theorem 2.8.
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(2) Let A be an uncountable set. Then, N" and its homeomorph Z" are not only typical examples of N-compact spaces but also the maximal N-compactifications of certain proper subspaces. To show this, let Z["] be the subspace of Z" consisting of all x for which x, = 0 except for at most countably many A's. We show that flN(Z["])= Z". Let YE C(Z["], N).By Theorem 2.5, it suffices to prove thatfextends continuously to Z". For a basic open set B in Z["', put AB = {A E A : e l ( B ) # Z}; then IllB(is at most finite. For each n E N, take a maximal collection A?,, of disjoint, nonempty, basic open sets in Z["] included inf-'(n), and put U,, = U { B :B E a,,}. Then by the maximality of a,,,f - ' ( n ) = C1 U,,, where CI stands for the closure in Z["]. Put M = U{AB: B E a,,,n E N}. By [Na, Corollary 2 to Theorem VIII.9, p. 4751, the cardinality of a collection of disjoint, nonempty, open sets in Z" is at most countable. Since Z["] is dense in Z", 1 A?,, I is at most countable, and hence so is I MI. We now claim that, for every n, m E N with n # m, e M ( f - ' ( n ) )n e M ( f - ' ( r n ) )= 8, where eM is the projection from Z" to Z M . Suppose the contrary, then there exist x , y E Z["' such that f ( x ) = n, f ( y ) = m and e,(x) = e M ( y ) . For a basic open nbd V of x in Z", put W = e,l(eM(V)) n Z["'; then Wis a nbd ofy in Z["l. Sincef-'(rn) = C1 U,,,, W n B # 8 for some B E A?,,,. Since As c M, B = e;'(eM(B))n Z["], which implies that V n B # 8. Since V is arbitrary, x E CI U,,, = f -'(m), a contradiction. Consequently, there exists a g E C(ZM,N) such that f = g o & , where e& is the restriction of eM to ZW.Hence, goeM is a continuous extension off to +". (3) Consider the subspace Z(") of Z" consisting of all x for which x, = 0 except for at most finitely many A's. Then Z(") is N-compact. To show this, Put X" = {xEZ":I{Il:X, # O)( < n}
for each n E N. Since A',, is closed in Z" and Z(") = UnsNXn, Z(") is G,-closed in Z". Hence it follows from Lemma 2.7 that Z(") is N-compact. 2.14. Notes. The origin of E-compact spaces goes back to the classical Tychonoff embedding theorem; however, the explicit definition was given by Mr6wka in [1958]. From then, extensive researches have been made by himself and others. In particular, E-compactness has become more interesting in relation with category theory; the reader is referred to Chapter 14 by Nakagawa on this aspect. Theorem 2.2 and Proposition 2.4 were proved by Mrowka [1968], and Theorem 2.5 by Mr6wka and Engelking [1958], who introduced the notion of an E-regular space under the name of an E-completely regular space. Remark 2.3 was mentioned by Mr6wka [1971]. That
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D-regularity is just 0-dimensionalityfollows from Theorem 6.2.16 of Engelking [1977]. N-compact spaces behave very much like realcompact spaces, and the relation of PNXto PDX is analogous to that of PRXto PIX. Lemma 2.7 was proved by Mrowka [1957] for realcompact spaces. Theorem 2.8 and Corollary 2.9 are due to Mrowka [1966]. We owe Theorem 2.10 and Corollary 2.11 to Stone's work in [1937]. Corollary 2.12 was proved by Herrlich [1967]. Examples 2.13 (2) and (3) are taken from Corson [1959]; ZIA1 and are known as the );-product and a-product of Z, respectively. For more information about 0-dimensional spaces, see Nyikos [1976]. 3. N-compactness vs. realcompactness
'The results in this section are motivated by two questions: What kinds of spaces are realcompact?When is a realcompact space N-compact? Recall the following theorem and its corollary from [Na, Section IV.6, pp. 174-1841. If d is a collection of all zero-sets of a topological space X , then an &-filter is called a zero-jilter. As usual, we use PX for PIX. 3.1. Theorem. For a completely regular space X , the following conditions are equivalent : (1) Xis realcompact. (2) X i s G,-closed in PX. (3) Every zero-ultrafilter on X with c.i.p. is f x e d . 3.2. Corollary. Every regular, Lindelof space is realcompact.
The above corollary is an answer to the first question; however, many spaces, beside regular Lindelof spaces, are realcompact. To show this, we need the following definition. For a set A, B(A)denotes the power set of A.
3.3. Definition. An Ulam measure on a set A is a map p from B(A)to {0, l} such that (1) P ( A ) = 1, (2) p ( { a } ) = 0 for each a E A, and (3) if A, E B(A),n E N , are pairwise disjoint, then p(UnENAn) = ZnpNp(An). A cardinal number a is measurable if there exists an Ulam measure on a set of cardinality a. The assumption that all cardinal numbers are nonmeasurable is known to be consistent with ZFC, the usual Zermelo-Fraenkel Set Theory together
K . Eda T. Kivosawa H . Ohra
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with the Axiom of Choice. On the other hand, the consistency of the opposite assumption has been believed by many set theorists, but the proof of it seems to be far away (see Jech [1978] or Hrbacek and Jech [1984]).
3.4. Theorem. A discrete space X is realcompact measurable.
if and only if I XI
is non-
Proof. If p is an Ulam measure on X , then { A c X : p ( A ) = I} is a free zero-ultrafilter with c.i.p. Conversely, if p is a free zero-ultrafilter on X with c.i.p., then we get an Ulam measure p on X by setting p ( A ) = 1 if A E p , and p ( A ) = 0 otherwise. Hence this follows from Theorem 3.1. 0 Clearly, o is nonmeasurable. The following lemma shows that a measurable cardinal number must be very large if it exists. 3.5. Lemma. Let N be the class of all nonmeasurable cardinal numbers. Then the following are valid: (1) I f a E N and fl < a, then fl E M . (2) If { ~ l ~ . } ~ ~ ,c, N , and I A I E N , then X ; , E h ~Ej , N . (3) I f a E N , then 2“ E N . Proof. (1) If B c A and p is an Ulam measure on B, then the map v defined by v ( S ) = p ( S n B), S E B ( A ) , is an Ulam measure on A . (2) Let ( A , } , , , be acollection of disjoint sets with I A , I = a;,.Suppose that there is an Ulam measure p on U I , , A L . Then, p ( A , ) = 0 for each A E A. For, otherwise, the restriction of p to B ( A , ) is an Ulam measure on A , , contradicting our assumption. Therefore, the map v defined by v ( M ) = p ( U I E M A j , ) , M E B(A), is an Ulam measure on A, a contradiction. (3) Let A be a set of cardinality a, and suppose that there is an Ulam measure p on B(A). For each x E A , put d,= { S E B ( A ) :x E S } and dx’= B ( A ) - d’.Moreover, we put B = { x E A : p ( d x ) = I}. Then, p ( d x ’ ) = Oforeachx E B a n d p ( d r ) = 0 foreachx E A - B. Sincea E N , the same argument as in the proof of(2) implies that the union of a many sets of p-measure zero is p-measure zero. Hence ~ ( ( U ~ E B ~ X LJ ’ )O
J
~
~
A
-
B
~
= ~ ) 0. )
(or+Bd,)
Put 9? = n (flreA Then, by De Morgan’s laws, p(9R = 1. but 99 = {B}.This contradicts the fact that p is an Ulam measure on B ( A ) .
0
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A completely regular space X is called submetrizable if there exists a continuous bijection from X to some metric space. A topological space X i s called pseudo-m-compact if every discrete collection of nonempty open sets in X is of nonmeasurable cardinality. All realcompact spaces are pseudo-mcompact. In fact, if Q is a discrete collection of nonempty open sets in a realcompact space X , then X has a closed discrete subspace Y of the same cardinality. Since Y is realcompact, 1 % I must be nonmeasurable by Theorem 3.4. 3.6. Theorem. For a submetrizable space X , the following conditions are equivalent : ( I ) X is realcompact. ( 2 ) X is pseudo-m-compact. ( 3 ) I XI is nonmeasurable.
Proof. That (1) implies (2) has already been proved. To prove the remaining implications, let f be a continuous bijection from X to a metric space T, and fix a base B = UnENBn of T, where each anis a discrete collection (cf. "a, Theorem VI.4, p. 2561). If X i s pseudo-m-compact, then so is T, and hence I Bn1 is nonmeasurable for each n E N. Since I X 1 = I T 1 < 2'"', it follows from Lemma 3.5 that I X 1 is nonmeasurable. Thus (2) implies (3). It remains to prove that (3) implies (I). First, we show that T is realcompact. If not, then there exists a free zero-ultrafilter p on T with c.i.p. For each n E N , put
UZ, = ( B E B , , : B n F = O f o r s o m e F E p } . Then UnE,,,Wncovers T , because p is free. By the c.i.p. and maximality of p , U{CI,B: B E %,} ~p for some m E N. We write W m = {Bj,}j.E,,. Define a map p : 9 ( A ) -, (0, l } by p ( M ) = 1 if u,,,CI,B, E p , and p ( M ) = 0 otherwise. Then it is easily checked that p is an UIam measure on A. Since I A 1 < I TI = I X 1, this contradicts (3), thus T is proved to be realcompact. Finally, suppose that X i s not realcompact; then there exists a point q E pRX - X. Since T is realcompact, f has a continuous extension bRf : pRX + T . Since f is bijective, there exists unique r E X such that ( b R f ) ( q = ) (pRf ) ( r ) .Select g E C(pRX,W) for which g ( q ) = 0 and g(r) = I , and define h(x) = Ig(x) I + d(f ( x ) , f ( r ) ) for x E X , where d is a metric on T. Then h E C ( X , W) and h(x) > 0 for each x E X . But, since (pRh)(q)= 0, l / h cannot be extended continuously to q, which is a contradiction. 0 3.1. Corollary. Let X be a closed subspace of a product of submetrizable spaces. Then X is realcompact if and only if it is pseudo-m-compact.
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Proof. We only need to prove the "if" part. Let X be a closed subspace of the product nlOA Y, of submetrizablespaces. If X is pseudo-m-compact, then so is each e l ( X ) , and hence Q ~ ( Xis)realcompact by Theorem 3.6. Since Xis closed in II,,,en(X), it is realcompact. 0 3.8. Corollary. A paracompact, Hausdorfl space is realcompact if and only if it is pseudo-m-compact.
Proof. A paracompact, Hausdorff space X is complete with respect to the strongest uniformity compatible with the topology of X , i.e., the uniformity consisting of all normal covers of X . Thus, by p a , Section VI.5.1, p. 2951, X is embedded in the product of metric spaces as a closed subspace. Hence this 0 follows from Corollary 3.7. We will now consider the second question. A completely regular space X is called strongly 0-dimensional if for every disjoint zero-sets A and B in X , there exists U E CO(X) such that A c U and U n B = 8. All strongly 0-dimensional spaces are 0-dimensional. 3.9. Lemma. A completely regular space X is strongly 0-dimensional if and only if f3 X is 0-dimensional.
Proof. This is immediate from the following fact: If A and B are disjoint zero-sets in X , then Cl,,A n CI,,B = 0.The details are left to the reader. 0 3.10. Theorem. A strongly 0-dimensional,realcompact space X is N-compact.
Proof. By Theorem 3.1, Xis G,-closed in f3X. By Lemma 3.9 and Corollary 2.6, /.?J = f3X, so it follows from Corollary 2.9 that X is N-compact. 0 3.11. Corollary. A 0-dimensional,Lindelof space X is strongly 0-dimensional, and hence it is N-compact.
Proof. Let A and B be disjoint zero-sets in X . For each x E X , choose a clopen nbd U, of x satisfying either U, n A = 8 or U, n B = 8. Since X is Lindelof, there is a countable subcover {Un}nsN of {Ux}x,,. Putting V , = U, - ui<.U,for each n E N, we get a disjoint, open cover { V , j n P N of X . Put V = U{ V , : V , n B = 8,n E N}. Then V n B = 0,A c V and V E CO(X) hold, thus proving that X is strongly 0-dimensional. By Corollary 3.2 and Theorem 3.10, X is N-compact. 0
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3.12. Remark. There exist various examples of N-compact spaces which are not strongly 0-dimensional. See Mr6wka [1974], E. and R. Pol [1977], Wage [ 19781, Przymusinski [ 19791, and Mysior [ 19841. 3.13. Corollary. Both Q-compactness and (R cal to N-compactness.
-
62)-compactnessare identi-
Proof. By Corollary 3.11, Q and R - Q are N-compact, and hence Q-compact spaces and (88 - Q)-compact spaces are N-compact. The converse is obvious, because both Q and R - Q include copies of N as closed subspaces. 0 It is natural to ask whether a 0-dimensional, realcompact space is N-compact. In his papers of [1971] and [1973], Nyikos answered this question in the negative by proving that Roy's space A, which is a 0-dimensional, but not strongly 0-dimensional, metric space, is not N-compact. Here, we take up Mysior's solution which is much simpler and rather interesting in its categorical context.
3.14. Theorem. For every 0-dimensionalspace E of non-measurable cardinality, 0-dimensional realcompactness is not E-compactness. To prove the theorem, we need a definition and a lemma. For an infinite cardinal number u, a 0-dimensional space X is called 0-dimensionally u-compact if every clopen ultrafilter p on X with a-i.p. (i.e., the intersections of less than u members of p are nonempty) is fixed. By Corollary 2.12, N-compact spaces are precisely 0-dimensionally o, -compact spaces. The proof of the following lemma is a routine.
3.15. Lemma. An arbitrary product of 0-dimensionally u-compact spaces is 0-dimensionally a-compact. Every closed subspace of a 0-dimensionally a-compact space is 0-dimensionally u-compact.
Proof of Theorem 3.14. Let E be an arbitrary 0-dimensional space with
IE I
=
/3, where /3 is nonmeasurable. Define
.
u = /3+2~+2zp+2z2p+...
Then by Lemma 3.5, u is nonmeasurable, and it is easily checked that am = 2". Since I E I < a, E is 0-dimensionally u-compact, and hence so is each E-compact space by Lemma 3.15. Thus, it suffices to show the existence
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of a 0-dimensional, realcompact space which is not 0-dimensionally a-compact. To do this, let H be a generalized Hilbert space with the index set A of cardinality a, i.e., the underlying set of H is the subset of 64" consisting of all x for which &.,,x~ < +a,and the topology of H is the metric Since am = 2",I HI = 2", topology induced by the norm 11 x II = (ZiEA~i)"2. and H has a dense subset Y of cardinality a. We now define a finer topology on H as follows. Let d = {A c Y:ICl,A n Cl,(Y - A ) [ = 2").
Since I d I < 2",we can choose inductively a point x, E C1,A n C1,( Y - A) for each A E d in such a manner that A # A' implies x, # x,, . Moreover, for each x E H - Y, choose a sequence S, in Y, converging to x, such that if x = x, for some A E d ,then both S, n A and S, - A are infinite. Define a new topology on H by agreeing that all points of Yare isolated and nbds of x E H - Ycontain all but finitely many points of S,. Denote the resulting space by X. Clearly, Xis 0-dimensional. Since the topology of X is finer than that of H , Xis submetrizable, and hence it follows from Theorem 3.6 that X is realcompact. It remains to show that X is not 0-dimensionally a-compact. We need the following fact about the space H: For each disjoint, nonempty open sets U and V in H , 1H - ( U U V)I = 2".
(3.1)
To show (3. l), observe that H = H I x H,, where H I and H , are copies of H, and let el : H + H , be the projection. Put F = H - (U u V ) , U , = e , ( U ) , and V, = e , ( V ) . Suppose that IF1 < 2"; then U1 u V, = H I . Since H2 is connected, U , n V, c e , ( F ) , which implies that U , n V, = 0,because the cardinality of every nonempty open set in HI is 2". This contradicts the connectedness of H I ,thus a contradictionoccurs. Next, we prove the following: ForeachUECO(X),(UI < 2 " o r l X - UI < 2".
(3.2)
If U E CO(X), then I Cl,(U n Y) n Cl,(Y - U ) l < 2". For, the negation implies U n Y E d,and so x(,,~),, E C1,U n Cl,(X - U ) . This contradicts the fact that U is clopen in X. On the other hand, Y being dense in X , U=Cl,(UnY)cCl,(Un Y)andX- U=Cl,(Y- U)cCl,,(Y- U ) hold, so we get that C1,U = Cl,(U n Y) and Cl,(H - U ) = Cl,(Y - U ) . Consequently, I C1,U n Cl,(H - U ) l < 2". This combined with (3.1) implies that either U or H - U is dense in H. If U is dense in H , then IX - UI = IH - U ( < 2", because H - U c C1,U n Cl,(H - U ) . Similarly, if H - U is dense, then I U I < 2", thus (3.2) is proved. Finally, put p = { U E CO(X) :I X - U I < 2");then by (3.2), p is a clopen ultrafilter on
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X. Since I X 1 = 2", p has a-i.p. Since each point of X has a countable clopen nbd, p is free. Hence, X is not 0-dimensionally a-compact. 0 The problem of characterizing those realcompact spaces which are also N-compact is unsolved, and there seems to be no simple answer (cf. Mrbwka [1972b], Broverman [1977], and Mysior [1978b, 19841). On the other hand, it is not difficult to isolate what is to be added to 0-dimensional spaces to produce N-compact spaces: 3.16. Theorem. A space Xis N-compact ifand only i f X is 0-dimensional and (I x N)-compact.
Proof. The "only if" part is obvious. To prove the "if" part, let X be a 0-dimensional, (I x N)-compact space and p a free clopen ultrafilter on X. By Corollary 2.12, it suffices to show that p has not c.i.p. Choose a zero-ultrafilter q on X including p. With q regarded as a point of fix, we have (4)
= n{Cl,,F: F E 4) c
fix - X .
Since X is (Ix N)-compact, there exists an embedding f of X to a closed subspace of (I x N)" for some index set A. Let BX = Cl,f(X), where Y = (fi(I x N))". Then, BX being compact, f has a continuous extension fif :fix + BX. Since fif maps fix - X to BX - X (cf. ma, Corollary 2 to Theorem IV.2, p. 128]), (flf)(q)E BX - X , so ( e l o f l f ) ( q ) E fl(I x N ) (I x N ) for some I E A, where el is the Ith projection of Y. For each n E N, put Fn = (eio/3f)-'(I x {n}); then F, is clopen in fix. Since q $ F,, X - F, E p. Since n,,,(X - F,) = 8,p has not c.i.p. 0 3.17. Remark. (I x N)compactness is an intermediate between compactness and realcompactness, and (I x N)-regularity is just complete regularity. The above proof shows that a completely regular space X is (I x N)-compact if and only if every zero-ultrafilter p on X such that p n CO(X) has c.i.p. is fixed. 3.18. Notes. Realcompact spaces were defined and investigated, under the name of Q-spaces, by Hewitt [1948], who proved the equivalence of (1) and (3) in Theorem 3.1 as well as Corollary 3.2. The equivalence of (1) and (2) in Theorem 3.1 was proved by Mrowka [1957]. Definition 3.3, Theorems 3.4 and 3.6, Lemma 3.5 and Corollaries 3.7 and 3.8 are taken from Gillman and Jerison [1960], which is a standard text on this area. Historically, Theorem 3.4 was proved by Mackey [1944] and Lemma 3.5 by Ulam [1930]. Theorem
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3.6 and Corollaries 3.7 and 3.8 are due to Katgtov [1951]and Shirota [1952a]. The name of a pseudo-m-compact space was given by HuSek [1971]. In terms of dimension theory, a strongly 0-dimensional space is a completely regular space X with dim X = 0. Lemma 3.9 is a special case of Theorem 7.1.17 of Engelking [1977] asserting that dim X = dim /3X. Theorem 3.10 was proved by Mr6wka [1966], and the first statement of Corollary 3.1 1 is Theorem 6.2.7 of Engelking [1977]. Corollary 3.13 is taken from Walker [1974, p. 2641. Theorem 3.14 was proved by Mysior [ 1978al; however, the proof given here uses a simpler method developed by Mysior [1984]. For other examples of 0-dimensional, realcompact spaces which are not N-compact, see Nyikos [1971, 19731, Broverman [1978] and E. Pol [1979]. Lemma 3.15 was proved by Hong [1973]. Theorem 3.16 and Remark 3.17 are due to Ohta [1982].
4. N-compactness of k,X We present certain technical results and examples which play an important role in Sections 6 and 7. For a topological space X , K(X) denotes the collection of all compact subsets of X . A map f from X to a topological space Y is called k-continuousif the restriction off to every K E K(X)is continuous. The set of all k-continuous maps from X to Y is denoted by C,(X, Y). Let E be an arbitrary fixed Hausdorff space. If each f E C,(X, E ) is continuous, then Xis called a k,-space. The k,-leader k,X of a space Xis the set X with the finest topology making eachf E C,(X,E ) continuous. In other words, the topology of kEX is that having as a subbase the collection of all sets of the formf - ' ( U ) , wheref E C,(X, E ) and Uis an open set of E. The identity map from k,X to X is also denoted by the same notation k,.
If X is an E-regular space, then (1) k,: k,X -, X is continuous, (2) each K E K(X)is homeomorphic to k,'(K), and (3) k,X is an E-regular k,-space.
4.1. Proposition.
This is immediate from the following lemmas; the first one is well known (cf. Engelking [1977, 1.4.1 and 3.12.21). 4.2. Lemma. Let X be a topological space with a subbase 9'.Then, a map from a topological space to X is continuous if and only if inverse images of all members of Y are open, and X is compact if and only if every open cover of X by members of Y has afinite subcover.
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4.3. Lemma. A Hausdorfs space X is E-regular if and only if 9' = { f - ' ( U ) :f E C(X, E ) , U E 6},where d is the topology of E, is a subbase of x. Proof. Suppose that X is E-regular, i.e., X is a subspace of E" for some index set A. Then, el I E C(X, E ) for each Iz E A, which implies that Y is a subbase of X . Conversely, if Y is a subbase of X , then one can easily check that the diagonal map of C(X, E ) is an embedding. By Proposition 4.1, an E-regular space X and k,X have the same compact subspaces and the same topology on those subspaces. We write kEX = X when k, is a homeomorphism. For every E-regular k,-space X , k,X = X . Recall that a topological space Xis a k-space if every subset, whose intersection with every K E K ( X ) is open in K , is open in X. Locally compact spaces and first countable spaces are known to be k-spaces (cf. [Na, Section IV.5, pp. 172-1741). Since a k-space is a k,-space, we have the following proposition. 4.4. Proposition.
If X is an E-regular k-space, then kEX = X .
The next lemma allows us to define a natural bijection from Ck(X, Y) to C(k,X, Y) for each E-regular space Y.
4.5. Lemma. Let X be a topological space and Y an E-regular space. Then, for each f E ck(x, Y ) ,f 0 k, E C(kEX, Y ) . Proof. Assume that Y is a subspace of E" for some index set A. Let E A, el 0f E Ck(X, E ) , SO el 0f k , E C(kEX, E ) . Hence it follows from Lemma 2.1 that f kE E C(kEX, Y). 0
f E Ck(X, Y). For each A
0
0
4.6. Proposition. Let E and F be Hausdorfl spaces for which E-regularity coincides with F-regularity. Then, for each E-regular space X , k E X = kFX, more precisely, k,' 0 k, is a homeomorphism. Proof. By Proposition 4.1, k;' E Ck(X,k F X )and k,X is F-regular. Hence, by our assumption and Lemma 4.5, k;' 0 k, is continuous, and similarly so is k,' 0 k,. This implies that k,' k, is a homeomorphism. 0 0
If 2 is a 0-dimensional space with 12I 2 2, then, as we noted in Section 2, 2-regularity coincides with 0-dimensionality. Hence, it follows from Proposition 4.6 that k Z X = k N Xfor each 0-dimensional space X. The topology of k,X consists of all sets whose intersection with every K E K ( X ) is clopen in K. Actually, this special case is all we need.
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4.7. Proposition. Assume that X and Z are O-dimensional spaces and I Z I 2 2. Then, k N X = X ifand only ifthe map x X :C(X, Z ) + C ( k N X ,Z ) defined by icX( f) = f0 kN is surjective.
Proof. As was mentioned above, Z-regularity coincides with O-dimensionality. If x X is surjective, then C ( X , Z) = C,(X, Z) by Lemma 4.5, so k,X = X , and hence k,X = X , because k,X = k Z X . The converse is obvious. 0 The rest of this section is devoted to examples of spaces, which will be used to construct those of abelian groups and non-Archimedian Banach spaces in later sections. By Corollary 2.1 1, points of BNA, where A is a discrete space, are identified with countably complete ultrafilters on A. Under this identification, we can assert the following:
4.8. Lemma. Let A be a discrete space and { P , } , ~ , , a countable subset of distinct points of BNA. Then there exists a partition { A n } n p Nof A such that A, E pnfor each n E N. The preceding lemma shows that any countable subset of BNA has no accumulation point, from which the following corollary is deduced.
4.9. Corollary. Let A be a discrete space and K E K(BNA). Then I K 1 isfinite. 4.10. Example. Let A be a discrete space of measurable cardinality. Then kN(BNA) is not N-compact.
Proof. By Corollary 4.9, Theorem 3.4.
kN(BNA)
is discrete. Hence this follows from
0
An N-compact space X for which k,X fails to be N-compact is of special interest in the subsequent discussions. To make the one of nonmeasurable cardinality, we need the next lemma. A subset of w , is called stationary if it meets every closed, unbounded (= c.u.b.) set in wl.
4.11. Lemma. There exist two disjoint stationary sets in w , .
Proof. Let { x u: a < a,}be a set consisting of irrational numbers, where xu # xg for distinct a, < wl. For each q E Q, put SJO) = { a : x , c q } and SJ1)
= {a:x, > q}.
N-compactness and its Applications
411
Suppose that for all q E Q, there is iq E (0,I } such that Sq(iq)includes a c.u.b. set in o l .Then, since the intersection of countably many c.u.b. sets in c a b . , C = nq,,Sq(iq) includes a c.u.b. set, while I CI < 1. This contradiction implies that S,(O) and S,(l) are disjoint stationary sets for some q E Q.
0 4.12. Example. There exists an N-compact space X of nonmeasurable cardinality for which k,X is not N-compact.
Proof. By Lemma 4.1 I , there exists a stationary set Y such that o1- Y is also stationary. Define X to be the subspace Y u {aI} of o1+ 1. Since X is G,-closed in o, 1, it is N-compact by Lemma 2.7. We prove the following: ( I ) For each U E CO(Y), either I UI < w or I Y - U ( < o. (2) For each K E K ( X ) , I K I < w . Suppose that both U and Y - U are uncountable for some U E CO(Y). Then the closures CI(U) and CI( Y - U ) in uIare c.u.b. sets, and hence so is Cl(U) n Cl( Y - U). But this c.u.b. set does not meet Y, which is a contradiction, thus ( I ) is proved. Next, suppose that K E K ( X ) is uncountable. Then K n Y is a c.u.b. set in w I , while it does not meet w1 - Y, a contradiction. Thus (2) is proved. By (I), p = { U E CO(Y): 1 Y - U I < o}is a free clopen ultrafilter with c.i.p. on Y, so Y is not N-compact. It remains to show that k,X is homeomorphic to Y. Since Y satisfies the first axiom of countability, k, Y = Y by Proposition 4.4. On the other hand, the characteristic function of Y on X is k-continuous by (2). Hence, k,X is obtained from X by making wIan isolated point. Since Y contains infinitely many isolated points, k, X is homeomorphic to Y. 0
+
4.13. Example. There exists an N-compact space X such that k,X # X , but k N X is N-compact.
Proof. Pick a point p E BN - N,and consider the subspace X = N u { p} of BN. By Corollary 3.1 1, X is N-compact. If A and B are disjoint infinite subsets of N,then CI,,A n CI,,B = 8,so at least one of them is discrete and closed in X. This means that X includes no infinite compact set, and hence k N X is homeomorphic to N. 0 Let Z", Z["] and Z(") be the same as in Example 2.13. It is well known (cf. Engelking [1977, Exercise 3.3.E.1) that Z" is not a k-space when A is uncountable; however, as seen in the following, all of these are k,-spaces:
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4.14. Example. For each index set A, (i) k , ( ~ ( " ) = ) Z"), (ii) kN(Z["])= Z["), and (iii) k N ( Z A )= Z".
Proof. In case A is countable, Z" and its subspaces satisfy the first axiom of countability, and hence (i)-(iii) follow from Proposition 4.4. So assume that A is uncountable. For each x E Z" and each finite subset A of A, put
U ( x , A ) = { y E Z" : y(A) = x(A) for each A E A } ; then the collection of all U ( x , A) forms a base of Z". To prove (i), let f~ Ck(Z("),N). All we have to show is the continuity off. Let x E Z'"), and suppose that there is no finite subset A of A such that U ( x , A) n Z(") c f - ' ( f ( x ) ) .Then we can choose a point xIE Z'") - f-'(f(x)) and, inductively, points x.+I E ( U ( x , A n ) n Z'"') - f-'(f(x)), n E N , where A, = { A i j : 1 < i < n, 1 < j < n} and the A i j , j = I , 2, . . . , is an enumeration of { A E A :x(A) # x,(A)}. For each basic nbd U ( x , B ) of x, {A,}nsNincreasing, there exists a k E N such that B n ( U n E N A ,c ) Ak, which implies that x,, E U ( x , B) for each n > k. Thus {x,,},,,, converges to x. However, since f ( x , , ) # f ( x ) for each n E N, f is not continuous on {x} u {x,,}~,,, contradicting the k-continuity off. Hencefis proved to be continuous. The proof of (ii) is the same as that of (i) if Z'") is replaced by Z["]. Finally, to prove (iii), letg E Ck(Z",N ) and s E Z". Then,just as above, U(v, V ) n Z(") c g-'(g(s)) for some finite subset V of A. It suffices to show that U(o, V) c g-'(g(s)). Suppose the contrary, and pick a point w E U ( s , V ) - g-'(g(v)). Define K = { u E Z" : u(A)
=
v(A) or w(A) for each A
E A};
then K is compact. Since g is k-continuous, there exists a finite subset W of A such that U(w, W) n K c g-'(g(w)). Since (U(t.9, V ) n Z"))n (U(w,W ) n K) # 0, g-'(g(s)) n g-'(g(w)) # 0, which contradicts the fact that g(s) # g(w). Hence, g is continuous, proving (iii). 0 4.15. Remark. All of the foregoing examples remain true if N is replaced by R. In particular, Example 4.12 provides the example in ZFC of a realcompact space X such that k,Xis not realcompact. The existence of such a realcompact space was shown by Blasco [I9771 under assumption of the continuum hypothesis.
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4.16. Notes. The definition of a k,-space for arbitrary E can be found in
Beckenstein, Narici and Suffel [1977, p. 1161, where Proposition 4.4 was stated. The case of E = R appears in Ptak [I9531 and that of E = N in Van der Put [1970]; in those cases, Proposition 4.1 is well known. The second statement of Lemma 4.2 is known as Alexander’s theorem. Corollary 4.9 and Example 4.10 were mentioned by Comfort in [1968]. Lemma 4.1 1 is due to Banach; however, Ulam [ 19301 proved much stronger result: There exist o, many disjoint stationary sets in 0 , . Example 4.12 is due to Eda and Ohta [ 19871. Example 4.13 is taken from Gillman and Jerison [ 19601, Problem 4M. Example 4.14 was proved by Noble in [1970].
5. Rings and lattices of continuous functions In case of a realcompact space X, the ring structure or the lattice structure of C(X, R) determines the topology of X (cf. Hewitt [1948] and Shirota [1952b], also p a , Section IV.4,6, pp. 157-168, 174-1841). Just as this, for an N-compact space X the ring structure or the lattice structure of C(X, Z) determinesthe topology of X. We prove this as a consequenceof a little more general result. To state it some definitions are necessary. Let X be a 0-dimensional space, R a ring with 1 and L a lattice. We regard R and L to be endowed with their discrete topologies. Then, C(X, R) and C(X, t)are a ring and a lattice under pointwise operations, respectively. We define generalized ultraproducts C(X, R)/p and C(X, L)/p for a clopen ultrafilterp on X. Forf, g E C(X,R), letf g if { x E X : f ( x ) = g(x)} E p; then is an equivalence relation on C(X, R). Denote the equivalence class containing f by [f], , and define
-,
-,
[fl, + [gl, = [f+ gl, and [fl, * [gl, = [f-gl, forf, g E C(X, R). This ring { [f], :f E C(X, R)} is denoted by C(X, R)/p and [ 1, is a ring homomorphism. For the case of C(X, L), the equivalence relation and the map [ 1, are defined just as above. Define
-,
[fl, v [gl, = [fv gl, and [fl, A [gl, = [fA gl, for f, g E C(X, L). Then the lattice {[f], : fC(X, ~ L)} is denoted by C(X, L)/p and [ 1, is a lattice homomorphism. In what follows, we identify a point p E B D X ( B N Xand ) a clopen ultrafilter (with c.i.p.) p on X such that {p} = nuepClsDxU(cf. Theorem 2.10 and Corollary 2.1 1); thus p E U and U E p mean the same thing for p E X and U E CO(X). We denote the characteristic function of a subset U by e,, and write the function constantly a simply as a.
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480
5.1. Theorem. Let X be a 0-dimensional space and R a ring with I , equipped with the discrete topology. I f c p : C ( X , R ) + Z is a ring homomorphism, i.e., rp preserves and and rp(1) = I , then there exist unique p E B N X and a ring homomorphism (p : C ( X , R ) / p + Z such that cp = (p [ I,.
+
-
0
-
,,
Proof. Let p = { U E C O ( X ) :rp(e,) = I}. Since e, e,, = e,, and e, + ex-" = I , it is easily checked that p is a clopen ultrafilter on X , i.e., p E flJ. We show that p satisfies c.i.p. Suppose the contrary; then there c C O ( X ) of X such that U, $ p for each n E N. exists a partition { Un}neN Since 2" and 3" are prime to each other, there are i,j E Zwith 2"i + 3'7 = 1. Let d,, = - 2 9 ; then 2"1dn and 3"1dn + 1. Define g E C ( X , R ) by g ( x ) = d, 1 for x E U,,. Then, for each n E N ,
-
2"icpw and 3 w g
+
(5.1)
1).
To show this, put A = U k 2 , , uk;then A E p by the maximality ofp. For each x E X , let s ( x ) = 2-" g ( x ) e,(x); then s E C(X, R ) by the fact that 2" I g ( x ) for x E A. Since cp(e,) = I , cp(g) = cp(g) cp(e,) = cp(g e A ) = 2"cp(s), and hence 2" I cp(g). Applying this argument to g + 1 and 3" instead of g and 2", we get 3"lcp(g + 1) and hence (5.1). Since (5.1) holds for each n E N , cp(g) = 0 = cp(g + I), and so cp(l) = 0, a contradiction. We have shown p E BNX.Define a map (p : C ( X , R)/p + Zby @([A,)= q(f) f o r f s C(X, R ) . If [ f ] , = 0, then U ~ p where , U = f - ' ( O ) , so cp(e,) = I , and hence q(f) = c p ( f . e,) = cp(0) = 0. Now, (p is well defined and consequently a ring homomorphism. The uniqueness of p and (p is clear. 0 9
-
-
-
The generalized ultraproduct C(X, Z)/p is a ring and a lattice at the same time; actually, C ( X , Z ) / p is isomorphic to Z. Because, i f f € C(X, Z ) , p E BNX and i = ( P N f ) ( p )then , [ f ] , = [i],, and hence the map assigning ( B N f ) ( p ) to [f], is an isomorphism. 5.2. Corollary. Let X be a 0-dimensional space and cp : C ( X , Z)-, Z a ring homomorphism. Then there exists p E BNX such that cp( f) = (BNf ) ( p )for e a c h f e C(X, Z).
Proof. By Theorem 5.1, there exist p E P N X and a ring homomorphism (p : C(X, Z ) / p --+ Z such that cp = (p 0 [ 1,. With C ( X , Z)/p regarded as Z, (p must be the identity of Z. Hence, as we have noted above, c p ( f ) = 0 [ f l , = ( P N f ) ( P ) for each f E C(X, Z). 5.3. Theorem. For N-compact spaces X and Y , the rings C(X, Z) and C( Y , Z)are isomorphic if and only if X and Y are homeomorphic.
N-compactness and its Applicarions
48 1
Proof. The “if” part is obvious. To prove the “only if” part, let 8:C(X,Z) C ( Y , Z) be a ring isomorphism. For each p E Y , [ Ip08 is a ring homomorphism. Since C(Y , Z)/p N Z, it follows from Theorem 5.1 that -+
there exist g ( p ) E X and a ring homomorphism qp making the following diagram commute:
C(X,Z) [
Ism
I
2 C(Y,Z)
I
[Ip
(PE
Y)
(5.2)
ax, Z ) M P ) ‘pp C(y, ZYP Conversely, for each q E X , there exist h(q) E Y and a ring homomorphism $q with the following commutative diagram: C(X, Z) [Iq]
C(X9 Z ) / q
0-1
‘
+q
C(Y,
I
INq)
(4 E X )
(5.3)
C(Y9 H M q )
Since both cpp and $, are the identities of H , g ( p ) = q if and only ifp = h(q), which implies that g : Y -+ Xis bijective and g-’ = h. It remains to show the continuity of g and h. For this end, let U E C O ( X ) .Then, g ( p ) E U if and only if [eo]g(pl= 1, and this is equivalent to [O(e,)], = 1 by Diagram (5.2). Hence, g - ’ ( U ) = { p E Y :(8(e,))(p) = l } E CO( Y ) , which means that g is continuous. Similarly, for each V E CO(Y), h - ’ ( V ) = {q E X : (8-’(ev)) (q) = l } E CO(X), and hence h is continuous. 0 Next, we state the analogue of Theorem 5.1 for C(X, L). 5.4. Theorem. Let X be a @dimensional space and L a lattice equipped with the discrete topology. Ifq : C(X, L ) -+ Z is a nontrivial lattice homomorphism, i.e., the cardinality of the range is at least two, then there exist unique p E bDX and a lattice homomorphism cj :C(X, L)/p -+ Z such that cp = c j 0 [ Ip. Moreover, if the range of cp is unbounded, then p E b N X .
Proof. Let i E H be an integer such that there exist A, gi E C(X, L ) with q(f;.)< i < cp(gi). For such i E H , fixf; and gi, and define
< gl,andq(g) < iimply cp(f) < i for each f, g E C(X, L)},
pi = {UECO(X):fl,
K. E&, T. Kiyosawa, H. Ohfa
482
By the condition of i, 8 4 p i . For U, V E p i , let f l u n Y < g l U n Yand cp(g) < i for5 g E C(X, L). Define h E C(X,L) by h(x) = f ( x )if x E U,and h(x) = g(x) otherwise. Then, hl < gl and V E p i , so cp(h) < i, which implies that c p ( f ) < i, because f l u = h I and U E p i . Hence U n V E p i . Suppose that neither U nor X - U belongs to pi for some U E CO(X).Then there exist h,, hz, h3,h4 E C(X, L) such that hl I < h2I Ll,cp(hz) < i < cp(h,), h,I x - u < h41x-,, and cp(h4) < i < cp(h3).Since h, A h, < hz v h4,
,
i <
cp(hl) A
cp(h3) <
cp(h2)
v cp(h4)
< i,
a contradiction. Now, we have shown pi E B,X. Next, we show that pi and pi ( i < j ) are equal if they are defined. Suppose not; then there exists W E C O ( X ) such that W € p i and X - W Epi. Define r E C(X, L) by r ( x ) = X(x) if x E W , and r ( x ) = gj(x) otherwise. Then, r ( = f;.l and cp(X) < i, so cp(r) < i < j . This together with gjI x- = r I x- implies that cp(gj) < j , which is a contradiction. Now, let p be the defined p i , and define a map @ : C(X, L)/P --* Z by @([fl,) = c p ( f ) forfE C(X, L). If If], = [gl, and c p ( f ) = k < cp(g) for some k E Z, then cp(g) < k, because { x E X : f ( x ) = g(x)} E p = P k , a contradiction. Hence @ is well defined and consequently a lattice homomorphism. To show the uniqueness ofp, suppose that there exist a q E /3,X with q # p and a lattice homomorphism $ :C(X, L)/q --* Z such that cp = $ o [ 14. Then G E P and X - G E q for some G E C O ( X ) and, by the assumption on cp, there are sl, sz E C(X,L) with cp(sl) < cp(sz). Define s E C(X, L) by s(x) = sI(x) A sz(x) if x E G, and s(x) = sI(x)v sz(x) otherwise. Then
, ,,,
cp@l)
= cp(s1
=
A 82)
$([Sir) =
=
ml,) =
MSI
,,,
cp(s)
v szlq) = c p h v
$2)
= cp(sz),
which is a contradiction. The uniqueness of @ is obvious. Finally, in case the range of cp is unbounded, we may assume that it is unbounded upward. Hence there exists {t,,}neNc C(X, L) such that cp(t,,) < cp(t,,+,) for each n E N. Suppose that p does not satisfy c.i.p.; then there exists a partition { H,,}neNc C O ( X )of X such that H,, 4 p for each n E N . Define t E C(X, L) by t ( x ) = V,<,,t,(x) for x E H,,. Then, cp(t) = j < cp(t,,,) for somej E Z and m E N. Put H = U,,,,,,H,,; then H E p = pi. Since t,,, I < t I and cp(t) = j , cp(t,,,) < j by the definition of pi, which is a contradiction. Hence p E b N X . 0
5.5. Theorem. For N-compact spaces X and Y, the lattices C(X,Z) and C( Y, Z) are isomorphic if and only if X and Y are homeomorphic.
N-compactness and its Applications
483
Proof. Let 8 : C(X,Z) + C(Y, Z) be a lattice isomorphism. For each p E Y and q E X, using Theorem 5.4 instead of Theorem 5.1, we get the same diagrams as (5.2) and (5.3) concerning lattice homomorphisms. We show that g ( p ) = q if and only if p = h(q).
(5.4)
If p # h(q), then W Ep and X - W Eh(q) for some W EC O ( X ) . Since PI, < [ewlp, ‘p,(~~-’(O)lg(,O)) < ‘pp([~-’(eW)l,(,))by ( 5 3 , which implies that [O-’(O)],(,, < [8-’(ew)],(,). On the other hand, since [O],,,, = [eW]h(q), [8-’(0)], = [O-’(e,)], by (5.3). These imply that g ( p ) # q. The converse can be proved similarly. It follows from (5.4) that g is a bijection such that g - ’ = h and, moreover, ‘p, is also bijective and, ‘ p i ’ = JI, provided g ( p ) = q. It remains to show the continuity of g and h. To this end, let U E C O ( X ) . Then, g ( p ) E U if and only if [O],,,, < [eu],(,). Since ‘p, is a lattice automorphism on Z,this is equivalent to (P,([O]~(~)) < ‘p,([e,],(,,), which is also equivalent to [8(0)], < [O(e,)], because of (5.2). Therefore, g - ’ ( U ) = { p E Y: (8(O))(p) < (8(e,))(p)) E CO( Y), which means that g is continuous. Similarly it can be proved that h-’( V) = {q E X :(8-’(O))(q) < (O-’(e,,))(q)) E C O ( X )for each V E CO(Y), and hence h is continuous. The “if” part is obvious. 0 5.6. Notes. In [1965], Mrowka proved Corollary 5.2, Theorem 5.3 and Theorem 5.5 using partly different methods. See Pierce [1961] for more extensive studies of the ring C(X,Z).
6. Applications to abelian groups In this section we apply N-compact spaces to abelian groups using abelian groups C(X, Z). We shall investigate the interrelationshipbetween topological properties of a space X and group theoretic ones of the group C(X, H), with special emphasis on the reflexivity. For most cases, some parts of Fuchs’ books [1970, 19731 are sufficient as the preliminary knowledge of abelian groups. However, for the convenience of the reader, we recall several definitions and preliminaries about abelian groups. All groups considered here are abelian groups, so we simply say “group” instead of “abelian group” in the following. Let A l , A E A, be groups. Then, the product nlc,,Al is a group under . The coordinatewise addition, and is called the direct product of { AA}Ae,, direct sum @nc,,Al of { A A } l , , ,is the subgroup of nAs,,Alconsisting of all x such that xl = 0 for all but finitely many A’s. In case all Al are copies of A,
484
K . Eda. T. Kiyosawa. H.Ohta
IILE,, A, and @ j,E,, A , are denoted by A A and A'")', respectively. As generalizations of these, we introduce inverse limits and direct limits. Suppose that A is a directed set by order <, and that for any 1, p E A with A < p a homomorphism (pi,, :A,, + A, is defined in such a manner that cp, 0 ( p r y = (pi" if 1 < p < v and (pi, is the identity for each 1 E A. Then the system (A;,, (p,,,, A) is called an inverse system. The inverse limit of (A,, (pi,,, A), denoted by lim, A,, is the subgroup of n;.eAA, consisting of all x for which q,,(x,,) = xi, for each A, p E A with 1 < p. 6.1. Lemma. Let (Aj., cp;., A) be an inverse system of groups. IfA is a group and there are homomorphisms T, : A + A, with the commutative diagram
then there exists a homomorphism T : A
-, lim,
A, for which the diagram
commutes. Proof. For each x
E
A, define T ( X )
E
lim, A, by (~(x)),= T , ( x ) , 1 E A. 0
Suppose again that A is a directed set by order < ,and that for any 1,p E A with 1 < p a homomorphism (pi,, : A , + A,, is defined in such a manner that (pry 0 (pi,, = (ply if 1 < p < v and (pi, is the identity for each 1 E A. Then the system (Ai, (pi,,, A) is called a direct system. For each x E A, and A E A, define ( x ) , to be the element of Oish A, defined by ( x ) , ( p ) = x if p = I, and ( x ) , ( p ) = 0 otherwise. Then ( ),is an injective homomorphism and each element of @j,EhA, can be written as the unique form ( x i , ) i . ,+ . . . +
(xi,>j.,
for some x,, E A,,, li E A, i = 1, . . . , n, and n E N. Let B be the subgroup of OiE,,Aiconsisting of all elements as above such that (pi,,(xi,) . . . + q ~ ~ , ~ , (=x 0, ~for ) some 1 2 A,, . . . , A,,. The direct limit of (A,, (pi,,, A), denoted by lim,A,, is the quotient group (@,E,,A,)/B. For each 1 E A, the
+
N-compactness and its Applications
485
symbol n, is reserved to denote the homomorphism from A, to lim,A, defined by n,(x) = ( x), + B for x E A,. If I < p, then n, = n , , cp,~.,
6.2. Lemma. For a direct system (A,, ( p i p , A) of groups, the following are valid. (1) lim,A, = {(x), B : x E A,, I E A}. (2) I f A is a group and there are homomorphisms c,: A, + A with the commutative diagram
+
then there exists a homomorphism Q :lim, A,
+
A for which the diagram
commutes. Moreover, if each Q, is injective, so is 6.
+
+
Proof. (1) For each ((xAI),, * * * (x,~),,) + B E lim,A,, choose I 2 I,, . . . , I,, and put x = X;=,cpij,(xAi). Then, cpu being the identity, (x), - ((xA,)lI + * . * + (x,"),") E B, and hence ((x,l),l + * * * + (x,J,") B = (x)A B. (2) If (x), + B = ( y ) , + B, where x E A, and y E A,,, then cpl,(x) cp,"(Y) = 0 for some v 2 A P, so QAW = c"(cpAd.4) = Q"(cp,"(Y)) = Q J Y ) . Hence, we can define a homomorphism Q :lim, A, + A by c(( x), + B) = a,(x), as required. Assume that each eAis injective, and let (x ) , + B # 0. Then x # 0, so c((x), + B ) = Q,(x) # 0. Hence Q is injective. 0
+
+
For a group A, the dual A* of A is the subgroup of Z" consisting of all homomorphisms from A to E. For a homomorphism cp: A + B, the dual cp* :B* + A* of cp is defined by cp*(h) = h 0 cp for h E B*. 6.3. Theorem. If(A,, cp,, A) is a direct system of groups, then (A?, cp?,, A) is un inverse system and there exists an isomorphism T : (lim, A,)* --* lim, A: making the following diagram commute:
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= ,p:r nf for each 1,p E A with 1 < p, it follows from Lemma 6.1 that there exists a homomorphism z making (6.5) commute. To show that t is surjective, let h E lim-A:. Then, e,(h) rplr = e,(h) for 1 < p. Combining this with Lemma 6.2 (2), we get g E (lim,A,)* satisfying that en@) = gon, for each 1 E A. Since q,(z(g)) = n:(g) = gon, = e,(h), t ( g ) = h. To show that t is injective, let t ( f ) = 0, wherefE (lim,A,)*. Then, for each 1E A, f o n, = n:(f) = e , ( t ( f ) ) = 0 by (6.5), and hence f(n,(x)) = 0 for each x E A,. By Lemma 6.2 (l), this means that f = 0. Hence t is proved to be an isomorphism. 0
Proof. Since nj'
0
0
6.4. Corollary. For groups A,, nlch
1 E A, (@,,,Aa)* is naturally isomorphic to
A:.
Proof. In case A is finite, the map Q,: (@,,,A,)* + lI,,,A: defined by @(h) = (h ( ),, is an isomorphism. For the general case, let A<wbe the 0
collection of all finite subsets of A ordered by the set inclusion. Then there exists a natural injection:,pr @,,,A, + for each E, F E A<wwith E c F, and @,,,A, is isomorphic to the direct limit of ( @ , , , A , , qEF,A<w). Dually, lI,,,A: is isomorphic to the inverse limit of (lI,,,A:, eEF,A<w), where eEF: lIIEFA: -+ lI,,,A: is the projection. Since (@,,,A,)* 1: lI,,,A: and rp& = eEFfor each E, F E A<wwith E c F, the corollary follows from 0 Theorem 6.3. The validity of the dual form of Corollary 6.4 depends on the N-compactness of the discrete space A. To show it, we prepare two fundamental theorems together with some notation. Let { x , , } , , ~be~ a sequence of elements of a direct product lIrlsAAlsuch that, for each 1E A, x,,(A) = 0 for all but finitely many n's; then the element ZnENx,,of l I I E A A Iis defined by (ZnENx,,)(l) = ZnSNx,,(1) for 1E A. The elements e,, 1 E A, of Z" are defined by e a ( p ) = 1 if p = 1,and e l @ ) = 0 otherwise. 6.5. Theorem. For each h E (Z")*, h(e,,) = 0 exceptforjinitely many n E N.
Proof. Suppose the contrary. Then, by using a homomorphism from ZNto hN,we get a homomorphism h : ZN + Z such that h(e,) # 0 for all n E N. Choose an increasing sequence { u , , } , , ,c ~ Z such that a,,,,! does not divide a,!h(e,,)for each n E N. Now, think of the set M = {Z,,,,e,,a,,!e,,: E,, = 0 or l}. Since I MI is uncountable, there exist distinct elements x, y E M such that h(x) = h( y ) . Then, x - y = ZnGN6,,u,,!e,,, where 6, = - 1,O or 1. Let m be
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487
the least number such that 6, # 0; then = h(x - y ) = 0.
Since h(&,,,6,,an!e,,) = a ~ + , ! h ( ~ , ~ ~ 6 , , a , , ! ( a m + this , ! ) ~ implies ~ e , , ) , that G,a,!h(e,) can be divided by a,,,!, which contradicts the definition of {an 0 For a topological space X and a topological group B, C(X, B) is a subgroup of B X .Now, assume that X is 0-dimensional. For a fixed subgroup A of C(X, B) and h E A*, a subset S of /3,X is called a support for h if for any f~ A, S c Cl,f-'(O) implies h ( f ) = 0, where CI, indicates the closure in PDX.A support in X(resp. P N X )means a support included in X(resp. P N X ) . From now on, for an elementf E C(X, B), or more generallyfE nxEX B,, and U c X , f l U denotes the element defined by (fI U ) ( x ) = f ( x ) if x E U , and (fl U ) ( x ) = 0 otherwise. 6.6. Theorem. Let X be a 0-dimensional space, B a group equipped with the discrete topology and A a subgroup of C(X, B ) satisfying the following conditions: (1) I f f € A and U E CO(X), thenfl U E A . (2) If {fn}neNc A and { X - f.-'(O): n E N } is locally jinite in X , then L v f . E A. Then each h E A* has a compact support in PNX .
Proof. Let Y be a collection of all compact supports for h, and partially 9, Y # 8. Let 9'be any nonempty order Y by inclusion. Since ~ , X E chain in Y ,and put So = n{S' : S' E Y ' } .If So c Cl,f-'(O), then there is S' E 9' such that S' c Cl,f-'(O), for Cl,f-'(O) is clopen by Corollary 2.6. Thus h ( f ) = 0, which implies that SoE 9. Hence, by Zorn's lemma, there exists a minimal element S of 9. Now, we show that S c PNX.Suppose not, and pick p E S - PNX. Then, by Theorem 2.8, there exists a decreasing c co(P,X) such that p E n , , N u n c PDX - X . For sequence {Un}nsN each n E N , by the minimality of S, there exists f n E A such that S - U,, c Cl,f.-'(O)and h ( f . ) # 0. Define g,, = f.I (X n U,);then g, E A by (1). Since {X n IY,,},,~~ is locally finite in X , (2) allows us to define a homomorphism cp: ZN -,A by cp(a) = ZnrNang,,for a E ZN. Since (f.- g , ) - W ) = f.-'(O) u ( X n U A clD(f. - gn)-'(O) 3
clDf .- '( o)
clD(X
( S - U,) u
u,, 3 s.
un)
K. Edo. T. Kiyosawa. H.Ohta
488
Since S is a support for h, h ( f , ) - h(g,,) = 0. Hence, (h 0 cp)(e,,) = h(g,,) = h ( f , ) # 0 for each n E N, which contradicts Theorem 6.5. 0
6.7. Corollary. Let X be a O-dimensional space. Then each h E C ( X , Z)* has a compact support in X if and only if X is N-compact. Proof. The “if” part follows from the preceding theorem. To prove the converse, suppose that X is not N-compact; then there exists p E BNX - X . Define h E C(X, Z)* by h( f ) = (BNf ) ( p ) ,and let S be a compact support for h in X . Then there is U E C 0 ( B D X ) such that p E U and U n S = 8. Let f E C(X, Z) be a characteristic function of U n X; then 0 = h( f ) = ( f l N f ) ( p ) = 1, a contradiction. 0 We are now in a position to prove the dual form of Corollary 6.4. First, we prove it in the general form. Let A A ,A E A, be groups and consider A to be a discrete space. For a countably complete ultrafilter p on A, we define K, to be the subgroup of the direct product A = nlcA Al consisting of all x such that { A E A : x , = 0 ) E p . The ultraproduct Alp is the quotient group A/K,. Let [ 1,: A + A / p denote the quotient map. With p regarded as a point of BNArwe get the following theorem. 6.8. Theorem. For the direct product A of groups A A ,A E A, A* is naturally isomorphic to @,,,BN,,(A/p)*. Moreprecisely, the map Y :@,,,BN,,(A/p)* + A* defined by Y (h) = ZPPEBN,, h,, [ I,, is an isomorphism. 0
Proof. Since Y is a homomorphism clearly, we show the bijectivity of Y . First, to prove that \y is injective, consider h E @,,B,A(A/p)* with Y ( h ) = 0. Then, h, = 0 except for finitely many p’s, say p I , . . . , p,, E PNA, so Y ( h ) = Zy=,h,,, [ I,, . There exists a partition {Al, . . . ,A,,} of A such that A, E p, for each i = 1, . . . ,n. We letj E { 1, . . . , n } be fixed for a while. For each x E A, since [ xI A,],, = 0 unless i = j , 0
h,,,([xl,,,) = h,,,([x I A,],,,) + h,,,([X I (A -
A,)Ip,)
c hp,([xIA,lp,) + h,,,([Olp,) n
=
I=
I
= (+(h))(xlA,)
+0
=
0,
and hence hp, = 0. Sincej is arbitrary, h = 0, proving that Y is injective. Next, to prove that Y is surjective, let g E A*. The direct product A can be considered naturally as a subgroup of C(A, @lshAA), where @,,,,A, is
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equipped with the discrete topology. Since A satisfies the conditions of Theorem 6.8, g has a compact support S in PNA.By Lemma 4.9, S is a finite set, so we put S = {ql, . . . , q m } .Choose a partition {MI,. . . , M,} of A such that MI E q1for each i = 1, . . . , m. For each i, definef; E (A/q,)* by 1;([x],,)= g ( x l M I ) for x E A . To see that this is well defined, suppose [x],, = 0. Then, since x-'(O) E q,, (xlM,)-'(O) E ql. For j # i, since M, c ( X I M,)-'(O), ( X I Ml)-'(0) E q,. These imply that S c cl&,,,(xl Ml)-'(0), and hence g ( x 1 M , ) = 0, because S is a support for g. Finally, define JE if p = qi, and f , = 0 otherwise. Then, for each
6.9. Corollary. If1 A I is nonmeasurable, then (lliE,, A,)* is naturally isomor-+ (IIi,AAi)* defined phic to @i,,AX. Moreprecisely, the map Y : @,,,,AX by Y ( h ) = Cit,,hi el is an isomorphism. 0
Proof. Since 1A1 is nonmeasurable, it follows from Theorem 3.4 that the discrete space A is N-compact. Hence each countably complete ultrafilter p on A is fixed, i.e., p = {M :p E M c A} for some p E A, which implies that (lliEAAi)/p2: A,, and [ Ip = e,,. Hence this follows from Theorem 6.8.
0 6.10. Remark. Theorem 6.3 and Corollary 6.4 remain true with the same proofs if dual groups are replaced by the groups Horn(-, C) which consist of homomorphisms to any group C. However, this is impossible for Theorem 6.8 and Corollary 6.9 unless C is a so-called slender group. For information about slender groups, see Fuchs [1973]. As an application, we show the reflexivity of Z" and Z").For a group A , there exists a canonical homomorphism 8 : A + A**, i.e., (8(x))(h)= h ( x ) for x E A and h E A*. A group A is said to be rejlexive if 8 is an isomorphism. The easy proof of the following lemma is left to the reader.
6.11. Lemma. For groups A and B, the following are valid: (1) The canonical homomorphism 8 : A + A** is injective i f and only i f A is isomorphic to a subgroup of Z"for some A. ( 2 ) I f A is reflexive, then so is A*. ( 3 ) I f A @ B is reflexive, then so is A .
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6.12. Corollary. If1 A I is nonmeasurable, then both Z" and Z(") are reflexive. Conversely, if Z(") is reflexive, then I A I is nonmeasurable.
Proof. For each f E Z*, there exists unique c( f ) E H such that f ( 2 ) = [( f ) z for each z E Z. Since c : Z* + Z is an isomorphism, we can identify Z* with Z. Under this identification, Corollary 6.9 tells us that if I A ( is nonmeasurable, then the map \y : Z(") -, (Z")* defined by "(a) = Zlr,,a,e, is an isomorphism. By Lemma 6.1 1 the canonical homomorphism 0, :Z" + (Z")** is injective. To prove that 8, is surjective, let q E (HA)**.Define x = (q(ei)),,,, E Z". Then, for each a E if("), (O,(x))(Y(a))= Z,,,,a,e,(x) = q(Y(a)), and hence O,(x) = q . Thus 8, is surjective and therefore Z" is reflexive. Since Z'") N (HA)*, so is Z(") by Lemma 6.1 1. Conversely, assume that Z("is )reflexive. If 1 A I is measurable, then there exists p E BNA - A by Theorem 3.4. Define g E (Z")* by g( f ) = (PNf ) ( p ) for f E Z". Combining Corollary 6.4 with the early remark, we get a natural isomorphism 0 :(Z("))* + Z". Since g o 0 E (Z("))**, there exists y E Z(") such that O,( y) = g 0 0, where O2 : Z(") + (Z("))** is the canonical homomorphism. Since y # 0, y , # 0 for some p E A. Then, (O2(y))(e,) = e , ( y ) = y , # 0, while (go@)(p,) = 0, because 0 ( e , ) = ( c ( e , o ( = e,. This contradicts the fact that O,( y ) = go 0. 0 By the preceding corollary, the assumption that I A I is nonmeasurable cannot be removed from Corollary 6.9. We shall show later that the second statement remain valid if Z(") is replaced by Z". A free group is a group which is isomorphic to Z(") for some A. Since all vector spaces have Hamel bases, groups C(X, R) are isomorphic to R(") for some A. By contrast, C(X,Z) is not always free; for example, C(Z, Z) is not free (cf. Fuchs [1970, Theorem 19.21). Next, we consider a space Xfor which C(X, Z) is free. It is easily checked that a group A is free if and only if A includes a subset {e,},,, satisfying that (1) the equality Zy=,aiei,= 0, ai E Z, li E A, i = 1, . . . ,n, n E N,implies = a, = 0, and a, = (2) for any x E A, there exists a E Z(") such that x = C,,,aien. In this case, {el}ieAis called a free base of A . For a homomorphism cp : A + B, the subgroup { x :cp(x) = 0 } of A is denoted by ker(cp). A subgroup S of a group A is a summand of A if there exists a surjective homomorphism 6 :A -, S such that 006 = 6,i.e., A = S @ ker(6). 6.13. Proposition. Let cp be a surjective homomorphismfrom a group A to a free group B, then ker(cp) is a summand of A , and hence A N ker(cp) @ B.
49 I
N-compactness and its Applications
Proof. Let {eA}AG,, be a free base of B. For each 1E A, choose an element r(eA)E A such that cp(z(ei)) = e l . Then, T can be extended to an injective homomorphism from B to A by setting z(XiE,,aAeA)= XLE,,aiz(ei) for a E Z(’). Define Q ( X ) = x - z(cp(x))for x E A. Then, Q Q = Q and CJ maps A onto ker(cp). Since ker(a) = z ( B ) N B, A N ker(cp) @ B. 0 0
We prove Bergman’s theorem on the line of Hill’s proof. It implies Nobeling’s theorem [1968] asserting that the subgroup of Z” consisting of all bounded functions is free and consequently the group C(X, Z) is free for a compact space X. A group A is torsion free, if nx = 0 implies x = 0 for x E X and n E N . By R + , we mean the group structure of a ring R. An element e of a ring R is an idempotent if e e = e. Especially, if R is C(X, E), then its idempotents are just characteristic functions of clopen sets in X . For the next theorem the existence of 1 is not necessary as shown by Hill, but it simplifies the proof.
-
6.14. Theorem. Let R be an associative and commutative ring with 1 which is generated by its iakmpotents as a ring. If R is torsionfree, then R is free. +
Let R be the ring stated in the theorem. Before proving the theorem, we prove an auxiliary result on R. By the assumption, every x E R is of the form x = Z:,,a,e,, where a, E Z and e, is the finite product b,, . . . blm(,) of idempotents. Now, we show that e l , . . . , en can be chosen so as to be pairwise orthogonal, i.e., i # jimplies e,e, = 0. Let {bl, . . . , b,} be the enumeration of{b,,:j = 1, . . . , n(i), i = 1, . . . ,n}.Putb,, = 1 - b,,bIl = b,foreach i = 1, . . . ,m, and let Fbe the set of all functions from { 1, . . . , m } to (0, l } such that blf(l)* * bmf(,) # 0. Then the elements ef = blf(l). . * bmf(,),f E F, are pairwise orthogonal. Since b, = X { e f :f E F,f ( i ) = l } for every i, x is expressed as a linear combination of these e,’s. Moreover, this form of x is unique. To show this, it suffices to prove that Cf=,a,e,-= 0 implies a, = * * * = ak = 0. Multiply both sides by e,-;then ale,- = 0 because of the pairwise orthogonality. Since e,, # 0 and R + is torsion free, a, = 0. Proof of Theorem 6.14. In case R is generated by finitely many idempotents bl, . . . , b,, then the set {e,},EFof pairwise orthogonal idempotents made from them as above is a free base of R + . Otherwise, let {ei}u
{0}
=
Ro c R I c
c R, c
...,
K. Eda, T. Kiyosawa, H. Ohta
492
and R, = Us,,R, for a limit ordinal a. If we show that R;+,/R,+ is free for each a < 1,then RJ+, N R: 0 (R;+,/R;) by Proposition 6.13, and hence successively adding free bases we can get a free base of R + . To do this, let a -= 1. For simplicity of notation, let e, = e and R, = S. Then R,+, = S Se, since 1 belongs to S. By the homomorphism theorem, (S Se)/S is isomorphic to Se/(Se n S) as groups. Since Se n S is an ideal of the ring Se and Se is a ring with 1 generated by a set of idempotents of cardinality less than 1, Se/(Se n S ) is a ring with 1 generated by a set of idempotents of cardinality less than A. To show that (Se/(Se n S ) ) + is torsion free, let u E Sand nue E S for an n E N.It suffices to show that ue E S. As we have noted before the proof, there exist pairwise orthogonal idempotentsf;, and ai, bi E E , i = 1, . . . , k, such that
+
+
k
u =
1 aif;
k
and nue
=
1 bif;. i= I
i= I
Let J = { i : A e # 0 } and u' = XicJaif;.Then for eachj 4 J, b= j$ = 0 because of the pairwise orthogonality, so
1 bif; = isJ
nue =
1 naiJe.
ic J
Multiplying f;e, we get b,Ae = na,f;e for each i E J. Since (S + Se)+ is torsion free, bi = nai, and hence nue = nu'. Since S + is torsion free, ue = u' E S. Now, Se/(Se n S) satisfies the conditions of the theorem and is generated by a set of idempotents of cardinality less than 1. By our induction hypothesis, (Se/(Se n S ) ) + is free, and hence so is R,++,/R:. 0
6.15. Corollary. Let X be a compact space. Then C(X, Z) is free, and hence C(X, Z) is reflexive if and only if I X I is nonmeasurable. Proof. Since Xis compact, C ( X , E) is generated by the set of characteristic functions of clopen sets. Hence it follows from Theorem 6.14 that C(X, Z) 2: E(") for some A. By Lemma 3.5,l XI is nonmeasurable if and only if so is 112 1. Therefore the latter statement follows from Corollary 6.12. 0 6.16. Remark. In view of Remark 2.3, the preceding corollary can be slightly generalized as follows: If P N X is compact, then C ( X , Z) is free. It is known that the converse is also valid (cf. Eda and Ohta [1987]).
Unlike the ring structure or the lattice structure, the group structure of C(X, Z) does not determine the topology of an N-compact space X as follows.
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493
6.17. Corollary. Let X and Y be 0-dimensional, compact spaces. Then, the groups C ( X , Z) and C( Y , Z) are isomorphic if and only if w ( X ) = w( Y ) .
Proof. We only need to consider infinite spaces. Since the group structure of a free group is determined by its cardinality, it suffices to prove that I C ( X , Z) I = w(X)for any infinite, 0-dimensional, compact space X . Such a space X has a base 53, with 1531 = w ( X ) , consisting of clopen sets (cf. Engelking [ 1977, Theorem 1.1.151). Since X is compact, C(X, Z)is generated by the set of characteristic functions of members of 9?, and hence I C(X, Z) I = w(X). 0 The final purpose of this section is to determine for what kind of spaces X the group C(X, Z) or C ( X , Z)* is reflexive. To this end, we need some notation and lemmas. In the rest of this section, for brevity, we write C ( X ) instead of C(X, Z). Recall from Proposition 4.1 that K ( X ) = K ( k , X ) for any 0-dimensional space X . For each K , L E K ( X ) with K c L, let y K L : C(L) --* C ( K ) ,yx : C ( X ) + C(K) and rK: C ( k , X ) + C ( K ) denote the restrictions. The easy proof of the following lemma is left to the reader.
6.18. Lemma. I f X is a 0-dimensionalspace and K E K(X), then yK is surjective. Order K ( X ) by inclusion. Then (C(K)*,y&, K ( X ) ) forms a direct system. We denote the subgroup of C ( X ) * consisting of all elements having compact supports in X by C(X)’.
6.19. Lemma. Let X be a 0-dimensional space. Then there exists an injective homomorphism rs for which the diagram
I. \ C(K)*
lim C(K)* -+
(K E
(6.6)
C(X)*
commutes. Moreover, the range of rs is C(X)’,and hence C(X)‘ N lim,C(K)*. Proof. If K c L, then yK = y K L 0 yL, and hence y$ = yZ y z L . Since y K is surjective by the preceding lemma, y g is injective. Hence, it follows from Lemma 6.2(2) that there exists an injective homomorphism rs making (6.6) commute. Since the range of y g is included in C ( X ) ’ for each K E K ( X ) , the range of 0 is included in C(X)’.Conversely, let h E C(X)’ and K a compact support for h in X . Then there exists g E C ( K ) * such that h = go yK. Since h = $(g) = a(nK(g)),h is in the range of 0 . 0 0
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K. Eda. T. Kiyosawa, H . Ohia
For a 0-dimensional space X , K , : C ( X ) + C ( k , X ) denotes the injective homomorphism defined by I C ~f () = f k , for f E C ( X ) , where kN is the identity from k,X to X , and 8, denotes the canonical injection from C ( X ) to C(X)**. 0
6.20. Lemma. Suppose that X is N-compact and each K E K ( X ) is of nonmeasurable cardinality. Then there exists an isomorphism z making the following diagram commute.
Proof. Since X is N-compact, it follows from Corollary 6.7 that C(X)’ = C ( X ) * .Hence, the homomorphism CT in Lemma 6.19 is an isomorphism. By (6.6), the diagram
-
c(x)**
1(lim c(K)*)*
(KElvX))
(6.8)
C(K)** commutes. By Theorem 6.3, there exists an isomorphism v: (lim, C(K)*)* -+ lim, C(K)** making the left triangle in (6.9) below commute, where & is the projection. By Corollary 6.15 and our assumption, each C ( K ) is reflexive. Since 8, o yKL = yzf 0 8, for each K c L, there exists an isomorphism 8 for which the middle square in (6.9) is commutative. Since K ( X ) = K ( k , X ) , (I-,( f ) ) K E K ( X ) E lim, C ( K )for each f E C ( k , X ) . This allows us to define an isomorphism ( making the right triangle in (6.9) commute, where eK is the projection.
(K E K(X))
(6.9)
Now, define z = ( - I o ~ - ~ o v o c T * . We show that (6.7) is commutative. Since r, 0 IC, = Y,, it suffices to show that r, 0 T 0 8, = yK for each K E K ( X ) .
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By (6.8) and (6.9), rKoz =
(6.10)
ei'Oy;*.
On the other hand, it is easily checked that that r K o T O e= x yK.
y t * 6, 0
=
eKo
y K . These imply
6.21. Theorem. Let X be an N-compact space. Then, C ( X )is reflexive ifand only i f each K E K ( X ) is of nonmeasurable cardinality and k,X = X. Proof. Assume that each K E K ( X ) is of nonmeasurable cardinality. Then, by Lemma 6.20, C ( X )is reflexive if and only if K, is surjective, i.e., k N X = X by Proposition 4.7. It remains to show that if C ( X ) is reflexive, then each K E K ( X ) is of nonmeasurable cardinality. By Corollary 6.15, C ( K ) is free, so it follows from Proposition 6.13 and Lemma 6.18 that C ( K ) is a summand of C ( X ) . Therefore, if C ( X ) is reflexive, so is C ( K ) by Lemma 6.1 1 . Hence it follows from the second statement of Corollary 6.15 that I KI is nonmeasurable. 0 6.22. Corollary. r f Z" is reflexive, then I A I is nonmeasurable. Proof. Consider A to be the discrete space; then Z" N C(A) 1: C(BNA). If I A I is measurable, then k N ( B N h )# BNA by Example 4.10. Moreover, by Lemma 4.9, each K E K(fiNA) is finite. These combined with Theorem 6.21 imply that Z" is not reflexive. 0 6.23. Theorem. Let X be an N-compact space. Then C(X)* is reflexive ifand only i f each K E K ( X ) is of nonmeasurable cardinality and k,X is N-compact.
Proof. By Proposition 4.1, k,X is 0-dimensional and K ( k , X )
K(X). Hence it follows from Lemma 6.19 that there exist injective homomorphisms 0 , and n2 with the following commutative diagram. =
Here, the range of 6'is C(X)' and that of q is C(k,X)'. Since Xis N-compact, C(X)' = C(X)* by Corollary 6.7, and hence 0 , is an isomorphism. Now, assume that each K E K ( X ) is of nonmeasurable cardinality. Let z be the
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K. Eda, T. Kiyosawa, H . Ohta
isomorphism defined iq Lemma 6.20 and 8 the canonical injection from C(X)* to C(X)***.We show the following diagram is commutative.
-
+ I r*
lim c(K)*
(6.12)
02
C(k,X)*
C(X)***
Let h E lim,C(K)*. Then by Lemma 6.2(1), there exist K E K ( X ) and g E C(K)* such that h = z&). By (6.10) and (6.1 l), (z* az)(h)= z*(T,*(g)) = g o Oil o y;*. Hence, for each q E C(X)**, 0
((t* O 6 2 )(h))(rl) =
=
d o , I (rl
O
7;)
rl(a,(h)) =
= (rl O Y
X d
W I ( ~ ) ) ) ( r l ) 9
which proves that (6.12) commutes. Consequently, C(X)* is reflexive if and only if azis surjective. Since the range of azis C(k,X)', this is equivalent to the N-compactness of k,X by Corollary 6.7. Finally, assume that C(X)* is reflexive. For each K E K ( X ) , C ( K ) is free, i.e., C ( K ) N Z(") for some A. By Corollary 6.4, C(K)* II Z". Since C ( K )is a summand of C ( X ) , C(K)* is a summand of C ( X ) * ,and hence C(K)* is reflexive by Lemma 6.11. Therefore, it follows from Corollary 6.22 that I A 1, and hence I KI, is nonmeasurable. 0 6.24. Examples. By Theorem 6.21 and Proposition 4.4, both C(Q) and C(R - Q) are reflexive. Let X be the space in Example 4.13. Then it follows from Theorems 6.21 and 6.23 that C ( X ) * is reflexive, but C(X)is not. If X is the space in Example 4.12, then C ( X ) * is not reflexive. Moreover, consider the product space Z" and its subspace Z("). If I A I is measurable, they include compact subsets of measurable cardinality; for example, {el}lG,, u ((0, 0, 0, . . .)}. Hence, by Theorem 6.21 and Example 4.14, C(Z^), or C(Z(")), is reflexive if and only if I A I is nonmeasurable. 6.25. Notes. Lemmas 6.1 and 6.2 are Theorems 12.1 and 11.1 of Fuchs [1970], respectively. Theorem 6.3 is due to Cartan and Eilenberg [1956]. Corollary 6.4 is Theorem 43.1 of Fuchs [1970]. Theorem 6.5 is Theorem 1 of Specker [1950], and the proof here is due to Sqciada [1959]. Theorem 6.6 and Corollary 6.7 are modifications of Lemma and Theorem of Mrowka [1972a], respectively. Theorem 6.8 is Corollary 2 of Eda [1982], Corollary 6.9 is due to tos, and is Corollary 94.5 of Fuchs [1973]. Lemma 6.1 1 and Corollary 6.12 are folklores. Proposition 6.13 is Theorem 14.4 of Fuchs [1970]. Theorem
N-compactness and its Applications
491
6.14 is Theorem 1.1 of Bergman [1972]; the proof here is due to Hill 119731. Corollary 6.17, Lemmas 6.19 and 6.20, and Theorems 6.21 and 6.23 are due to Eda and Ohta [1987]. The existence of a non-reflexive group A* of nonmeasurable cardinality was proved around 1982 under some set theoretic hypothesis (i.e., 0-axiom) by Sageev and Shelah. Some improvement by Eklof, Mekler and Shelah followed. The existence of such a group in ZFC was firstly shown by the method developed in this section. Further developments on non-reflexive groups A* will be published in the forthcoming book of Eklof and Mekler. 7. Applications to non-Archimedean Banach spaces There are striking analogies between certain properties of various objects in mathematics. For example, we have already seen that both the ring structure and the lattice structure of C(X,Z) determine the topology of X . In this section, the reader will observe that Banach spaces over certain nonArchimedean valued fields have much similar properties on the reflexivity to those of abelian groups. In fact, roughly speaking, all the results in the preceding section remain true for such Banach spaces. It is an open question whether there exists a unified viewpoint under which we can regard them as same results. We begin with the definition of a valuation, which generalizes the absolute value on R or the complex number field.
7.1. Definition. A valuation on a field K is a map I I : D6 + R satisfying the following conditions, for all a, b E K: (1) la1 2 0, la1 = 0 if and only if a = 0, (2) lab1 = la1 * l b l , (3) la bl < la1 + Ibl. A non-Archimedean valuation is a valuation I I satisfying the strong triangle inequality: (4) la bl < max{lal, lbl}; a, b E K. A field K provided with a (non-Archimedean) valuation I I is called a (nonArchimedean) valuedfield and is denoted by (K, I I), or simply H.
+
+
An obvious example of a non-Archimedean valuation is the trivial valuation which can be defined on any field by setting I a I = 1 for a # 0 and I 0 I = 0. Let p be a positive prime. Then any non-zero x E Q can be written in the form x = p " ( k / l ) ,where n, k , I E Z and k , 1 are not divisible by p . Define 1x1, = p-" and 101, = 0. The map I I p is a non-Archimedean valuation, called the p-adic valuation, on Q.
K. Eda, T. Kiyosawa, H . Ohta
498
Every valued field (H, I I) is a metric space with the metric d(a, b) = la - b I. The closed ball with center a E H and radius E , i.e., { b E H : d(a, 6 ) < E } , is denoted by B(a, E). The easy proof of the following lemma is left to the reader.
7.2. Lemma. Let K be a non-Archimedean valued field, a, b E K, and E 2 6 > 0. Then, either B(a, E ) n B(b, 6 ) = 0 or B(b, 6 ) c B(a, E), and hence any closed ball is clopen in H. 7.3. Proposition. A non-Archimedean valuedjeld H is strongly 0-dimensional.
Proof. Let Eand F be disjoint zero-sets in H. For each a E E, choose E, > 0 such that B(a, E,) n F = 0,and put U = U{B(a,E , ) : a E E}. Then, E = U, U n F = 8, and U is clopen by Lemma 7.2. 0 7.4. Corollary. A non-Archimedean valuedjeld K is N-compact fi and only if 1 H I is nonmeasurable.
Proof. Combine Proposition 7.3 with Theorems 3.6 and 3.10.
0
A metric space Xis called spherically complete if every decreasing sequence of closed balls has a nonempty intersection. Every spherically complete metric space (X, d ) is complete. To show this, let { x , , } , , be ~ ~a Cauchy sequence in (X, d ) . Choose a subsequence { x , i } i E such N that for each i E N, m 2 nj implies d(x,,,, x,,) < 2-i-'. Then, {B(xni,2-i)}iENis decreasing, and hence nieNB(xn,, 2-i) # 0. Since { x , , } , , . converges ~ to the point of this intersection, (X, d ) is complete. The converse, however, is false as the foliowing example shows. 7.5. Example. Define the metric d on N by d(m, n) = max{l + m-I, 1 + n-I} if m # n, and d(m, n ) = 0 otherwise. Then every Cauchy sequence in (N, d ) is eventually constant. Hence (N, d ) is complete, while it is not spherically complete, because {B(n, 1 + n - l ) : n E N } is decreasing and n n E N B ( n1 , n - ' ) = 0. 17
+
A valued field H is called (spherically) complete if it is a (spherically) complete metric space. The p-adic number field Qp, the completion of (Q, I Ip ) , is the most important example of a spherically complete, non-Archimedean valued field. A complete, non-Archimedean valued field can fail spherical completeness, and this fact is crucial in the subsequent discussions.
N-compactness and its Applications
499
A typical example is the completion C, of the algebraic closure of Q,. For details of Q, and C pand more information about spherically complete metric spaces, see Schikhof‘s readable book [1984]. 7.6. Definition. Let A be a linear space over a non-Archimedean valued field K. A non-Archimedean norm on A is a map (1 (I : A + R satisfying the following conditions, for all x, y E A : ( I ) (1 x 11 2 0, 11 x (1 = 0 if and only if x = 0, (2) llaxll = l a l - l l x l l ; a E K , (3) IIX + YII G max{lIxll, IIYII). A linear space provided with such a norm is called a non-Archimedean normed space over K. A non-Archimedean normed space is a metric space with the metric d ( x , y ) = I( x - y I), and is called a non-Archimedean Banach space over K if it is a complete metric space. From now on, let K be an arbitrary, fixed, complete, non-Archimedean valued field. Unless otherwise stated, by a normed (Banach) space we mean a non-Archimedean normed (Banach) space over K. The term “subspace” always means a linear subspace. The field K is itself a Banach space, where the norm is the valuation. Let x,, n E N , be elements of a normed space. When the sequence { Z i q n x i : nE N } converges, the limit is denoted by C,,,x,. The following lemma will often be used without warning the reader.
7.7. Lemma. For elements x,, n E N , of a Banach space X , the following are valid: (1) r f I1 XI II < 1 I x2 II then It XI + x2 II = II x2 11. (2) r f ll x, II -+ 0 (n + 001, then ZnENx,exists and II LA, (I < maxnEN II x, II. 9
Proof. Since IIXIII < 11x1 + x2II G IIx2Il. If 11x1 + XZII < Ilx2II, then (1 x2 ( I < max{ I( xI x, (I, 11 - x I [I} < 11 x211, a contradiction. Thus ( I ) is proved. If )I x, 11 + 0, then it is easily checked that {Ziqnx,: n E N } is a Cauchy sequence. Since X is complete, the sum exists. The last inequality follows from (I). 0 1 1 ~ 2 1 1 9
+
Let A and B be normed spaces, and cp a linear operator from A to B. Throughout this section, all linear operators are assumed to be continuous. Just as in the usual functional analysis, cp is continuous if and only if there exists M 2 0 such that )I cp(x) 11 G M 11 x I( for each x E A . We define the operator norm of cp to be the infimum of the set of all such M’s, and denote
K . Eda, T. Kiyosawa. H. Ohta
SO0
it by 11 cp 11. As is easily seen,
IIcpII = s u P ~ I I c p ( x ) I l / l l x l l : ~ ~ Az, x0). When IIcp(x) )I = IIxIJ for each x E A , cp is called a linear isometry. An isomorphism is a surjective, linear isometry. If there is an isomorphism from A to B, then we call A and B isomorphic and write A N B. The easy proof of the following lemma is left to the reader.
7.8. Lemma. Let cp be a linear operatorfrom a dense subspace S of a Banach space A to a Banach space B. Then cp can be extended uniquely to a linear operator CP from A to B. Moreover, ifcp is an isometry, so is CP, and if@) is dense in B, then is surjective. Let A be a normed space. A linearfunctional on A is a linear operator from A to K. The dual A* of A is the space of all linear functionals on A , where addition and scalar multiplication are defined pointwise and the norm of h E A* is the operator norm. Since K is complete, it is easily checked that A* is a Banach space.
7.9. Theorem (Non-Archimedean Hahn-Banach theorem). Assume that K is spherically complete. Then, for each normed space A and its subspace S, f E S* can be extended to F E A* such that (1 f I( = 11 F (1. Proof. By use of Zorn’s lemma, we only need to settle the case where A = { x a y : x E S, a E K> and y 4 S. For each x E S, let r ( x ) = Ilfll II x - Y II * Then I f (4 - f (x’)I < II f II II x - x’ II < max{r(x), r < 4 > for x , x‘ E S. Thus by Lemma 7.2,
+
-
-
r(x) < r(x’) implies B( f ( x ) ,r(x)) c B( f (x’), r(x’)). Since K is spherically complete, there is z E n,,,B( f ( x ) , r(x)). Define F E A* by F(x ay) = f ( x ) + az. Then, since
+
IF(x
+ aY)I
= IaI.If((-a)-lX) -
< (I F II
< (1 f 11.
144-a)-1x)
ZI
=
Since F is an extension off, 11 F (1 2
I I S I I * I l X + VII, 1) f (I.
0
We shall show in Remark 7.15 that spherical completeness of K is essential in Theorem 7.9. From here we shall proceed parallel to the preceding section. Let A l , I E A, be Banach spaces. Then the product nle,,Al is a linear space under
N-compactness and its Applications
50 I
coordinatewise addition and scalar multiplication. The subspace X
iEAA,
= {x:
I1 x i )I <
+ a}
of lI,.,,A, is a Banach space under the norm defined by 11 x 11 = supiE,, 1) xi 11, and is called the direct product of { A j , } , € A . The direct sum of { A i } , € , , is the closed subspace @,,AA,
of X
is,,
=
{ x : for each E > 0, I (1: 11 xi. 11 2
A,. As is easily seen,
a
E}
I < a}
A, is the completion of the dense subspace
= { ~ : 1 { 1#: ~ 011 , <
O}
of it. In case all A, are copies of K, X i o A A i (resp. a i e A A j ~is )denoted by P ( A ) (resp. co(A)) and, in particular, P ( N ) (resp. c o ( N ) )is denoted by I" (resp. co). Next, we introduce inverse limits and direct limits of Banach spaces. Suppose that A is a directed set by order < ,and that for any 1,p E A with 1 < p a linear operator cpj,,, : A,, 4 A , is defined in such a manner that (pj,+ o (ppv = qlvfor 1 < p < v and (pj,j. is the identity for each 1 E A. Then the system (A,, cpplp, A) is called an inverse system. The inverse limit of ( A , , (pi,,, A), denoted by lim, A j , ,is the subspace of X i E A A j ,consisting of all x for which (p,,,(x,,) = xi for each 1 < p. Being closed in X j.E,,Aj,,lim, Aj, is a Banach space. A) be an inverse system of Banach spaces. &f A is a Banach space and there are linear operators T ~ ,A: + A , such that SUP,^,, I( zi 11 = M < co and 7, = (pj,,, 7,, for 1 < p, then there exists a linear operator T : A + lim, A, such that 11 T 11 < M and = ej. 7 for each 1 E A (cf. Diagrams (6.1) and (6.2)). 7.10. Lemma. Let ( A i , cp,,
+
0
0
Proof. For each x E A, define 7(x) E lim, A , by ( ~ ( x ) )=~ t,j , ( x ) ,1 E A; then II x II, II t II < M . 0 7, = e j , 0 7 . Since II T ( X ) II d S U P ~ ~ AII 7j. II * II x II = Suppose again that A is a directed set by order <, and that for each 1, A with 1 < p a linear operator (pi,,: A, + A,, with (1 (pi,, 11 < 1 is defined in such a manner that (pPv (pip = (pi" for 1 < p < v and (pU is the identity for each 1 E A. Then the system (A,, cpj,,,, A) is called a direct system. Similarly to the case of abelian groups, for each x E A , , 1 E A, define (x); E a j , € , , A j . by ( x ) , ( p ) = x if p = 1, and ( x ) , ( p ) = 0 otherwise. Then, ( ) j , : A , + a i E , , A i is a linear isometry. Let B be the subspace of @ i e A A jconsisting . of all ( x i , ) , , + . . . + ( x j , J i n , x j ~E, A i , , ;li E A, i = 1 , . . . , n , n E N , such
p
E
0
502
K. Eda. T. Kiyosawa. H . Ohra
that C;==, (pi,,(xi,) = 0 for some A 2 A=,,. . . , A,,, and let C be the closure of B in @,,,,A,. The direct limit of (A,, (pi,,, A), denoted by lim,A,, is the quotient Banach space AJC, i.e., the quotient linear space with the norm 11 x + C (1 = inf,,, 11 x + y 11 ;in general, the quotient A/Fof a Banach space A is known to be a Banach space for any closed subspace F. For each A E A, the symbol wi is reserved for the linear operator from A, to lim,A, ~ C for x E A i . If A < p, then wi = w , , o (pi,,. defined by w i ( x ) = ( x ) +
(a,,,,,
7.1 1. Lemma. For a direct system (A,, (pi,,, A) of Banach spaces, the following are valid: (1) S = { ( x ) ; , C : x E A i , A E A } is a dense subspace of lim, A i . (2) For each x E Aj., A E A, II ( x ) , C II = id,,, II cpi,(x) II. (3) If A is a Banach space and there are linear operators a, : A, + A such that SUP,,,, 11 ail[ = M < 00 and a, = o,,0 (pi,, for A < p, then there exists a linear operator a : lim, A, + A such that 11 a 11 < M and a, = o ni for each I E A. Moreover, ifeach aj.is an isometry, so is o (cf. Diagrams (6.3) and(6.4)).
+
+
+
0
Proof. (1) Similar argument to the proof of Lemma 6.2 shows that S is a subspace of lim, A,. For each x E aiE,,A,, x is a limit of a sequence { x,,},,~ in @,,,,Aj,. Since S is a subspace, x,, C E S for each n E N, and { x,, + C } , , sconverges N to x + C. Hence, x C is in the closure of S, which proves (1). (2)Note that 11 ( x ) , + C 11 = inf,,, (1 ( x ) + ~ y 11 by the continuity of the quotient norm. For any E > 0, there exists p > A such that 11 cpi,,(x)11 < inf,,, It cpi,(x) II + E . Since ( - x > , + (vi,,W),,E B,
+ +
inf II ( X ) i
Y ~ B
+ Y II < II (P&)
II < inf II Cpi,(X) II
+ E.
Conversely, for any E > 0, there is z = ( z,, + . . . + (z i n J AEmB such that 1I ( x ) , z 1 I < inf,,, II ( x ) ~ y II + E . If A 4 {Al, . . . , A n } , then, since I1 (pi,, II < 1, II (pip(x)I1 < 1) x II < II ( x ) , z I( for each p 2 A. Otherwise, assume that A = I l . Since z E B, C ; = ' = , ~ p ~ , ~= ( z ,0, )for some ( 2 A , , . . . , I,. Since ( > A=,= I ,
+
Thus, in any case, II (pie(x)II proves (2).
+
+
< inf,,, II (x ) , + y 11 + E for some ( 3 A, which
503
N-compactness and its Applications
(3) By (1) and Lemma 7.8, it suffices to show that there exists a linear operator o :S 4 A such that 11 o 11 < M and oi = o 0 ni for each A E A. Assume that ( x ) ; . C = (y ) , C , where x E A , and y E A,. Choose v 2 1, p, and put z = cpiv(x) - cp,,( y ) . Then ( z ) , E C, because ( z ) , (( x ) ~ ( y ) , ) E B and ( x ) ~ ( y ) , E C . Combining this with (2), we get
+
r inb!
A,,
+
II cp&)
-
cp,c(Y) II
< inf II cp,c(Z) II
= 0.
t 2 V
Since for each ( 2 1,p,
II C i ( X )
-
OJY) II =
<
II oc(cpir(4) - Qt(cp,dY))II M II cPi.r(x) - C P , ~ ( YII ) 0, +
o i ( x ) = c p ( y ) . Hence, we can define a linear operator o : S + A by e ( ( x ) ; . C ) = o , ( x ) , as required. To show that Iloll M , let
+
<
( x )+ ~ C E S . Then by (2),
and hence 11 o 11 < M . Finally, assume that each oj. is an isometry. Then so is each (pi,,, because oi = o,, ( p i p . Hence, for each ( x ) ~ + . C E S, it follows from (2) that II (x>i. + C II = II x I1 = I1 o;.(x>II = II c ( ( x ) i . + C ) 11. 0
The dual cp*: B* + A* of a linear operator cp: A cp*(h) = ho cp for h E B*.
+
B is defined by
7.12. Theorem. If (A;,, (pi,,, A) is a direct system of Banach spaces, then (A:, ,,p:c A) is an inverse system, and there exists an isomorphism t: (lim,A,)* + 1im-A: such that n: = e j . o z for each 1 E A, where ej,is the Ath projection from lim, AX (cf. Diagram (6.5)). Proof. For each A, p E A with 3, < p, nf = cpf, 0 and II n: I1 < II X i , II < 1. Hence, by Lemma 7.10, there exists a linear operator z : (lim,Aj,)* lim-A: such that 11 z 11 < 1 and n: = eioz for each 1E A. The proof that z is surjective goes just like that of Theorem 6.3. To prove that z is an isometry, let h E (lim,Ai)*. Since 11 z (1 < 1, (I z(h) 11 < I[ h 11. Conversely, for each x E A , , 1E A, it follows from Lemma 7.1 l(2) that
By Lemmas 7.8 and 7.1 1(1), these mean that II h II = II z(h) 11.
0
504
K. Eda, T. Kiyosawa, H . Ohta
The following corollary can be proved similarly to Corollary 6.4 if one uses Theorem 7.12 instead of Theorem 6.3. 7.13. Corollary. For Banach spaces A l , A morphic to X i . A A X .
E
A, ( a i E A A i ) *is naturally iso-
To prove the dual form of Corollary 7.13, analogues of Theorems 6.5 and 6.6 are necessary. The elements e;,,1 E A, of ["(A)are defined by e i ( p ) = 1 if p = 1, and e;.(p)= 0 otherwise. In the following we prove somewhat more than we need. 7.14. Theorem. The following are equivalent: (1) K is not spherically complete. (2) (I"lco)* = (0). (3) For each h E (1")*, h(e,) + 0 (n + a). Proof. (1)+(2): Suppose, on the contrary, that (Z"/c,)* # { O } ; then there exists nonzero h E (I")* which vanishes on c,. Let E = l"/ker(h), where ker(h) = {x E 1" :h(x) = 0 } , and cp the quotient operator. Since c, c ker(h),
for each x E 1". Since the algebraic dimension of E is 1, (1) implies that E is not spherically complete. Hence, there exists a decreasing sequence {B(a,, r n ) } n Eof N closed balls in E such that nnsNB(un, r,) = 8. By Lemma 7.2, we may assume that r, > r,,,. For each n 2 2, choosey, E I" such that cp( y,) = a,,, - a,. Since 11 cp( y,) [I 6 r, < r , - , , there is z, E ker(cp) with [I y , z, 11 < r , - , . Choose xI E I" such that cp(x,) = a,, and define inducz, + x,. Then, by induction, cp(x,) = a, for each n E N . tively x,,~ = y, Put M = max{rl, II x, 11, I1 x2[I}. If II x, I1 < M , then II x,+, II < M , because 11 xnil - x, 11 = 11 y , z, 11 < r,-, < M . Thus II x, 11 < M for each n E N , which enable us to define x E I" by x(n) = x,(n), n E N . For each n 2 2,
+
+
+
II cp(x)
- a,
II < lim sup I x ( j ) I+"
x,(j)
I 6 lim sup II xI - x, 11.
]>I
I-"
131
Since II xJ - xn II 6 maXnGI<, II x,+i - x1II f o r j > i , II ~ ( x ) an II 6 r , - , . Hence, by Lemma 7.2, cp(x)
E
n w a n ,r n - , )
n>2
which is a contradiction.
=
n B(~,-,,
rn-l),
n>2
505
N-compariness and iis Applications
(2)+(3): Suppose the contrary. Then, by using a linear operator from I" to I", we get h E (l"')* and E > 0 such that Ih(e,)l 2 E for each n E N. Define a linear operator v I : I" + 1" by ( v l ( x ) ) ( n )= h(e,)-'x(n), n E N , for x E 1". Then, for each x E c,, (hov,)(x) = h
(c ,EN
(h(e.)Yx(n))e.)
Next, define a linear operator v 2 :I" for x E I", and define g E (I")* by g(x) = xu)
-+
=
JN x(n).
1" by (v2(x))(n)= x ( n
+
I), n
E
N,
+ (hov,ov,)(x) - (hov,)(x)
for x E 1". Then g ( x ) = 0 for each x E c,, but g # 0, because g ( ( l , 1, I , . . .)) = I , which contradicts (2). (3)-+(1): Definef E (co)*by f ( x ) = C f l E N x ( nIf) .K is spherically complete, then by Theorem 7.9, f admits an extension F E (I")*, and F(e,) = 1 for each n E N. This contradicts (3). 0 7.15. Remark. The preceding theorem shows that the converse of Theorem 7.9 is also valid. Thus, we can say that a spherically complete, nonArchimedean valued field is precisely a non-Archimedean valued field making Hahn-Banach extension theorem hold. Let X be a topological space and B a Banach space. In this section, we use C ( X , B ) to denote the subspace of X ,ExB,r,where B, 2: B, consisting of all bounded (i.e., sup,,, 11 f ( x ) 11 < co), continuous maps from X to B. Since C(X, B ) is closed in X .rExB.r,it is a Banach space. Assume that X is 0-dimensional, and let A be a subspace of C(X, B ) . For E > 0 and h E A*, a subset S of 8,X is called an &-supportfor h if for anyf E A , S c CI, f '(0) implies Ih(f)l < E 11 f 11, where CI, indicates the closure in PDX. For an element f E C ( X , B ) , or more generally f E X .rExB,, and U c X , f I U denotes the element defined by ( f I U ) ( x ) = f ( x ) if x E U , and ( f I U ) ( x ) = 0 otherwise.
+
~
7.16. Theorem. Assume that K is not spherically complete, and let X be a 0-dimensional space, B a Banach space, and A a subspace of C ( X , B ) satisfying the following conditions: ( I ) I f f E A and U E C O ( X ) ,then f I U E A. (2) If{f,},,EN c A , { X - f,-'(O): n E N } is locallyfinite in X , andZ,,,f, E C ( X , B ) , then L s N fEfAl, where ( & t E ~ f n ) ( x ) = C f l p N f n ( X ) f o r x E X.' 'Only for this theorem and its proof, we use the symbol Z with different meaning from that of sum defined before Lemma 7.7.
K. Eda. T. Kiyosawa, H . Ohta
506
Then, f o r each included in PNX.
> 0 and h E A*, there exists a compact &-supportf o r h
E
Proof. We shall proceed similarly to the proof of Theorem 6.6. Let E > 0 and h E A*. The empty set is an &-supportfor h = 0, so we assume that h # 0. It suffices to show that for a chain 9' of compact &-supportsfor h, So = n{S': S' E 9")is an &-support for h, and that a minimal compact &-support S for h is included in BNX. To prove the former, assume that So c Cl,f-'(0), wherefc A andf # 0. Let 6 = E II f 11/11 h 11. Since B(0, 6) is clopen in [id by Lemma 7.2, there exists U E CO(P,X) such that U n X = f - ' ( B ( O , 6)). Since So c U, U includes some S, E 9".Define fo = fl(X - U ) ; then fo E A by (1). Since Cl,fo-'(O)= U 3 S , , Ihcf,)l < ~ l l f o l < l ~ l l f l l . Since Ilf-foII 6, l h c f - f o ) l < 6Ilhll = ~ I l f I l . These imply that I h ( f ) I < E 11 f 11, proving that So is an &-supportfor h. To prove the latter, suppose on the contrary that there exists p E S - PNX. Then, by Theorem 2.8, there exists a decreasing sequence { Un}nsN c CO(lpDX) such that p E U, c p , X - X. For each n E N , by the minimality of S, there existsf, E A such that
s-
clDf,-'(o)
un
and I h ( f , )I >
IIf, 11.
Now, consider any b E K with 0 < I b I < 1, and for brevity the number I b-'l as r . Then each nonzero g E A has a scalar multiple g' with 1 < 11 g' 11 < r. Hence we can assume that 1 < 11 f, 11 < r for each n E N. Define g, = f,l ( X n U,,); then g, E A by ( I ) . For each a E I" and x E X , 1) a,g,(x) 11 < (a,,] 11 f , ( x ) 11 < r 11 a 11. Since g ; ' ( x ) = 0 for all but finitely many n E N , 11 XnSNa,,g,(x) 11 < r 11 a 11. Since X - g;'(O) c X n U, and { X n Un}neNis locally finite in X , );nENa,g,E C ( X , B ) , and hence XnENangn E A by (2). Thus we can define a linear operator rp:l" + A by q ( a ) = Z,,,,a,g, for a E I". For each n E N , the same argument as in the proof of Theorem 6.6 implies that Cl,(f, - g,,)-'(O) 3 S. Since S is an &-supportfor h,
-
< &Ilf,II < I h ( f , ) l Combining this with Lemma 7.7( I), I h ( f , ) I < I h(g,,)I, and hence I (h Ih(L
-&)I
< EIlf,
-
gnII
rp) (en)I I1f,II 2 E for each n E N. Since [id is not spherically complete, 0 this contradicts Theorem 7.14. = I h(g,) I >
0
E
Let A i , , IE A, be Banach spaces and consider A to be a discrete space. For a countably complete ultrafilter p on A, i.e., p E BNA, define Kp to be the subspace of the direct product A = X i,,,Ai consisting of all x such that
507
N-compactness and its Applications
{A E A : xi = 0 } E p . To show that K, is closed in A , let { x , , } , , ~c~ K, be a sequence converging to x E A. Put M = {A :x,(A) = 0 for each n E N } ;then M E p by the countable completeness ofp. Since (1 x(A) II = 11 x(A) - x,(A) 11 < 11 x - x,, 11 + 0 for each A E M , M c { I E A :x(A) = 0 } , which implies that x E K,. Hence, K, is closed in A. The ultraproduct Alp is the quotient Banach space A/K,. Let [ 1, : A -,A/p denote the quotient operator. For each x E A, II [ X I , II
=
inf{lI Y II : Y
E
A , [YIP = [xl,)
by the definition of the quotient norm. The following theorem is a generalization of the dual form of Corollary 7.13. 7.17. Theorem. Assume that K is not spherically complete. Then,for a direct product A = X leAAi of Banach spaces Ai, I E A, A* is naturally isomorphic to @pEBNA(A/p)*. Moreprecisely, the map Y :aPSBN,,(A/p)* + A* defnedby Y ( h ) = ~ p C p E B N1,h hispan o [isomorphism.
Proof. Let h E a P E B N A ( A / pthen ) * ; h, = 0 for all but countably many p's, say p,, i E N , and W )= G N h p , [o I,,. Since 11 h ; [ I,, 11 < 11 A,, 11 + 0, this sum is defined by Lemma 7.7. We show that 11 Y ( h )11 = 11 h 11. For any E > 0, 11 Y ( h )11 - E < 1 (Y(h))(x)I/ 11 x 11 for some x E X.By Lemma 7.7 there exists k E N such that I (Y(h))(x)I < I h,,([xl,,) I, so
II W h ) II - E < I hPk([xIPk) 1/11 [XI,, II < II hPkII G II h II, and hence 11 Y ( h )11 < 11 h 11. T o prove the converse, choose I E N with 11 h,, 11 = 11 h 11. For any E > 0, there is x E A such that [XI,, # 0 and I h,,([x],,) I/ 11 [XI,, 11 > 11 h 11 - E , and hence there is y E A such that [ y],, = [x],,and I h,,([ y],,) I/ )I y 11 > 11 h 11 - E . By Lemma 4.8, there exists a partition { A , } r sof N A such that A, E p , for each i E N. Since [ y I A,],, = 0 unless i = 1, h,,([Yl,,) = h,,([Y I A/I,/) + h,,([Y I (A - A,)Ip,) =
1 h,JY
rsN
I A/],,)
=
W(h))(YI A,),
Therefore
II w4 II
> I (Y'(h))(Y I A/) I/ II Y I A/ II 2 I h,,([Yl,,) 1/11 Y II > II h II -
E9
and hence 11 Y ( h )11 2 11 h 11. Now, Y is shown to be an isometry. Next, to prove that Y is surjective, let g E A*. Since the direct product A can be considered naturally as a subspace of C(A, @iSAAl)and A satisfies the conditions of Theorem 7.16, g has a compact n-'-support S,, included in /?NA
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for each n E N. We may assume that S, c S,,, . By Lemma 4.9, each S,, is finite. Put S = UneNSn; and we write S = { q l } l e N . By Lemma 4.8, there exists a partition of A such that MI E q,, i E N . For each i E N , define f,, E (A/q,)*byf,,([x],) = g ( x I M I )for x E A . To see that this is well defined, suppose [XI,, = 0. Then, by the same argument as in the proof of Theorem 6.8, S c CI,,,,(xI M,)-'(O). Hence Ig(xl M I ) /d n-I 11 X I MI I1 d n-I I( x II for each n E N , and thus g ( x I MI) = 0. Define f = (fp)peSNhby f p = f,, if p = q,, and& = 0 otherwise. To show that f E @,,,,,,(A/p)*, we let x E A be fixed for a while. If q14 S,, then S, c Cl,,,,(y I M,)-'(O)for each y E A , and hence lf,,([xlq,)l/ll[XI,, II d SUP{If,,([Ylq,)l/II Y II : [ y I ,=
[XI,,}
d ~ ~ P { l ~ ~ Y l ~ , ~ l /:[Ylq, l l Y= l ~[xlq,) l I I
< n-I. This means that f E @,,,,,,(A/p)*. Finally, to show that Y (f ) = g, let z E A. Put z, = z - Clgn(zI M I ) ,n E N . For each n E N , there is m, E N such that S, c { q , : i < m,}. If m 2 m,, then S, c cl~,,,Z,'(o), so lg(z,)l < n-' II z, I[ d n-' [I z 11, which implies that g(C,,,(z I M, )) converges to g(z). Hence,
proving that Y is surjective.
0
Analogously to Corollary 6.9, and with the same proof, we have: 7.18. Corollary. Assume that K is not spherically complete and I A I is nonmeasurable. Then ( X j,s,,Ai)* is naturally isomorphic to &j.E,,Af. More precisely, the map Y : GiE,,Af + ( X j,€AAj.)*de$ned by Y ( h ) = Cj,E,,hj. Q;. is an isomorphism. 0
For a normed space A , the canonical operator 8 : A + A** is defined by (8(x))(h)= h ( x ) for x E A and h E A*. A normed space A is called reflexive if 8 is an isomorphism. The easy proof of the following lemma is left to the reader (cf. van Rooij [1978, Theorem 4.241). 7.19. Lemma. For Banach spaces A and B, the following hold: (1) I f A is a subspace ofl"(A), then 8 is an isometry. (2) I f A is reflexive, then so is A*. ( 3 ) I f A @ B is reflexive, then so is A .
N-compaciness and iis Applications
509
7.20. Corollary. Assume that 116 is not spherically complete. I f I A 1 is nonmeasurable, then both l"(A) and co(A) are reflexive. Conversely, if co(A) is rtlflexive, and if I W I is nonmeasurable, then I A I is nonmeasurable. Proof. The first statement can be proved similarly to Corollary 6.12, if one uses Corollary 7. I8 and Lemma 7.19 instead of Corollary 6.9 and Lemma 6.11, respectively. Now, assume that I 116 I is nonmeasurable. Then it follows from Corollary 7.4 that W is N-compact. Hence, eachfE l"(A) (= C(A, W)) can be extended continuously to PNA. Thus the rest of the proof goes just like that of Corollary 6.12. 0 We shall show later that co(A)can be replaced by lm(A) in the second statement. Next, we consider a space X for which C(X, W) is isomorphic to c,,(A) for some A. Let A be a Banach space. Then a subset E = {el}lshof A is said to be orthonormal if it satisfies the following conditions: ( I ) For each 2 E A, 11 ej,11 = 1. (2) If L I , . . . , il, are distinct elements of A and ai E W,i = 1, . . . , n, n E N , then II Xig,aiel,II = maxl,ignI a,I. In this case, the linear operator q E co(h) : + A defined by qE(a)= Xj.E,,ai,ej, for a E co(A)is an isometry. In fact, for each a E co(A),there is a countable subset of A, and
=
lim (max I aj,,I) = 11 a (I.
n-rm
lgidn
When qEis surjective, we call E an orthonormal base o f A.
7.21. Theorem. Let X be a compact space. Then, C(X, W) has an orthonormal base consisting of characteristic functions of clopen sets. To prove this, we need the following theorem due to Ostrowski (see Schikhof [1984, Theorem 10.11 for the proof).
7.22. Theorem. If1 1 is a nontrivial, non-Archimedean valuation on Q, then there exist s > 0 and a prime number p such that 1 I = (I Ip)s. ProofofTheorem7.21. LetR = { a E W : I a l < l } a n d Z = (aED6:lal < l}. Let 116, be the prime field of W, and let R,, = R n KOand Zo = I n 116,.
K . Eda, T. Kiyosawa. H.Ohta
510
Then, Z (resp. I,) being a maximal ideal of the ring R (resp. &), we get two quotient fields k = R/I and ko = &/Z,. Since the homomorphism carrying a + Z E k to a + Zo E ko is injective, k, can be regarded as a subfield of k. For each U E CO(X), let e, (resp. 8,) denote the M-valued (resp. k,-valued) characteristic function of U on X . By the definition of k and k,, we get the following claim:
Claim 1. Let U,E C O ( X ) , a, E R (resp. ai E &), ai = ai + Z E k (resp. a, = ai Zo E k,), i = 1,. . . ,n, n E N,and x E X . Then, Zicnaicy(x)= 0 if and only if I Zi,,,aie,(x) I < 1.
+
By Zorn's lemma, there exists a maximal collection A c CO(X)such that is linearly independent over k,. We show that E = {e,}UEAis an orthonormal base of C(X, M). Suppose that E is not orthonormal. Then there exist U, E A, a, E M, i = 1, . . . , n, and io < n such that 11 Z,,,a,e,, 11 < max,,,,,Ja,) = ~ u l o Let ~ . b, = a;'a, for each i. Then b,E R and IIXl,,,bleU,II< 1, so it follows from Claim 1 that ZlC,,flIc,,= 0, where PI = b, 1.Since ko is a subfield of k , the linear space k over kohas a Hamel base {q,}€,., For each j E J , let e,:k + ko be the j t h projection, i.e., @a,?,) = a,; then e, is k,-linear. Hence, for each x E X and j E J , Z,,,,e,(/3,)cU,(x) = 0. Since x is arbitrary, Z,,,q(/?,)c,, = 0, which implies that e,(BI) = * * = @,(fin) = 0, because { E , } , ~ , , is linearly independent over k,. Since j is also arbitrary, P, = . . = fl,, = 0, and hence b, E Z for each i = 1, . . . , n. This contradicts the fact that I b, I = 1. Next, we show that the linear isometry ' p E :co(A) + C(X,W) is surjective. Since the range of q E includes the linear hull H ( E ) of E, it suffices to show that H ( E ) is dense in C(X,K). For this end, we prove two claims. {E,},~,,
+
-
Claim 2. There exists 0 < r, < 1 such that for each V E C O ( X ) , dist(e,, H ( E ) ) (= infgoH(E) 1I e, - g II) < ro.
Proof. If V E C O ( X ) ,then by the maximality of A, there exist U, E A and ai E k,, i = 1, . . . , n, such that E , = Xi4,,aicU,.For each i < n, choose ai E R, with ai = a, + I,. Then, for each x E X , it follows from Claim 1 that 1 e,(x) - Xi,,,aieu,(x)l < 1. Since M, is prime, it is finite or is isomorphic to Q. Hence it follows from Theorem 7.22 that there exists r , < 1 such that 11 e, - Zicnaie,, (1 < r,. Define r = sup{r,: V E C O ( X ) } ; then r < 1 by Theorem 7.22 again. Finally, choose r, with r < r, < 1. The claim is proved. 0 Claim 3. For each f E C(X,W), dist(f, H ( E ) ) < r, (1 f (I.
N-compactness and its Applications
51 I
Proof. Let f E C(K, W). Since X is compact, there exist a partition . . . , V , } c CO(X) of X and ai E W, i = 1, . . . , n, such that II f - CiGnaie,:II < r~ II f 11. Define g = xi
{v,
G r ~ *I bmi ban x I a i I= rollgll
so
[If
- h 11
< max{ [If
-
g 11,
)Ig
-
h II}
< r, I l f l l .
r~llfll~ The claim is proved.
0 To complete the proof of Theorem 7.21, suppose that H ( E ) is not dense in C(X, W); then there existsfc C(X, W) such that dist(f, H ( E ) ) > 0. Since r, < 1, there is g E H ( E ) with [If - g 11 < rol * dist(f, H ( E ) ) . Therefore, r, Ilf - g 11 < distv, H(E)) < distv - g, H(E)), which contradicts Claim 3. 0
7.23. Corollary. Let X be a compact space. Then, if I XI is nonmeasurable, C(X, W) is rejlexive. Moreover, the converse is true when I 06 1 is nonmeasurable. Proof. This follows from Theorem 7.21 and Corollary 7.20. As the following corollary shows, a non-Archimedean Banach space C(X, W) fails to determine the topology of a 0-dimensional, compact space X. The reader should compare it with Banach-Stone's theorem (cf. "a, Section IV.1, p. 1671) asserting that if both X and Y are compact and the real Banach spaces C(X, W) and C(Y, W) are isomorphic, then X and Y are homeomorphic.
7.24. Corollary. Let X and Y be 0-dimensional, compact spaces. Then, Banach spaces C(X, W) and C( Y , W) are isomorphic if and oniy if w ( X ) = w( Y ) . Proof. We only need to consider infinite spaces. Let A c CO(X) be the same as in the proof of Theorem 7.21, i.e., it is the maximal collection for which { E ~ } ~ ~is, linearly , independent over k,. First, we show that I A I = w ( X ) . Since Xis compact, 1 CO(X) 1 < w ( X ) , and hence 1 A 1 < w ( X ) . Let A, be the collection of all intersections of finitely many members of A u {X - U : U E A}. Then A,, is a base of X. For, if x E V E CO(X), then
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K . Eda, T. Kiyosawa, H . Ohta
by the maximality of A, E,, = C f d n t l r ~for u , some U, E A and a, E k , , i = 1, . . . , n, so there is a function g from { I , . . . , n> to ( 0 , I } such that x E nfdBU,,,,,c V , where U,, = U, and U,, = X - U,. Since IA,I = I A I , 1 A I 2 w ( X ) .Now we have proved that 1 A 1 = w ( X ) , which implies the “if” part is valid, because C ( X , W) 1: c,(A). Next, we prove that c,(A,) N c0(A2) implies I A, I = I A21, from which the “only if” part is deduced. To this end, suppose that co(A,) 1: c0(A2) and I All = t,, i = I , 2. Then, co(AI)has an orthonormal base E of cardinality t2,and each x E E can be written in the form x = X,,,,a,e, for some a E co(Al). Put M, = {A E AI :a, # 0}, and let M = U,,,M,. Then I MI < t2, and y , = 0 for each y E H ( E ) and A E A1 - M , where H ( E ) is the linear hull of E. Since H ( E ) is dense in co(AI), the latter implies that M = A,, and hence t1 < t2.The proof that t12 t2is similar.
For more information about orthonormal bases, see van Rooij [ 1978, Chapter 51. We now turn to different applications of Theorem 7.21, which are needed in the proof of later th,porems. 7.25. Corollary. Let S be a compact subset of a @dimensional space X , and let y,: C ( X , W) + C(S, W) be the restriction. Then the following are valid: (1) There exists a linear isometry 6,: C(S, W) + C ( X , W) such that ys 0 6, = id, and hence y, is surjective. (2) C ( X , W) = kedy,) 0 C(S, W. ( 3 ) y: is an isometry. Proof. (1) By Theorem 7.21, there exists A c CO(S) such that (eu}uE,,is an orthonormal base of C(S, W), where e,, is the characteristic function of U on S. For each U E A, there is V ( U ) E CO(X) such that V ( U ) n S = U . Define 6, by 6,(Cu,,,aueu) = Zu,,,a,,e,,(,,) for a E co(A). Then, clearly, y s o 6, = id. Let f E C ( S , W); then f = C,,,,aUneq for some a E c,(A) and (U,},,,, c A. Since {e,,}u,,, is orthonormal, it follows from Lemma 7.7 that
On the other hand, 6,( f ) being an extension off, II 6,( f ) 11 2 II f 11. Hence 6, is a linear isometry. (2) The linear operator q :C ( X , W) + ker(y,) 0 C(S, W) defined by V ( f ) = ( f - 6,(Y,(f )I, rs(f)) is an isomorphism.
N-compactness and its Applications
513
7.26. Remark. The above corollary also shows the essential difference existing between real Banach spaces and non-Archimedean Banach spaces. In their paper [1960], Gqba and Semadeni proved that there is no continuous linear operator 6, from a real Banach space C ( S , W) to C(X, W) for which y s o 6 , = id, even if S is a closed subspace of a 0-dimensional, compact space X. For h E C(X, W)*, a subset S of X is called a support for h in X if for any f~ C(X, W), S c f - ’ ( O ) implies h ( f ) = 0. We denote the subspace of C ( X , K)* consisting of all elements having compact supports in X by C(X, K)’.
7.27. Corollary. Assume that W is not spherically complete, and let X be a 0-dimensional space. I f X is N-compact, then C ( X , W)’ is dense in C ( X , W)*. The converse is also true when either 1 K 1 or 1 X 1 is nonmeasurable. Proof. Assume that Xis N-compact, and let h E C(X, W)*. For each E > 0, h has a compact &-support S included in B N X by Theorem 7.16. Since X = BNX,S c X. Define ho = h 6 , . yS, where ys and are the same as in Corollary 7.25. Then, S being a support for ho, h, E C(X, K)’. For each f E C(X9 W), since s = (f- (6soY,)(f))-’(o), 0
I1
-
h o ) ( f )II = G
II h ( f E
- ( 6 s O Y s ) ( f ) )II
I1f - (6s. Y s ) ( f ) II
Therefore (1 h - ho 11 < E , which implies that C(X, W)’ is dense in C ( X , W)*. To prove the converse, suppose that X is not N-compact and pick a point p E B N X - X. For each f~ C ( X , W), If(,%’) is nonmeasurable by our assumption. Hence it follows from Proposition 7.3 and Theorems 3.6 and 3.10 that f ( X ) is N-compact. Therefore, f admits a continuous extension B N f E C(BNX, Define go E C(,%’,W>*by go(f) = ( B N f X P ) . We show that 11 g - go11 2 1 for each g E C(X, W)’. Let G be a compact support for g in X. Since p 4 G, there exists f E C ( X , W) such that ( B N f ) ( p )= 1 and f(G) = { O } . Sincego(f) = 1 andg(f) = 0, IIg - go11 2 1. Consequently, C(X, W)’ is not dense in C(X, W)*. 0
w.
514
K. Eda. T. Kiyosawa. H.Ohla
We now come to counterparts of Theorems 6.21 and 6.23 in non-Archimedean Banach spaces, which determine when the Banach space C(X, W) or C(X, W)* is reflexive. In the remainder of this section, for brevity, we write C ( X ) instead of C(X, W). For each K, L E K ( X ) with K c L, define y K L , y , and r, to be the same as in the preceding section. Then, (C(K)*, K ( X ) ) forms a direct system of Banach spaces. First, we prove two lemmas analogous to Lemmas 6.19 and 6.20. 7.28. Lemma. Let X be a 0-dimensional space. Then, there exists a linear isometry r~ : lim, C ( K ) * + C(X)* such that yX = CT n, for each K E K ( X ) . Moreover, the range of r~ includes C(X)’ (cf. Diagram (6.6)). 0
Proof. If K c L, K , L E K ( X ) , then y: = y: yXL. By Corollary 7.25, y ; is a linear isometry. Hence, it follows from Lemma 7. I I that there exists a linear isometry CT : lim, C(K)* C ( X ) * such that y; = r~ n, for each K E K ( X ) . Let h E C(X)’ and K a compact support for h in X . Define g = h 0 6 , 0 y , , where 6 , is the same as in Corollary 7.25. Then, h = yX(ho 6,) = a(a,(h o a,), and hence h is in the range of CT. 0 0
--f
0
For a 0-dimensional space X , I C ~ C: ( X ) 4 C ( k N X )denotes the linear isometry defined by rcX(f)= f k,, where k, is the identity from k,X to X , and 8, denotes the canonical linear isometry from C ( X ) to C(X)**. 0
7.29. Lemma. Assume that W is not spherically complete, X is N-compact, and each K E K ( X ) is of nonmeasurable cardinality. Then there exists an isomorphism z :C(X)** + C ( k N X )such that x X = T 8, (d. Diagram (6.7)). 0
Proof. Since X is N-compact, it follows from Corollary 7.27 that C(X)’is dense in C(X)*.Hence, the linear isometry cr in Lemma 7.28 is an isomorphism. Observe that our task is strikingly similar to that presented by the proof of Lemma 6.20. The only changes are to replace the references to Theorem 6.3 and Corollary 6.15 by ones to Theorem 7.12 and Corollary 7.23, respectively. 0 Now, the following theorem and its corollary can be proved similarly to Theorem 6.21 and Corollary 6.22, if one uses Corollary 7.25 instead of Proposition 6.13 and Lemma 6.18, and Corollary 7.23, Lemmas 7.19 and 7.29 instead of Corollary 6.15, Lemmas 6.11 and 6.20, respectively. 7.30. Theorem. Assume that K is not spherically complete and X is N-compact. Zfeach K E K ( X ) is of nonmeasurable cardinality and k,X = X , then C ( X ) is
N-compactness and its Applications
515
reflexive. Moreover, the converse is also true when either 1 K 1 or I XI is nonmeasurable.
7.31. Corollary. Assume that K is not spherically complete. Then, $1" (A) is reflexive, I A I is nonmeasurable. 1.32. Theorem. Assume that K is not spherically complete and X is N-compact. If each K E K(X)is of nonmeasurable cardinality and k,X is N-compact, then C ( X ) * is repexive. Moreover, the converse is also true when either 1 K I or I XI is nonmeasurable. Proof. Similarly to the proof of Theorem 6.23, we can establish the following Diagram (7.1); one needs only replace the references to Corollary 6.7 and Lemmas 6.19, 6.20 by ones to Corollary 7.27 and Lemmas 7.28, 7.29, respectively.
Here, (T, and T* are isomorphisms and 8 is the canonical linear isometry. By Lemma 7.1 I , S = { n K ( h )h: E C ( K ) * , K E K ( X ) } is dense in lim,C(K)*. Hence, if 8 6 ,= T* 0 n2on S, then the above diagram would be commutative. Therefore, the proof goes just like that of Theorem 6.23, if one uses Corollaries 7.13, 7.27, 7.23 and 7.31 instead of Corollaries 6.4, 6.7, 6.15 and 6.22, respectively. 0 0
7.33. Examples. By the foregoing results and examples, all the statements in Examples 6.24 remain valid for Banach spaces C ( X ) except that the final one requires nonmeasurability of I K I. In particular, if X is the space in Example 4.12, then it follows from Theorem 7.32 that C ( X , K)* is not reflexive for any non-spherically complete field K. This is the first example of a Banach space A of nonmeasurable cardinality for which A* fails to be reflexive (cf. Van der Put [ 19701, Gruson and Van der Put [ 19741, and Van Rooij [ 1978, Problem 4.61).
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K. Eda. T. Kiyosawa, H . Ohra
7.34. Remark. In Theorems 7.17, 7.30, 7.32, Corollaries 7.18 and 7.20, the assumption that K is not spherically complete is essential. In fact, if K is spherically complete, then c, is not reflexive. To show this, considerf E (co)* defined byf(x) = XneNx,,.If K is spherically complete, then by Theorem 7.9, f admits an extension F E (I")*. By Corollary 7.13, there exists a natural isomorphism 0 :(co)* -+ I". Hence, F . 0 E (co)**. Observe that for each g E K*, there exists unique ( ( g ) E K such that g(u) = [ ( g ) u for each u E K. Let 8: c, -+ (co)** be the canonical linear isometry. For any x E c,, (8(x))(ei) = xi -+ 0, while (FoO)(ei)= 1 for each i E N , because O(ei) = ( ( ( e i o ( ) n ) ) n E N = ei. Thus, 8 ( x ) # F . 0 , and hence c, is not reflexive. As a matter of fact, Fleischer [1954] proved a much stronger result: If K is spherically complete, then no infinite dimensional normed space is reflexive (for the proof, see Gruson and Van der Put [1974], or Van Rooij [1978], Theorem 4.16). In addition, we cannot remove the same assumption from Theorem 7.16 and Corollary 7.27. To see this, let F E (I")* (= C(N, W)*) be the same as above. Then, since F(e,,) = 1 for each n E N , it is easily checked that Fhas no compact 1/2-support included in N , and that Fis not contained in the closure of C(N, K)'.
7.35. Notes. Lemma 7.2, Proposition 7.3, Corollary 7.4 and Example 7.5 are taken from van Rooij [1978, Chapter 21 and Lemma 7.7 is Lemma 3.1 and Exercise 3K, ibid. Theorem 7.9 is well known as Ingleton's theorem and was proved in his paper [1952]. Inverse limits and direct limits of real Banach spaces were discussed by Semadeni and Zidenberg [ 19651; however, nonArchimedean counterparts Lemmas 7.10 and 7.11, and Theorem 7.12 cannot be found in the literatures. Corollary 7.13 appears in van Rooij [1978, Exercise 3Q]. Corollaries 7.18 and 7.20 as well as Theorem 7.14 were proved by van der Put [1968] without using Theorem 7.17. Theorem 7.21 is due to Gruson and van der Put [1974], and the proof here is taken from van Rooij [1978, pp. 187-1881. van der Put [1970] proved Corollary 7.25 (l), the first part of Corollary 7.27, Lemma 7.29, Theorem 7.30, Corollary 7.31 and Theorem 7.32 under the assumption that all cardinal numbers are nonmeasurable. However, the proofs given here entirely differ from his ones except for the one of Corollary 7.25. For the whole theory of non-Archimedean functional analysis the reader is referred to van Rooij [1978].
8. Problems
The following problems are unsolved as far as the authors know.
N-compaciness and its Applicaiions
517
(1) (Woods [1975]). What 0-dimensional spaces are homeomorphic to a closed subspace of a product of P-spaces? Here, a P-space is a completely regular space in which every G,-set is open. (2) (van Douwen [ 19841). Is there a (preferable metrizable) nonlocally compact, N-compact space Xwith Exp,(X) c Exp,(X)? Here, Exp,(X) = min{a : X embeds as a closed subspace in E” with I A I = a}. (3) (Blair [1976]). Is a perfectly normal space of nonmeasurable cardinality realcompact? (Ostaszewski [ 19761 gave a negative answer under assuming “Axiom Club”.) (4) (Mrbwka [1971]). Suppose that X is a completely regular (resp. 0-dimensional)space such that every completely regular (resp. 0-dimensional) continuous image of X is realcompact (resp. N-compact). Then, is X a Lindelof space? ( 5 ) Show that abelian groups C(Q, Z) and C(W - Q, Z) are not isomorphic. There are many problems which concern the measurable cardinality. Out of set theory not much attention has been paid to phenomenons above the least measurable cardinal. However, more attention should be paid to them, when we look at progresses in set theory, e.g., the cardinality collapsing argument using forcing, a reduction from 0-1 measure to stationary sets and so on. If A is a discrete space of measurable cardinality, then kN(/INA) is not N-compact. Therefore, Example 4.12 can be seen as a reduction of the second kind. The following ones are related to both measurable cardinality and contents of this chapter. (6) (Herrlich [ I98 I]). Does Theorem 3.14 remain true if the assumption of nonmeasurability of I E I is removed? (7) (Hulek [1972]). Is the cardinality of a realcompact k-space X such that each K E K ( X ) is of nonmeasurable cardinality nonmeasurable? (8) (Eda and Ohta [1987]). Is the cardinality of an N-compact k,-space X such that each K E K ( X ) is of nonmeasurable cardinality nonmeasurable? (9) Let X be an N-compact space such that the group C ( X , Z) is isomorphic to Z”for some A. Is X the maximal N-compactification of a discrete space? (In case X is of nonmeasurable cardinality, X is discrete; cf. Eda and Ohta [1987].) (10) Show the existence of a complete, non-Archimedean valued field K such that the dual Banach space Im(A)* over K is naturally isomorphic to co(A), where A is a discrete space of the least measurable cardinality. (Of course, K is of measurable cardinality and not spherically complete.)
518
K . Eda. T . Kiyosawa, H. Ohru
References Beckenstein, E., L. Narici and C. Suffel [I9771 Topological Algebras (North-Holland, Amsterdam). Bergman, G. M. [I9721 Boolean rings of projection maps, J . London Marh. SOC.4, 593-598. Blair, R. L. [I9761 Spaces in which special sets are z-embedded, Canad. J . Math. 28, 673-690. Blasco, J. L. [I9771 On p-spaces and k,-spaces, Proc. Amer. Marh. SOC.67, 179-186. Broverman, S. [I9771 N-compactness and weak homogeneity, Proc. Amer. Math. SOC.62, 173-176. [1978] Another realcompact, 0-dimensional, non-N-compact space, Proc. Amer. Marh. SOC. 69, 156-158. Cartan, H.and S. Eilenberg [I9561 Homological Algebra (Princeton University Press, Princeton). Comfort, H. H. [I9681 On the Hewitt realcompactification of a product space, Trans. Amer. Marh. SOC.131, 107-1 18. Corson, H. H. [I9591 Normality in subsets of product spaces, Amer. J. Murh. 81,785-796. van Douwen, E. K. [I9841 The integers and topology, in: Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) pp. 1 1 1-167. Eda, K. (19821 A Boolean power and a direct product of abelian groups, Tsukuba J. Murh. 6,187-194. Eda, K.and H. Ohta [19871 On abelian groups of integer-valued continuous functons, their Zduals and Z-reflexivity, in: Abelian Group Theory, Proc. Confer. Obenvolfach, 1985 (Gordon and Breach, London) pp. 241-257. Engelking, R. [1977] General Topology (Polish Scientific Publishers, Warszawa). Fleischer, I. [1954] Sur les espaces normbs non-archimbdiens,Zndag. Muth. 16, 165-168. Fuchs, L. [I9701 Infinite Abelian Groups, Vol. Z (Academic Press, New York). [I9731 Infinite Abelian Groups, Vol. IZ (Academic Press, New York). Gpba, K. and Z. Semadeni [I9601 Spaces of continuous functions (V), Studia Murh. 19,303-320. Gillman, L.and M. Jerison [I9601 Rings of Continuous Funcrions (Van Nostrand, New York).
Gruson, L. and M. van der Put [I9741 Banach spaces, Bull. SOC.Math. France, MPm. 39-40, 55-100.
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Herrlich, H. [I9671 C€-kompakte Raume, Mafh. Z. 96, 228-255. [I98 I] Categorical topology 1971-198 I , in: General Topology and its Relations to Modern Analysis and Algebra V, Proc. Fifth Prague Symp. 1981 (Heldermann Verlag, Berlin) pp. 279-383. Hewitt, E. [I9481 Rings of real-valued continuous functions, I, Trans. Amer. Math. SOC.64, 54-99. Hill, P. [I9731 The additive group of commutative rings generated by idempotents, Proc. Amer. Math. SOC.38, 499-502. Hong, S. S. [I9731 On f-compactlike spaces and reflective subcategories, General Topology Appl. 3 , 3 19-330. Hrbacek, K. and T. Jech [I9841 Introduction to Set Theory (Marcel Dekker, New York, 2nd ed.). Hukk, M. [I9711 Pseudo-m-compactness and w(P x Q), Indag. Math. 33, 320-326. [I9721 Realcompactness of function spaces and v ( P x Q). General Topology Appl. 2, 165-179. Ingleton, A. W. [ 19521 The Hahn-Banach theorem for non-Archimedean valued fields, Proc. Cambridge Phil. SOC.48.41-45. Jech, T. [I9781 Set Theory (Academic Press, New York). KatBtov, M. [I9511 Measures in fully normal spaces, Fund. Math. 38,73-84. Mackey, G. W. [I9441 Equivalence of a problem in measure theory to a problem in the theory of vector lattices, Bull. Amer. Math. Soc. 50, 719-722. Mrowka, S. [I9571 Some properties of Q-spaces, Bull. Acad. Polon. Sci. 5, 974-950. [I9581 A property of Hewitt extension V X of topological spaces, Bull. Acad. Polon. Sci. 6, 95-96. [I9651 Structures of continuous functions. 111. Rings and lattices of integer-valued continuous functions, Indag. Math. 27, 74-82. [I9661 On E-compact spaces. 11, Bull. Acad. Polon. Sci. 14, 597-605. [I9681 Further results on E-compact spaces, I, Acta Math. 120, 161-185. [ 19711 E-complete regularity and E-compactness, in: General Topology and its Relations to Modern Analysis and Algebra, I l l , Proc. Confer., Kanpur, 1968 (Academia, Prague) pp. 207-214. [1972a] Structures of continuous functions. VIII. Homomorphisms of groups of integer-valued continuous functions, Bull. Acad. Polon. Sci. 20, 563-566. [ 1972bI Recent results on E-compact spaces and structures of continuous functions, Proc. Univ. Oklahoma Topology Confer., 1972, 168-221. [I9741 Recent results on E-compact spaces, in: T O P 0 72-General Topology and its Applications, Second Pittsburgh Internat. Confer., 1972, Lecture Notes Math. 378 (Springer, Berlin) pp. 298-301. Mrowka, S. and R. Engelking 119581 On E-compact spaces, Bull. Acad. Polon. Sci. 6, 429-436.
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Mysior, A. [1978a] The category of all zero-dimensional realcompact spaces is not simple, General Topology ~ p p ia, . 259-264. [1978b] On two problems of Mrowka, in: Topology Colloq Math. Soc.Janos Bolyai, Budapest (North-Holland, Amsterdam) pp. 873-875. [I9841 Two easy examples of zero-dimensional spaces, Proc. Amer. Math. SOC.92, 615-617. Nagata, J. [I9851 Modern General Topology (North-Holland, Amsterdam, 2nd rev. ed.). Nobeling, G. [1968] Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math. 6, 41-55. Noble, N. [I9701 The continuity of functions on Cartesian products, Trans. Amer. Math. SOC.149, 187-1 98. Nyikos, P. [I9711 Not every 0-dimensional realcompact space is N-compact, Bull. Amer. Math. SOC.77, 392-396. [I9731 Prabir Roy’s space A is not N-compact, General Topology Appl. 3, 197-210. [I9761 A survey of zero-dimensional spaces, in: Topology, Proc. Memphis State Univ. Confer., 1975 (Marcel Dekker, New York) pp. 87-1 14. Ostaszewski, A. [I9761 On countably compact, perfectly normal spaces, J . London Math. SOC.14, 505-516. Ohta, H. [I9821 Topological extension properties and projective covers, Canad. J. Math. 34, 1255-1275. Pierce, R. S. [1961] Rings of integer-valued continuous functions, Trans. Amer. Math. SOC.100, 371-394. Pol, E. and R. Pol. [I9771 A hereditarily normal strongly zero-dimensional space with a subspace of positive dimension and an N-compact space of positive dimension, Fund. Math. 97, 43-50. Pol, E. [I9791 Some examples in the dimension theory of Tychonoff spaces, Fund. Math. 102,2943. Przymusinski, T. C. [I9791 On the dimension of product spaces and an example of M. Wage, Proc. Amer. Math. SOC.76, 315-321. Ptak, V. [I9531 On complete topological linear spaces (in Russian, with English abstract), Czech. Math. J. 3, 301-364. van der Put, M. [I9681 Espaces de Banach non-archimkdiens, Bull. SOC.Math. France 97, 309-320. [ 19701 Reflexive non-Archimedean Banach spaces, lndag. Math. 32, 341-352. van Rooij, A. C. M. [I9781 Non-Archimedean Functional Analysis (Marcel Dekker, New York). Sgciada, E. [I9591 Proof that every countable and reduced torsion-free abelian group is slender, Bull. Acad. Polon. Sci. 7. 143-144.
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Schikhof, W. H. [ 19841 Ultrametric Calculus (Cambridge Univ. Press, Cambridge). Semadeni, Z. and H. Zidenberg [I9651 Inductive and inverse limits in the category of Banach spaces, Bull. Acad. Polon. Sci. 13, 579-583. Shirota, T. [1952a] A class of topological spaces, Osaka Math. J . 4, 2340. [1952b] A generalization of a theorem of I. Kaplansky, Osaka Math. J. 4, 121-132. Specker, E. [I9501 Additive gruppen von Folgen ganzer Zahlen, Portugaliue Math. 9, 131-140. Stone, M. H. [I9371 Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. SOC.41, 375-381. Ulam, S . [I9301 Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16, 140-150. Wage, M. [I9781 The dimension of product spaces, Proc. Nut. Acad. Sci. U.S.A. 75, 46714672. Walker, R. C. [ 19741 The Stone-tech Compactijcations (Springer, Berlin).
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K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 13
TOPOLOGICAL GAMES AND APPLICATIONS
Yukinobu YAJIMA Department of Mathematics, Faculty of Engineering, Kanagawa University, Yokohama 221, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The games and winning strategies . . . . . . . . . . . . . . . . . . . . . . K-scatteredspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure-preserving covers by compact sets . . . . . . . . . . . . . . . . . . Covering properties of product spaces . . . . . . . . . . . . . . . . . . . . The games in product spaces . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications to dimension theory . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. 2. 3. 4. 5.
523 524 529 536 543 549 553 560
Introduction The concept of topological games G(K,X )was introduced and studied by TelgBrsky [1975a]. The games are played by two persons called Player I and 11. In this chapter, we only consider the case that Player I has a winning strategy in the game. First, we must state that winning strategies for Player I in G(K,X )can be strengthened in terms of stationary ones. Secondly, we deal with K-scatteredness, which is useful to represent the topological structure of a space X such that Player I has a winning strategy in G(K,X). The main theme of this chapter is the applications of topological games. The first is to use the game for studying spaces which have a closure-preserving closed cover by (countably) compact sets. The second is to study some covering properties of product spaces one factor of which is defined by the game. The third is to generalize the countable sum theorems for dimension
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theory in terms of the games, so that we can obtain closure-preserving sum theorems for dim and Ind. The games treated here are only one side of the whole of topological games, of which there are quite various sorts. Telgarsky [1987] seems to be a very convenient source to study them. This survey does not contain proofs, but many cross-references which we cannot quote here. All spaces are assumed to be T I .In particular, compact spaces and paracompact spaces are assumed to be T,. For a space X , 2' denotes the family of all closed subsets of X . The letters n, m, k, i a n d j denote nonnegative integers. The letter o denotes the set of natural numbers. Let K be a class of spaces. We always assume that K is nonvoid and X E K implies 2' c K. For other classes K, and K2,we make the same assumptions. For a class K, we denote by DK ( F K ) the class of all spaces which have a discrete (finite) closed cover by members of K. We denote by C the class of all compact spaces. So, DC is the class of all T,-spaces which have a discrete cover by compact sets. The other notations and undefined terminologies follow Nagata's book [1985]. We quote some results from this book and Engelking's book [1977], the former of which is abbreviated by [Na]. We often use the finite sequences in the proofs below. So we adopt the following notations for them: For a finite sequence a = ( a , , . . . , a,,), let a 0 a = (al, . . . , a,,, a) and a - = (a,, . . . , a,-,). Moreover, let a l k = ( a , , . . . , a k ) for 1 6 k 6 n and a10 = 8. So we have a- = a l n - 1 for n 2 2 a n d a - = 8 f o r n = 1. Let coo = (8). Then U,20conis the set of all finite sequences of natural numbers and the empty set 8. According to the above, for each z = (kl, . . . , k,) E con and k 2 1, we have t 0 k = (kl, . . . , k,, k ) and T- = ( k , , . . . , k , - , ) . The author wishes to thank Professor Telgarsky for his helpful advice and suggestions.
1. The games and winning strategies For a class K and a space X , the topological game G ( K , X ) is defined as follows: There are two players I and I1 (the pursuer and the evader). They alternatively choose consecutive terms of a sequence (El, Fl,E 2 , F, , . . .) of subsets in X . When each player chooses his term, he knows K , X and their previous choices. A sequence (El, Fl, E,, F,, . . .) of subsets in X is a play of G ( K , X ) if it satisfies the following conditions; for each n 2 1,
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(1) En is the choice of Player I, (2) F, is the choice of Player 11, (3) E, E 2' n K, (4) F n E 29' ( 5 ) E, v F, c F,-,, where Fo = X , (6) En n F, = 8. Player I wins this play if On,,F, = 8 (Player I1 has no place to run away). Otherwise, Player I1 wins it. A finite sequence ( E l , F , , . . . , Em,F,) is admissible if it satisfies the above conditions (1)-(6) for each n 6 m. Let s' be a function from U,,,(2')" into 2' n K. Let Yo= { X } and
Y;
=
{ F E 2' 1 ( s ' ( X ) , F ) is admissible for G(K, X ) } .
Moreover, we can inductively define
. I ( E l , F,, . . . , En,F,) is admissible for G(K, X ) , where Fo = X and E, = s'(Fo, . . . , 4- ,) for each i d n}.
= {(F,, . . , F,)
Then the restriction s of s' to Un,oYnis called a strategy for Player I in G(K, X ) . We say that the strategy s is a winning one if Player I wins every play ( E l , F,, E,. F2,. . .) such that En = s(F,, . . . , F,-,) for each n 2 1. Next, we can define another (winning) strategy for Player I in the game, which depends only on the preceding choice of Player 11. A function s from 2' into 2' n K is called a stationary strategy for Player I in G(K, X ) if s(F) c F for each F E 2'. We say that the s is winning if he wins every play ( s ( X ) , F , , s ( F , ) , F,, s(F,),. . .). That is, a function s from 2' into 2' n K is a stationary winning strategy for Player I in G(K, X ) if and only if it satisfies (i) s(F) c F for each F E 2', and (ii) if {F,ln 2 l} c 2'satisfies thats(X) n F, = 8 ands(F,) n F,,, = 8 for each n 2 1, then F, = 8. If F E 2'meets s ( X ) , we always assume s(F) = s ( X ) n F. Hence it should be noted that s(F) is nonempty for each nonempty F E 2'. The following theorem shows that there is no essential difference between the winning strategy and the stationary winning strategy for Player I in G(K, X I .
n,,
1.1. Theorem (Galvin and Telgarsky [1986]). Player I has a winning strategy in G(K, X ) if and only fi he has a stationary winning strategy in it.
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Proof. The sufficiency is obvious. Let s be a winning strategy for Player I in G ( K , X). Well-order 2'\{0} by <. Take any H E 2'\{Q}. Claim (i) (ii) (iii)
1. There is some F ( H ) = (F,, . . . , F,) E (2')", satisfying s(Fo, F,, . . . , 4)n H = 8 for 0 < i < m - 1, s(Fo, F,, . . . , F,) n H # 8, F,+, = min{FE 2'IH c F c 4 and F n s(Fo, . . . , F,) = S} for 0 < i < m - 1, where Fo = X and F ( H ) = 8 may occur.
Assume the contrary. Then we can inductively choose some (F,, F,, . . .) E (2')" such that s(Fo, F,, . . . , Fk) n H = 8 and Fk
=
min{FE 21' H c F c Fk-,and S(F0,. . . , Fk-1) n H =
8}
for each k 2 1. Then ( E l ,F,, E2, F2, . . .), where Ek = s(F,, F,, . . . , Fk-l) for each k 2 1, is a play of G(K, X). By the definition of s, we have n k r I F k = 8. This contradicts 8 # H c Fk. Now, we put s*(8) = 8 and s*(H) = s(Fo, F,, . . . , F,) n H , where F ( H ) = (F,, . . . , F,), for each H E 2'\{0}. Then s* is a function from 2' into 2' n K such that s*(H) c H for each H E 2'. We prove that s* is a stationary winning strategy for Player I in G(K, X). Let ( E l , H , , E2, H 2 , . . .) be a play in it such that El = s * ( X ) and E, = s*(H,-,) for each n b 2. We show that On,,Hn = 8. For the above F ( H ) = (Fl, . . . , F,) and n < m, let F ( H ) l n = (F,, . . . , F,) and I F(H)I = m.
nk,,
Claim 2. There are some (Fl, F,, . . .) E (2')" and a sequence k(1) < k(2) < * * . such that k b k(n) implies (Fl, . . . , F,) = F(Hk)ln for each n 2 1. Let Fo = X.Assume that (F,, . . . , F,) E (2')" and { k ( i )I i < n} has been already chosen. We first show that IF(H,)I > n for each k > k(n). Let k > k(n). The inductive assumption implies F(Hk)ln = F((Hk(,))In = (F,, . . . , F,). In case that s(Fo, . . . , F,) is disjoint from Hk(,,), by Hk c Hk(,), it is also disjoint from H k . In case that s(Fo, . . . , F,) meets Hk(,), by (ii), we have F(ffk(,))= (Fl, . . . , F,), SO that S * ( H k ( , ) ) = S(F0, . . . , F,) n Hk(,). Hence, ~ F o. ., . F,) f-7 Hk = s*(Hk(n)) n Hk Ek(,)+i n H k ( , ) + i = 8. In both cases, s(Fo, . . . , F,) is disjoint from H A .By the choice of F(Hk),this means IF(Hk)I > n. Let F,+,(k) be the (n 1)st term of F(Hk)for each k > k(n). The existence of it is assured by I F(Hk) I > n. Now, we put F,,, = min{F,+,(k) I k > k(n)}. 1) > k(n) such that F,+, = F,+,(k(n + 1)). Let Choose some k(n 3
+
+
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+
k 2 k(n 1). Clearly, &+, < F,+,(k). Note again F(Hk)In = F(Hk,+,))ln = (F,, . . . , F,) and Hk c Hk(,+,). By the choice of (iii), we obtain F,+,(k) < F,+,(k(n + 1 ) ) = F,,,. Hence, F,+, = F,+,(k) whenever k 2 k ( n + 1). This means (F,, . . . , F,, F,,,) = F ( H k ) l n + 1 for each k 2 k(n + 1). Claim 2 has been proved. Note that each ( E l , F,, . . . , En,F,) such that Ei = s(Fo, F,, . . . , 4- ,) for 1 < i < n, where Fo = X , is an admissible sequence in G(K,X). Hence the sequence ( E l , F,, E,, F2, . . .) is a play in G(K,X). The definition of s yields F. = 8. By Claim 2, each F, is a term of some F(Hk).By (iii), each F, contains the Hk. Therefore, we conclude 0 = I;, 3 H,. 0
n,,
nnBl nngl
Some of the following propositions will be used in the latter sections.
1.2. Proposition. Let K,and K, be classes with K, c K,.If Player I has a winning strategy in G(K,, X ) , then he has a winning strategy in G(K,,X ) . 1.3. Proposition. Let Y be a closed subspace of X . I f Player I has a winning strategy in G(K,X ) , then he has a winning strategy in G(K,Y ) . Proof. Lets be a stationary winning strategy for Player I in G(K,X). Then it is easy to verify that the restriction s* of s to 2' is a stationary winning strategy in G(K,Y) 0 1.4. Proposition. IfPlayer I has a winning strategy in G(FK,X ) , then he has a winning strategy in G(K,X ) . Proof. Lets be a stationary winning strategy for Player I in G(FK,X). For each F E 2', there exists a finite collection {D(F, i ) E 2' n K J1 < i < k ( F ) } whose union is s(F). Let ( E l , F,, . . . , Fk(,))be an admissible sequence in G(K,X), where En = D(X, n) n F,- for 1 < n < k ( X ) = k(1) and Fo = X. Continuing in this manner, for each m 2 1, we can obtain an admissible sequence ( E l , F,, . . . , Ek(,,,),Fk(,,,))such that
,
En
=
t(Fo,4 , . . . , F,-,) = D(Fk(i),n - k ( i ) ) n F,-,
for k ( i ) < n < k ( i + l), where k ( i + 1) = k ( i ) + and i < m. Consider the play ( E l , F,, E2, F2, . . .) in G(K,X). Note that s ( F ~ ( n ~)) Fk(i+l)= 8 for each i 2 1. Hence ni21Fk(i) = 0,so that n n a l F n = 8. Thus, t is a winning strategy for Player I in G(K,X). 0
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1.5. Proposition. Zfa space X has a countable closed cover { X , I n 2 I } such that Player I has a winning strategy in G(K, X,) for each n 2 1, then he has a winning strategy in G ( K , X ) . Proof. Let s, be a stationary winning strategy for Player I in G(K, X,) for each n 2 1. Let s ( X ) = sl(Xl). Assume that ( E l , F,, . . . , E,, F,) is an admissible sequence in G(FK, X) such that Ei = s(Fo, F,, . . . , K - , ) for each i < n, where Fo = X. Here, put E,,, = s(Fo, F,, . . . , F,) = UkQn+,sk(Fn (7 X k ) , which belongs to 2' n FK. Consider the play ( E l , F,, E2, F2, . . .) in G(FK, X) such that E, = s(Fo, F,, . . . , F,-,) for each n 2 1. Pick an m 2 1. Since E,,, n F,,, = 8, it follows that s,(F, n X,) n (F,,, n X,) = Qforeachn 2 m.Hencen,,,F, n X, = 8. This means F, = 8. Thus, Player I has a winning strategy in G(FK, X), so that Proposition 1.4 completes the proof. 0
n,, ,
1.6. Proposition. Let X be a space with a subset E such that E E 2' n K . If Player I has a winning strategy in G ( K , F )for each F E 2'disjoint from E, then he has a winning strategy in G(K, X ) . Proof. For each F E 2' disjoint from E, let sF be a stationary winning strategy for Player I in G(K, F). Let s ( X ) = E. Assume that ( E l , F , , . . . , E,, F,) is an admissible sequence in G ( K , X) such that Ei= s(Fo, . . . , for each i < n, where Fo = X. Put E,,, = s(Fo, . . . , F,) = sF,(F,). Then it is easy to verify that s is a winning strategy for Player I in G(K, X ) . 0
c-,)
1.7. Proposition. Zfa space X has a a-discrete closed cover { X , I u E A } such that Player I has a winning strategy in G(K, X,) for each u E A, then Player I has a winning strategy in G(DK, X ) . Proof. By Proposition 1.5, we may assume that { X , I u E A} is a discrete closed cover of X. Let s, be a stationary winning strategy of Player I in G(K, X,) for each u E A. Put s ( F ) = U{s,(F n X,) I u E A } E DK for each B E 2'. Then it is easy to check that s is a stationary winning one in G(DK, X ) . 0 1.8. Remark. Let a K be the class of all spaces which have a countable closed cover by members of K . Clearly, K c FK c a K . Hattori [I9851 showed that FK cannot be extended to a K in Proposition 1.4.
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1.9. Remark. Let 1 be the class of all one-point spaces. Galvin [1978] introduced the point-open game G(X), which is equivalent to G(1, X). He showed under CH that Player I has not a winning strategy in C ( X ) even if he has one in G(a1, X), where X is a subset of the real line R. We can similarly define a (winning) strategy for Player I1 in G(K, X), though we do not take this up here. In fact, considering the applications, it seems not so important as a winning strategy of Player I. 2. K-scattered spaces In this section, we discuss the relations between the existence of winning strategies for Player I in G(K, X) or G(DK, X) and spaces which have a countable closed cover by K-scattered sets. Let X be a space. For each F E 2', let
F") =
(~~FInonbdofxinFbelongstoK}.
Let X = X(O)and X('+') = (X('))(')for each ordinal 1.If I is a limit ordinal, let X(') = r)p
2.3. Theorem (Telgarsky [1975a]). If X is a subparacompact K-scattered space, then Player I has a winning strategy in G(DK, X).
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Proof. There is nothing to prove for ( ( X ) = 0 since X = 0.Assume that the statement is true for all spaces Y with ( ( Y ) < ( ( X ) . It suffices from Proposition 1.7 to show that X has a a-discrete closed cover 9 such that Player I has a winning strategy in G(DK, F) for each F E 8.It should be noted by the inductive assumption that each F E 8 with ( ( F ) < c ( X ) has this property. Let ((X)= I 1. For each x E X, we choose an open nbd U, of x in X such that U, n X(’) = 0 if x E X\X(‘) and U, n X(’) is contained in some N, E K if x E A’(,).Since X is subparacompact, the open cover { U‘I x E X} has a a-discrete closed refinement 8.Pick an F E 9. If F meets X(’), then F n X(’) E K. Moreover, if H E 2F is disjoint from F n X(’), then ( ( H ) < <(X\X(”) = R < ( ( X ) . It follows from the inductive assumption and Proposition 1.6 that Player I has a winning strategy in G(DK, F). If F is disjoint from X‘”, then ( ( F ) < I < <(A’). Next, let < ( X ) be a limit ordinal. Since { X \ X ( ” l I < g ( X ) } is an open cover of X , it has a a-discrete closed refinement 9. Since each F E 8 is contained in some X\X(”’, we have ( ( F ) < I < ( ( X ) . 0
+
In particular, a space Xis said to be C-scattered if K = C in Definition 2.1. 2.4. Corollary. r f X is a subparacompact C-scatteredspace, then Player I has a winning strategy in G(DC, X ) . Recall that a space Xis said to be hereditarily metacompact if each (open) subspace of X is metacompact.
2.5. Proposition. A space X is hereditarily metacompact if and only if every L, is open in X for each R < q disjoint cover { L , I I < q} of X such that has a point-finite open expansion { U, 11 < q} (i.e.,L, c U,for each I < q). Proof. (Necessity): Let X be a hereditarily metacompact space and { L, I I < q} a cover described as above. The statement is true for q = 1. Let q be a fixed ordinal. Assume that the statement is true for each [ < q. If q is a non-limit ordinal, let q = 1. Since U,,rL, is hereditarily metacompact, it follows from the inductive assumption that {L,I I < [} has a pointfinite open expansion {UllR < (}. Put U, = X . Then {U,lI < q} is a point-finite open expansion of { L , I 6 < q}. Assume that q is a limit ordinal. Since Xis metacompact and {U,,,L, 16 < q} is an open cover of X , there exists a point-finite refinement {U,(6 < q} such that U, c U l < , L , for each 6 < q. Applying the inductive assumption to U, and { L , n U , l I < S},
+
53 I
Topological Gomes and Applications
there exists a point-finite open expansion { W;,,,I 1 < S} of { L, n U, : 1 < S} such that w,,, c U6 for each i < 6. Now, we put y.= I 1 < 6 < q} for each 1 < q. Then it is not difficult to verify that {w,l1 < q} is a point-finite open expansion of { L;,I I. < q}. (SufJlciency):Let 4 = {U,,11 < q } be any collection of open sets of X. Let L,. = Uj,\u,,,Up for each I < q , and L, = X \ u 4 . Then U,,,L, = u,,
u{w..,
+
w.
Let X be a set and 'id! a collection of subsets of X. The order of 42 at a point X , denoted by ord,@, is the number of members of o& which contain x . The following lemma is well-known and the proof of it is quite standard. For example, see Pears [1975, Lemma 3.5.91.
.Y E
2.6. Lemma. Let 42 = { U , 11 < q } be a point-finite open cover of X and X,, = { x E X I ordv% < n } for each n b I . Then {TI[n b I } is a countable closed cover of X and = {E(E., . . . A,,)I I I < . . . < 1, < q} is a discrete (dopen) cover ofXn\Xn I f o r each n 2 I , where E(1, . . . 1,) = n , < , , U j n , (Xfl\Xfl. I ) .
We say that a class K isfinitely additive if FK = K, that is, every space with a finite closed cover by members of K belongs to K.
2.7. Theorem (Junnila, Smith and Telgarsky [1986]). Let K be a finitel?, additive class. If a hereditarily metacompact space X is K-scattered, then Player I has a winning strategy in G(DK, X ) . Proof. Let 9 = { L,, I 1 < q } be a K-scattered partition of X. The existence of it is assured by Proposition 2.2. It follows from Proposition 2.5 that there exists a point-finite open expansion 42 = { U ,I I < q} of 2'.Since Y is a n 2 1, be the same as described cover of X , so is %!. For the 42, let X,, and in Lemma 2.6. Pick a nonempty F E 2'". Let k ( F ) = min{k 2 1 IF n X, # 8). Here we set R ( F ) = { E ( I , . . . I,) n F I E ( I , . . . 1,) E 8,and k = k ( F ) }
and S(0) = (0).Then it is easily checked by Lemma 2.6 that, for each F E 2'", (i) each member of b ( F ) is closed in X , (ii) b ( F ) is discrete in X.
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Note that E(A, . . . A,) c UiakLj,,. Since K is finitely additive, by (i), we have E(1, . . . A,) n F E K. It follows from Lemma 2.6 and (ii) that U b ( F ) = (Xk\Xk-,) n F E DK n 2', where k = k ( F ) . Hence we can obtain a stationary strategy s: 2' -+ DK n 2' for Player I in G(DK, X) defined by s ( F ) = (X,,,,\X,,,,-,) n F for each F E 2'. We show that s is a winning one. Let ( s ( X ) , F,, s(F,), F,, s(F,), . . .) be a play in G(DK, X). In order to show n,,,F, = 8, it suffices to prove that F, n X, = 8 for each n 2 0, where F, = X and X , = 8. Assume F, n X, = 8. Since F,+, c F,, we can also assume F, n X,+, # 8. Observe k ( F , ) = n + I . Then we have s(F,) n F,+, = (Xl+l\Xn)n F,,, = 8. Since X, n F,+, c X, n F, = 8, we obtain %+I Fn+I = 8. 0 2.8. Lemma (Konig). Let d,, be a nonemptyfinite collection of subsets of a set S for each n 2 0. Let cp, he a function from d,into d,-for each n 2 1. Then there exists a sequence { A,},,o such that A,, E d, and cpn+ , ( A , + I = A, for each n 2 0.
,
This is quite well known as Konig's Lemma. One can consider it as a special case of Engelking [1977, Theorem 3.2.131. 2.9. Theorem (Junnila, Smith and Telgarsky [1986]). Let X he a regular hereditarily metacompact space. If Player I has a winning strategy in G ( K , X ) , then X has a countable closed cover by K-scattered sets. Proof. Let s be a stationary winning strategy for Player I in G ( K , X). Let A, = (8) and @, = ( U ( 8 ) ) = {X}. For each n 2 0, we construct a collection @, = { U(a)I a = ( a , , . . . , a,,) E A , } of open sets in X satisfying, for each a E A,,, (1) a-
E
A,-,,
(2) { U ( b ) I b L A , + , with b- = a } is a point-finite open cover of 3\4U(ah (3) U(b) c U(a)\s(U(a)) whenever b E A,,,, with b- = a, Assume that @ i , i < n, have been constructed. Pick an a E A,. For each x E U(a)\s(V(a)), choose an open nbd V, in X such that Vv c U(a)\s(Cr(a)). Since Xis hereditarily metacompact, thereexists a point-finite open refinement { U ( a @ a ) I a ~ A ( a ) } o f { V , lU(a)\s(U(a))}.PutA,+, x~ = {a 0 a I a ~ A ( a ) and a E A , } . Then (i) and (ii) are satisfied. Claim 1. @
=
U,,,,@, is a point-finite open cover of X .
Topological Games and Applications
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Clearly, 4 is an open cover of X. Pick an x E X . Let 4,(x) = { U E 4 , l x E U }for each n 2 0. It follows from (1) and (2) that each 4, is point-finite in X. So, each 4 , , ( x ) is finite. Now, assume that each 4 , , ( x ) is nonempty. Since each member of %,(x) is contained in some member of %,-l(x),it follows from Lemma 2.8 that there exists some (a,, a2, . . .) such each n2 1, where that x E U(an)for -a, = (a,, . . . , an) E A,. By (3), the sequence ( s ( X ) , U(a,),s(U(a,)),U(a,), . . .) is a play in G(K, X). Hence, we U(a,) = 8, which contradicts x E U(a,). Thus, some 4 % ( x ) have final must be empty. Since u 4 n + I ( x )c U 4 , ( x ) , it follows that 4,(x) is empty for each n 2 no. This implies that 4 is point-finite at x. For each k 2 1, let X, = { x E XI ord,4 < k}. It follows from Claim 1 that {&lk 2 I } is a countable closed cover of X. Since a0= {X} and = x \ s ( X ) , we have XI = s(X)EK. Assume that is K-scattered. We show that Xk is K-scattered. Let F be a nonempty closed set in X, . We may assume that F meets X,\X,- I . Pick x E F n (X,\X,- I ). Since 4 is point-finite in X , one can choose the maximal b , , . . . , b, E A = Un,oAnsuch that, for each i < m, (i) x E U(bi), (ii) if a E A with a- = bi, then x # U(a).
nnz,
u%!l
Claim 2.
x, n (ni,,U(bi))
c ni,,s(U(bi)).
-
Assume that there exists some y E (X, n (n,,,U(bi)))\s(U(bn)), where n < m. By (2), one can choose some c E A such that c- = b, and y E U(c). By (ii), we have x $ U(c).By the maximality of {b,I i < m } , we conclude that .ord,4 2 k 1. This is in contradiction with y E X,. By Claim 2, V = ni,,U(bi) is an open nbd of x E F in X such that V n X, E K. Hence V n F E K. Thus, X, is K-scattered. 0
+
Since DK-scattered spaces are K-scattered, Theorems 2.7 and 2.9 and Proposition 1.5 yield the following corollary. 2.10. Corollary. Let K be a finitely additive class. A regular hereditarily metacompact space X has a countable closed cover by K-scattered sets i f and only if Player I has a winning strategy in G(DK, X ) . 2.11. Remark. Let SK be the class of all K-scattered spaces. Then K c DK c SK. However, the game G(SK, X) is not so different from G(DK, X). In fact, for a subparacompact space X , Player I has a winning strategy in G(SK, X) if and only if he has one in G(DK, X) (cf. Telghrsky
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[I9831 and Yajima [1984]). It is not known whether this is true for a regular O-refinable space X. Recall that a space X i s said to be scattered if each nonempty subset of X has an isolated point in it. Now, let Y be a disjoint collection of subsets of a space X. We say that Y is scattered if, for any nonempty subset H of UY, there exists an S E Y such that S n H is a nonempty open set in H.
2.12. Lemma. Let X be a completely regular space. Let Y and Z be closed subsets in X with Z c Y . Then there exists a sequence { Y , ( Z , Y ) } of scattered collections of closed sets in X such that Y\Z = U n B I ( U Y n ( Z Y ,) ) . Proof. We define a transfinite sequence {Yj,I rl < q } of collections of subsets I I < q } of closed sets in Y as follows: of Y\Z and a transfinite sequence (0;. Let 90 = (0) and Do = Y . Let 1 be an ordinal and assume that 3 ' , and D, have been defined for each p < I . If 1is non-limit, let Yi be a maximal disjoint subcollection of the collection of all open F,-sets in Dj,-, disjoint from Z . Moreover, let D, = Dj,-,\UYj,. If 1 is limit, let Yj. = (0) and D;, = n,.;,D,. By the choice of Dj., there exists some ordinal 9 such that D, = Z and D, # Z for each I < q . Let Y = Uic,,Yi. For each F,-set G E 9,we choose a sequence { F , ( G ) } of closed sets in X such that G = U,,,F,(G).Here,wesetY,(Z, Y) = {F,(G)IGE%}foreachn 2 1. Then it is easy to verify that the union of Un,,Y,(Z, Y ) is exactly Z\ Y . Pick n 2 1 and a nonempty subset H of U Y n ( Z ,Y). There exists the least ordinal I < q such that F,,(G) n H # 0 for some G E gj,.Note that 1 is non-limit and G is an open set in D , - I disjoint from D , . Since Yi is a disjoint collection and F,(G) c G, F,,(G) is an open set in n ( U Y n ( Z ,Y ) ) . Since H c Dj.-,,it follows that F,(G ) n H i s a nonempty open set in H. Thus each Y,(Z, Y) is scattered. 0 2.13. Theorem (Telgarsky [1983]). Let X be a completely regular space. r f Player I has a winning strategy in G ( K , X ) , then X is the countable union of K-scattered subspaces.
Proof. Let s be a stationary winning strategy for Player I in G(K, X). We inductively construct countably many collections {Y(t)I t E U n B , , ~ " } of closed sets in X. They are defined by Y(0) = {X} and 9(t6 k) = U(Yk(s(F)F , ) 1 F E Y(T)}for each T E UnbOmn and k 2 1, where Yk(s(F),F) is described in Lemma 2.12 for Y = F and Z = s ( F ) .
Topological Games and Applications
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Claim 1. Each Y ( 7 ) is scattered. It is clear that each Y ( t )is a disjoint collection of closed sets in X. Assume that Y ( r ) is scattered. Pick a k 2 1. Let H be a nonempty subset of U 9 ( t @ k). Since uY(z @ k) c ~ Y ( Tone ) , can choose F, E Y(7)such that F, n H i s nonempty and open in H. Moreover, since Fl n H c Fl n (UY(7 @ k)) = uYk(S(Fl), FI), there exists an F, E Yk(S(F,), F1) c y(t @ k ) such that F2 n (Fl n H ) = F, n H is nonempty and open in H. We set X ( z ) = U { s ( F )I F E Y ( T )for } each 7 E (JaZOwn.
Claim 2. { X ( T )I z E U n Z 0 w ncovers } X. Assume the contrary. Pick some x E X\X(r) for all 2. Let k, = 0 and F, = X. Then one can inductively choose some ( k , , k,, . . .) E o"'and F,) and x E F, for each {F,,1 n 2 0} c 2' such that F,,,, E 9kn+l(s(Fn), n 2 0. Since F,,+I c F, and s(F,) n F,,, = 8 for each n 2 0, we obtain F, = 0.This contradicts x E nnZIF..
nnZl
Claim 3. Each X ( t ) is K-scattered. Let H be any nonempty closed set in X(z). Since Y ( r ) is scattered and X ( t ) c uY(z),there exists an F, E Y(7)such that Fo n H i s nonempty and open in H. Then F, n H = Fo n X ( t ) n H
=
s(F,) n H.
Hence, s(F,) n H i s nonempty and open in H. Moreover, since s(Fo) n H is closed in s(F,) E K , we have s(F,) n H E K . 0
2.14. Example (Junnila, Nogura and Telgarsky [1987]). There exists a regular (Lindelof) space X such that Player I has a winning strategy in G ( X ) (see Remark 1.9), but X has no countable closed cover by C-scattered sets.
Proof. Let T = {ti I I < w , } be a subset of the closed interval [0, I], where ti # t, if I # p. Consider the subspace Y = { ( t i , p ) 1 I < p < w , } u T x {aI} of [0, I] x [0, o,], where [0, w , ] has the usual order topology. The regular space X is obtained from Y by contracting T x {a1} to the single point x,. Note that the complement of any nbd of x, in Xis at most countable. It follows from Proposition 1.6 that Player I has a winning strategy in C ( X ) . Assume that X has a countable closed cover { X ,I n 2 l} such that each X,, is C-scattered. For each n 2 1, let
T,, = { t E TI ( t , A)
E
X,, for uncountably many I < w , } .
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Then some T, must be uncountable. Indeed, if not so, take some t , E T\Uns, T,. Moreover, we can take some p < o,such that ( t l , p) E X\ X, for each n 2 1. This contradicts that ( X , 1 n 2 I} covers X. Since a scattered separable metric space is at most countable, there exists a countable, dense in itself subspace D (i.e., D has no isolated point in it) of T,. Since X, is closed in X , for each t E D,Af = {A < wI I ( t , A) E X,} is closed and unbounded in [0, q).Hence so is n { A f l t E D}.Pick A,, E n { A , I t E D}. Let F be the closure of { ( t ,A,)( t E D } . Then F is a countable, dense in itself closed subspace of X,. Since F is homeomorphic to the space of rational numbers, X , is not C-scattered. This is a contradiction. 0
3. Closure-preserving covers by compact sets Tamano [1971] asked whether a completely regular space X with a closurepreserving cover by compact sets is paracompact. If the “closure-preserving” is strengthened by “locally finite” or “hereditarily closure-preserving” (cf. Chapter 7, Definition l.lO), then the space X is paracompact. Indeed, we have the following propositions. 3.1. Proposition. A T2-space X has a locallyfinite cover by compact sets if and only if X is locally compact and paracompact.
3.2. Proposition. A T2-space X has a hereditarily closure-preserving cover by compact sets if and only if X is a closed continuous image of a locally compact, paracompact space. Proof. The “if” part is easy to see. Let {K, I a E A ) be such a cover as above. K,. Then Y is a locally compact, paracomLet Y be the topological sum OacA pact space. The continuous mapf: Y + Xis defined byf(x, a ) = x for each x E K, and a E A. It is easy to check thatfis a closed map.
o
Tamano’s question was solved negatively by Potoczny [1972]. On the other hand, the question initiated the study of spaces which have a closure-preserving cover by (countably) compact sets. One of the most important properties of these spaces is stated in terms of topological games. The next theorem improves the result of Potoczny [1973]. 3.3. Theorem (Telgarsky and Yajima [1980]). If a space X has a closurepreserving closed cover % by countably compact sets, and K is a class with % c K, then Player I has a stationary winning strategy in G(DK, X ) .
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Proof. For each F E 2', we assign a maximal disjoint subcollection 9 ( F ) of {C n FI C E %}. Since each 9 ( F ) is closure-preserving and disjoint, it is discrete. So we can define a function s : 2' + 2' n DK by s ( F ) = U 9 ( F )for each F E 2 x . We show that s is a stationary winning strategy for Player I in G(DK, X). Let {F, 1 n 2 I } be a decreasing sequence of closed sets in X such that s ( X ) n F, = 8 and s(F,) n F,+, = 8 for each n 2 1. Assume that F, # 8. We can find a C, E 3 ' which meets all F,'s. Then C, n F, 4 9(F,) for each n 2 1. Indeed, if C, n F, E 9 ( F , ) for some n, then we get a contradiction with
n,, ,
COn Fn+, = (COn Fn) n Fn+, c
U 9 K ) n Fn+, = s(F,)
n F,+, =
8.
Pick an n 2 1. Since 9 ( F , ) is maximal and disjoint and C, n F, 4 9 ( F n ) we , can take a C, E % such that C, n F, E 9(F,) and C, n C, n F, # 8. Pick some x, E C, n C, n F, . By x, E s(F,), we have x, 4 F,+ I . Hence, { x,} is a sequence of distinct points in C,. Since C, is countably compact, {x,} has a cluster pointy in C,.Then we have y E IF,. Indeed, if y 4 F, for some n, we can choose an m > n with x, 4 F,,. This is in contradiction with x,, E F, c F,. Now we claim that y 4 U, I C, . Indeed, suppose that y E C, for some n. Since C, n F, E 9 ( F , ) and y E F,+, ,we have a contradiction with Y E (Cn n F n ) n Fn+, c s(Fn) n & + I = 0. On the other hand, since % is closure-preserving, we have y E {x, I n 2 I } c UnrI C, = U, C,.This contradicts y 4 Un2I C, . 0
n,,
,
Theorem 3.3 and Proposition 1.5 yield the following corollary.
3.4. Corollary. Zfa space X has a a-closure-preserving closed cover by compact sets, then Player I has a winning strategy in G(DC, X ) . 3.5. Definition. A space X is called a P-space if there exists a function p : u,,,(2')" -+ 2', satisfying (i) p ( F l ,. . . , F,) n = 8 for each ( F l ,. . . , F,) E (2')" and n 2 1, (ii) u,,Ip(Fl, . . . ,F,) = X for each (F,, F2, . . .) E (2')" with nn,,Fn= 8. Morita [1964] defined P-spaces for the characterization of normal spaces such that their products with metric spaces are normal. The above representation is due to Telgarsky [1975b]. As is easily seen, countably compact spaces are P-spaces. More generally, we have the following proposition. 3.6. Proposition. Z f a space X has a a-closure-preserving closed cover by countably compact sets, then X is a P-space.
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Proof. Let U,, be a closed cover of X by countably compact sets such that each W, is closure-preserving. It suffices to show that there exists a function p* : 2’ x w -+ 2’, satisfying (i’) p*(F, n) n F = 8 for each F E 2’ and n 2 1, (ii’) if {F,} is a decreasing sequence of closed sets in X with n n 2 1 F ,= 0, then Un3,p*(F,, n) = X . For each F E 2’ and n 2 I , put p*(F, n) = U { C E Wil C n F = 0 and i < n } . Note that each p*(F, n) is closed in X . Let {F,} be a decreasing sequence of closed sets in X with On*,F, = 8. Take any C E Wm. Since it is countably compact, one can choose n 2 m such that C n F, = 8. Then we have C c p*(F,, n). Thus (i’) and (ii’) are satisfied. 0
3.7. Proposition. Let X be a space and V a closure-preserving closed cover of X by compact sets. Then every open cover 4 of X has an open refinement Y such that each C E V meets at most finitely many members of Y . Proof. It follows from Corollary 3.4 that there exists a stationary winning strategy s for Player I in G(DC, X). We construct a sequence { ~ , , } ) n 2 0 of collections of open sets in X satisfying that, for each n 2 1, (1) s ( F , - , ) c Ug,, where F,-I = X\Ui,,(UYi) and F,, = X , (2) for each C E V, { G E g,,I G n C # 8 ) is finite, (3) each G E 9,is contained in some U E 4. Let go = (8). Assume that {gi1i < n} have been constructed. Let F, = X\UiG,,(Ugi).Since s(F,) E DC,there exists a discrete collection 9, of compact sets such that s(F,) = U9, . For each D E 9,,,we can choose a finite subcollection 9 ( D ) of 9 such that D c U 4 ( D ) . For each U E 4 ( D ) ,we put G(D, U ) = U\U{C E Wgl D n C = S}, which is an open set in X. Here, we set = { G ( D , U ) l U E 9(D)and D E 9,}.Then (1) and (3) are clearly satisfied. Pick a C E V. Since C is compact and 9, is discrete, C meets only finitely many members DI,. . . ,D,,,of 9,.Hence C meets only finite members G(Di, U), where U E 9 ( D i ) and i < m , of g,,+l. Thus (3) is satisfied. By ( l ) , Y = Un,13,,is an open cover of X. By (3), Y is an open refinement of 9.Put L, = U { C E %‘I C c UiS,,(Ugi)} for each n 2 1 and Lo = 8. We setY, = {G\L,-,IG~%’,}foreachn 2 1 , a n d Y = U,,,Y,.SinceeachL, is closed in X , each V E Y is open in X.Pick an x E X . We choose the smallest ‘m 2 1 such that x E UYm.Then we have
Hence, Y is an open cover of X , so that it is an open refinement of 4. Take any C E W . Since C is compact and 3 is an open cover of X , we can choose
Topological Games and Applications
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a k 2 1 with C c Lk.Hence C n (UVn)= 8 for each n > k. Moreover, by (2), C meets at most finitely many members of UnckV,,. 0 3.8. Theorem (Katuta [19741, Potoczny and Junnila [ 19751). Zfa space X has a (a-)closure-preservingclosed cover by compact sets, then X is metacompact (6-refinable). 3.9. Remark. Junnila [1979, 1978b] proved the following nice theorem: A space Xis metacompact (Srefinable) if and only if every directed open cover of X has a (a-)closure-preserving closed refinement. This immediately yields Theorem 3.8. Indeed, if X has such a cover 5# as in Theorem 3.8, then every directed open cover of X has the refinement 5#.
Since every collectionwise normal, O-refinable space is paracompact (cf. m a , p. 2211, we get the following corollary. 3.10. Corollary (Potoczny [1973]). Zfa collectionwise normal space X has a a-closure-preserving cover by compact sets, then X is paracompact. 3.11. Example. The space P of irrational numbers with the usual topology has no cr-closure-preservingcover by compact sets.
Proof. Let X be a closed subspace of P and 5# a closure-preserving cover of X by compact sets. Since Xis separable, there exists a countable dense subset {x,,ln 2 l} of X. Taking C,, E %' with x,, E C,, for each n 2 1, we can see X = U,ls,Cn.Hence X is a-compact. Now, assume that P has a o-closurepreserving cover by compact sets. Then P is also a-compact. Hence P is an F,-set in the real line R with the usual topology. This contradicts Baire's theorem (cf. m a , Theorem 111.71). 0 We shall consider what kinds of spaces have a closure-preserving cover by compact sets. More generally, we give two sufficient conditions for a space to have a closure-preserving cover by members of K. 3.12. Fact. I f a n open cover Q of a space X is interior-preserving (i.e., isopeninXforeachV c % ) , t h e n 9 = { X \ U { U E % ~ ~U$} I x E X } i s a closure-preserving closed cover of X . 3.13. Lemma. Let K be afinitely additive class. Suppose that a space X has a countable closed cover {X,,(n2 1) and, for each n 2 1, there exists a point-finite open cover %, of X such that X,\U(%,,\V) E K for each finite Y c a,,. Then X has a closure-preserving closed cover by members of K.
Y . Yajima
540
Proof. Foreachn 2 l,letWn = {U\U,,,X,lUE@,}.ThenW = Un,iWn is a point-finite open cover of X . Put K, = X\u{ W E W I x 4 W } for each x E X . Since every point-finite open cover is interior-preserving, it follows from Fact 3.12 that {K, I x E X } is a closure-preserving closed cover of X . Pick an x E X and let n, = min{n I x E X , } . Observe that ny < n, whenever y ~ K , . I t i s e a s y t o s e e t h a t K , c U { X , \ U { U E @ ~ ! , I X ~
to K . 3.14. Theorem (Junnila, Smith and Telgarsky [1986]). Let K be ajinitely additive class. r f a metacompact (%-refinable)space X has a countable closed cover by locally K subspaces, then X has a (a-)closure-preserving closed cover by members of K.
Proof. Let X be a metacompact space and {X,I n 2 l} a closed cover of X such that each X, is locally K. Then, for each n b I , there exists a point-finite open cover 92"of X such that U n X, E K for each U E 42,. Take n 2 1 and a finite subcollection V of a,,.Since %, covers X,we have X,\U(%,\V)
c (J{ U n X,I U E V } c
u{ U n X , I U
E
V }E K.
Since X,\U(%,\V) is closed in X , it belongs to K. Hence it follows from Lemma 3.13 that X has a desired cover. For the parenthetic part, we may assume that X i s O-refinable and locally K. Let @ be an open cover of X such that 0 E K for each U E @. There exist a countable closed cover {Xnln b l} o f X and a sequence {V,}of open refinements of % such that { V n X , I V E V,} is point-finite in X, for each n b 1. Then it also follows from Lemma 3.13 that each X , has a closurepreserving closed cover by members of K. 0 3.15. Corollary.
I f a metacompact (B-rejinable)space X is 0-locally compact, that is, X has a countable closed cover by locally compact subspaces, then X has a (a-)closure-preserving closed cover by compact sets. 3.16. Theorem (Junnila, Smith and Relgarsky [1986]). Let K be aJinitely additive class. I f a hereditarily metacompact space X has a countable closed cover by K-scattered sets, then X has a closure-preserving closed cover by members of K.
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Proof. Let { X , In 2 l} be a countable closed cover such that each X, is K-scattered. Pick an n 2 1. By Proposition 2.2, there exists a K-scattered partition {L,,j.IO < 1 < q,} of X,. Let Ln,o= X\X,. Put 9,= {L,.,IO < 1 < q,}. Then it follows from Proposition 2.5 that there exists a point-finite for each n 2 1. Take a finite open expansion @, = { U,,;,I 0 < 1 < q,,} of 9, V . c @, Since 9, is a cover of X and L,,;, c U,,,;.for each 0 < 1 < q,,, we have F n ( v ) = Xn\U(@n\v) c U{Ln,i E I un,;.E Y and A > 01. Hence, F , ( Y ) is contained in some member of K . Since F , ( Y ) is closed in X , it belongs to K. By Lemma 3.13, X has a desired cover. 0 3.17. Question. Is the converse of Theorem 3.16 true for a regular hereditarily metacompact space X ? 3.18. Remark. A closure-preserving closed cover 9 of a space Xis said to be special if there exists a point-finite collection @ of open sets in X such that, for each F E 9, X\F = U { U E 92 I U n F = 0).Junnila, Smith and Telgarsky [1986] also showed that the converse of Theorem 3.16 is true if the space X has a special closure-preserving closed cover by members of K. In case K = C, Question 3.17 is answered affirmatively. In fact, from Corollaries 2.10 and 3.4 and Theorem 3.16, we obtain the next corollary. 3.19. Corollary. A regular hereditarily metacompact space X has a countable closed cover by C-scattered sets if and only i f X has a closure-preserving cover by compact sets. However, the following question remains open. 3.20. Question (Telgarsky [ 19741). Let X be a (paracompact) space which has a closure-preserving cover by compact sets. Does X have a countable closed cover by C-scattered sets? In the particular case of a closure-preserving cover by finite sets, we can obtain a complete characterization of spaces with such a cover. 3.21. Theorem. The following are equivalent for a space X . (a) X has a closure-preserving cover by finite sets. (b) X is hereditarily metacompact and has a a-closure-preserving cover by finite sets.
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Y. Yajima
(c) X is hereditarily metacompact and has a countable cover { X , I n 2 l } by discrete subsets such that UkGnXkis closed in X for each n 2 1 . (d) X is hereditarily metacompact and has a countable closed cover by scattered subsets.
Proof. Assume that X has a closure-preserving cover by finite sets. Then note that every subspace of X has such a cover. Hence it follows from Theorem 3.8 that X is hereditarily metacompact. (a)+(b): Obvious. (a)+(c): Let 9 be a cover of X as in (a). For each x E X , let U ( x ) = X \ U { F E 9 1 x .$ F}, and let 9 = { U ( x ) I x E X}. Moreover, let X, = { x E XI ord,9 = n } for each n 2 1 . Assume x E X\U,,lX,. Then there exist infinitely many x , , x 2 , . . . E Xsuch that x E V ( x , )for each n 2 1. Take F, E 9 with x E F,. By the choice of U(x,,), all x,’s are in F,,. This is in contradiction with the finiteness of F,. Hence, {X,,ln 2 l } is a cover of X. Since UiGnx = { x E XI ord,9 < n } and 9 is an open cover of X,UiGnx is closed in X for each n 2 1. Pick an n 2 1 and an x E X,. There exist distinct x I ,. . . ,x, E X such that { x ’ E XI x E U(x’)} = { xI,. . . , x,,}. Let y E U ( x ) n X,,. It should be noted that y E U ( x ) implies U( y ) c U ( x ) .So we have y E U( y ) c U ( x ) c V ( x i ) .By ord,,9 = n,y must be some x i . Hence, U ( x ) n X, c {xI,. . . , x,,}. Since X is T I , it follows that X, is a discrete subset of X for each n > 1 . (b)+(c): Let {X, I n 2 l } be a countable closed cover of X such that each X, has a closure-preserving cover by finite sets. By (a)+(c), each X, has a countable cover {X,,k 1 k 2 l } by discrete sets such that UkGmx,k is closed in X, for each m 2 1. Then we can arrange the countable cover 1n, k 2 l} of X such that it satisfies (c). (c)+(d): Let Y, = UkanXkfor each n 2 1. Then { Y,ln 2 I } is a countable closed cover of X. It is easy to see that each Y, is scattered. (d)+(a): This is an immediate consequence of Theorem 3.16.
ni4,,
In Theorem 3.21, the equivalence of (a), (b) and (c) was proved by Yajima [1977], (a)+(d) was proved by Telgarsky [I9741 and (d)-+(a) was proved by Nyikos [19771 and Junnila [ 1978al. We conclude this section with a negative answer to Tamano’s question. 3.22. Example (Peregudov [1974]). There exists a normal space X with a closure-preserving cover by finite sets, but not paracompact. Proof. We shall show that Bing’s example [1951] modified by Michael [1955] is such a space X. Let P be an uncountable set and B the set of all
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subsets of P. Consider the product space Y = IIQEpDerwhere each DQ = (0, I } denotes the two-point discrete space. For eachp E P, takef, E Y defined asfp(Q) = 0 i f p $ Q and&@) = 1 i f p E Q. Let F = { f , l p E P}. Let X = F u {YEY l f ( Q ) = 0 except for finitely many Q E 9 } be the set. The topology of X is then defined by (i) eachf, E F has the nbd base as the subspace of Y, (ii) eachfc X \ F is an isolated point in X. Let Vp = { f ~ X I f ( { p }=) l} for each P E P . Each EX satisfies f({p } ) = I for at most finitely many p E P, so { Vplp E P} is a point-finite collection of open sets in X. Let S c X and $2 an open cover of S. For each f , E S n F, assign UpE $2 withf, E Up.Then (Up n
6 l f p E 5-" FJu { { f } I f E S\FJ
is a point-finite open refinement of $2. Hence, Xis hereditarily metacompact. Since Vp n F = { f,}, it follows that F is a closed discrete subset of X. Since X \ F i s an open discrete subset of X,it follows from Theorem 3.21 that X has a closure-preserving cover by finite sets. It is well-known that Xis normal and that F cannot be separated by open sets, thus X is not collectionwise normal (cf. Engelking [1977, Example 5.1.231). 0
4. Covering properties of product spaces Tamano [1962] raised a question of whether one can characterize the class of those spaces whose product with every paracompact space is paracompact. This question is still open, but some partial results were obtained by Katuta [1971] and Telghrsky [1971, 1975al. Here we take up one of the latter, which is stated in terms of topological games, and we also prove analogical results for subparacompactness and metacompactness. For the proofs of this section, we need the concepts of various rectangles in product spaces. Let X x Y be a product space. A subset of the form A x B in X x Y is called a rectangle. Such a rectangle R is denoted by R = R' x R", that is, R' and R" are the projections of R into X and Y respectively. Let R be a rectangle in X x Y. We say that R is a closed (open) rectangle if R' and R" are closed (open) in X and Y respectively. We also say that R is a cozero rectangle if R' and R" are cozero sets in X and Y respectively. Moreover, R is called a closed x open rectangle if R' is closed in X and R" is open in Y.
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4.1. Theorem (Telgarsky [1975a]). If X is a paracompact space and Player I has a winning strategy in G(DC, X ) , then X x Y is paracompact f o r every paracompact space Y .
Proof. Let s be a stationary winning strategy for Player I in G(DC, X). Let Y be any open cover of X x Y. From Michael’s Theorem (cf. ”a, Theorem V.11) it suffices to prove that Y has a a-locally finite open refinement. However, for the sake of the last section, we prove a little more: Y has a a-locally finite refinement by cozero rectangles in X x Y. Let e0= {8},A. = (8)and R(0) = H(0) = X x Y. For each n 2 1, we shall construct a collection 4V, of cozero rectangles and a collection { { R ( a ) ,H ( a ) } I a = ( a l , . . . , a,) E A,} of pairs consisting of a closed rectangle R(a) and an open rectangle H(a), satisfying the following conditions (l)-(7): For each n 2 1, (1) 9,is locally finite in X x Y, (2) each U x V in 4Vn is contained in some G E Y, ( 3 ) { H ( a )I a E A,} is locally finite in X x Y . For each a E A, and n 2 1, (4) a- E An-,, (5) R ( 4 c N u - ) n H ( 4 , (6) @ ( a _ ) ’ ) n R(a)’ = 8, (7) R(a)\U@n+l c U{R(a @ a)Ia @ a ~ A n + [ } * Assume that and {R(a),H(a)}, a E Ai,satisfying (1)-(7) have been constructed for each i < n. Pick an a E A,. Since s(R(a)’)E DC, there exists a discrete collection { C, I y E T ( a ) }of compact sets in R(a)’ such that @(a)’) = U{ C, I y E T ( a ) } . Since Xis collectionwisenormal and R(a)’ c H(a)’, we can take two discrete collections { W, 1 y E T(a)} and {O,l y E T ( a ) } of open sets in X such that C, c W, c c 0, c H(a)‘ for each y E T(a). Since Y is paracompact, it is easy to see that there exists a locally finite collection
my
%, = {U6,ix V,li = 1, . . . , m6 and 6
E
A(y)}
of cozero rectangles, satisfying (i) C, c U, = U{ U6,iI i < m6}c W, (cH(a)’) for each 6 E A(y), (ii) each 17,,~ x V, is contained in some G E 9, (iii) { & 16 E A(y)} is locally finite in Y, (iv) R(a)” c U{V,lG E A(y)} c H(a)”. Moreover, there exists a closed cover {La16 E A(y)} of R(a)” such that La c V, for each 6 E A(y). Here, we set a,+,= u{%,lyE T(a) and a E A,} and A,+I = { a @ 6 1 6 E A ( y ) , y E T ( a ) a n d a E A n )u { a @ O I a E A , } .
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Topological Games and Applicaiions
Pick an a @ a E A,, I . In case of a = 6 for some 6 E A(y) and y E r ( a ) ,we Put R(a @ 6) = (( q \ U , ) n R(a)’) x L6 and H(a @ 6) = 0,x
b.
In case of a = 8, we put
R(a @ 8) = (R(a)’\U{W,ly
E
I-@)}) x R(aY’ and
~ ( @a e) = ~ ( a ) .
It is easily checked by the above construction that and {R(a), H ( a ) } , a E A,+,, satisfy (1)-(5). Pick an a @ a E A,+, again. If a = 6 for some 6 E A(?) and y E T(a), by (i), we have s ( R ( 4 ’ ) n R(a @
4’
= (U{C,lS E r ( a ) } )n =
C, n ( q \ U , )
=
< n
0.
If a = 8, we have
s ( R ( 4 ’ ) n R(a 0 0)’
=
U{C,lS E r(a)>\U{W,IB E
=
0.
W)}
Hence, (6) is satisfied. Pick an a E A, and an (x, y) E R(a)\U4,,+,. In case 4 U { W, I y E r@)}, clearly (x, y) E R(u 0 8). So we can assume x E W, for . an i < ma some y E T(a). Take 6 E A(y) with y E L,. Assume x E 17,Find with x E UaSi.Then we have
x
ua,i x La c U6.i x Va E c @,+I, which contradicts (x, y ) 4 U@,+l. Hence it follows that (x, Y ) E
(x, Y ) E ((W,\Vd n
W)’) x L6 = R(a 0 6).
Thus (7) is satisfied. Now, we set 4 = U,,, I 4,,. By (1) and (2), 4 is a c-locally finite collection of cozero rectangles such that each U x V E % is contained in some G E Y. It remains to show that 4 covers X x Y.
Claim. I f {a,} is a sequence such that a,, E A , and (u,,)- = a,-, for each n 2 1, where a, = 8, then nn,IR(a,,)’= 8. It follows from ( 5 ) and (6) that R(a,,)’ = R(a,,-,)’ and s(R(u,,-,)’)n R(a,)’ = 0 for each n 2 1. This implies that the sequence ( s ( X ) , R(a,)’, s(R(a,)’),. . . , s(R(u,,-,)’),R(a,)’, . . .) is a play of G(DC, X). Player I necessarily wins this play. Hence, we have n,,,,R(a,)’ = 8. Assume that 4 does not cover X x Y. Pick some (x, y ) E X x Y\U%. Notice that (x, y ) E X x Y = R(0).Since (x, y ) 4 U%,,+,for each n 2 0, by
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Y. Yajima
(7), one can inductively choose an infinite sequence (aI,a*, . . .) such that a, = (aI,. . . , a,) E A, and ( x , y ) E R(a,) for each n 2 1. So we have x E I R(a,)’, which contradicts the above claim. 0
n,,
By Corollary 3.4 and Theorem 4.1, we have this corollary. I f X is a paracompact space with a a-closure-preserving cover by compact sets, then X x Y is paracompact for every paracompact space Y.
4.2. Corollary.
4.3. Theorem (Yajima [1983b]). I f X is a regular subparacompact space and
Player I has a winning strategy in G(DC, X ) , then X x Y is subparacompact for every subparacompact space Y.
Proof. Lets be a stationary winning strategy for Player I in G(DC, X). Let Y be any open cover of X x Y. It suffices to prove that Y has a a-locally finite refinement by closed rectangles in X x Y. Let So= {O}, A . = (8) and R(O) = X x Y. For each n 2 1, we shall construct two collections 9, and { R ( a )I a = (a,, . . . , a,) E A,} of closed rectangles, satisfying the following conditions (1)-(7): For each n 2 I , (1) 9, is a-discrete in X x Y, (2) each D x L in 9, is contained in some G E Y, (3) { R ( a ) l aE A,} is a-discrete in X x Y. For each a E A, and n 2 I, (4) a- E A,-l, (5) R ( 4 = W - ) , (6) s(R(a-)’) n R(a)’ = 8, (7) R(a)\U~,+, = U@(a 0 @ ) l a6 a E A,+, 1. Assume that and { R ( a )1 a E A i } satisfying (1)-(7) have been constructed for each i < n. Pick an a E A,. Let { C, I y E T ( a ) } be a discrete collection of compact sets in R(a)’ such that @(a)’) = U {C, I y E r ( a ) } . Since R(a)’ is subparacompact and {R(a)’\U{C, I B E T ( a )with /I# y } I y E T ( a ) }is an open cover of R(a)’, ‘thereexists a a-discrete closed cover I y E T ( a )and k 2 l} of R(a)’ such that each Fyqk is disjoint from C, if fl # y , where {Fy,k I y E T ( a ) }is discrete in X for each k 2 1. Using subparacompactness of R(a)”, it is easily seen that for each y E T ( a ) and k 2 1 there exists a a-discrete collection SYsk =
{o,,ix L61i = 1, . . . , maand 6
of closed rectangles, satisfying (i) each U6,iis open in FYsk, (ii) C, n Fy,k c U, = U{U,,il i
E
A(y, k ) }
< m,} for each 6 E A(?, k),
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541
(iii) each Dh,ix L6 is contained in some G E 9, (iv) {La16 E A(y, k)} is a a-discrete closed cover of R(a)”. Here, we set 9,+1 = U { 9 J y E T(a), a E A, and k 2 l} and A,+I = { a @ 616 E A(y, k), y E T(a), a E A, and k 2 l}.
For each a @ 6 E A,+I, where 6 E A(y, k), we put R(a @ 6) = (F7,k\UJ x La. Then it is similarly checked that 9,+1 and {R(a)la E A,,,} satisfy (1)-(7). Now, we set 5 = UnrI 9”. By (1) and (2), 9is a a-discrete collection of closed rectangles such that each D x L E 9 is contained in some G E Y. Moreover, in the same way as above, we can show from (5)-(7) that 9covers X x Y. 0 4.4. Corollary. If X is a paracompact (regular subparacompact) C-scattered space, then X x Y is paracompact (subparacompact)for every paracompact
(subparacompact) space Y.
This follows from Theorems 2.3, 4.1 and 4.3. 4.5. Theorem (Yajima [1983b]). I f X is a regular metacompact P-space and Player I has a winning strategy in G(DC, X ) , then X x Y is metacompact for
every metacompact space Y .
Proof. Lets be a stationary winning strategy for Player I in G(DC, X ) and p be a function from u,s1(2x)” into 2’ described in Definition 3.5. Let B be any open cover of X x Y. It suffices to prove that Y has a point-finite refinement by open rectangles. Let 42, = {O}, A, = (0)and R(0) = H(0) = X x Y. For each n 2 1, we shall construct a collection @n of open rectangles and a collection {{R(a), H(a)}I a = (aI, . . . , a,) E A,} of pairs consisting of a closed x open rectangle R(a) and an open rectangle H(a), satisfying the following conditions (1)-(9): For each n 2 1, (I) a,, is point-finite in X x Y, (2) each U x V in 42, is contained in some G E Y, (3) {H(a)laE A,} is point-finite in X x Y, (4) U@n c U{H(a)Ia E An-,}. For each a E A, and n 2 1, ( 5 ) a- E A,- I , (6) R(a) c R ( a - ) and R(a) c H(a) c H(a-), (7) s(R(a-)’) n R(a)’ = 0, (8) R(a)\U%+i c U{R(a @ a) l a @ E An+t 1,
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Y . Yajima
(9) p(R(a1 l)’, . . . ,R(a I n - l)’, R(a)’)n H(a)’ = 8 (see the Introduction for the notation of a 1 i). Assume that 9i and {R(a), H ( a ) } , a E A i , satisfying (1)-(9) have been constructed for each i < n. Pick an a E A,. Let { C, I y E T(a)} be a discrete collection of compact sets in R(a)’ such that @(a)’) = U{C,I y E T(a)}.Since X is regular metacompact, it follows from Chapter 5, Theorem 7.17 that there exist two point-finite E T(a)} of open sets in X such that collections { W, I y E T ( a ) }and (0, c 0, c H(a)’\U{Csl p E
C, c W, c
T ( a ) with p # y}
for each y E T(a). Since Y is metacompact and R(a)” is open in Y , for each y E T ( a ) there exists a collection x &li = I,
%? =
. . . , m, and 6 E A(y)}
of open rectangles, satisfying (i) C, c U, = U{U,,ili < m,} c W, for each 6 E A(?), (ii) each U,,i x & is contained in some G E Y, (iii) { & 1 6 E A(y)} is a point-finite open cover of R(a)”. Here, we similarly set 9,+, = U ( 9 , I y E r(a)and a E A,} and A,+I = { a @ 616EA(y),yET(a)andaEA,} u { a @ O I a E A , } .
Pick an a @ a E A,,,. Observe that a1 i some 6 E A(y) and y E r(a),we put R(a 0 6 ) =
E
Ai for each i
<(q\u,> n R(a)’) x
< n. In case a
= 6 for
&,
H(a 0 6 ) = (O,\p(R(aI I)’, . . . , R(aln)’, R ( a 8 6)’)) x
6.
In case a = 8, we put
~
(
o el o e) 0
=
(RW\U{
w,I Y E r(m x RW”,
= ( ~ ( a ) ’ \ p ( ~I iy, (a
. . . ,~
o
( in)’, a ~ ( a e)’)) x ~ ( a ) ” .
Then it can be verified similarly that and { { R ( a ) ,H(a)}I a E A,,,} satisfy (1)-(9). Now, we set $2 = Un2,9,. By (2), 9 is a collection of open rectangles in X x Y such that each U x V E 9 is contained in some G E Y. In the same way as in the proof of Theorem 4.1, we can show from (6)-(8) that 9covers X x Y. It remains to show that 9 is point-finite in X x Y. Claim 1. If {a,} is a sequence such that a, E A , and n 2 1, where a,, = 8, then H(a,)’ = 8.
(0,)-
=
for each
Topological Games and Applications
549
By the proof of the Claim in the one of Theorem 4.1, we have T\,> ,R(a,,)’= 0. So U,,,,p(R(a,)’,. . . , R(a,)’) = X. By (9), we obtain the desired one. Claim 2. nnbl(UXfl) = 8, where Xn = { H ( a ) l a E A,,}. Assuming the contrary, pick a zo E T\,,,,(UX,,). Let A,(z,) = { a E A , I zo E H(a)}for each n 3 0. By (3), each A,,(z,) is finite. By (5) and (6), a E A,,(zo)implies a- E A,,-, (z,). It follows from Lemma 2.8 that there exists an infinite sequence (a,, a2, . . .) such that a,, = (cq, . . . , a,) E A,(z,) for each n 2 1. Then we have zo E H(a,,),which contradicts Claim 1. Pick any z E X x Y. By Claim 2, find an m 2 1 such that z r$ UXm. Observe from (5) and (6) that UX,,, c UXnfor each n 2 1. Since z 4 UX,, for each n 2 m, it follows from (4) that z $ U9,, whenever n m. Therefore, by (l), 9 is point-finite at z. Thus, 9 is point-finite in X x Y. 0
n,, , ,
=-
From Corollary 3.4, Proposition 3.6, Theorems 4.3 and 4.5, we get the following corollary. 4.6. Corollary. If X is a regular metacompact (subparacompact)space with a a-closure-preserving cover by compact sets, then X x Y is metacompact (subparacompact)for every metacompact (subparacompact)space Y. Unfortunately, we do not have an analogical result to include the 8-refinability in Corollary 4.6. So we raise the following question. 4.7. Question. Let X be a regular space with a (a-)closure-preservingcover by compact sets. Is X x Y O-refinable for every O-refinable space Y? 4.8. Remark. Quite recently, Gruenhage and Yajima [198-] have given an affirmative answer to this question. In fact, they have proved that the analogue of Theorem 4.3 is valid for O-refinability (and metacompactness).
5. The games in product spaces In this section, for two classes K , and K2 of spaces, we define a new class K , x K2. Assuming that Player I has winning strategies in G ( K , ,X) and G(K2,Y). We consider the question of whether he has a winning strategy in G(D(K, x K2),X x Y). For this question, we introduce the concept of
550
Y. Yajima
D-products. The result obtained here will be applied to dimension theory in the next section. For two classes K,and K2 of spaces, we put Kl x K2 = U { 2 X x Y I XKl~ and Y E Kz}.
That is, it consists of all product spaces X x Y such that X E K,and Y E K2 and all closed subsets of them. Here, the concept of closed rectangle, which is defined in the previous section, plays an important role.
5.1. Definition. A product space X x Y is called a D-product if for any disjoint pair {E, F } of a closed rectangle E and a closed set Fin X x Y there exists a a-discrete collection W of closed rectangles such that F c UW c X x Y\E. It is easy to see that each closed rectangle in a D-product is also a D-product. As a typical D-product, from the proof of Theorem 4.3, we get the following proposition.
5.2. Proposition. If X is a regular subparacompact space and Player I has a winning strategy in G(DC, X ) , then X x Y is a D-product for any subparacompact space Y. Here we need some more notations: Let Ni = (3n + il n 2 0 } for i = 0, 1, 2. So, w u (0) is decomposed into No, N I and Nz. For each z = ( k , , . . . , k,,) E d"' n 2 1, let Xz = kl * * * k,,. Identify z = 8 with T = (0), so that Xp = 0. For each z = ( k , , . . . , k,,) E a",n 2 1, with k, 4 No,we put z* = (kl, . . . , ki)if k,,, . . . , ki+,E N, EN^) and ki4 N, (#Nz)for some i c n. If k,,, . . . , k, E N, EN^), put z* = 8 = (0).
+
+
5.3. Theorem (Yajima [1983a]). Let X x Y be a D-product. ZfPlayer I has winning strategies in G ( K l ,X ) and G(Kz, Y ) , then he has a winning strategy in G(D(K, x K2),X x Y ) .
Proof. Let s, and s2 be stationary winning strategies for Player I in G(K,, X ) and G(Kz, Y ) respectively. From Proposition 1.4, it is sufficient to construct a winning strategy t for Player I in G(L, X x Y ) , where L = x Kz)l*
mwl
55 I
Topological Games and Applications
Let El = t(X x Y ) = sl(X) x s,(Y). Let AB = {8},B(8) = {X x Y} and Fo = R(8) = X x Y. Player I1 chooses any F, E 2’“ disjoint from E l . Now, assume that we have already constructed an admissible sequence ( E l , F,,. . . , Em, F,) in G(L, X x Y) such that Ei = t(Fo, F,,. . . ,K - , ) for each i < m and there exists a discrete collection {R(a)I a = (a,, . . . , a,,, a,,+,) E A,@&}of closed rectangles for each t E w“, n 2 0, with Ct < m - 1 and k 2 1, satisfying the following conditions (1)-(3): (I) a @ a E Are&implies a E A,. (2) For each a E A,, R(a) n q X r ) + l
= U { R ( a @ a)la @ a E Are&and k
3 I } c R(a).
(3) For each a @ a E ATek,where t = ( k , ,. . . , k,,)E w“, n 2 0, and k 2 1, (i) s,(R(a)’)n R(a €I3 a)‘ = 8 if k,, k E Nou N,, (ii) s2(R(a)”)n R(a €I3 a)” = 8 if k,,, k E Nou N,, (iii) s,(R(a*)’) n R(a a)’ = 8 if k, E N, and k E Nou N,, (iv) s,(R(a*)”)n R(a @ a)” = 8 if k,, E N, and k E Nou N,, where a* = a I i E A,. if a E A, and t* E w’. In case of t = 8, consider k, = 0 E No. Pick a t = ( k , , . . . , k,,) E w”, n 3 0, with Ct = m. Note that R(r) has been constructed since t = T- @ k,, and Ct- = m - k,, < m - 1. For each a E A,, we put E(a) = s,(R(a)‘)x s,(R(a)”)
if k,, E No,
E(a) = s,(R(a)’)x (s,(R(a*)”)n R(a)”) if k,, E NI,
E(a)
=
(s,(R(a*)’)n R(a)’) x s,(R(a)”) if k,, E N,.
Then E(a) E K, x K,.We set Em+, = t(Fo, Fl,
. . . , F,)
= (U{E(a)I a E A,
and t E Un20w“with CT = m})n F,.
Since {R(a)I a E A,} is discrete in X x Y, Em+,belongs to L. Player I1 chooses any F,+, E 2’” such that F,,, c F, and Em+,n F,+, = 8. Pick again a t E U,,20wn, with CT = m,and an a E A,. Since R(a)is a D-product and E(a) is a closed rectangle in R(a) with E(a) n F,+, = 8,there exists a collection {R(a @ a) I a E B(a, k) and k 2 l} of closed rectangles in R(a) such that { R ( a a) I a E B(a, k)} is discrete in R(a) for each k 3 1 and R(a) n F,,, c U { R ( a @ a) I a E B(a, k) and k 2 l} c R(a)\E(a).
552
Y. Yajima
Since E(a) and R(a @ a) are disjoint rectangles in X x Y, we may assume that, for each a E B(a, k),
E(a)’ n R(a @ a)’ = 0 and E(a)” n R(a @ a)” = 0 implies k
E
No,
E(a)’ n R(a @ a)’ = 0 and E(a)” n R(a @ a)” # 0 implies k
E
N, ,
E(a)’ n R(a @ a)’ # 0 and E(a)” n R(a 8 a)” = 0 implies k
E
Nz.
We set AT@k= { a 0 aim E B(a, k) and a E A,} for each k 2 1. It is easy to check that {R(b)I b E A,,,} is discrete in X x Y and (1)-(3) are satisfied. We show that t is the desired winning strategy. Since Player I1 has arbitrarily chosen each F,, it suffices to show that &,, I F, = 0. Assuming the contrary, pick (x, y ) E Om,,F,. Notice that (x, y) E X x Y = R(0). By (2), one can inductively choose some ( k i ,k z , . . .) E wo and ( a I ,a2, . . .) such that a, = ( a I ,. . . , a,) E A,,, where t, = (kl, . . . , k,), and ( x , y) E R(a,) for each n 3 1. Then x E I R(a,)’ and y E I R(a,)”. Assume that (kl, k,, . . .) contains the infinite subsequence (ki(l),ki(,),. . .) consisting of all natural numbers k, belonging to Nou N l . Let r(0) = a(0) = 0. Let t ( j ) = t i ( j = ) ( k i , . . . , k i ( j ) )and a ( j ) = q j )= (a1,. . . , a i ( j ) )for eachj 2 1. Let us show
n,
(4)
s , ( R ( a ( j ) ) ’ )n R ( a ( j
n,,
+ 1))’ = 0
for eachj 2 0.
+
Case ki(j)+l E Nou N,: Since ki(j)+l= ki(j+l), we have ~ ( j 1) = ~ ( j@) k i ( j + l ) .Hence a ( j 1) = a ( j ) @ ai(j+I) E A , ( j + l ) .So, (4) follows from
+
(3) (9. Cuseki(j)+lE N,: Letm = i ( j + 1). Leta = t ( j + 1)- = (kl, . . . , k,-l) and b = a ( j 1)- = ( a l , . . . , a,-1). Then t ( j + 1) = a @ k,, k, E Nou N, and a ( j 1) = b 0 a,,,. Since k,-l, . . . , ki(j)+lE N,, we have . a* = (kl, . . . , k i ( j ) )= ~ ( j ) Hence, b* = (a1,. . . , aiCj,)= a ( j ) E An.. Thus (4)follows from (3) (iii). BY (2) and (4), the sequence (sl(X), R(u(l))’,sl(R(a(l))’), R(a(2))’, sl(R(a(2))’),. . .) is a play of G(K,, X). Hence we have n j , l R ( a ( j ) ) ’ = 0, which contradicts x E n j a 1 R ( a ( j ) ) ’On . the other hand, if ( k , , k,, . . .) contains the infinite subsequence &(,,, ki(2),. . .) consisting of all natural numbers k, belonging to Nou N,, the same way of reasoning yields a contradiction. In this case, (3)(ii) and (3)(iv) are used instead of (3)(i) and (3)(iii), respectively. 0
+
+
A product space X x Y is called a C-product if for any disjoint pair {E, F} of a closed rectangle E and a closed set F i n X x Y there exists a countable COkCtiOn {Rklk 2 1} Of Closed reCbngleS such that F C UkalRk C X X Y\E.
Topological Games and Applications
553
5.4. Proposition. Let X x Y be a C-product. If Player I has winning strategies in G ( K l ,X ) and G(K2,Y ) ,then he has a winning strategy in G(K, x K,, X x Y ) . The proof is quite parallel to that of Theorem 5.3. In fact, we only have to replace a discrete collection {R(a) I a E A , } of closed rectangles with a closed rectangle R ( T )for each T E U n r O w n .
5.5. Remark. Let X = [0, wl) be the discrete space. Let Y = [0, wI]be the space such that w1has the nbd base {(x, w , ]I x < wl}and each x < wIis an isolated point in Y. Then it is easy to verify that the product space X x Y is a D-product but not a C-product. The author has been informed of the following question by Telgarsky. Let X and Y be completely regular spaces. If Player I has winning strategies in G(DC, X) and G(DC, Y), does he have a winning strategy in G(DC, X x Y)? 5.6. Question.
If X and Y are regular subparacompact spaces, the above question is answered in the affirmative. Since D(DC x DC) = DC, this immediately follows from Proposition 5.2 and Theorem 5.3. Next, note that if X x Y is regular Lindelof, then it is a C-product. So it follows from Telgarsky [1975a, Corollary 14.141and our Proposition 5.4 that if Player I has winning strategies in C(C, X) and G(C, Y), then he has one in G(C, X x Y ) .
6. Applications to dimension theory
In this section, we deal with the application of topological games to the sum and product theorems in dimension theory. We omit (by the restriction on the number of pages) the proofs of several basic theorems and lemmas in dimension theory. First, we give a closure-preserving sum theorem for dim and Ind, and then we deal with the product theorem for dim and Ind. 6.1. Definition. For a normal space X , we say that dim X < n (the covering dimension G n) if every finite open cover 42 of X has a (finite) open refinement 9-such that ord,V < n + 1 for each x E X .
5 54
Y. Yajima
6.2. Theorem (The countable sum theorem for dim). Zfa normal space X has a countable closed cover {Xk I k 2 I } such that dim Xk < n for each k 2 I , then dim X < n.
6.3. Theorem (The locally finite sum theorem for dim). I f a normal space X has a locallyfinite closed cover 9such that dim F < n for each F E 9,then dim X < n. 6.4. Lemma. Let X be a normal space and E a closed subset of X with dim E < n. For each finite open cover { U, I i < k} of X, there exist a finite open cover Y = { V, I i < k} of X and an open set G containing E such that (i) V, c V, for each i < k, and (ii) ord,TY< n 1 for each x E G.
+
Proof. It follows from Pears [1975, Proposition 3.1.61 that there exists a closed cover 9 = (41i < k} of E such that each 4 is contained in U, n E and o r d , 9 < n I for each x E E. Moreover, it follows from Engelking [1977, Theorem 7.1.41 that there exists a finite collection W = { i < k} of open sets such that 6.c 4 c Q for each i < k and ord,W < n + 1 for each x E X. Take an open set G in X such that E c G c G c U i G k K. Put V, = u (U,\G) for each i < k. Then Y = { V,l i < k} and G are desired ones. 0
+
vl
Let Dim, denote the class of all normal spaces with covering dimension that X E Dim, implies 2' c Dim, (cf. Pears [1975, Proposition 3.1.51). We can generalize the countable sum theorem in terms of topological games as follows.
< n. Note
6.5. Theorem (Telgarsky and Yajima [1980]). Let X be a normal space. I f Player I has a winning strategy in G(Dim,, X ) , then dim X < n.
Proof. The proof is a modification of that of Engelking [1977, Lemma 7.2.11. Lets be a stationary winning strategy for Player I in G(Dim,, X). Let { V, I j < k} be a finite open cover of X. For each i 2 0, we construct a finite open cover 4?, = { U,,jlJ < k} of X and an open set Gi in X , satisfying the following conditions: For each i 2 I , ( I ) V;.j c U,.-,,j c V,, wherej = 1 , . . . , k, (2) ord,42i < n + 1 for each x E G,, (3) s(X\G,-,) u Gi-, c G,.
Topological Games and Applications
555
Let Uo,j = V, for eachj 6 k and Go = 8. Assume that aiand Gi have been constructed for each i 6 m. Let E = s(X\G,,,). Then dim E < n. Applying Lemma 6.4 to E and Q,, we can take a finite open cover 9,,,+ I = { Um+l.j Ij 6 k} of X and an open set G containing E such that Um+l,jc Umvjfor each j 6 k and ord,Qm+I 6 n + 1 for each x E e. Since Xis normal, it may be assumed that~,,,+,vj c U,,,,jforeachj 6 k.LetG,,,+, = G,,, v G.Then9,,,+,andG,,,+, satisfy (1)-(3). Let 4 = q.,jfor each j 6 k. Pick an x E X . Since each ajis a finite cover of X , we can choose j(x) 6 k such that x E Q,jcx, for infinitely many i. By (I), we have x E U,jcx,= F/c,,. Hence 9 = (41j 6 k} is a closed cover of X such that 5 t V, for each j 6 k. From (3), {Gjli 2 l } covers .'A Take i,, 2 1 such that x E Gb. By (2), we have o r d x 9 6 ord,eb 6 n + 1. Hence o r d , f 6 n + 1 for each x E X . It follows from Pears [1975, Proposition 3.1.61 that dim X 6 n. 0
ni,,
ni,,
By Theorem 6.3, we have D(Dim,) = Dim,. So Theorems 3.3 and 6.5 yield the following corollary. 6.6. Corollary (Closure-preservingsum theorem for dim).
Ifa normal space
X has a closure-preservingclosed cover % such that C is countably compact and dim C 6 n for each C E V, then dim X 6 n. 6.7. Definition. Let Ind 8 = - 1. For a normal space X , we say that Ind X 6 n (the large inductive dimension < n ) if for each closed set E and each open set Vwith E c V there exists an open set U such that E c U c V and Ind aU 6 n - 1, where aU denotes the boundary of U. 6.8. Definition. A normal space X is said to be totally normal if for each open set U in X there exists an open cover 9 of U such that (i) each member of Q is an F,-set in X , (ii) 9 is locally finite in U .
Clearly, perfectly normal spaces are totally normal. Note that every subspace of a totally normal space is totally normal (cf. Pears [1975, Remark 1.4.121). 6.9. Theorem (The countable sum theorem for Ind). If a totally normal space X has a countable closed cover {xk I k 2 l } such that Ind Xk 6 n for each k 2 1 , then Ind X 6 n.
556
Y. Yajima
6.10. Theorem (The locally finite sum theorem for Ind). Zfa totally normal space X has a locally finite closed cover 9 such that Ind F < n for each F E 9,then Ind X < n. Let Ind, denote the class of all normal spaces with the large inductive dimension < n. Again, note that X E Ind, implies 2' c Ind, (cf. Pears [1975, Proposition 4.2.51). 6.1 1. Lemma. Let X be a totally normal space and E a closed subset in X with Ind E < n. Zf Ind F < n for each closed subset F in X disjoint from E, then Ind X < n. For example, this is found in Pears [1975, Proposition 4.4.111.
6.12. Theorem (Yajima [1979]). Let X be a totally normal space. I has a winning strategy in G(Ind,, X ) , then Ind X < n.
If Player
Proof. Lets be a stationary winning strategy for Player I in G(Ind,, X). Let A, = (0) and F0= (F(0)) = {X}. For each T E U n B O w n we, construct a collection 9, = {F(a)I a = (a,, . . . , a,,) E A,} of closed sets in Xsatisfying, for each a E A,, (1)
a- € A , - ,
(2)
{F(b)I b E A,,, each k 2 1,
(3)
U { F ( b ) ( bE A,,,
with b- = a } is locally finite in F(a)\s(F(a)) for and k 2 1 with b-
=
a } = F(a)\s(F(a)).
Assume that 9, has been constructed as above for each T E UiGmwi.Pick and an a E A,. Since F(a) is totally normal and F(a)\s(F(a)) is open a T E urn in F(a), there exists a collection @(a) = { U , I /I E B(a)} of open F,-sets in F(a) such that @(a)is locally finite in F(a)\s(F(a)) and (J@(a)= F(a)\s(F(a)). Since U pis an F, in F(a) E 2', there exists a sequence {Fp,k}of closed sets in X such that U, = U k 3 FB,, , for each 8 E B(a). Here we set A,,,
= {a
0 a Ia
=
(8, k), 8 E B(a) and a E A,}
for each k 2 1. Moreover, we set F(a 0 a ) = F,,, for each a 0 a E A,,, with a = (8, k) and Frek = {F(a 0 a)la 0 a E A,,,} for each k 2 1. Then (1)-(3) are satisfied. For each T E w m , m 2 0, we set X ( T ) = u { s ( F ( a ) ) l aE A T l i and 0 < i < m}.Using induction on m,it is not difficult to show from (1) and
Topological Games and Applications
551
(2) that
(4)
9, is locally finite in X \ X ( r - ) and UF, c X \ X ( t - ) for each t E wn.
u
nT I
So, in particular, we have (4')
{s(F(a))I a E A,} is locally finite in X\ X ( t _ ) for each T
E
u w".
" 2I
Note that X ( 8 ) = s ( X ) and X ( r ) = u{s(F(a))Ia E A,} u A'(?-) for each t E U n 2 1 ~ Assuming ". that X ( L ) is closed in X , by ( 4 ) ,X ( t ) is closed in X . Hence, (5)
X ( r ) is closed in X for each t E
u w".
n20
Clearly, Ind X(8) = Ind s ( X ) < n. Assume Ind X ( K ) < n. Let H be a closed set in X ( t ) disjoint from X ( t - ) . By (4'), {s(F(a))n HI a E A,} is a locally finite closed cover of H. Since Ind(s(F(a)) n H) < n, it follows from Theorem 6.10 that Ind H < n. Therefore, by Lemma 6.1 I , we have (6)
Ind X ( r )
< n for each t E
u w".
"20
Assume that we can pick some x E X \ U { X ( t ) I t E UnbOwn}. By Fo= {X} and (3), one can Inductively choose some (k,,k,, . . .) E wo and ( a I ,a,, . . .) such that a,,, = ( a l , . . . , a,) E A,, where z, = ( k l ,. . . , k,), and x E F(a,) for each m 2 1. By (3) again, we have F(a,+,) c F(a,) and s(F(a,,,)) n F(a,+,) = 8 for each m 2 0, where a, = 8. Hence nrnblF(a,,,)= 8. This contradicts x E n,,,,,F(a,). We have shown
If follows from (5)-(7) and Theorem 6.9 that Ind X
< n.
0
By Theorem 6.10, we have D(Ind,) = Ind,. So, Theorems 3.3 and 6.12 yield this corollary. 6.13. Corollary (Closure-preservingsum theorem for Ind). r f a totally normal space X has a closure-preserving closed cover V such that C is countably compact and Ind C < n for each C E V, then Ind X < n.
6.14. Question.
lary 6.6 or 6.13?
Is it possible to remove the "countably compact" in Corol-
558
Y. Yajima
6.15. Remark. Under the CH assumption, Ciesielski [I9851 gave a negative answer to this question. On the other hand, we can show the following: If a normal, semistratifiable space X has a closed cover {FAI A < 11) such that U,
+
Various kinds of product spaces are rectangular; see Pasynkov [1975]. In particular, the following are typical rectangular products. 6.18. Proposition. A product space X x Y is rectangular i f either (a) X is compact, or (b) X is nondiscrete metrizable and X x Y is normal. Taking together the above, Theorem 2.3, Corollary 3.4 and the proof of Theorem 4.1 get the following proposition. 6.19. Proposition. For paracompact spaces X and Y, the product space X x Y is rectangular i f either (a) X is C-scattered, or (b) X has a closure-preserving cover by compact sets. Thus, Theorem 6.17 can unify a number of product theorems for dim which had been proved before. Recall that a closed set L in a space X is called a partition between E and F if there exist disjoint open sets U and V such that E c U, F c V and X\L = u u v.
Topological Games and Applications
559
The following lemma is well-known (cf. Engelking [1978, Lemma 2.3.161); one can prove it with a standard method. 6.20. Lemma. Let E and F be disjoint closed sets in a space X . If V is a a-locally finite open cover of Xsuch that P n A = 8 or P n B = 8for each V E V , then there exists a partition L between E and F such that L c U{av: V E V } .
We say that the a-locallyfinite finite) sum theorem for Ind holds in a normal space X if Ind(U5) f sup{Ind F I F E F}for every a-locally finite (finite) collection F of closed sets in X . Note that is also normal. Concerning rectangular products, the product theorem for Ind was proved earlier than Theorem 6.17. Using the above lemma, we can give a simple proof of it. 6.21. Theorem (Nagata [1967]). Let X x Y be a normal and rectangular product. If the a-locally finite sum theorem for Ind holds in X x Y, then Ind X x Y f Ind X Ind Y.
+
Proof. We prove it by induction on Ind X + Ind Y. If Ind X + Ind Y = - 1, then it is clear. Assume that it is true if Ind X + Ind Y f n + m - 1. Let Ind X f n and Ind Y f m. Let E and F be disjoint closed sets in X x Y. Since X x Y is normal and rectangular, there exists a a-locally finite cover { U, I a E A} of X x Y of cozero rectangles such that U, n E = 8 or U, n F = 8 for each a E A. It is not difficult to see that there exists a a-locally finite cover { V, I B E B} of X x Y of open rectangles such that, for each B E B, (i) 5 is contained in some U,, (ii) Ind 8% f n - 1 and Ind 8V; f m - 1, where we recall that V, = y x Vs”. By (i), we have V, n E = 0 or 5 n F = 8. It follows from Lemma 6.20 that there exists a partition L between E and F such that L c U{8V ,I /3 E B}. Note that 8 5 = (8v x Ti) u (Pi x 8Vs”). By (ii) and the inductive assumption, we have Ind 8 V, f n + m - 1. Note that { 8V, I fi E B} is also a-locally finite in X x Y. So the assumption of our theorem gives Ind L f Ind(U{8Valb E B}) f
Hence Ind X x Y
f
sup{Ind 8 V , ( pB~) f n
n
+ m.
+ m - 1. 0
560
Y . Yajima
6.22. Remark. Pasynkov [1975] announced that “the 0-locally finite sum
theorem” in Theorem 6.21 can be replaced by “the finite sum theorem”. As far as the author knows, the proof has not been published. However, Basmanov [1981] proved that if the finite sum theorem for Ind holds in normal spaces X and Y, then ind(X x Y) < Ind X + Ind Y. Applying Theorem 5.3, we can obtain another product theorem. 6.23. Theorem (Yajima [1983a]). Let X and Y be subparacompact spaces such that the product space X x Y is (totally) normal. If either (a) X is C-scattered, or (b) X has a closure-preserving cover by compact sets, then dim X x Y < dim X dim Y (Ind X x Y < Ind X Ind Y).
+
+
Proof. Let dim X < n and dim Y < m. Now, assume that X x Y is normal. It follows from Corollaries 2.4 and 3.4 that Player I has a winning strategy in G(DC, X). So we can consider that he has winning strategies in G(DC n Dim,, X) and G(Dim,, Y). By Proposition 5.2, X x Y is a D-product. It follows from Theorem 5.3 that he has a winning strategy in G(DL, X x Y ) , where L = (DC n Dim,) x Dim,,,.Take a product E x F E L . Let {C,1 ct E A} be a discrete closed cover of E by compact sets. So Theorem 6.17 and Proposition 6.18(a) give dim C, x F
< dim C, + dim F < n + m.
Since {C, x F I tl E A} is a discrete closed cover of E x F, we have dim E x F < n m. Hence L c Dim,,,. This implies DL c Dim,,,,,. Thus, Player I has a winning strategy in G(Dim,+,,,,X x Y). By Theorem 6.5, we have dim X x Y < n m. The proof for dim is complete. In the same way as above, we can show the parenthetic part for Ind. 0
+
+
6.24. Remark. Ohta [1981] proved the following: If a (perfectly) normal space Y is not paracompact, then there exists a paracompact scattered space X with a closure-preserving cover by finite sets such that X x Y is (perfectly) normal but not rectangular. Since there exists a subparacompact, perfectly normal and nonparacompact space (e.g., Bing [1951, Example HI), Theorem 6.23 cannot be reduced to Theorem 6.17.
References Basmanov, V. N. [I9811 Inductive dimensions of products of spaces, Vesrnik Moskov. Univ. Ser. I Mat. Mekh. 36. 17-20.
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Bing, R. H. [I9511 Metrization of topological spaces, Canad. J. Math. 3, 175-186. Ciesielski, K. [I9851 L-space without any uncountable 0-dimensional subspace, Fund. Math. 125, 231-235. Engelking, R. [ 19771 General Topology (Polish Scientific Publishers, Warszawa). [ 19781 Dimension Theory (North-Holland, Amsterdam). Galvin, F. [I9791 Indeterminacy of point-open games, Bull. Acad. Polon. Sci. Ser. Math. 26, 4 4 5 4 9 . Galvin, F. and R. Telgfrsky [I9861 Stationary strategies in topological games, Topology Appl. 22, 51-69. Gruenhage, G. and Y. Yajima (198.1 A filter property of submetacompactness and its application to products, Topology Appl., to appear. Hattori, Y. [1985] A note on infinite dimensional spaces defined by topological games, Proc. A M S 94, 360-362. Junnila, H. J. K. [1978a] Stratifiable pre-images of topological spaces, in: A. Csaszar, ed., Colloquia Mathematica Societas Janos Bolyai 23 (North-Holland, Amsterdam) 689-703. [1978b] On submetacompactness, Topology Proc. 3, 375-403. [ 19791 Metacompactness, paracompactness and interior-preserving covers, Trans. A M S 249, 373-385. Junnila, H. J. K., T. Nogura and R. Telgarsky [1987] A singular space related to the point-open game, Proc. AMS 99, 568-570. Junnila, H. J. K., J. C. Smith and R. Telgarsky [ 19861 Closure-preserving covers by small sets, Topology Appl. 23, 237-262. Katuta, Y. (19711 On the normality of the product of a normal space with a paracompact space, General Topology Appl. 1, 295-319. [ 19741 On spaces which admit closure-preserving covers by compact sets, Proc. Japan Acad. 50, 826-828. Michael, E. [I9551 Point-finite and locally finite coverings, Canad. J. Math. 7 , 275-279. Morita, K. [1964] Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. Nagami, K. [I9571 Some theorems in dimension theory for non-separable spaces, J. Math. SOC.Japan 9, 80-92. Nagata, J. (19671 Product theorems in dimension theory, Bull. Acad. Polon. Sci. Ser. Math. 15, 4 3 9 4 8 . [1985] Modern General Topo/ogy (North-Holland, Amsterdam). Nogura, T. [I9831 A compact-like space which does not have a countable cover by C-scattered closed subsets, Proc. Japan Acad. 59, 83-84.
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Nyikos, P. [ 19771 Covering properties on o-scattered spaces, Topology Proc. 2, 509-542. Ohta, H. [I9811 On normal, non-rectangular products, Quart. J. Math. Oxford 32, 339-344. Pasynkov, B. A. [I9751 On the dimension of rectangular products, Soviet Math. Dokl. 16, 344-347. [I9811 Factorization theorems in dimension theory, Russian Math. Surveys 36, 175-209. Pears, A. R. [ 19751 Dimension Theory of General Spaces (Cambridge Univ. Press). Peregudov, S. A. [I9741 On normal spaces which admit a closure-preserving cover by bicompact sets, Vestnik Moskov. Vniv. Ser. I Mat. Mekh. 5, 40-42. Potocny, H.B. [I9721 A non-paracompact space which admits a closure-preserving cover of compact sets, Proc. A M S 32, 309-3 11. [19731 Closure-preserving families of compact sets, General Topology Appl. 3, 243-248. Potocmy, H.and H. Junnila [ 19751 Closure-preserving families and metacompactness, Proc. A M S 53, 523-529. Tamano, H. [I9621 On compactifications, J. Math. Kyoto Univ. 1, 161-193. [I9711 A characterization of paracompactness, Fund. Math. 72, 189-201. Telgarsky, R. [I9711 C-scattered and paracompact spaces, Fund. Math. 73, 59-74. [ 19741 Closure-preserving covers, Fund. Math. 85, 165-1 75. [1975a] Spaces defined by topological games, Fund. Math. 88, 193-223. [l975b] A characterization of P-spaces, Proc. Japan Acad. 51, 802-807. [I9761 Concerning two covering properties, Colloq. Math. 36, 57-61. [I9831 Spaces defined by topological games 11, Fund. Math. 116, 189-207. [I9871 Topological games: On the 50th Anniversary of the Banach-Mazur game, Rocky Mountain J . Math. 17, 227-276. Telgarsky, R. and Y. Yajima [I9801 On order locally finite and closure-preserving covers, Fund. Math. 109, 21 1-216. Yajima, Y. [I9771 On spaces which have a closure-preserving cover by finite sets, Pacific J. Math. 69, 571-578. [I9791 On order star-finite and closure-preserving covers, Proc. Japm. Acad. 55, 19-21. [I9791 A note on order locally finite and closure-preservingcovers, Bull. Acad. Polon. Sci. Ser. Math. 27, 401-405. [I9811 Topological games and products I, Fund. Math. 113, 141-153. [1983a] Topological games and products 11, Fund. Math. 117, 47-60. [1983b] Topological games and products 111, Fund. Math. 117, 223-238. [I9841 Notes on topological games, Fund. Math. 121, 31-40.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V.(1989)
CHAPTER 14
CATEGORICAL TOPOLOGY
Ryosuke NAKAGAWA Institute of Maihemaiics, Universiiy of Tsukuba, Ibaraki, 305 Japan
Contents I. 2. 3. 4. 5.
6. 7. 8. 9. 10.
1 I. 12. 13.
Introduction . . . . . . . . . . . . Categories and functors . . . . . . . Monomorphisms and epimorphisms . . Diagrams and limits. . . . . . . . . Complete categories . . . . . . . . . Factorizations of morphisms . . . . . Reflective subcategories . . . . . . . Characterization theorem . . . . . . (E, M)-categories. . . . . . . . . . Epireflective vs. bireflective in Top. . . Separation axioms and connectedness . Simplicity of epireflective subcategories Topological functors. . . . . . . . . Topological categories . . . . . . . . References. . . . . . . . . . . . .
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563 565 569 575 579 582 588 592 598 602 606 61 I 614 618 622
Introduction In this chapter we will show the first steps of categorical topology. We start from the definitions of category and functor (Section 1) and proceed to the characterization theorem of epireflective subcategories (Section 7). In the course of proving this theorem, several standard methods in categorical topology will be given. Roughly speaking, the characterization theorem is a categorical version of the theory of Stone-Cech compactification. As is well known, the notions of embedding and product play the essential role in the construction of the compactifications of Tychonoff spaces. Hence we first
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define the notion of extremal monomorphism in abstract category which corresponds to that of embedding in Top, the category of all topological spaces (Section 2). Then we consider the notions of diagram and limit, which include the notion of product (Sections 3 and 4). For a given continuous map f :X + Y, there is a factorization (Z, g, h ) o f f such that g : X + 2 is a surjection, h : Z + Y is an embedding andf = h 0 g. This fact is also used in the construction of the compactifications. Hence factorization structures of categories are discussed in Section 5 , and in Section 8 again with more general treatments. The universality of the Stone-Cech compactification corresponds to the notion of reflective subcategory, which will be introduced in Section 6. Then we arrive at the characterization theorem (Sections 7 and 8). Although the proof is essentially easy, the result is very powerful. In fact, other universal type theorems, such as abelianization of groups, completion of uniform spaces and Hewitt realcompactification of Tychonoff spaces are also special cases of the above result. The purely categorical treatment provides this power. Moreover, it follows by this treatment that the dual statement is also valid. We add some topics in the succeeding sections. In Section 9, we deal with the structure of thc collection of all epireflective subcategories of Top. In Section 10, we introduce a new notion which induces both separation axioms and connectedness. Section 11 is devoted to the classical problem concerning the special epireflective subcategories of Top. In recent developments of categorical topology, the investigations of topology functors and topological categories are most important. To conclude this chapter we shall demonstrate their definitions and several fundamental results (Sections 12 and 13). There exist good text books for category theory, e.g., Freyd [1964], Mitchell [1965], MacLane [1971] and Herrlich and Strecker [1979]. The last one is standard for students who want to learn categorical topology. There are also some surveying papers, e.g., Herrlich [1971], Wyler [I9711 and Nel [1976]. Herrlich [1983a] gave a wide survey of this area with a complete list of papers published in 1971-1980. Although I intend to give a brief and rudimentary introduction for those who are not familiar with category theory, the contents may be too narrow. For example, though the notion of reflective subcategory is a special case of that of adjoint functor, the general theory of adjoint functors is not discussed. The references are also incomplete. They do not contain the valuable papers which contributed to the theorems treated here. I would like to express my gratitude for indebtedness to all authors in this area. I hope that readers will obtain some interests in categorical topology and proceed to study it.
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1. Categories and functors We consider the collection of all sets or the collection of all topological spaces. As is well known, these collections cannot be considered as sets. Hence we need the notion of classes. However, it is not necessary for us to take a strictly formal axiomatic treatment of set and class theory. A class is considered to be a collection of sets. If P is a “property”, then there exists a class consisting precisely of those sets which have the property P. This is called the class of all sets with property P and is denoted by { x I x is a set and P ( x ) } . Sets can be considered as classes and are called small classes. Those classes which are not sets are called proper classes. For given families of classes, we can construct several kinds of new classes, such as, unions, intersections, Cartesian products and disjoint unions. We can also consider functions between classes, relations and equivalence relations on classes and so forth. The axiom of choice for classes is also assumed, i.e., every equivalence relation on a class has a system of representation. 1.1. Definition. A category A consists of two classes 0 and M, two functions dom and cod from A4 into 0 and a function 0 from D = {(f, g)lf, g E M and cod f = dom g} into M satisfying the following conditions: ( I ) If (f,g) E D, then dom(g 0f ) = dom f and cod(g 0f ) = cod g, where gof denotes o ( f , g) and is called the composition off and g. (2) If (f,g), (g, h) E D,then h 0 (g of 1 = (h 0 g) of. (3) For each X E 0, there exists an element e E M such that dom e = cod e = X , f o e = f, whenever dom f = X and eog = g, whenever cod g = X . (4) For any pair ( X , Y) with X , Y E 0 the class [ X , Y] = { f I f € M , d o m f = X a n d c o d f = Y}isaset. Elements of 0 are called objects of A (or A-objects) and those of M are called morphisms of A (or A-morphisms). The class 0 is denoted by Ob A and M by Mor A . If it is needed, we use the notations dom,, Cod,, o , , DA and [ , 1., For an object X of A the morphism e given in (3) is uniquely determined and is called the A-identity of X and denoted by I,. For a morphismf, dom f is called the domain off and cod f is called the codomain off. f E [ X , Y], is expressed by f : X --+ Y. It is noted that if A is a category, the class Ob A can be considered as a subclass of Mor A consisting of all identities, and that categories can be defined as classes of morphisms satisfying several conditions.
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1.2. Definition. There is a category A such that Mor A is the empty class, which will be called the empty category. If a category A has only one object X , Mor A is a set which has the structure of a semigroup with the unit 1,. Conversely, a semigroup with a unit can be considered as a category with only one object. A category A is said to be small when Mor A is a set (or, equivalently, Ob A is a set). A category A is said to be discrete when every morphism in A is an identity of some object in A, in other words, when Ob A = Mor A. 1.3. Example. (1) Set: the category of sets. Ob Set is the class of all sets, Mor Set is the class of all functions between sets, and the functions dom, cod and are defined as usual as domains, codomains and composition of functions between sets respectively. (2) Grp: the category of groups. Ob Grp is the class of all groups, Mor Grp is the class of all homomorphisms between groups, and dom, cod and are equal to those in (1). (3) Top: the category of topological spaces. Ob Top is the class of all topological spaces, Mor Top is the class of all continuous functions between topological spaces, and dom, cod and are equal to those in (1). (4) pTop: the category of pointed topological spaces. Ob pTop is the class of all pairs of the form ( X , x,), where X is a topological space and x, is an element of X called base point, Mor pTop is the class of all base-point preserving continuous functions, and dom, cod and are equal to those in (I). ( 5 ) Products of categories: Let A and B be categories. Then the product category A x B of A and B consists of Ob(A x B) = Ob A x Ob B, Mor(A x B) = Mor A x Mor B, dom = dom, x dorn,, cod = cod, x cod, and is defined by (h, k) V; g ) = (h o A f , k 0,g) for (f,g), (h, k) E Mor(A x B) with (f, h) E DA and (8, k ) E D,. 0
0
0
0
0
0
1.4. Definition. Let A and B be categories. B is said to be a subcategory of A if the following conditions are satisfied: (1) There exists an inclusion i: Ob B --t Ob A. (2) There exists an inclusion j : Mor B --t Mor A. (3) i*dom, = dom,*j, i*cod, = cod,*j and j - 0 , = o A - k , where means compositions of functions between classes and k is a function from DB into DA defined by k ( f , g ) = (j(f), j(g)). (4) For each X E Ob B, j(B-identity of X) = A-identity of i ( X ) . A subcategory B of a category A is said to be full if, for all X , Y E Ob B, j(iX,
ylB)
=
rX,
y],4.
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1.5. Example. (1) The category of abelian groups, denoted by Ab, is the full subcategory of Grp whose class of objects is that of all abelian groups. (2) The category of q-spaces, denoted by Top,, is the full subcategory of Top whose class of objects is that of all T,-spaces for i = 0, 1, 2, 3. (3) Intersections of categories: Let A be a category, A a class, and A, be a subcategory of A for each 1 E A. Then we can define an intersection niE,,A, of the class ( A l ) l , A of subcategories such that Ob(n,,,A,) = n,,,Ob Ai and Mor(nAe,,Al) = nlphMOr A,. 1.6. Definition. Let C and D be categories. D is said to be a quotient category of C if the following conditions are satisfied: (1) Ob D = Ob C. (2) There exists a surjection m: Mor C --+ Mor D. (3) dom,*m = dom,, cod,-m = cod, and m.0, = o D * n , where n is a function from D, into Do defined by n(f, g) = (m(f), m(g)).
The homotopy category of topological spaces, hTop, is a quotient category of Top with Ob hTop = Ob Top and Mor hTop = the quotient class of Mor Top by the equivalence relation “homotopic”. 1.7. Definition. Let A and B be categories. B is said to be a dual (or an opposite) category of A if the following conditions are satisfied: (1) Ob B = Ob A. (2) Mor B = Mor A. (3) For the identity i : Mor B --* Mor A, dom, = COdA*iand cods = dOmA i. (4) For the function j : Ds DA defined by j ( f , g) = (i(g), i( f)), --+
i*oB
= oAmj.
For any category A, the dual category B of A can be constructed by the = A. above conditions and will be denoted by A’. Obviously, (A’‘)’ 1.8. Definition. Let A and B be categories. A functor Ffrom A to B is a function F from Mor A into Mor B satisfying the following conditions: (1) If cod, f = domAg,then cod,F(f) = dom,F(g). (2) F * o A = oB*F’,where F’ is a function from DA into DB defined by F’(f, g) = (F(f), F(g)). (3) If e is an A-identity of X E Ob A, then F(e) is a B-identity of Y = dom,F(e).
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A functor from A to B will be denoted by F : A + B and A and B are called a domain and a codomain of F respectively. For any X E Ob A , dom,F(l,) is denoted by F(X). Then we have a function F from Ob A into Ob B. A functor F : A + B may be defined as a pair of two functions F : Ob A + Ob B and F : Mor A 3 Mor B satisfying the following conditions: (1) F o r f : X + Y E Mor A , F(f): F(X) + F(Y) E Mor B. (2) F o r f : X + Y a n d g : Y + Z i n M o r A , F(g0,f) = F(g)o,F(f)in Mor B. (3) W,) = IF(,). For two functors F : A 3 B and G : B + C,we can define a functor G 0 F : A + C by the function G F : Mor A + Mor C.The functor G F will be called the composition of F and G . If B is a subcategory of A with the inclusions i: Ob B + Ob A and j : Mor B .+ Mor A , j satisfies the conditions (l), (2) and (3) in Definition 1.8. Then we have a functor j : B + A , called the inclusion functor. If B = A , the inclusion functorj: B 3 A will be called an identityfunctor and denoted by 1A . 0
1.9. Example. Letf : X Y be a continuous function between topological spaces X and Y and I f 1 : I XI + I Y I be a function between their underlying sets. Then a function F: Mor Top + Mor Set defined by F(f) = If1 induces a functor F : Top -, Set, which is called the forgetful functor.
1.10. Definition. Let A and B be categories. A contravariant functor F : A + B is a function F from Mor A into Mor B satisfying the following conditions: (1) If Cod, f = dom,g, then dom,F(f) = cod,F(g). (2) F * o A = oB*F”,where F” is a function from DA into DB such that F“(Lg) = (F(g), F(f)>. (3) If e is an A-identity of X E Ob A , then F(e) is a B-identity of Y = dom,F(e). If F : A + B and G : B -P C are contravariant functors, then the composition G.F of the functions F : Mor A + Mor B and G :Mor B + Mor C induces a functor Go F : A -, C.When one of F and G is a functor and another a contravariant functor, we have a contravariant functor G 0 F. If Aop is a dual category of A , an identity function i :Mor A + Mor Aop induces a contravariant functor i : A --* Aop. For any contravariant functor F : A .+ B, we have a functor F’ : Aop + B such that F = F 0 i.
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1.11. Example. Let A be an arbitrary category, X E O b A and g E [ Y , Z ] , . For YE [ X , Y ] , , let F,(g)(f) = go, f :X + Z . Then we have a function F,(g): [ X , Y ] , + [ X , Z ] , , which is a morphism in Set. Thus we obtained a functor F, : A + Set. If we define a function G * ( g ) : [Z, X I , + [ Y , X I , by G X ( g ) ( h )= h 0 g : Y + X for h E [ Z , XI,, we obtain a contravariant functor G X :A + Set.
2. Monomorphisms and epimorphisms 2.1. Definition. A morphism f :X + Y of a category A is said to be an isomorphism in A (or A-isomorphism) if there exists an A-morphism g : Y + X such that gof = 1, andfog = 1,. f : X + Y is said to be a section in A if there exists an A-morphism g : Y + Xsuch thatgof = I,. f:X + Y is said to be a retraction in A if there exists an A-morphism g : Y + Xsuch thatfog = 1,.
If f : X + Y is an isomorphism, a morphism g : Y -+ X satisfying g c>f = 1, andfog = 1 is uniquely determined. Hence g is called an inverse ojfand denoted b y f - ’ . Iff is both a section and a retraction, thenfis an isomorphism. An object X of A is said to be A-isomorphic with an A-object Y (denoted by X x Y ) if there exists an A-isomorphismf: X -, Y . The relation “A-isomorphic” is an equivalence relation on the class Ob A. 2.2. Definition. A morphism f:X + Y of A is said to be a monomorphism in A if, for any pair of A-morphisms u, v : Z + X such that fo u = f o v, it follows that u = v. A morphism f :X + Y of A is said to be an epimorphism in A if, for any pair of A-morphisms u, v : Y + Z such that u o f = v o f , it follows that u = v.
2.3. Proposition. Suppose that f : X + Y , g : Y -, Z and h :X + Z are morphisms of a category A such that h = g o $ (1) Iff and g are monomorphisms in A, then h is a monomorphism in A. ( 2 ) Iff and g are epimorphisms in A, then h is an epimorphism in A. ( 3 ) If h is a monomorphism in A, then f is a monomorphism in A. (4) Ifh is an epimorphism in A, then g is an epimorphism in A.
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2.4. Definition. A morphism of A is said to be a bimorphism in A if it is both a monomorphism and an epimorphism in A. In any category each isomorphism is a bimorphism, but the converse does not hold in general. A category A is said to be balanced if each bimorphism in A is an isomorphism in A. By the propositions below, we have that the categories Set, Grp and Ab are balanced and that Top and Top, are not balanced for each i = 0, 1, 2, 3. 2.5. Proposition. For the categories Set, Grp and Ab, (1) a morphism f : X + Y is a monomorphism i r it is injective; ( 2 )f is an epimorphism it is surjective.
Proof. We will show that epimorphisms in Grp are surjective. Other implications are easily shown. Suppose thatf: X + Y is not surjective. Then f (X) is a proper subgroup of Y. Let S be a set { y f (X) 1 y E Y } u { P }consisting of all left f (X)-cosets of Y and another element P, and let Z be the set of all bijective functions a : S + S. For a, /l E 2, define a multiplication a/l by the composition a 0 8. Then Z becomes a group with this multiplication. Define elements cto, r( y), s( y ) E Z for y E Y as follows: ao(P) = f W ) ,
ao(f(X)) = p,
ao( Y ’ f ( X ) ) = y ’ f ( X ) for Y’
~ ( Y ) ( P= ) P, S(Y)
=
# f( X ) ,
r ( y ) ( y l f ( X ) ) = Y Y X X ) fory’ E Y and
aor(Y)G’.
Then we have homomorphisms r, s : Y + Z such that r 0f = s ofand r # s. This implies that f is not an epimorphism. 0 2.6. Proposition. In any full subcategory of Top, (1) a morphism is an isomorphism iff it is a homeomorphism; ( 2 ) a morphism is a monomorphism iflit is injective. The proof is easy. 2.7. Proposition. ( I ) In Top and Top,, a morphism is an epimorphism iff it is surjective. (2) In Top, and Top,, a morphism f : X + Y is an epimorphism it is a dense map, i.e., f ( X ) is a dense subset of Y.
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(3) In Top,, a morphismf : X + Y is an epimorphism ifffor any y E Y, every neighborhood of y intersects f ( X ) n C1( y} (Baron [1968]). Proof. We shall prove (3). (1) and (2) can be shown similarly. Letf :X -, Y be a morphism in Top, and b( f (X)) = { y E Y I every neighborhood of y intersects f ( X ) n Cl{ y}}. Then b( f (X))3 f ( X ) . Suppose that b ( f ( X ) )is a proper subset of Y. Let {P, Q} be a discrete space with two points, Z the product space Y x (P, Q } and j , k : Y -,Z the continuous maps defined byj( y) = ( y, P) and k( y) = ( y , Q). Consider an equivalence relation on Z such that
-
Let Z’ = Z / - be the quotient space and q : Z + Z’ be the quotient map. Then we can show that Z’ is a T,-space. For morphisms r = q o j , s = q k : Y + Z’ in Top,, we have that r f = s f and r # s. This implies that f is not an epimorphism. Next, suppose that b(f(X)) = Y and u, v : Y -+ W are Top,-morphisms with u f = v of. If there exists an element y E Y with u( y ) # v( y), we may assume that there exists an open set U in Wsuch that u( y) E U and v ( y) 4 U . Then u - ’ ( U ) is a neighborhood o f y and there is an element y’ E u - ’ ( U ) n f(X) n CI{ y}. Hence, u ( y ’ ) = v ( y ’ ) E v(Cl{ y}) c Cl{v( y)}. On the other hand, u( y’) E U and v ( y) 4 U and this is a contradiction. Hence, u = v and f is an epimorphism. 0 0
0
0
0
For an injective continuous map f : X + Y between topological spaces, let Z be a subspace of Y with the underlying set f (X). Then we have a surjective continuous map g : X --f Z and an embedding h :Z + Y such that f = h 0 g. fcan be considered as an embedding if and only if g is a homeomorphism. This induces the following definition. + Y of a category A is said to be an extremal monomorphism in A when f is such that iff = h 0 g with an epimorphism g : X + Z and a morphism h : Z + Y , then g must be an isomorphism. An epimorphism f : X + Y of A is said to be an extremal epimorphism in A whenf is such that iff = h 0 g with a morphism g : X + Z and a monomorphism h : Z + Y, then h must be an isomorphism.
2.8. Definition. A monomorphism f : X
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In balanced categories, every monomorphism is an extremal monomorphism and every epimorphism is an extremal epimorphism. 2.9. Proposition. (1) In Top and Top,, a morphism is an extremal monomorphism iff it is an embedding. (2) In Top, and Top,, a morphism is an extremal monomorphism iff it is a closed embedding. (3) In Top,, a morphism f : X + Y is an extremal monomorphism i f f it is a front closed embedding, i.e., an embedding satisfying the following condition: Ifany neighborhood of y E Y intersects f ( X ) n C1{ y } ,y E f ( X ) (Nel-Wilson [ 19721). (4) In Top and Top,for i = 0,1,2,3, a morphism is an extremal epimorphism iff it is a quotient map. Proof. (3): Suppose that f : X + Y is a front closed embedding in Top,, i.e., b( f ( X ) ) = f ( X ) ,and suppose that f = h g with an epimorphism g : X -, Z and a morphism h : Z + Y in Top,. For z E Z and for any neighborhood U of h(z) in Y , there exists an element z’ E h.’(U) n g ( X ) n Cl{z}. Then h(z’) E U n f ( X ) n Cl{h(z)}and hence, h(z) E b( f ( X ) ) = f ( X ) . Since f is injective, we have a function k : Z + X between sets such that h = f o k . It is easily shown that k is a morphism in Top, and k o g = Ix. Hence, g is an isomorphism and f is an extremal monomorphism in Top,. Next, suppose that f : X + Y is an extremal monomorphism in Top,. For a subspace Z = b( f ( X ) ) of Y , there exist an epimorphism g : X 4 Z and a morphism h : Z + Y such that f = hog. Then g is an isomorphism and hence b ( f( X I ) = f (XI. (I), (2) and (4)are proved similarly. 0 0
2.10. Remark (Duality Principle). Let B be a dual category of a category A and denote the identities by i : Ob B 4 Ob A and i : Mor B + Mor A . It is easily verified that f : X + Y is an epimorphism in B if and only if i ( f ): Y + X is a monomorphism in A . In fact, the definition of epimorphisms can be obtained by reversing all arrows of morphisms in that of monomorphisms. In general, for a statement S concerning objects and morphisms, we can construct a statement Sopby reversing all arrows of morphisms in S. If a statement S defines a property P for objects and morphisms, then Sopalso defines a property for objects and morphisms, which is denoted by Popand is called a dual property of P . Sometimes, the dual property of P is named by co-P, e.g., sections will be called coretractions from now on. If a statement S holds for all categories, then Sopalso holds for all categories. These facts
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are called the duality principle for categories. For example, the notion of extremal epimorphism can be given by stating that it is the dual notion of extremal monomorphism. After this, the dual of each notion will be given only by its name, and the dual of a proposition or a theorem will be frequently omitted but used freely. 2.11. Definition. An equalizer in a category A of morphisms u, v : X -+ Y is a pair ( Z ,f ) (or A in brief) consisting of an object Z and a morphism f : Z -, X satisfying the following conditions: (1) uof = v o f ; (2) for any pair ( Z ’ ,f ‘) of an object Z ’ and a morphism f ’ : Z’ + X with uof ‘ = v of’, there exists a unique morphism cp : Z’ + Z such that f‘ =fop
z‘ Dual: coequalizer. 2.12. Proposition. If(Z,f ) is an equalizer of u and v , thenf is a monomorphism. This follows from the uniqueness of cp in the definition of equalizers. 2.13. Definition. A morphism f : X + Y of a category A is said to be a regular monomorphism if there exist morphisms u and v such that ( X ,f ) is an equalizer of u and v . Dual: regular epimorphism. 2.14. Proposition. For a morphism in any category thefollowing implications hold: isomorphism * coretraction => regular monomorphism * extremal monomorphism, isomorphism * retraction * regular epimorphism * extremal epimorphism.
Proof. Suppose that f : X -+ Y and g : Y + X satisfy g f = 1,. Then we have that f is an equalizer of 1 and f g . This shows that any coretraction is a regular monomorphism. Next, we will show that any regular monomorphism is an extremal monomorphism. Let f : X -+ Y be an equalizer of u, v : Y + Z . It was shown in Proposition 2.12 that f is a monomorphism. Let 0
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f = h 0 g with an epimorphism g : X + Wand a morphism h : W + Y. Then u h o g = v 0 hog. Since g is an epimorphism, u o h = v 0 h. Since f is an equalizer of u and v , there exists a morphism cp : W + Xsuch thatfo cp = h. Then f 0 cp 0 g = f. By the definition of equalizers, cp og = l X and hence, go cp og = g. Since g is an epimorphism, g o cp = 1w. Thus we have that g is an isomorphism and that f is an extremal monomorphism. 0
W.
’
2.15. Corollary. In any category the following statements on a morphism f are equivalent: (1) f is an isomorphism. (2)f is both an epimorphism and a coretraction. ( 3 )f is both an epimorphism and an extremal monomorphism. (4) f is both a monomorphism and a retraction. ( 5 ) f is both a monomorphism and an extremal epimorphism. 2.16. Proposition. In Top and Top,for i = 0, 1, 2, 3, each extremal monomorphism is a regular monomorphism. This follows from Proposition 2.9 and the proof of Proposition 2.7. 2.17. Definition. For an object X of a category A, a monomorphism f : Y --* Xis called a monomorphism into Xor sometimes a subobject of X. Let M ( X ) be a class of all monomorphisms into X, andf : Y + X and g :Z + X be two elements of M ( X ) . f i s said to be isomorphic with g if there exists an isomorphism rp : Y + Z such that f = go cp. It is clear that “isomorphic” is an equivalence relation on M ( X ) . (Each equivalence class is called a subobject of X by some authors.) A subclass R of M ( X ) is said to be a representative set of monomorphisms into X if it is a set and for any f E M ( X ) there exists an element g E R such that g is isomorphic withf. A category A is said to be well-powered (or locally small) if each object X of A has the representative set of monomorphisms into X. If for each object X of A the class of all extremal monomorphisms into X has a representative set of extremal monomorphisms into X,then A is said to be (Exmono)-wellpowered. Well-powered categories are (Ex mono)-well-powered.
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Dual: representative set of epimorphismsfrom X , co-well-powered, (Ex epi)-cowell-powered. Small categories are obviously well-powered and co-well-powered. Set, Grp, Ab, Top and Top, for i = 0, 1 , 2 , 3 are not small, but we can show that they are well-powered and co-well-powered.
3. Diagrams and limits 3.1. Definition. A functor D : K + A from a small category K to a category A is called a diagram D in A over K.
LetK,beacategorywithObK,= {1,2}andMorK2= { l , , 1 2 , a : l+2}. For a morphism f : X + Y in a category A, define a functor D : K2 + A by D(l) = X , D(2) = Y and D(a) = f. Then we have a diagram D : K2 + A, which is expressed by X
L
Y
Sometimes we do not need the specific contents of objects and morphisms of a small category K. In this case, K is called a diagram scheme of D. We shall express a small category with a few morphisms by showing all objects and all morphisms except identities, e.g., the category K2 is expressed by "l2 The diagram scheme K2 is also expressed by
3.2. Definition. Let D : K + A be a diagram in A over K. A pair (X, (ai)iGObK) of an A-object Xand a family of A-morphisms a i : X + D ( i )for i E Ob K is said to be a natural source for D if for any K-morphism a :i +j , D(a)Oa, = a j .
(natural source)
(natural sink)
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Dual: A pair ( X , ( a i ) i & b K ) of an A-object X and a family of A-morphisms a i : D ( i ) + X for i E Ob K is said to be a natural sink for D if for any K-morphism a : i + j , ai = uj0 D(a). 3.3. Definition. For a diagram D in A over K, a natural source ( X , ( a i ) i e O b X ) for D is said to be a limit for D provided that the following holds: For any natural source ( Y , ( f i , ) i e O b X ) for D , there exists a unique morphism cp : Y + X in A such that pi = uio cp for any i E Ob K.
a connecting morphism from the natural source ( y , ( f i i ) i s O b X ) to the limit (x, (ai)ieObK)* Dual: colimit for a diagram D . cp will be called
The following example shows the motive for this naming: For a directed set (A, <), we shall construct a category. For each 1 E A, consider an object 1 and for each pair (1,p) of A, p E A such that A < p, consider a morphism pi.p:p --* 1.Then classes of these objects and morphisms form a small category, which will be denoted by A. Let A be the category Set, Grp or Top. A functor D : A -+ A defines an inverse system ( X i , A+) in A by X , = D(1) and & = D(p,,). Then ( X , ( f l ) i & b A ) is a limit for the diagram D if and ony if ( X , (A);,€,,) is an inverse limit of the inverse system ( X i ,A,+)in A. 3.4. Proposition. For a diagram D in a category A over a small category K, a limit ( X , ( a i ) i E O b K ) is uniquely determined up to isomorphism. Proof. Suppose that ( X , ( r x i ) i E O b K ) and ( Y , ( f i i ) i s O b K ) are limits for D. Then there exist connecting morphisms cp: Y + X and $ : X + Y such that fii = aiocpandui = f i i 0 $ f o r i ~ O b K . H e n c e , u i o c p o $= a i f o r i ~ O b K and this implies that cp $ is a connecting morphism from ( X , ( t l i ) i & b K ) to itself. By the uniqueness of connecting morphism, we have that cp $ = 1,. Similarly, cp = I ,,and this means that ( X , ( a i ) i & b K ) and ( Y , ( f i i ) i & K ) are isomorphic natural sources. 0 0
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3.5. Proposition. For a category A each notion of thefollowing can be deJined us the limit for a suitable diagram D in A over a suitable diagram scheme K: equalizer, pullback, product, terminal object, intersection, multiple pullback, multiple equalizer.
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Proof. ( I ) Equalizer: Let K be a category expressed by 1 2. For a pair of morphisms J g : X + Y in A, let D : K -+ A be a functor such that D(a) = f and D(b) = g. Then the limit for D is an equalizer off and g defined in Definition 2.1 1. (2) Pullback: Let K be a category with Ob K = { I , 2, 3} and Mor K = { I I , 1 2 , I,, a : 1 + 3, b : 2 -+ 3). For a pair of morphisms f : X -+ 2 and g : Y + 2 in A, let D :K -+ A be a functor such that D ( a ) = f and D(b) = g. Then the limit ( W , (a,, a2, a j ) ) for D is called a pullback of the pair ( J g). Since a3 = f al = go a2,a3is usually omitted. In other words, ( W , (aI,a 2 ) ) is a pullback of ( J g ) if and only if the following hold: (a)fo@l = goa2; (b) for any triple ( V , (/II, /I2)) such thatfo /II = g /I2, there exists a unique morphism rp : V -, W such that /I, = a, rp and p2 = a2 rp. 0
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( 3 ) Product: Let I be a set and K be a discrete category with Ob K = I. For a family of objects in a category A, define a diagram D :K -+ A by D ( i ) = X, for i E I. Then the limit ( X , ( p i ) i c lfor ) D is called a product of a family (A&, and each morphism p i : X -+ is called a projection. X will be denoted by l I i s l X , . (4) Terminal object: In the definition of product, suppose that I is the empty set. Then the limit for D consists of an object X such that for any object Y in A, there exists exactly one morphism rp : Y -+ X. X is called a terminal object of A. ( 5 ) Intersection: Let I be a set, 0 4 I and K be a category with Ob K = I u {0} and Mor K = { l i l i E I } u ( 1 , ) u { a i : i + O l i ~ l } For . a family (f;:X, -, Y ) i s l of monomorphisms in a category A, define a diagram D : K -+ A by D ( i ) = D(0) = Y and D(ai) = f;. Then the limit (X,((gi:X-+ g,:X + Y ) ) (or go) for D is called an intersection of
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( f ; ) i s l . By the uniqueness of connecting morphism, we have that go is an A-monomorphism. X is denoted by nislX,.
(6)Multiplepullback: In the above definition, if we do not assume that each A-morphismf; is a monomorphism, the limit ( X , ((gi)iEI, go))for D is called It is noted that if I is a set { 1, 2}, the multiple a multiple pullback of (&,. pullback of a pair (f,,fi) coincides with its pullback. (7) Multiple equalizer: Let I be a set and K be a category with . a family Ob K = (1, 2) and Mor K = { l , , 1 2 } u {ai:1 + 2 1 i ~ I ) For ( f ; : X + Y ) i p lof A-morphisms, define a diagram D : K + A by D(ai) = f;, i E I. Then the limit (Z, ( g :Z + X , h :Z + Y)) for D (or (Z, g), in brief) is called a multiple equalizer of ( j J i s , . 0 3.6. Proposition. Suppose that ( W , (a,, a2)) is a pullback of ( f : X + Z, g : Y + Z ) and that f is a monomorphism in A. Then a2 is a monomorphism in A, which is called an inverse image off by g. The proof follows from the uniqueness of connecting morphism.
Dual: coequalizer, pushout, coproduct, initial object, cointersection, multiple pushout, multiple coequalizer, co-(inverse image). Notions of unions and images in category theory are also defined (Mitchell [1965]), but they are not the dual notions of intersections and inverse images
respectively. The following shows that it is useful to construct new categories from given data. The proof is easy and so omitted. 3.7. Proposition. Suppose that D : K + A is a diagram in a category A over a small category K and &fine a category (0)as follows: Ob (0)h the class of all natural sources ( X , (ai)i&bK)for D, andfor ( X , (ai)),( Y , (pi)) E Ob ( D ) the set of (D)-morphisms [(X, (ai)), ( Y , (pi))](D,consists of all A-morphisms f:X -, Y such that ai = pioffor each i E Ob K. Then ( X , (ai)ieObK) is a limit for D i f it is a terminal object in ( D ) .
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3.8. Example. ( I ) In Ab, the product is the direct product and the coproduct is the direct sum, while in Grp the product is the direct product and the coproduct is the free product. In both categories, the terminal object coincides with the initial object (called zero object, in general). Set and Top do not have zero objects. (2) In Top, the product is the topological product and the coproduct is the topological sum. For morphismsf, g : X + Y, let Z = { x E XI f ( x ) = g ( x ) } be the subspace of X and h : Z + X be the embedding map. Then (2,h) is the equalizer off and g . Next, for morphismsf, g : X + Y, define a relation on Y as follows: For y, y’ E Y, y y’ iff there exists an element x E X such that y = f ( x ) and y’ = g ( x ) . Then the relation generates an equivalence relation on Y. The quotient map h : Y + Z = Y / - is the coequalizer of f and g . The terminal object is a space consisting of exactly one point, and the initial object is the empty space. For morphismsf : X + Z and g : Y + Z, let (X x Y , ( p x ,p , ) ) be the product of X and Y and (W. (a, : W + X , a*: W Y)) be the equalizer of f o p x and g o p , . Then (W, (a,, a*)) is the pullback of (f,g). We can show that in any category, pullbacks are constructed from products and equalizers in this way. In the next section we will show several relations among special limits in general categories.
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4. Complete categories 4.1. Definition. Let K be a small category. A category A is said to be K-complete if for any diagram D: K + A in A over K, there exists a limit for D. A is said to be complete if, for any small category K,A is K-complete. A is called finitely complete if for any finite category K (i.e., Mor K is a finite set), A is K-complete. Dual: K-cocomplete, cocomplete,Jiniteb cocomplete. If A is K-complete for a diagram scheme K expressed by m A is said to have equalizers (see Proposition 3.5(1)) and so forth. 4.2. Theorem. For a category A , the following are equivalent. ( I ) A is complete. ( 2 ) A has multiple pullbacks and terminal objects. ( 3 ) A has products and pullbacks. (4) A has products and inverse images. (5) A has products andjnite intersections.
i
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(6) A has products and equalizers. (7) A has products and intersections. Proof. (1)=42) follows from Proposition 3.5. (2)*(3) follows from Lemma 4.3 below, which is easily verified. be 4.3. Lemma. Let T be a terminal object in A, I a nonempty set and T ) )is a multiple pullback a family of A-objects. ( X , ( ( p i :X -+ X , ) i E l , tx : X of ( t i : X ,+ T ) i c l fi and only if ( X , ( p i ) i s l )is a product of (X,)iEI, where ti : X, + T, i E I and t,: X -+ T are morphisms determined uniquely by the property of terminal object. -+
Proof of Theorem 4.2 (continued). (3)*(4) is obvious. (4)=45): For a family ( f ; : X , -+ Y ) i c l of A-monomorphisms with I = { I , 2}, an intersection ( X , ( ( g i : X X,)iEI, g,:X Y ) ) of ( J , f , )is obtained by the formation of a pullback of ( J ,fi), i.e., by that of an inverse can be obtained image offi byf,. For a finite set I, an intersection of by iterating the formation of inverse images. (5)-(6) follows from Lemma 4.4 below, which is easily verified. -+
-+
4.4. Lemma. Suppose that A hasfinite products. For a pair CI; g ) of morphisms -+ Y there exists a pair ( f ’, g‘) of morphisms f ’, g‘ : X -+ X x Y such that pxOf’ = l,, pYOf’ =f, pxog’ = 1, and puog‘ = g. I f ( Z , ((h, k :2 X ) , 1 : 2 -+ X x Y ) )is an intersection of ( f ’, g’), then h = k and ( Z , h) is an equalizer of (f,g). I f ( Z , h) is an equalizer of (f,g), then (2, ((h, h),f ’ h)) is an intersection of ( f ’, g’).
f, g :X
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X g f X x Y
In order to prove (6)=.(1) lemma.
of Theorem 4.2, we shall show the following
4.5. Lemma. I f A has products and equalizers, it has intersections of regular monomorphisms. Proof. Let Z be a set and, for each i E Z,let hi:Zi + X be an equalizer off;, gi:X -+ Y,. For a product (Y, ( p i :Y -+ Y,)iEI)of the family there
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exist morphismsf, g : X + Y such that pi"f = f; and pi 0 g = gi for each i E I. Let ( W, h : W -+ X) be an equalizer of (f,8).Then we have thatf; 0 h = gi 0 h and that there exists a morphism cpi: W --+ Z, such that h = hio (pi for each i E I. Thus we can verify that (W, ( ( ~ p ~ ) ~ ~ ,h)) , is an intersection of (hi)is,. 0
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Proof of Theorem 4.2 (conclusion). Now we will return to the proof of (6)*(1). Let K be an arbitrary small category and D :K A a diagram in A over K. We shall construct a limit for D. Let (X, ( p i :X D(i)),,,,,) be a product of the family (D(i))isObK. For each K-morphism a : i -+ j , let ( Z o ,h, :Z , -, X) be an equalizer in A of D(a)" p i and p,. Then there exists ~ ~ of ~ , (hu)aEMorK. We can show that an intersection (Z, ( ( ( P , ) , ~ ~h)) (Z, (pi" h : Z + D(i))icObK) is a limit for D. ( 1 ) 4 7 ) and (7)*(5) are obvious and we complete the proof of Theorem 4.2. 0 -+
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4.6. Theorem. For a category A , the following are equivalent. (1) A is finitely complete. (2) A has pullbacks and terminal objects. ( 3 ) A has finite products and pullbacks. (4) A has finite products and inverse images. ( 5 ) A has finite products and finite intersections. (6) A has finite products and equalizers. 4.7. Example. The categories Set, Grp and Top are complete and cocomplete. The full subcategory of Set consisting of all finite sets is finitely complete and finitely cocomplete but it is neither complete nor cocomplete.
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5. Factorizations of morphisms For a continuous mapf :X + Y between topological spaces, let Z = f ( X ) be a subspace of Y. Then we have a continuous and surjective map g :X -+ Z and an embedding h :Z + Y such thatf = h 0 g. In other words, a morphism fin Top has a factorization (Z,g, h ) with an epimorphism g and an extremal monomorphism h. If Z’ = f ( X ) is a quotient space of X,we have a factorization (Z’, g’, h’) with an extremal epimorphism g’ and a monomorphism h’. From the point of view of category theory, these are very important properties of the category Top. In this section we discuss general categories which have such properties.
5.1. Proposition. Suppose that a category A is well-powered and has intersections and equalizers. Then, for any morphism f : X -+ Y, there exists a triple ( Z , g, h ) consisting of an object Z, an extremal epimorphism g :X -+ Z and a monomorphism h :Z -+ Y such that f = h 0 g. Proof. Let J be a class of all triples j = (Z,, g,, h,) consisting of an object Z,, a morphism gj:X + Z, and a monomorphism h, : Z, -+ Y such that f = hjogj. If h, = hi. for j , j’ E J , then gj = gi. and hence j = j’. Since A is well-powered, the class {hjlj E J } has a representative set {hiI i E I } , which contains 1 y : Y -+ Y. Let ( Z , ((ki:Z -+ Z i ) i a fh, : Z -+ Y ) )be an intersection of (hi:Zi -+ Y ) i , f .Then h is a monomorphism and there exists a morphism g : X -+ Z such that f = hog. In order to show that (Z, g, h) is a desired triple, it is sufficient to prove that g is an extremal epimorphism. Suppose that r, s :Z + Ware morphisms such that r 0 g = s 0 g. For an equalizer I : V + Z of r and s, there exists a morphism cp : X + V such that g = 1 0 cp. Then f = h 10 cp. Since h 0 1 is a monomorphism, the triple ( V , cp, h 0 I ) belongs to J and there exist an element i = ( Z i ,gi, hi) E I and an isomorphism m : Z i + V such that hi = h0l.m. Then h o l o m o k i = hioki = h = ho I, and hence 10 m 0 ki = I,. By Corollary 2.15, I is an isomorphism and hence r = s. Thus we have obtained that g is an epimorphism. Similarly, we can show that g is an extremal epimorphism. 0 0
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5.2. Proposition. Suppose that a category A has pullbacks and that an extrema1 epimorphkm f:W + X, morphkms g : W + Y, h :X + Z and a monomorphism k : Y + Z satisfy that h f = k 0 g. Then there exists a morphism d : X + Y such that g = dof and h = k o d . 0
k
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’Z
Proof. Let (V, ( I : V + X , m : V + Y)) be a pullback of (h, k). Then 1 is a monomorphism and there exists a morphism cp : W + V such that f = 10 cp and g = m cp. Sincefis an extremal epimorphism, I is an isomorphism. Thus we can obtain a desired morphism d : X + Y putting by d = m I - ’ . 0 0
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5.3. Definition. A category A is said to be (Ex epi, Mono)-factorizable if, for each morphism f :X + Y , there exists a triple (Z, g , h ) consisting of an object Z, an extremal epimorphism g : X + Z and a monomorphism h : Z + Y such that f = h og. ( Z ,g , h ) is called an (Ex epi, Mono)-factorization off. A is said to be uniquely (Ex epi, Mono)-factorizable if it is (Ex epi, Mono)-factorizable and for two (Ex epi, Mono)-factorizations (Z, g, h) and (Z’, g’, h‘) of each morphismf, there exists an isomorphism cp :Z + Z’ such that h = h’ 0 cp (or, equivalently, g’ = cp 08).
A is said to have the (Ex epi, Mono)-diagonalization property provided that whenever an extremal epimorphism f :W + X , morphisms g: W + Y , h :X + Z and a monomorphism k : Y + Z satisfy h of = k o g, there exists a morphism d : X + Y such that g = dof and h = k d. d is called a diagonal 0
morphism (or a diagonal) of the square (f,g, h, k ) .
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5.4. Proposition. I f a category A is (Ex epi, Mono)-factorizable and has the (Ex epi, Mono)-diagonalization property, then it is uniquely (Ex epi, Mono)factorizable.
Dual: (Epi, Ex mono)-factorizable, (Epi, Ex mono)-factorization, uniquely (Epi, Ex mono)-factorizable, (Epi, Ex mono)-diagonalization property. 5.5. Theorem (Factorization Theorem I). If a category A is well-powered and has intersections andfinite products, then A is uniquely (Ex epi, Mono)factorizable. This follows from Propositions 5.1, 5.2, 5.4 and Theorem 4.6. Dual (Factorization Theorem co-I): If a category A is co-well-powered and has co-intersections and finite coproducts, A is uniquely (Epi, Ex mono)factorizable. Since Top is well-powered, co-well-powered, complete and cocomplete, it is uniquely (Ex epi, Mono)-factorizable and uniquely (Epi, Ex mono)factorizable. In the following, we will deduce the unique (Epi, Ex mono)factorizability o f A under the assumption similar to the one in Theorem 5.5 (not the dual type). 5.6. Lemma. Suppose that A is well-powered and has intersections and equalizers. Let f : X + Y be a morphism and M be a class of monomorphisms satisfying the following conditions: ( 1 ) Closed under the formation of intersections: i.e., i f ( U , ((ui: U + q.)iel, u,, : U -, V ) )is an intersection of a family (vi : U, + V ) i s lof monomorphisms into V and if each vi E M for i E I, then uo E M. (2) r f a morphism q : X + U , a regular monomorphism r : U + V and a morphism s: V + Y satisfy that f = s o r o q a n d s E M , then sor E M . (3) 1 E M . Then there exists a triple ( Z ,g , h) of an object Z , an epimorphism g : X -, Z and a monomorphism h : Z + Y belonging to M such that ( a ) f = hog; (b) if a triple ( Z ‘ ,g’ : X + Z ’ , h‘ : Z ’ + Y ) satisfies that f = h’og’ and h E M , then there exists a morphism k : Z -, Z‘ with h = h o k, (c) if a triple ( W , 1 : X + W , m : W + Z ) satisfies that g = m 1 and h m E M , then m is an isomorphism. 0
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This is a generalization of Proposition 5.1 and the proof is obtained in a similar way to that of Proposition 5.1.
5.7. Lemma. Suppose that a category A is well-powered and has intersections and equalizers. Then A has the (Epi, Ex mono)-diagonalization property. Proof. Let an epimorphism f:W X, morphisms g : W + Y , h : X -+ Z and an extremal monomorphism k : Y + Z satisfy that h of = k og. We shall apply Lemma 5.6 for the morphism k : Y + Z and the class M of all monomorphisms n : V -, Z such that there exist morphisms I : X + V and m : Y -+ V with n 0 I = h and n om = k. To show that M satisfies condition (2) of Lemma 5.6, let k = soroq with a morphism q : Y + U, an equalizer r : U + V' of u, 'u: V' + V" and a morphism s : V' -+ Z belonging to M. Then there is a morphism I' : X + V' such that s I' = h and I' of = r 0 q og. Hence, u o l ' o f = u 0 r o q . g = v o r o q o g = v o l ' o f . Sincefis an epimorphism, u I' = 'u I'. Since r is the equalizer of u and 'u, there is a morphism I": X -+ U such that s r I" = h. This means that s r E M and that M satisfies (2). It is easy to show that M satisfies (1) and (3) of Lemma 5.6. Hence there exists a triple (Vo, m,:Y + 6 , no: 6 -, Z ) satisfying the conditions (a), (b) and (c) in Lemma 5.6. Since no E M , there exists a morphism lo: X + 6such that n,,0 I, = h. Since m, is an epimorphism and k is assumed to be an extremal monomorphism, m, is an isomorphism. A diagonal morphism of the square (Lg , h, k ) is given by d = mi' lo. 0 -+
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5.8. Lemma. Zf a category A has the (Epi, Ex mono)-diagonalization property, then the following hold in A : The composition of extremal monomorphisms is an extremal monomorphism. The intersection of extremal monomorphisms is an extremal monomorphism. The inverse image of an extremal monomorphism is an extremal monomorphism. The product of extremal monomorphisms is an extremal monomorphism.
Proof. First, we show (3). Let ( W , f , g ) be a pullback of ( h : X -+ 2, k : Y + 2) and let k be an extremal monomorphism. We already showed that f is a monomorphism. Let f = mo I with an epimorphism I: W -+ V and a morphism m : V + X.Then there exists a diagonal morphism n : V -+ Y of the square (I, g, h 0 m, k). Since ( W,f,g ) is a pullback of (h, k), there exists a morphism cp : V + Wsuch thatf cp = m and g 0 cp = n. By the uniqueness of connecting morphism, we have that cp I = 1 and hence, I is an isomorphism. This shows that f is an extremal monomorphism. Next, we will make sure the meaning of (4). Let (L: X, -+ YJiElbe a family of morphisms and ( X , ( p i ) i e l )and ( Y , (qJiC,) be products of families ( & ) i s , and ( K)iE,respectively. Then there exists a unique morphism f :X + Y such that qi 0f = f; " p i for each i E I. The statement (4) claims that if eachJ is an extremal monomorphism for i E I, then f is an extremal monomorphism. The proofs of (l), (2) and (4) are similar to that of (3) and so omitted. 0 0
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If a category A has two extremal monomorphisms f and g whose composition h = g 0f is not an extremal monomorphism, then A contains morphisms I and k such that h = 10 k and k is not an isomorphism but an epimorphism. Since f and g are not epimorphisms, A contains morr # s, u o g = v o g and u # v. phisms r, s, u and v such that rof = SOL Thus we have the following example of categories (see below) in which compositions of extremal monomorphisms are not necessarily extremal 5.9. Example.
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monomorphisms. All objects are shown by dots and all morphisms except the identities are shown by arrows.
5.10. Theorem (Factorization Theorem 11). Zfa category A is well-powered and has intersections and equalizers, then it is uniquely (Epi, Ex mono)factorizable. Proof. For a morphism f : X + Y and the class M of all extremal monomorphisms of A, we can apply Lemma 5.6 and then obtain an (Epi, Ex mono)-factorization of$ Hence A is (Epi, Ex mono)-factorizable. By Lemma 5.7 and Proposition 5.4, A is uniquely (Epi, Ex mono)-factorizable. 0
5.1 1. Theorem. ZfA is well-powered, (Ex epi)-co-well-poweredand complete, then A has coequalizers. Proof. Let f, g :X + Y be A-morphisms. Then the class of all extremal epimorphisms h from Y with the property h of = h 0 g has a representative set (hi:Y + Zi)iel. Let (Z, ( p i ) i s l )be the product of ( Z j ) i eThen l. there exists a morphism k: Y -, Z such that pjok = hi for i e I. Let ( W , I : Y + W , m : W + Z) be an (Ex epi, Mono)-factorization of k . We will show that I is a coequalizer off and g . It is easily shown that lof = log. Let n : Y + V satisfy that nof = nog and let ( U , r : Y + U, s: U + V) be an (Ex epi, Mono)-factorization of n. Then r of = r g and hence there exist an element i E Z and an isomorphism t i :Zi + U such that r = t i o hi. Thus we obtain a connecting morphism q : W + V from (W, I ) to ( V , n) by putting q = tiopiom.The uniqueness follows from the fact that I is an epimorphism. 0
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6. Reflective subcategories 6.1. Definition. A subcategory B of a category A is said to be rejective in A (or, a reflective subcategory of A) if any A-object X has a B-reflection r x :X + Y , i.e., there exist an object Y and a morphism r x :X + Y in A satisfying the following conditions: ( I ) Y E Ob B; (2) for any A-morphism f : X + Z with Z E Ob B, there exists a unique B-morphism f : Y + Z such that f rx = f. f will be called an extension off.
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0
6.2. Proposition. Let B be a rejective subcategory of A and let rx : X + Y be a B-reflection of an A-object X . Then we have the following. ( I ) IfA g : Y + Z are B-morphisms and i f f 0 rx = g r x , then f = g. ( 2 ) rx is essentially unique. 0
Proof. ( 1 ) follows from the uniqueness of extension. (2): Let s x : X + W be another B-reflection of X . Then there exist B-morphisms (sX)O: Y + Wand (rX)':W + Y such that (sX)'orx = sx and (rX)'osx = r x . Hence, ( s x ) o ~ ( r x ) o ~=s xsx and ( r x ) o o ( s x ) o ~ r=x r x .
By ( I ) we have that (sX)' (rX)' = 1 and (rX)' (s,)' rx is essentially unique. 0
0
=
1 y . This means that
0
6.3. Definition. Let B be a reflective subcategory of a category A and denote the B-reflection of each A-object X by r x :X + RX. For an A-morphism f : X + Y , there exists a unique B-morphism Rf : RX + R Y such that Rf = (ryef )', i.e., Rf rx = ry0f . By assigning Rf tof, we obtain a functor R : A + B, which will be called a B-rejector. 0
6.4. Remark. Let B be a subcategory of a category A with the inclusion functor E: B + A. Then B is reflective in A with the B-reflector R iff E has the left adjoint functor R : A -, B. Hence reflectors are considered as special cases of adjoint functors. The theory of adjoint functors is the most important
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theme treated in every text book of category theory. However, considering applications for general topology, we do not discuss here the general theory but restrict ourselves to the theory of reflective subcategories.
6.5. Proposition. Let B be a reflective subcategory of A. Then B is full in A i f f o r each B-object X the identity 1 : X + X is a B-rejection. 6.6. Proposition. Let B be a subcategory of A and C be a full subcategory of Bsatisfying the following condition: For any X E Ob B, there exist Y E Ob C and a B-isomorphism c p : X + Y . Then C is reflective in A if B is reflective in A.
These statements are easily proved. Most of reflective subcategories which we shall consider are full and isomorphism-closed, where a subcategory B of a category A is said to be isomorphism-closed whenever if X E Ob B and cp: X + Y is an A-isomorphism, then Y E Ob B. 6.7. Convention. From now on, all subcategories will be assumed to be full and isomorphism-closed. 6.8. Definition. Let E be a class of morphisms of a category A which is closed under composition with isomorphism. A reflective subcategory B of A is said to be E-reflective in A if each B-reflection rx of each A-object X belongs to E. For the case E = Epi A, the class consisting of all epimorphisms in A, B is called epireflective in A and similarly we use notations Mono A, Bi A, Ex epi A and terminologies monoreflective, bireflective, (Ex epi)-reflective and so forth.
If B is monoreflective in A, then B is epireflective in A.
6.9. Proposition.
Proof. Let X E Ob A and r x :X -, RX be the B-reflection of X . If = v o r x for A-morphisms u, v : R X + Y , then r y o u o r x = r y o v o r x . Since B is full in A, we have that r Y o u = r Y o v by Proposition 6.2. Since r y is assumed to be a monomorphism, we have that u = v . This implies that r x is an epimorphism in A. 0 # O r X
. x i
7[ry
RX-
RY
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Dual: corefective, coextension, corefector, monocorefective and so on. 6.10. Definition. An object S of a category A is called a separator whenever i f f , g : X + Y are different A-morphisms, then there exists a morphism x : S + Xsuch that f o x # g o x . In Set or Top, all nonempty objects are separators and in Grp or Ab, the additive group Z of all integers is a separator. 6.11. Proposition. Suppose that a category A has a separator S, and B is a subcategory of A which contains S. Then B is bicorefective in A ifl it is corefective in A. Proof. It is sufficient to show that if B is coreflective in A, B is epicoreflective in A. Let r,: RX + X be the B-coreflection of an A-object X and let u, v : X + Y be A-morphisms such that u o rx = er 0 r x . If u # 01, there exists an A-morphism x : S -, X such that u 0 x # v 0 x . Since S E Ob B, there exists a morphism y : S + RX such that rxOy = x . Then u o x = uorx o Y = v 0 r,o y = v 0 x and this is a contradiction. Thus we obtain that rx is an epimorphism and that B is epicoreflective in A. 0
Dual: coseparator. 6.12. Remark. Proposition 6.1 1 has some applications to Top and its subcategories, but the dual version has few. In fact, a topological space X is a coseparator in Top iff it contains a nontrivial indiscrete subspace. 6.13. Example. ( I ) Let X be a completely regular Hausdorff space. The Stone-Cech compactification / J x : X + /J X of X is a dense embedding from X into a compact Hausdorff space /J X with the "universal property". This means that the category CompT, is an E-reflective subcategory of the category Tyc, where Tyc (respectively CompT,) is the full subcategory of Top consisting of all completely regular Hausdorff spaces (respectively all compact Hausdorff spaces) and E is the class of all dense embeddings in Tyc. (2) Tyc is epireflective in Top. The reflector R : Top + Tyc is known as the Tychonoff functor. (3) Ab is an epireflective subcategory of Grp. The reflection rx : X + RX of a group X is the abelianization a,: X + X / [ X ,X I . (4) Let n be an integer > 1 and let Ab'") be the full subcategory of Ab consisting of all objects X such that nX = 0. Then Ab'"' is epireflective in Ab and the reflection of X is the quotient homomorphism r x :X + X/nX.
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( 5 ) The full subcategory Tor of Ab which consists of all abelian torsion groups is monocoreflective in Ab. The coreflection r x :RX + X of X is the inclusion homomorphism from the torsion subgroup Tor(X) into X. (6) The full subcategory TF of Ab which consists of all torsion-free abelian groups is epireflective in Ab. The reflection of Xis the quotient homomorphism r x :X + X/Tor(X). (7) The full subcategory Ind of Top which consists of all indiscrete spaces is bireflective in Top. (8) The full subcategory Dis of Top which consists of all discrete spaces is bicoreflective in Top. Let X be a topological space, RX a discrete space with the same underlying set to X and rx : RX + X a continuous map induced by the identity map on the underlying sets. Then rx is the coreflection of X . By Proposition 6.1 1 any coreflective subcategory of Top is bicoreflective. Hence, in general, the coreflection of X is obtained by the modification of topology of x.
We conclude this section by showing the relations between reflective subcategories and colimits. 6.14. Proposition. Suppose that B is a reflective subcategory of a category A with inclusion functor E :B + A and reflector R : A + B, and suppose that D : K + B is a diagram in B over a small category K. Then if there exists a colimit for E 0 D : K + A , then there exists a colimit for D.
Proof. Let ( X , ( a i :D ( i ) + X);&K) be a colimit for Eo D and let r x : X
+ RX the B-reflection of X . Then we can show that ( R X , (rxoai:D ( i ) + R X ) ) is a colimit for D . 0
6.15. Corollary. I f a category A is cocomplete and B is a reflective subcategory of A , then B is cocomplete. 6.16. Proposition. Suppose that B is a reflective subcategory of A with inclusion functor E : B + A and reflector R ; A + B and that D : K + A is a diagram in A over a small category K. Then if ( X , ( a ; : D ( i ) + X)icObK) is a colimit for D, then ( R X , ( R E , :R D ( i ) + RX)icObK) is a c o h i t for R o D : K + B. 6.17. Remark. Suppose that B is a coreflective subcategory of Top with coreflector R :Top + Band that (X,)lclis a set of topological spaces X,. Then by the dual of Proposition 6.16, we have that R(n,,,X,) = nl’,,RX,, where
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ll and I'I' denote the products in Top and B respectively. If B is a reflective subcategory of Top with reflector R : T o p + B, the formula R ( n i , , x ) = II,;,RX, does not hold in general (Glicksberg [1959]).
7. Characterization theorem
In this section we always assume that A is a category and B is a full and A. isomorphism-closed subcategory of A with the inclusion functor E : B 7.1. Proposition. Let D : K + B be a diagram in B over a small category K. Then E 0 D is a diagram in A over K. ( I ) A limit ( X , ( a i :X --* D(i))icObK) for E o D is a limit for D i@X E Ob B. (2) I f B is reflective in A, a limit ( X , ( a i :X + D(i))i,ObK) for D is a limit for E o D. Proof. ( I ) is obvious. (2): Suppose that B is reflective in A and ( Y , (B,: Y --* D(i)),,obK) is a natural source for Eo D. Then there exists a B-morphism (fl,)': RY -+ D ( i ) such that (PI)' 0 r r = P, for each i E O b K , where r r : Y + RY is the B-reflection of Y. We have that ( R Y , ( < f l , ) ' ) > , , ~ b ~ ) is a natural source for D and hence there exists an A-morphism cp: RY + X such that a , cp~ = (PI)' for each i E Ob K. Thus we obtain a connecting morphism cp 0 r r from ( Y, (fl,),sObK) to ( X , ( a , ) , , O b K ) . The uniqueness can also be shown. 0
7.2. Definition. Let K be a small category. B is said to be closed in A under K-limits if, for any diagram D :K + B in B over K, a limit for Eo D is a limit for D. 7.3. Proposition. If B is reflective in A, then B is closed in A under K-limits for any small category K.
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Proof. Let D : K + B be a diagram and ( X , (ai : X D ( i ) ) i s O b Ka )limit for Eo D . We will show that X E Ob B. Let r,: X RX be the B-reflection of X. Then there exists a B-morphism (ai)' : RX -+ D ( i ) such that (ai)' 0 rx = ai for each i E I and (RX, ( ( C Q ) ' ) ~is~a~natural ~ ~ ) source for E o D . Hence there exists a morphism cp: RX + X such that a i o c p = (ai)' for i E I. Thus ai cp o rx = tli for i E I and, by the uniqueness of connecting morphism, q o r x = 1,. By Proposition 6.2(1), r,ocp = l,, and we have that rx is an isomorphism and that X E Ob B. 0 -+
-+
0
7.4. Corollary. r f A is a complete category and B is a reflective subcategory of A, then B is complete.
7.5. Definition. Let K be a small category. A diagram D :K A is said to be initially in B if for any K-object i there exist a K-objectj and a K-morphism a : j -+ i such that D ( j ) E Ob B. B is said to be strongly closed in A under K-limits if for any diagram D :K A which is initially in B a limit ( X , (ai)rcObK) for D satisfies that X E O b B. -+
-+
7.6. Example. B is closed in A under equalizers if whenever u, v : Y -+ Z are A-morphisms with Y, Z E Ob B, the equalizer of (u, v) in A belongs to B. B is strongly closed in A under equalizers if whenever u, v : Y Z are A-morphisms with Y E Ob B, the equalizer of (u, v) in A belongs to B. It is noted that the strong closedness under products coincides with the closedness under products. -+
7.7. Proposition. ZfB is epireflective in A , then B is strongly closed in A under K-limits for any small category K.
Proof. Let D : K -+ A be a diagram which is initially in B and ( X , (a,),,ObX) a limit for D. We will show that the B-reflection r x :X -+ R X of X is an isomorphism. Let S = { i E Ob KI D ( i ) E Ob B}. For i E S, there exists a B-morphism (a,)' : RX -+ D ( i ) such that ( a , ) ' ~rx = a,. For i # S , there exist a K-object j E S and a K-morphism a : j -+ i. Define (a,)':RX -+ D ( i ) by (a,)' = D ( a ) o (a,)'. Since rx is an epimorphism, we have that (a,)' does not
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depend on the choice o f j and a, and that ( R X , ((ai)o)icObX) is a natural source for D. Then we can show that rx is an isomorphism in a way similar to the one in the proof of Proposition 7.3. 0
7.8. Definition. B is said to be closed in A under extremal monomorphisms (respectively monomorphisms) whenever i f f : X + Y is an extremal monomorphism (respectively monomorphism) and Y E O b B, then X E Ob B. It is obvious that if B is closed in A under extremal monomorphisms, then B is strongly closed in A under equalizers. We will show that the converse of Proposition 7.7 holds under some conditions on A. This result plays a fundamental role in categorical topology.
7.9. Theorem (Characterization Theorem of Epireflective Subcategories). Suppose that a category A is complete, well-powered and co-well-powered and B is a full and isomorphism-closed subcategory of A. Then the following are equivalent. (0) B is epirepective in A. (1) B is strongly closed in A under K-limits for any small category K. (2) B contains a terminal object of A and is strongly closed in A under multiple pullbacks. (3) B is strongly closed in A under products and pullbacks. (4) B is strongly closed in A under products and inverse images. ( 5 ) B is strongly closed in A under products andjnite intersections. (6) B is strongly closed in A under products and equalizers. (7) B is strongly closed in A under products and intersections. ( 8 ) B is closed in A under products and extremal monomorphisms. Proof. (0)==-(1) is proved in Proposition 7.7. The equivalence of conditions (1) through (7) can be proved by the similar ideas as those of (1) through (7) in Theorem 4.2. Hence we will give the proof of (6)*(1). We need the following lemmas. 7.10. Lemma. Zf B is strongly closed in A under equalizers, then B is closed in A under intersections of regular monomorphisms.
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Proof. Let I be a set and hi:Zi + X an equalizer off;, g i :X + Y, for each i E I. By Lemma 4.5, an intersection ( W , ((qQicl, h)) of (hJielcan be constructed Y, such that p i o f = f;, from an equalizer ( W, h) of morphismsf, g : X + njEl p , g = gi for i E I. Hence, if hi E Mor B for each i E I, then W E O b B. 0
Proof of Theorem 7.9 (continued). The rest of the proof of (6)*(1): Suppose that D : K + A is a diagram which is initially in B. Let S = { i E Ob K 1 D ( i ) E Ob B} and let ( X , ( p , : X + D ( i ) ) i E sbe ) the product in A of a family ( D ( i ) ) i c sBy . the assumption, X E Ob B. Let A be a set of all pairs (a, a’) of K-morphisms a :j + i and a’ :j ’ + i such that b o t h j and j’ belong to S. For each 1 = (a :j + i, a’ :j ’ + i ) E A, let ( Yj,,J , : Yj, -, X) be an equalizer in A of D(a)op, and D(a’)op,,. Then Y, E Ob B. Let (Z, ((&&A, f))be an intersection of (A),€,,.Then we have that Z E Ob B by Lemma 7.10. For i E S, let ai = p i o f : Z + D ( i ) . If i 4 S, there exist a K-objectj E S a n d a K-morphism b : j + i. Let ai = D ( b ) o p , o f : 2 + D ( i ) . Then ai does not depend on the choice o f j and b, but is determined by i. We is a limit for D and that Bis strongly closed under can show that (Z, (ai)iEObK) K-limits.
(1)*(8): Suppose thatf: X + Y is an extremal monomorphism in A and Y E Ob B. Let M be a class of A-monomorphisms h : Z -, Y with Z E Ob B such that there exists an A-morphism g : X + Z satisfyingf = h o g . Then A, f and M satisfy the conditions (1)-(3) of Lemma 5.6. Hence there exists a triple ( W , k : X + W , 1 : W -, Y ) of a B-object W , an A-epimorphism k and 1 E M satisfying the conditions (a)-(c) of Lemma 5.6. Sincefis an extremal monomorphism, k is an isomorphism and hence X E Ob B. (8)*(0): Suppose that X is an A-object and let (f;:X + Y , ) i E lbe a representative set of epimorphisms from X whose codomains belong to B. Let (Y, ( p i : Y + q ) i c lbe ) a product of in A. Then Y belongs to B and there exists a morphismf : X + Y such that p j o f = f; for each i E I. By the
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Factorization Theorem I1 (5.10), there exists a factorization (Z, g : X Z, h : Z + Y) off such that f = h og, g is an epimorphism and h is an extremal monomorphism in A. By condition (8), Z E Ob B. We shall show that the epimorphism g is a B-reflection of X. Suppose that k : X -, W is an A-morphism with W E Ob B and ( V , I: X -P V , m : V + W ) is a factorization of k such that k = m 0 I, I is an A-epimorphism and m is an extremal monomorphism in A. Then V E Ob B and there exist an element i E I and an isomorphism n : r, -+ V such that noA = 1. Thus we have an extension ko = m o n 0pio h of k. The uniqueness of extension follows from the fact that g is an epimorphism. 0 -+
7.11. Corollary. Suppose that A is complete, well-powered and co-wellpowered. Then we have the following. ( 1) If Bj, is an epirejlective subcategory of A for each element I. of a class A, then the intersection B,: is epirejlective in A. ( 2 ) For any subcategory B of A, there exists an epirejlective subcategory B* of A satisfying the following conditions: (a) B* 3 B, (b) ifB' is an epirejlective subcategory of A and B' 3 B, then B' 3 B*. ( 3 ) The class of all epirejlective subcategories of A is a complete lattice.
niEA
7.12. Definition. In Corollary 7.11(2), the category B* will be called an epirejlective hull of B in A. For an A-object X or a class C of A-objects, the epireflective hulls of X or C are also considered. 7.13. Proposition. Suppose that A is complete, well-powered and co-wellpowered and B is a subcategory of A. Then the following conditions,for an A-object X are equivalent. ( I ) X belongs to the epirejlective hull B* of B in A. (2) There exist a set (z.)iE, of B-objects X, and an extremal monomorphism f : X -+ n;,,x,. Proof. By using Lemma 5.8, we can show that the full subcategory of A which consists of all A-objects X satisfying the above condition (2) is closed under products and extremal monomorphisms. Thus, this proposition follows from the Characterization Theorem 7.9. 0
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7.14. Example. (1) Top, is epireflective in Top. For a topological space X , define a relation on X as follows: x y for x, y E X iff any neighborhood of x contains y and any neighborhood of y contains x. Then we have an on X. The quotient map q : X + X / - is the Top,equivalence relation reflection of X. (2) Top, is epireflective in Top for i = 1,2,3. This is shown by the fact that Top, is closed under products and extremal monomorphisms. However, the reflections cannot be obtained in a way similar to the above. We can find an example for Top, in Sharpe, Beattie and Marsden [1966]. (3) Tyc is epireflective in Top by the same reason stated above. As is well known, every completely regular Hausdorff space can be embedded in a product space of bounded closed intervals. Hence we can say that Tyc is the epireflective hull of the closed (or open) unit interval in Top. (4) Since every compact Hausdorff space is homeomorphic to a closed subspace of a product of bounded closed intervals, we can say that CompT, is the epireflective hull of the closed unit interval in Top,. (5) The epireflective hull of the open unit interval in Top, is the category of all realcompact spaces. (6) For a Hausdorff space E, an object of the epireflective hull of E in Top is called an E-regular space and an object of the epireflective hull of E in Top, is called an E-compact space. Many interesting studies on these spaces are given by S. Mrowka and others, see e.g., Engelking and Mrowka [1958] and Mrowka [1968].
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Dual: B is closed in A under K-colimits, D : K + A is Jinally in B, B is strongly closed in A under K-colimits, B is closed in A under extremal epimorphisms, Characterization Theorem of Monocoreflective Subcategories, monocorefective hull. 7.15. Theorem. Suppose that A is a cocomplete, well-powered and co-wellpowered category and B is a full and isomorphism-closed subcategory of A . If A has a separator and B contains the separator in A , then the following are equivalent. (1) B is corefective in A. (2) B is bicorefective in A. (3) B is strongly closed in A under K-colimits for any small category K. (4) B is strongly closed in A under coproducts and coequalizers. ( 5 ) B is closed in A under K-colimits for any small category K. ( 6 ) B is closed in A under coproducts and coequalizers. ( I ) B is closed in A under coproducts and extremal epimorphisms.
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Proof. By Proposition 6.1 I and the dual of Theorem 7.9 it is sufficient to show that (6)*(4). Suppose that f:X + Y is a coequalizer of g, h :Z + X and that X E Ob B. We will show that Y E Ob B. Let S E Ob B be a separator in A. Denote a set of all A-morphisms from S to Z by (k, : S + ZI i E I} and let ( W , (q, :S, -, W ) , € , )be a coproduct of (Si),€,with S, = S for each i E I. Then W E Ob B and there exists a morphism k : W + Z such that ki = k 0 qi for i E I. If morphisms I, m : Z + V satisfy that 10 k = m 0 k and I # m, then there exists an element i E I such that l o ki # m k,. This contradicts the fact that 0
lok, = I o k o q ,
=
mokaq, = mok,,
and hence we have that k is an epimorphism in A. Now suppose that f : X + Y’ satisfies thatfogok = f o h o k . Thenfog =fob and since f:X + Y is a coequalizer of g and h, there exists a unique morphism cp: Y + Y’ such that f’ = c p o f . This implies that f is a coequalizer of 0 B-morphisms g 0 k and h 0 k in A and hence we have that Y E Ob B.
7.16. Example. (1) Comp,,
the category of compactly generated spaces, is bicoreflective in Top. For a topological space X , let RX be a topological space with the same undying set to X and a topology defined as follows: G c I XI is open if and only if G n C is open in C for each compact subspace C of X. Then the identity map between underlying sets induces a continuous map r, : RX + X , which is a Comp,-coreflection of X. Comp,, is a coreflective hull in Top of Comp, the full subcategory of Top consisting of all compact spaces. (2) Seqgea,the category of sequential spaces, is bicoreflective in Top. The Seq,-coreflection of a topological space X can be obtained in a similar way as above. These examples are unified to the theory of A-generated spaces for some subcategories A of Top or Top,. 8. (E, M)-categories
In this section, let E and M be classes of morphisms in a category A which are closed under composition with isomorphisms. We will generalize the notions of (Ex epi, Mono)-factorizable etc., treated in Section 5.
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8.1. Definition. A category A is said to be (E, M)-jiactorizable if for each A-morphism f : X + Y there exists a triple (Z,g, h ) consisting of an object Z and morphisms g : X + Z in E, h : Z + Y in M such thatf = hog. The triple (2,g, h) is called an (E, M)-factorization off. An (E, M)-factorizable category A is said to be uniquely (E, M)-factorizable if for two (E, M)-factorizations (Z, g , h) and (Z’,g‘, h’) of any morphism f :X + Y , there exists an isomorphism cp : 2 + Z’ such that g’ = cp og and h = h‘ocp. A category A is said to have the (E, M)-diagonalizationproperty whenever if morphisms f:W -, X , g : W + Y , h :X + Z , k : Y + Z satisfy h 0f = kog, f E E and k E M , then there exists a morphism d : X + Y such that g = dof and h = k 0 d. The morphism d is called a diagonal of the square (Lg, h, k). A category A is said to be an (E, M)-category if it is (E, M)-factorizable and has the (E, M)-diagonalization property.
8.2. Proposition. If E c Epi A or M c Mono A, then the following are equivalent: (1) A is an (E, M)-category. (2) (a) E and M are closed under the compositions. (b) A is uniquely (E, M)-factorizable. Proof. (1)=.(2)(a): Let f :X + Y and g : Y + Z be morphisms in E and (W, h, k ) be an (E, M)-factorization of g of. Then there exists a diagonal d : Y + Wofthesquare(Jh,g,k)suchthath = d o f a n d g = k o d . F o r the square ( g , d, l,, k ) there exists a diagonal d‘: Z + W such that d = d‘og and 1, = kod‘. Hence, kod’o k = k. On the other hand, d’okoh = d‘ogof = d o f = h. Since k E Mono A or h E Epi A , d’ 0 k = I and hence k is an isomorphism. Thus we have that g of belongs to E.
(1)=.(2)(b): Suppose that ( Z ,g , h) and (Z’, g’, h’) are two (E, M ) factorizations o f f : X + Y. Then there exist morphisms d : Z + Z’ and d’:Z’ + Z such that g’ = dog, h = h’od, g = d’og’ and h’ = hod’.
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We can show that d is an isomorphism by the assumption that E c Epi A or M c Mono A. The proof of ( 2 ) 4 1 ) is easy and so omitted. 0
8.3. Proposition. I f A is an ( E , M)-category with E c Epi A, the following conditions for a morphism k : Y + Z are equivalent. (1) k E M . (2) For any morphisms f : W + X , g : W + Y, h : X + Z such that h f = k g and f E E, there exists a diagonal d : X + Y such that 'g = dof andh = k a d . ( 3 ) I f k = m I with I: Y + W E E andm : W + Z, then 1 is an isomorphism. 0
0
0
8.4. Corollary. I f A is an (E, M)-category with E c Epi A, M satisfies the following. (1) I f ( W, (aI: W + X , a2: W + Y ) )is apullback of ( f :X + Z , g : Y + Z ) and i f g E M , then a1E M . ( 2 ) M is closed under intersections and products.
8.5. Example. (1) By Theorem 5.5, Lemma 5.8 and Theorem 5.10, we have that if a well-powered category A has intersections and finite products, then it is an (Ex epi, Mono)-category and if A has intersections and equalizers, then it is an (Epi, Ex mono)-category, where Ex epi denotes the class Ex epi A of all extremal epimorphisms in A and so on. (2) Let E be a class of all dense maps in Top and M a class of all closed embeddings in Top. Then Top is an (E, M)-category. (3) A continuous map f : X + Y between topological spaces X and Y is said to be an initial map provided that any subset A of X is open iff there exists an open set E of Y such that A = f - ' ( E ) . It is shown that a continuous map f : X + Y is initial iff Cl,(A) = f -'(Cly(f ( A ) ) )for any subset A of X , where CI, denotes the closure operator in X . Denote the class of all initial maps in Top by Initial Top or Initial. Then we have that Top is a (Bi, Initial)-category. Obviously, Epi Top =I Bi Top and Ex mono Top c Initial Top. It is noted that i f f : X + Y is an initial map and X is a To-space, then f is an embedding.
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8.6. Definition. An ( E , M)-category A is said to be E-co-well-powered if for each object X of A the class of all morphisms belonging to E which have a domain X has a representative set.
8.7. Theorem (Characterization Theorem o f E-reflective Subcategories). Suppose that A is an E-co-well-powered ( E , M)-category with E c Epi A and that A has products. Then,for a full and isomorphism-closed subcategory B of A, the following conditions are equivalent: ( I ) B is E-reflective in A. ( 2 )B is closed under products and satisfies that iff: X + Y belongs to M and Y E Ob B, then X E Ob B. Proof. (1)-(2): It is shown in Proposition 7.3 that B is closed under products. Let f : X + Y belong to M and Y E Ob B. For the B-reflection r, : X + R X o f X , there exists an extensionf' : RX -+ Y such thatf = f 0 r,. Then there exists a diagonal d : RX + X of the square (r,, I,, So, f) such that do r, = I , and f O = f d. Since r, E E c Epi A, r, is an isomorphism and hence X E Ob B. The proof of (2)=41) is similar to that of (8)*(0) in Theorem 7.9. 0 0
8.8. Corollary. Zf A is an E-co-well-powered (E, M)-category with E c Epi A which has products, then the following hold. (1) The intersection of any class of E-reflective subcategories of A is E-reflectivr in A. ( 2 ) For any subcategory B of A, there exists an E-reflective subcategory B* of A satisfying the following conditions: (a) B* 3 B. (b) ZfB' is an E-reflective subcategory of A and B' 3 B, then B' 3 B*. of B-objects X, and (3) An A-object X belongs to B* ifthere exist a set a morphism f :X + niElX, belonging to M . The category B* will be called an E-rejective hull of B in A and denoted by A,(B). (The suffix E is omitted if no confusion is expected.) For an A-object X , the E-reflective hull AE(X) o f X in A is also defined. An A-object Y belongs to AE(X) iff there exist a set I and a morphism f : Y + lIrelX, belonging to M with X, = X for each i E I. 8.9. Example. (1) Letf :X + Y be a monomorphism in Top, i.e., an injective continuous map and let Y be a To-space (respectively, TI-space, T,-space). Then X is a To-space (respectively, Tl-space, T,-space). Since Top is
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(Ex epi, Mono)-category, Top,, Top, and Top, are (Ex epi)-reflective subcategories of Top respectively. (2) Letf: X Y be an initial map in Top and Y be a regular (respectively completely regular) space. Then X is regular (respectively completely regular). Hence the categories Reg and Creg are bireflective in Top, where Reg (respectively, Creg) is the full subcategory of Top consisting of all regular (respectively completely regular) spaces. --f
9. Epireflective vs. bireflective in Top
For bireflective subcategories Reg and Creg of Top mentioned in the last section, Reg n Top,, = Top, and Creg n Top, = Tyc are epireflective but not bireflective in Top. The bireflective hulls of Top, and Tyc in Top are Reg and Creg respectively. In this section we shall consider the relationship between epireflective subcategories and bireflective subcategories of Top in general. First, we recall the properties of To-spaces and show that Top, has a special property in Top. Let Do, D, and D, be topological spaces with the same underlying set consisting of two points P, Q and with topologies (8, {P,Q}}, (8, { P } , (P,Q } } and (8,{P},{ Q } , { P , Q } } ,respectively. 9.1. Proposition. For a topological space X , the following are equivalent. (1) X is a To-space. ( 2 ) There exist a set A and a monomorphismf : X nlEh X , with X , = D , for each 1 E A. ( 3 ) Any continuous map f : Do + X is constant. (4) Any initial map with the domain X is an embedding. --f
9.2. Proposition. For an epirejlective subcategory A of Top the following hold. (1) A is birejlective in Top iffit contains Do. ( 2 ) Zf A is not birejlective in Top, then A c Top,,. (3) ZfA is birejlective in Top, then A Q Top, and Top, n A is epireflective but not birejlective in Top.
Proof. (1): It is shown that i f f : X Y is an initial map, there exist a set + Y x llis,Ziwith Zi = Dofor each i E Z. Hence if A is epireflective and contains Do,it is closed under products and initial maps. Hence it is bireflective. Conversely, if A is bireflective, then the A-reflection of Domust be Do. --f
Z and an embedding g : X
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(2): Suppose that there exists a topological space X such that X E Ob A and X # Ob Top,. Then X contains a subspace Doand this contradicts (1). (3) is obvious. 0 9.3. Theorem. Let G be a collection of all bireflective subcategories of Top and H a collection of all subcategories of Top which are epireflective but not bireflective in Top. Then there exists a one-to-one correspondence between G and H.
Proof. Let ts : G -+ H and T : H -+ G be the functions defined by o ( A ) = Top, n A , A E G and T ( B ) = T o b , ( B ) ,B E H. We shall show that they are mutually inverse bijections. First, let X E Ob A for A E G and let q :X + X / be the Top,,-reflection of X (see Example 7.14(1)). Then we have a function 5 : X / -+ X between the sets such that q 0 5 is the identity of X / . We can show that 5 is continuous and is an embedding and hence X / - E Ob(Top, n A). Since q is an initial map, X E Ob(Top,,,(Top, n A)). Hence, A c Top,(Top, n A). The converse is obvious and we have that A = Toh,(Top, n A), i.e., 7 . 0 = 1,. Next, let X E Ob(Top,, n Top,,(B)) for B E H. Then there exists an initial mapf : X Y with Y E Ob B. Since Xis a To-space,f is an embedding and hence X E Ob B. The converse follows from Proposition 9.2 and we have that B = Top,, n Tobi(B), i.e., C O T = 1,.
-
-
-+
0 9.4. Corollary. Suppose that B = Top,, n A (or equivalently,A = Toh,,(B)) for A E G and B E Hand that r,: X + R X , s,: X -+ H a n d t,: X -+ TXare the A-rejection, the Top,-reflection and the B-rejection of an object X E Ob Top respectively. Then (r,, s R x ) is a (Bi, Initial)-factorization oft,.
Now we shall show another property of bireflective subcategories of Top, that is, bireflective subcategories correspond to separation axioms which involve separation of pairs consisting of a point and a closed subset.
I be an operator which associates with a topological space X a function I,: P(X)-+ P(X)between the power set P(X)of X . If I satisfies the following conditions (1)-(4),it is called an upper limit operator and if 1 satisfies the conditions (l)’, (2)-(4), it is called a lower limit operator. (1) CI,(Z) c 1,(Z) for Z E P ( X ) . (1)’ Z c l , ( Z ) c CI,(Z) for Z E P ( X ) . LJ Z2) = l , ( Z , ) u IX(Z2) for z , ,z 2 EP W . (2)
9.5. Definition. Let
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(3) I,(@) = 0. (4) For any morphism f:X + Y in Top and for any Z E P ( X ) ,f(lx(Z)) c Ids( a ) . For an upper limit operator I, a topological space Xis said to be I-separated provided that I , = C1, and for a lower limit operator k, X is said to be k-separatedwhenever if 2 = k,(Z) for Z E P ( X ) , then Z = Cl,(Z). For an upper or a lower limit operator I, the full subcategory of Top consisting of all I-separated spaces will be denoted by S(1). Then we have the following proposition. 9.6. Proposition. Let A be afull and isomorphism-closedsubcategory of Top. Then, (1) A is birejlective in Top iff there exists an upper limit operator I such that A = S(1); (2) A is bicorejlective in Top iff there exists a lower limit operator k such that A = S(k).
Proof. (1): For an upper limit operator I, we can show that S(1) is closed under products and initial maps. Hence, S(1) is bireflective in Top. Conversely, let A be a bireflective subcategory of Top and let X be a topological space with the A-reflection r,: X -P R X . For Z E P ( X ) , let l,(Z) = (rX)-'(CIRX(rX(Z))). Then we have an upper limit operator 1 such that A = S(1). (2): For a lower limit operator k , we can show that S(k) is closed under coproducts and quotient maps. Hence, S(k) is monocoreflective in Top and this means that it is bicoreflective in Top. Conversely, let A be a bicoreflective subcategory of Top and X be a topological space with A-coreflection r,: RX + X . For Z E P ( X ) , let k(Z) = r,(CIRX((r,)-'(Z))). Then we have a lower limit operator k such that A = S ( k ) . 0 9.7. Example. (1) Let X be a topological space and Z E P ( X ) . Define functions & : P ( X ) + P ( X ) for i = 0, 1, 2, 3, as follows. 1%) = { x E ~ l C l , ( { x } ) n Cl,(Z) z 0}. l i ( Z ) = { x E XI there is a pointy E Cl,(Z) such that all open subsets U , V with U 3 x, V 3 y have a nonempty intersection}. l i ( Z ) = { x E XI any open subsets U , V with U 3 x, V 3 Z have a nonempty intersection}. l i ( Z ) = {x E XI there is no continuous map f :X + [0, 11 with f(x) = 0 and f ( Z ) = l } . Then we obtain an upper limit operator I' for each i = 0, 1, 2, 3.
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X is /'-separated iff it is an R,-space, i.e., it satisfies the following axiom: (R,)
If x E CM{ y } ) for x, y E X , y E CM{ x}).
X is /'-separated iff it satisfies the following condition:
(R,) If F is a closed set in X and if x 4 F, y
E F, then there exist open sets U , V such that U 3 x, V 3 y and U n V = 8.
(R,) and (R,) are equivalent to the axioms (S,) and ( S , ) stated in the book of Csaszar [1978]. Obviously, S(Z2) = Reg, S(Z3) = Creg and we have S(1') n Top, = Top,,
S ( / ' ) n Top,
S(12)n Top, = Top,,
S(Z3) n Top, = Tyc.
=
Top,,
Top, and Top, are (Ex epi)-reflective in Top, while Top, and Tyc are not. We shall discuss (Ex epi)-reflective subcategories of Top in the next section. (2) Let X be a topological space and 2 E P ( X ) . Define functions k': P ( X ) + P ( X ) for i = 0-5, as follows.
k;(Z) =
z,
k k ( 2 ) = { x E XI there exists an indiscrete subspace S of X with x E Cl,(Z n S)},
ki(2)
=
{x E XI there exists a finite subspace E of X with x E Cl,(Z n E)},
k i ( 2 ) = { x E XI there exists a sequence ( x , ) in 2 which converges to x}, ki(2)
=
{ x E XI there exists a compact subspace K of X with x E C1,(Z n K ) } ,
ki(2)
=
Cl,(Z).
Then each k' is a lower limit operator and we have that Ob S(k') consists of all spaces satisfying each of the following conditions.
S(k'): discrete space, S(k' ): topological sum of indiscrete space, S(k2): finitely generated space, S(k3):sequential space,
S(k4): compactly generated space, S(ks): topological space.
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Limit operators were studied by Herrlich [1968, 19691 for the first time. Only lower limit operators were treated there and called limit operators. Upper limit operators were discussed in Nakagawa [1976].
10. Separation axioms and connectedness Various kinds of separation axioms for topological spaces are studied by many authors and systematic investigations are also given. There are three basic types of separation axioms depending on whether they involve separation of pairs of points, pairs consisting of a point and a closed subset, or pairs of subsets. Axioms of the second type can be defined as I-separatedness for some upper limit operator I discussed in the last section and Proposition 9.6 is a general result on separation axioms of this type. For a separation axiom of the third type, the structure of the full subcategory of Top consisting of all separated spaces is not so good, e.g., products of normal spaces are not necessarily normal. This yields one of the most important problems in general topology as is treated in the other chapters of this book. However, a similar result to Proposition 9.6 is obtained in the category Sep, which consists of all separation spaces and all continuous maps between separation spaces defined by Wallace [1941] (Nakagawa [1976]). It is noted that sometimes a subcategory of Top has better structures in a supercategory A of Top than in Top. (If B is a subcategory of A, A is called a supercategory of B.) The category Near of nearness spaces introduced by Herrlich [1974b] is a very useful supercategory of Top in this sense. Separation axioms of the first type were characterized in terms of category theory by Wyler [1973]. We shall discuss this in the following. 10.1. Definition. A natural relation e in Top is an operator which associates with a topological space X a relation ex on X satisfying the following naturality condition. ( ( x , y ) E ex c X x X will be denoted by x ex y.) For any continuous mapf: X -, Y and any points x, y E X,x ex y means f ( x ) e f( Y). If ex is reflexive, symmetric of transitive for any X , e is called reflexive, symmetric or transitive, respectively. If ex is an equivalence relation for any X , e is called a natural equivalence relation. In the following, we shall always consider reflexive natural relations and hence call them natural relations in brief.
10.2. Definition. For two natural relations e and T, define e < T if for any topological space X and any points x, y E X , x ex y means x t xy.
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For a family {ei 11 E A} of natural relations, we can define nj.EAe'. and U j . p A @ ' by ( n r e n e i ) x = n j . , A ( e " ) x and ( u j . E A @ " ) X = u/.EA(@")A'* Then the collection of all natural relations in Top is a complete lattice by this order < . The natural relation d and e defined by dx = { ( x , x ) I x ~ X }and ex = X x X
foreachXEObTop
are the minimum and the maximum elements respectively. For a natural relation e, we have a natural relation e' such that { ( x , y)l ( y , x ) E e x } . Denote e n e' by e A and e u e' by e".
(e'Ix =
10.3. Proposition. For a natural relation e in Top, (1) e " and e " are symmetric natural relations; ( 2 ) if@' is a symmetric natural relation with e' < e, then e' is a symmetric natural relation with e < e', then e v < ( 3 ) the following are equivalent: (a) e is symmetric, (b) e" = e, ( 4 ev = e, (d) e" = e".
Proof. Obvious.
< e" and if@' e';
0
10.4. Definition. For a natural relation e in Top, a topological space X is said to be e-separated iff ex = dx. The full subcategory of Top consisting of all e-separated spaces will be denoted by X(e).
If @ < z, C(@) =I X(z). Hence X ( n i . E A @ j , )=I u j . e A Z ( @ j ' ) , while we have that X(U,,A@^) = nj.,,,C(e"). As a special case, X ( e v ) = Z(e) c X(e"). X ( d ) = Top and Z(e) consists of all one-point spaces and the empty space. 10.5. Proposition. Let B be a full and isomorphism-closed subcategory of Top. Then B is (Ex epi)-reflective in Top fz there exists a natural relation e in Top such that B = X(e).
Proof. It is easily verified that X(e) is closed under products and monomorphisms and hence it is (Ex epi)-reflective in Top. For (Ex epi)-reflective subcategory B and X E Ob Top with the B-reflector r x :X RX, let ex = { ( x , y ) I rx(x) = r x ( y ) } .Then we have a natural equivalence relation 0 e in Top such that Z(e) = B.
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10.6. Definition. The natural equivalence relation @ obtained in Proposition 10.5 from B will be denoted by @(B).For any natural relation e we have that e < e(Z(e)). @ is called maximal provided that @ = @(Z(@)). We shall see many examples such that @ # e(Z(e)) and this means that different natural relations may associate with the same (Ex epi)-reflective subcategory of Top.
Now we proceed to consider separation axioms in general categories. First we recall the notion of natural transformation.
10.7. Definition. Let F, G : A + B be functors. A natural transformation q : F + G is an operator which associates with each A-object X a B-morphism q,: F ( X ) + G ( X ) satisfying the following condition: For each A-morphism f : X + Y, G ( f ) o ~ x = r t u o F ( f ) For functors F, G , H : A + B, natural transformations q :F + G , [ : G + Hand X E Ob A , define (, : F ( X ) + H ( X ) by t, = l x o q,. Then we have a natural transformation (, which will be called the composition ofq and and denoted by i q. 0
Suppose that A is a category with products of pairs. For X E Ob A, let S ( X ) = X x X be the product with projections p i , p i and d,: X + S ( X ) be the morphism such that pied, = pied, = 1,. Then we have a functor S : A + A and a natural transformation d : 1, + S . 10.8. Definition. A natural relation in A is a pair (R, e) of a functor R : A + A and a natural transformation @ : R + S satisfying the following two conditions: (1) For any X E Ob A , ex : R X + SX is a monomorphism. (2) There exists a natural transformation i : 1, --* R such that e i = d. Natural equivalence relations and an order < between natural relations are also defined. (l,, d ) is the minimum natural relation in A and ( S , e ) with e, = Is(,) is the maximum one. For a natural relation (R, e) in A , an A-object X is said to be @-separated iff e x :R ( X ) + S ( X ) = d,: X + S ( X ) . 0
The full subcategory of A consisting of all @-separatedobjects will be denoted by Z (e). The dual notions are also defined. In these settings we can prove that the following statement and its dual are valid: Suppose that A is a complete,
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well-powered and co-well-powered category and B is a full and isomorphism-closedsubcategory of A. Then B is (Ex epi)-reflective in A i f there exists a natural relation (R, e) in A such that C(e) = B. 10.9. Example. (1) Let X E Ob Top and x, y E X. Define efY,for i = 0-5 as follows: e! = {<x,y)lx E C M Y > > ; eZ = {(x, Y)I C M x ) n CIX{Y } z 01; e: = {(x, y)l for any open subsets U, V with U 3 x, V 3 y, U n V # S}; e: = {(x, y) I there exist an integer n and points xi E X for 0 < i < n such that x, = x, x, = y, (xi, xi+,)E e: for 0 < i < n - l}; et = {(x. y)l for any open subsets U , V with U 3 x, V 3 y , CI,(U) n CMV) z 0); e: = {(x, y) I x and y belong to same connected component of X}. Then we have that (Q’))” < el, (e’))” < e2 < e’, e’ < e4, e2 < e4, and W e o ) ) ^ )= TOP,; X(eo) = X((eo))”)= C(e’) = Top,; X(e2) = Z(e3) = Top,; I;(e4) = the full subcategory of all Urysohn spaces; X(eS) = the full subcategory of all totally disconnected spaces. (2) In Grp and Ab, any natural relation is shown to be a natural equivalence relation. For X E Ob Ab, let R ( X ) = {(x, y) I there exists an integer n such that n(x - y) = 0} and ex : R ( X ) -+ S ( X ) be the inclusion homomorphism. Then (R, e) is a natural relation in Ab and C(e) is the full subcategory of Ab consisting of all torsion-free groups. Let Sop(X) = X 0 X , direct sum, R ’ ( X ) = {(x, -x)I there exists an integer n such that nx = 0}, and (eop)X:Sop(X)+ R o p ( X ) = P P ( X ) / R ’ ( X be ) the quotient homomorphism. Then (RoP, cop) is a co-(natural relation) in Ab and an abelian group X is eoP-co-separatediff it is a torsion group.
By considering natural relations we can clarify the relationship between the notions of separatedness and connectedness. 10.10. Definition. For a natural relation e in A, an A-object X is said to be e-connected iff
ex: R ( X ) + S ( X ) = ex:S ( X ) + S ( X ) . The full subcategory of A consisting of all e-connected objects will be denoted by Ue).
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If@< T , then I-(@) c T(z). T ( d ) = Z(e) and T(e) = X ( d ) = A . In Top, we have that r(n@') = nr($)), r((J@') 3 Ur(@')and r(@") 3 r(@) =
r(eA1. 10.11. Proposition. For a natural relation @ in Top, the following hold. (1) I f f : X -+ Y is an epimorphism in Top and X is @-connected,then Y is @-connected. ( 2 ) An A-object X is @-separated and @-connected simultaneously if X E T(d). ( 3 ) Zf f :X + Y is a morphism in Top from a @-connectedspace X to a @-separatedspace Y, then f is constant. (4) When @ is maximal, X is @-connectedfi C(e)-reJlection sx :X + Z ( X ) is constant. 10.12. Example. (1) Let ei be the natural relations in Top defined in Example 10.9 for i = 0-5. Then r((e0)>^) = Ind. X E Ob Top is (e')" -connected iff q ( X ) has a total order < such that any open set G of q ( X ) satisfies that if x E G and x < y , then y E G, where q ( X )
is a space obtained by Topo-reflectionq : X + q ( X ) of X. @'-connectednessis called ultraconnectedness and @* -connectednessis called hyperconnectedness by some authors. $-connectedness = connectedness. Thus we have the following implications: ultraconnected (@')"-connected
* hyperconnected
3
*
+
path connected Q connected i
e4-connected 47
(2) Let @ and Qop be the natural relation and the co-(natural relation) in Ab defined in Example 10.9. Then X E Ob Ab is @-connectediff it is a torsion group and X is eoP-co-connectediff it is torsion-free. Thus we have that @-separatedness= QoP -co-connectedness and @-connectedness= QoP -coseparatedness in this case. (3) Let E be a class of topological spaces and Top@) be the (Ex epi)reflective hull of E in Top. Then we have a natural equivalence relation eE = @(Top@))defined in Definition 10.6. By Proposition 10.11, the following conditions for a topological space X are equivalent: (a) Xis @,-connected. (b) The Top(E)-reflection s x : X + C ( X ) is constant. (c) Any morphism f : X -+ Y with Y E E is constant. Thus the notion of @,-connected coincides with that of E-connected introduced by Preuss [1970]. @,-separatedness is also treated in detail by Preuss.
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The natural relation e in Top induces a natural relation eo in pTop, the category of pointed topological spaces. Then r(eo) is an (Ex mono)coreflective subcategory of pTop. This follows from the dual of the statement in Definition 10.8. If a category A has zero object and satisfies some nice conditions and e is a natural relation in A, e-separatedness can be considered as the dual of e-connectedness (Nakagawa [ 19851).
11. Simplicity of epireflective subcategories 11.1. Definition. Let A be a complete, well-powered and co-well-powered category. An epireflective subcategory B of A is said to be simply generated (or, simple) in A if there exists an A-object X such that B is the epireflective hull A ( X ) of X.
11.2. Example. ( I ) Top(Do) = Ind, Top(D,) = Top, and Top(&) are simply generated in Top. It is easily shown that Top@,) consists of all zero-dimensional spaces. If X is a discrete space containing at least two points, then Top(X) = Top(D2).Especially, Top(N) = Top@,), where Nis a topological space consisting of all natural numbers with discrete topology. If X is a To-spacewhich is not a TI-space, X contains D,as a subspace and henceTop(X) = Top(&) = Toh. (2) Let T be a topological space with three points P, Q, R and with a topology (8,{P},{P,Q, R}}.Then Top(T) = Top@, x 0,) = Top, that is, Top is simply generated in Top. (3) Tyc = Top([O, I]) = Top((0, I)) is simply generated in Top, where [0, I] and (0, 1) denote the closed and the open unit intervals respectively. (4) CompT, = Topz([O, 11) and Top,((O, 1)) are simply generated in Top,. An object of Topz((0, 1)) is known as a realcompact space or a Q-space. ( 5 ) Obviously, a subcategory of Top, is a simply generated epireflective subcategory of Top iff it consists of all E-regular spaces for some Hausdorff space E and an epireflective subcategory of Top, is simply generated in Top, iff it consists of all E-compact spaces for some Hausdorff space E (see Example 7.14(6)). (6) Obviously, Top,(D2) c Top,(N) c Top,((O, I)). A topological space X is D2-compact iff it is a zero-dimensional compact Hausdorff space and hence Top,(N) is strictly larger than Top@,). If X is an N-compact space which is not D,-compact, then Top,(X) = Top,(N). Spaces of all rational numbers and that of all irrational numbers with the usual topology satisfy
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this condition. N-compact spaces are studied in detail in Chapter 12 by Eda, Kiyosawa and Ohta. 11.3. Proposition. Suppose that A is a birejlective subcategory of Top and
B = Top, n A. Then B is simply generated in Top i f A is simply generated in Top. Proof. Let A = Top(X) with X E Ob Top and let q : X + X / - be the Topo-reflection of X. Then X / - E Ob B, as is shown in Theorem 9.3. For any Y E Ob B, there exist a set A and an embedding f :Y + llj,,,,Xj, with X , = X for 1 E A. Then we obtain the initial map g = (l-Ij,E,,qj.)o f : Y + l l i E , , Z jwhere , Z , = X / - and qj, = q : X , + 2,for each 1 E A. Since Y E Ob Top,, g is an embedding and this implies that B = Top(X/-). Conversely, let B = Top(X) with X E Ob Top. Since A is bireflective in Top, A contains Doand hence, A
=J
Top({X, Do})= Top (X x Do)I> B.
Since Top(X x Do)contains Do,it is bireflective in Top. By Theorem 9.3, A = Top,,(B). Thus we have that A = Top(X x Do). 0 It is natural to ask whether Top,, Top, and Top, are simply generated in Top or not. The answer is negative and the proof is based on the following lemma due to Herrlich [1965], which is a generalization of the classical results of a problem of Urysohn. 11.4. Lemma. For any Tl-space Y , there exists a regular TI-spaceX such that any continuous map f :X -+ Y is constant.
Proof. We will construct the space X step by step as follows. (1) Suppose that card( Y ) < K,, where card( Y) denote the cardinality of the underlying set of Y . Define a topological space R, with a fixed point r, and card(R,) = Nu+, for each i = 1,2, having the following topology: B c R, is open iff r, $ B or card(& - B) < K,+,.Then we can verify that each R, is a regular TI-spaceand has the followingproperty: If B, is a nbd of r, in R, for eachj E J ( i ) , where J ( i ) is a set with cardinality K,+,-I, then nlEJC,,B, is also a nbd of r,. Letf: R, + Y be a continuous map. Then, for y , = f (r,),
n i f - v- W ) I YE
y - {YJ} and this is shown to be a nbd of r, by using the above fact. In other words, any continuous map from R, to Y is constant on a some nbd of r, for i = 1,2. f-l(y,) =
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( 2 ) Let T = R, x R, - {(rl,r , ) } . Then for any continuous map f : T -+ Y, we can find nbds U, of r, for i = 1, 2 such thatfis constant on UI x u2 - ((rl, r 2 ) ) . (3) Let T, = T for any integer n and R be a topological space obtained from the topological sum of the family (T,),,z by adding two points P , Q with the topology defined as follows: A subset B of R is a nbd of P iff there exists an integer n such that B 3 7‘’ I rn 3 n} u { P } and B is a nbd of Q iff there exists an integer n such that B 3 T,I m < n} u { Q } . Then R is a regular TI-space. Any point of R except P, Q is expressed by (x, y , n) with x E R,, y E R,, n E Z . (4) Let S be a topological space obtained from R by identifying the following points:
u{
( x , r,, 2n)
-
( x , r2, 2n
( r , , y , 2n - 1)
-
u{
+ 1)
(rlry, 2n)
for x
E
R, - { r , } and n E Z,
for y E R, - ( r 2 }and n E Z.
Then S is a regular TI-space and, for any continuous map f : S -+ Y, f ( P ) =f(Qh ( 5 ) For an arbitrary topological space Z, let S ( Z ) be a topological space with the same underlying set as the product Z x S and with the topology defined as follows: A subset B of S ( Z ) is an open nbd of ( z , P ) iff there exists a nbd U of z in Z such that U x ( P } c B and B is an open nbd of ( z , s) for s # P iff there exists a nbd U of s in S such that { z } x U c B. Then let S ’ ( Z ) be a topological space obtained from S ( Z ) by identifying Z x { Q} as one point. Then if Z is a regular TI-space, S ’ ( Z ) is also a regular TI-space.2 is canonically embedded in S ’ ( Z ) by the map h :Z -+ S ’ ( Z )such that h(z) = ( z , P ) . (6) Let X , be a one-point space and let X,,, = S’(X,) for each natural number n. Then we have a family (A:) such that X, c X,,,. Let X = UX,with the weak topology. Then X is a regular TI-space. If f : X -+ Y is a continuous map, f (X,) = f ( P ) for each n and hence, f is constant. 0 11.5. Theorem. Let A be a full and isomorphism-closed epirejlective subcategory of Top such that Top, 3 A 3 Top,. Then A is not simply generated in Top. Proof. Suppose that A is simply generated, i.e., A = Top(Y) with a TI-space Y. Then there exists a regular T,-space X such that any continuous map f : X -+ Y is constant. Since X E Ob A , there exist a set A and an
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embedding g : X + ITIle,, Y, such that Y, = Y for each 1E A. Then p n0 g : X + Y is constant for each I E A and hence, g is constant. Since X contains at least two points, this is a contradiction. 0 The above result was strengthened by Hajek and Wilson [1973]. There are many results on the simplicity of epireflective subcategories in Top and in various kinds of categories. 12. Topological functors Let A be a set, {X;.I I E A} a family of topological spaces, Y a set and : Y + I X , I be a function between the sets for each I E A. Then we can define the smallest topology on Y which makes everyh. continuous, where A can be taken as a proper class or the empty set. If g , : I X, I + Z is a function between the sets for each I E A, we can also define the largest topology on Z which makes every gj,continuous. These properties play a fundamental role in the category Top and also in some other categories, which have properties similar to Top. In other words, these properties yield many important properties of topological spaces, uniform spaces, nearness spaces and so forth. Hence we proceed to the investigation of general categories with these properties. By the following observation, we can get a result on the relationship between the existence of a smallest topology and that of a largest topology in a general setting. 12.1. Definition. For a category A, a pair ( X , ( J . : X + X;.)j.E,,) of an A-object X and a family of A-morphismsh, :X + X , with an indexing class A is called an A-source. An A-source is a natural source in A for a discrete category A, and a natural source is a source if the naturality condition is forgotten. For a functor F: A + B, a B-source ( Y , (gj.: Y + F(Xj.))j,e,,)with A-objects X , for each 1 E A is called an F-source. An A-source ( X , (h.)j,E,,) is said to be a monosource whenever if a pair of A-morphisms u, v :Z + X satisfiesh, 0 u = 0 v for each I E A, then u = v. An A-morphism can be considered as a source, and a monomorphism is a monosource. If ( X , ( p j . : X+ Xj.)j.e,,)is a product of A-objects (Xj,)j,e,,, then it is a monosource. In general, if ( X , (a,)) is a limit for a diagram D : K + A, then it is a monosource. Let E be a class of A-morphisms and M be a collection of A-sources which are closed under composition with isomorphisms. A category A is said to be (E, M)-factorizuble if for each A-source (X, (A. : X + X i ) i e h )there exists a triple (2,g , ( Z , ( h j . : Z + Xilie,,)) consisting of an object Z, a morphism
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g : X + Z in E and an A-source ( Z , ( h , : Z + X,.)j,EA)in M such that J , = hj, g for each 1E A. Notions of uniquely (E, M)-factorizable, (E, M ) 0
diagonalization property and (E, M)-category are also defined in a way similar to Definition 8.1.
Dual: A-sink ( X , (fn : X, + X),,,,), F-sink ( Y , (g, :F(X,) + Y),€,,),episink, (E, M)-fuctorizuble with a collection E of A-sinks and a class M of A-morphisms, etc. 12.2. Example. Let (X, (&,:A' + Xj,),.,,,) be a Set-source and let (Zj,,g r , h,) be the (Epi, Mono)-factorization o f h , for each 1 E A. Since Set is co-wellpowered, there exists a representative subset {g,. I I E M} of {g,. 11 E A}. Let (Y, (pi.: Y -+ Z,.)j.EM) be the product of ( Z j . J r EkM : X, -+ Y be the morphism such that pj.ok = g , for 1 E M and let (W, I, m) be the (Epi, Mono)factorization of k. Then (W, I, W, (n,: W + is the (Epimorphism, Monosource)-factorization of (X, (J,)),where n, = h,. " p i m for 1 E M and n , = h,, 0 cp., op,, 0 m for 1 E A - M with p E M and the isomorphism c.p,, : Z,, 4 Z , . Thus we have that Set is an (Epimorphism, Monosource)-category. Also Set is an (Episink, Monomorphism)-category. 0
12.3. Definition. A functor F : A -,B is said to be topological if, for any F-source (Y, (gl : Y + F(Xi))l.A), there exists a unique A-source satisfying the following conditions: ( X , (fi: X + (1) F ( X ) = Y and F(fn) = g, for 1 E A (F-lift). and a B-morphism k :F ( Z ) + (2) For an A-source ( Z , (h, :Z + F ( X ) such that F(fl)0 k = F(hi), I E A, there exists a unique A-morphism h : Z + X such that F(h) = k and fn h = hAfor 1E A (F-initiality). The A-source ( X , (J.))is called an F-initial l f l of ( Y , (gj,)). 0
A
F
rB
R. Nakagawa
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Dual: co-topological, F-final lift. The forgetful functor I 1 :Top -+ Set is topological and co-topological. -+ Set is not topological, but can be similarly treated by using the notion of (E, M)-topological functors (Herrlich [ 1974a1).
1 I :Top,
12.4. Proposition. I f a functor F : A -+ B is topological, then it is faithful, i.e., the associated function F : [U, VIA + [F(U), F(V)IB is injective for any U , V E O b A. Proof. Suppose that r, s : U + V are different A-morphisms such that F(r) = F(s). Let A be a proper class, X , = V and
g,:F(U)
-+
F(X,) = F(r):F(U)
-+
F(V)
for each I E A. Then there exists an F-initial lift ( X , (A,: X -, X , ) ) of the F-source (F(U), (g;.)). For a E A, define an A-source ( U , (h, : U -,X,)) by h, = r and hj, = s for A # a. Then g,, 0 = F(h,) for every I E A and, by the F-initiality of ( X , (A.)),there exists an A-morphism h : U X such that F(h) = I F ( u )andJ.0 h = h , . If A # u, h , # h, and hence,J. # f,.Thus we have that (J,)ren is a proper class. For each I E A,J, belongs to [X, V I A , which is a set, and we have a contradiction. 0 -+
12.5. Proposition. If F: A -+ B is a topologicalfunctor and if D :K -+ A is a diagram in A over a small category K,thefollowing conditions on an A-source ( X , (xi : X D(i))ioObK) are equivalent: ( I ) ( X , (a,)) is a limit for D. (2) ( X , (a,)) is F-initial and ( F ( X ) , (F(a,))) is a limit fur F o D. -+
Proof. (1)=42): Let (Z, (hi:Z + D(i))) be the F-initial lift of (F(X), (F(ai))). Then there exists an A-morphism h :X -+ Z such that hio h = ori and F(h) = 1F ( x ) . For any K-morphism a : i --* j , F(D(a)oh,)
=
F(D(a))oF(a,)
=
F(D(a)oa,)
=
F(aj) = F(hj).
Since F is faithful, we have that D(a) 0 hi = h, and that (Z, (hi)) is a natural source for D. Hence there exists an A-morphism k : Z -+ X such that a, k = hi. By the uniqueness of F-initial lifts and limits, we have that h is an isomorphism and that (X, (a,)) is F-initial. In order to show that (F(X), is a (F(a,))) is a limit for F a D, suppose that (Y, (g,: Y -+ F(D(i)))ipObK) natural source for F o D. Then there exists the F-initial lift ( W , W + D(i))) of (Y, (gi)), which is shown to be a natural source for D. Hence there is an 0
(x:
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A-morphism cp : W + X such that a, 0 cp = 1;. Thus we have the connecting morphism F(cp) from ( Y , (g,)) to ( F ( X ) , (F(a,))).The uniqueness follows from the facts that ( X , (a,)) is F-initial and that F-initial lifts and limits are unique . (2)=41): Since F is faithful, ( X , (a,)) is shown to be a natural source. Let (Z, (P,:Z+ D ( i ) ) ) be a natural source for D. Then ( F ( Z ) ,(F(P,))) is a natural source for Fo D and hence there exists a B-morphism cp :F ( Z ) + F ( X ) such that F(a,)cp = F(b,). By F-initiality of ( X , (a,)), we obtain the connecting morphism h : Z + X such that a, h = 8,. The uniqueness is also shown. 0
0
0 12.6. Corollary. Any topological functor preserves limits and monosources.
12.7. Theorem. A functor F : A
+
B is topological iff it is co-topological.
Proof. Suppose that F: A + B is topological and that (Y, (gi : F ( X , )+ Y)j.e,,) is an F-sink. Consider the following property of B-morphism h : Y + F ( Z ) : hOg,:F(X,) + F ( Z ) has an F-lift, i.e., there exists an A-morphism k , : Xi + Z such that F(k,) = h o g , for every 1E A. Then denote the class of all B-morphisms with this property by (h,,: Y -+ F(Z,,)),,,,,,and the F-lift of h,, g , by k,,,: X , + Z,,. Since F is topological, the F-source (Y, ( I I , , ) , , , ~ ) has a unique F-initial lift ( X , (I,,: X + Z,,),,,,,) such that F ( X ) = Y , F(l,,) = h,,, p E M. 0
For each 1,consider a pair of an A-source (A';,,(k,,;,: X , + Z,,),,,,,) and a B-morphism g , : F ( X , ) + Y. Then, by the initiality of ( X , (I,,)), there exists a unique A-morphismh, : X , + Xsuch that F(h.) = g , , l,,.oJ, = k,,;.for p E M. Thus we obtained an F-lift ( X , (h,)) of (Y, (g;,)).In order to show the finality of (X, (h.)), suppose that (W, (m;,: X , + W)) is an A-sink and n : Y + F( W) is a B-morphism such that n o g , = F(mJ for , IE A. Then there exists an element p E A4 such that h, = n, W = Z,, and m, = k,,, and hence 1, satisfies that F(I,,) = n and I,,oJ. = m , for 1E A. The uniqueness is obvious. Therefore we have that F is co-topological. The converse is the dual. 0
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12.8. Corollary. Any topologicalfunctor preserves colimits and episinks. 12.9. Corollary. Suppose that F : A + B is a topologicalfunctor. Then ifB is complete, A is complete and if B is cocomplete, A is cocomplete. Suppose that F : A + B is a topological functor and Y E Ob B. Let (G( Y ) , 0)be the F-initial lift of the F-source (Y, 8) and let ( H ( Y ) , 8) be the F-final lift of the F-sink ( Y , 0).Then we have the right adjoint functor G and the left adjoint functor H of F. If F: Top + Set is the forgetful functor, G( Y ) is the indiscrete space and H(Y ) is the discrete spaces with the underlying set Y.
13. Topological categories 13.1. Definition. Let F : A + B be a functor. A pair ( A , F ) is said to be a topological category if the following conditions hold: ( 1 ) B = Set. (2) F is a topological functor. (3) For any Y EOb B, the class F-'( Y) = { X I SOb A I F(X) = Y } is a set. (4) For a B-object {P}consisting of precisely one point P, F-'({P}) consists of precisely one A-object. A pair ( A , F ) of a category A and a faithful functor F : A + Set is called a concrete category, in general. By Proposition 12.4, topological categories are concrete. However, some authors consider topological categories without condition (1).
13.2. Proposition. If ( A , F ) is a topological category, the following hold: ( 1) A is complete, cocomplete, well-powered and co-well-powered. ( 2 ) An A-morphism f is a monomorphism if F( f) is injective and f is an epimorphism if F( f ) is surjective. ( 3 ) For any A-morphism f the following are equivalent: (a) f is an embedding, i.e., an F-initial monomorphism in A ; (b) f is an extremal monomorphism; (c) f is a regular monomorphism. (4) For any A-morphism f the following are equivalent: (a) f is a quotient map, i.e., an F-final epimorphism in A ; (b) f is an extremal epimorphism; (c) f is a regular epimorphism.
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( 5 ) A is a (Quot, Monosource)-category and an (Episink, Emb)-category, where Quot and Emb are the classes of all quotient maps and all embeddings respectively. (6) Let B be a full and isomorphism-closed subcategory of A and F :B + Set a functor obtained from F by restricting the domain to B. I f B is bicorejlective or birejlective in A , then (B, F‘) is a topological category.
These properties are easily shown. Concerning the last result (6), we have the fact that the following are equivalent (Miiller [ 19741, Herrlich [1976]): (i) (B, F‘)is topological. (ii) Bis a bicoreflective subcategory of some bireflective subcategory of A . (iii) B is a bireflective subcategory of some bicoreflective subcategory of A. Many important properties of Top hold in topological categories. For example, the relationship between bireflective subcategories and epireflective subcategories, discussed in Section 9, hold for “universal” topological categories (Marny [ 19791). Roughly speaking, properties of topological spaces which are interesting for categorical topology are those of topological categories. Nowadays, the main object of categorical topology seems to be the investigation of topological categories. Now let ( A , F ) be a topological category and X be an A-object. For any A-object Y , let X x ( Y ) = X x Y and, for any A-morphism f : Y + Z , let X x ( f ) = 1, x f : X x Y -+ X x Z. Then weobtain afunctorX x : A -+ A. 13.3. Proposition. For a topological category ( A , F ) , the following are equivalent . (1) For any A-object X , the functor X x preserves colimits. ( 2 ) For any A-object X , the functor X x preserves coproducts and quotient maps. ( 3 ) A satisfies the following conditions (a) and (b): (a) For any A-object X a n d a family of A-objects ( Yj,Jle,,with an indexing set A, X x (HIE,,Y i ) is isomorphic to LI;.€,,(X x Y;,),where H denotes the coproduct in A . ( b ) Z f f : X + Y i s a q u o t i e n t m a p , t h e n s o i s 1 , x f : Z x X + Z x Y. (4) A satisfies condition (a) above and the following condition (b‘). (b‘) Zf f : X -+ Y and g : Z + W are quotient maps, then so is f x g : X x Z + Y x w. ( 5 ) For any A-object X , the functor X x preserves F-final episinks. (6) For any A-objects X , Y, there exists an A-object Y x satisfying the following conditions:
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(i) F ( Y x ) = [ X , Y ] . (ii) For the Set-morphism ex.,: F(X) x [ X , Y ] + F( Y ) defined by e,,(x, f ) = F( f )(x) for x E F ( X ) ,f E [A', Y ] ,there exists an A-morphism d,.,: X x Y x + Y such that F(d,,,) = ex.,. (iii) For any A-objecf X , Y , Z, the Set-morphism c : [Z, Y"] + [X x Z, Y ] defined by c(f ) = d,. (1, x f ) for f E [Z, Y x ]is an epimorphism. (7) For any A-object X , the functor H = X x has a right adjoint, i.e., for any A-object Y there exist an A-object G ( Y ) and an A-morphism q , : H(G( Y ) ) + Y satisfying the following condition: For each A-object Z and each A-morphism f : H ( Z ) + Y , there exists a unique A-morphism f : Z + G( Y) such that q , 0 (H(f O)) = f.
, 0
Proof. (1)*(2)*(3) are obvious. (3)*(4): For f : X -, Y and g : Z + W, f x g = (1, x g ) o ( f x IT). Hence, (b) implies (b'). (4)*(5): Let ( Y , ( J . :Y, + Y);,€,,)be an F-final episink and let (Z;,, g , , h , ) be the (Ex epi, Mono)-factorization off;, in A for each 1.Then there exists a representative subset {hj.lA E M } of {h,11 E A). Let (UieMZj.,(q;,);.€,,,) be the coproduct and let k:LIiEMZj,+ Y be an A-morphism such that k q;, = h , for A E A. Then we have that 0
I
=
ko(
g;,): ;.EM
Yj, + Y /.EM
is an F-final epimorphism, i.e., a quotient map. Hence I, x I: X x (Uj.EM Y;.)+ X x Y is a quotient map and this implies that the sink (X x ( Y ) , (X x ( f ; ) : X x Y, + X x Y);.€,,)is an F-final episink. (5)*(6): Let X , Y, Z be A-objects and let f : F(Z) + [ X , Y ] be a Setmorphism. Then c'( f ) = ex., 0 (( I , , , ) x f ) is a Set-morphism from F(X x Z ) to F( Y ) . Let A be the indexing class of the class { J ;: F(Z,) -, [ X , Y ] I there exists an A-morphism g,: X x Z, + Y such that c ' ( f ; ) = F(g,)}. For any h E [ X , Y ] ,we can findJ. such that h belongs to the image off; by taking Z, with F(Z,) = {P}and g , = h opx, where {P}is a one-point set and p, is the projection. Thus we have a Set-episink ([X, Y ] , (J,:F(ZJ + [ X , Y]);,,,,). Define YXsuchthat ( Y x , (h;,: Z , -, Y'),,,,) is an F-final lift of ([X, Y ] , (J.)). Then F( Y x ) = [ X , Y ] and ( Y x , (h,)) is an F-final episink. By the assumption, (X x Y x , (1, x h , : X x Z;. + X x Y")) is also an F-final episink. For the A-sink ( Y , (g;.: X x Z, + Y ) ) , Qi) = ~ X . Y O ( ( ~ F [ X )xh.1 ) = ex.yoF(1x x hi). By the finality of (X x Y x , (1, x hi,)), we have an A-morphism d,,, such that F(d,,,) = ex,,. In order to show that c is an epimorphism, suppose that
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k : X x Z + Y is an A-morphism. For any z E F ( Z ) , there exists an A-morphism z o :F - ’ ( P ) -, Z such that F ( z o ) ( P )= z. Let k, = k (1, x z o ) o i : X + Y , where i : X + X x F - ’ ( P ) is an isomorphism p i ’ , and define l : F ( Z )-, [ X , Y ] by l(z) = k;. Then c’(1) = F(k). By the definition of Y x , there exists an A-morphism m : Z + Y x such that F(m) = 1 and c(m) = k . (6)*(7): For any A-object Y, let G ( Y ) = Y x and q y = d,.y. Then the condition is satisfied. (7)*(1): Let D : K + A be a diagram in A over a small category K and ( Y , (ai :D ( i ) + Y ) ) be a colimit for D. Then (H( Y), ( H ( a i ) )is a colimit for H O D .This can be shown in a similar way as in the proof of Proposition 7.1(2). (In fact, this is one of the properties of adjoint functors.) 0 0
13.4. Definition. A category with products of pairs is said to be cartesianclosed if it satisfies condition (7) in Proposition 13.3. 13.5. Proposition. IfA is a Cartesian-closedtopological category, the following hold: (1) X Y X Zx (XY)”; ( 2 ) (nxj.)Y = n(X;.)‘; (3) XUY’x n(x)‘;.; (4) X x (LIY,) x LI(X x Y,). 13.6. Remark. Top is not Cartesian-closed. The full subcategories of Top consisting of all spaces satisfying each condition below are Cartesian-closed topological categories: (1) finitely generated space, (2) sequential space, (3) (compact Hausdorff)-generated space. The following are also Cartesian-closed topological categories: (4) The category Simp consisting of all simplicia1complexes and simplicia1 maps. (5) The category Born consisting of all bornological spaces and bounded maps. The investigations of Cartesian-closed categories were started in order to obtain “convenient categories” for algebraic topology, theory of topological algebra, etc. and were continued by many authors (see Nel [1976]). The category of all (compact Hausdorff)-generated spaces is a Cartesian-closed topological category but it is not closed under finite products in Top. Hence, we cannot call this category convenient. Herrlich [ I983b] showed that there
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is no “sufficiently convenient” subcategory of Top. However, there are several convenient categories containing Top. In this direction we have many nice results (see, e.g., Herrlich and Stretcker [1986]).
References Baron, S. [I9681 Note on epi in To, Canad. Math. Bull. 11, 503-504. Caszir, A [1978] General Topology (Adam Hilger, Bristol). Engelking, R. and S. Mr6wka [I9581 On E-compact spaces, Bull. Acad. Polon. Sci. 6, 429-435. Freyd, P. [I9641 Abelian Categories (Harper 8t Row, New York). Glicksberg, 1. [ 19591 Stone-Cech compactifications of products, Trans. AMS 90, 369-382. Hajek, D. W. and R. G. Wilson [ 19731 The non-simplicity of certain categories of topological spaces, Math. Zeitschr. 131, 357-359. Herrlich, H. (19651 Wann sind alle stetige Abbildungen in Y konstant? Math. Zeitschr. 90, 152-154. [I9681 Topologische Reflexionen und Corejlexionen, Lecture Notes in Mathematics 78 (Springer, Berlin). [I9691 Limit-operators and topological coreflections, Trans. AMS 146, 203-210. [I9711 Categorical topology, General Topology Appl. I , 1-15. [1974a] Topological functors, General Topology Appl. 4, 125-142. [1974b] A concept of nearness, General Topology Appl. 4, 191-212. [I9761 Initial completions, Math. Zeitschr. 150, 101-1 10. [1983a] Categorical Topology 1971-1981, General Topology and its Relations to Modern Analysis and Algebra V (Heldermann Verlag, Berlin) 279-386. [1983b] Are there convenient subcategories of Top? Topology Appl. 15, 263-271. Herrlich, H. and G. E. Strecker [I9791 Category Theory (Heldermann Verlag, Berlin, 2nd ed.). [I9861 Cartesian closed topological hulls as injective hulls, Quest. Math. 9, 263-280. Marny, T. [ 19791 On epireflective subcategories of topological categories, General Topology Appl. 10,
175-181. Mitchell, B. [ 19651 Theory qf Categories (Academic Press, New York). Mrowka, S. [I9681 Further results on E-compact spaces I, Acta Math. 120, 161-185 Nakagawa, R. [I9761 Separation axioms and subcategories of Top, J. Austral. Math. Soc. 22, 476-490.
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[ 19851 Natural relations, separations and connections, Questions Answers General Topology 3, 28-46.
Nel, L. D. [ 19761 Cartesian closed topological categories, in: Lecture Notes in Mathematics 540 (Springer, Berlin) 439451. Nel, L. D. and R.G.Wilson [1972] Epireflections in the category of T,-spaces, Fund. Math. 75, 69-74. Preuss, G. [ 19701 Trennung und Zussammenhang, Monatsch. Math. 74, 70-87. Sharp, R., M. Beattie and J. Marsden [I9661 A universal factorization theorem in topology, Canad. Math. Bull. 9, 201-207. Wallace, A. D. (19411 Separation spaces, Ann. Math. 42, 687-697. Wyler, 0. [1971] Top categories and categorical topology, General Topology Appl. 1, 17-28. [I9731 Quotient maps, General Topology Appl. 3, 149-160.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V.(1989)
CHAPTER 15
TOPOLOGICAL DYNAMICS
Nobuo AOKI Department of Mathematics, Tokyo Metropolitan University, Tokyo, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orbit structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansive behaviours . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansivity and dimension. . . . . . . . . . . . . . . . . . . . . . . . . Pseudo-orbit-tracing property. . . . . . . . . . . . . . . . . . . . . . . . Coordinate systems and topological stability . . . . . . . . . . . . . . . . . Representations of maps with hyperbolic coordinates . . . . . . . . . . . . . Chain components and decompositions . . . . . . . . . . . . . . . . . . . Markov partitions and subshifts . . . . . . . . . . . . . . . . . . . . . . 9. Topological entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I. 2. 3. 4. 5. 6. 7. 8.
625 627 635 658 667 680 687 695 707 727 736
Introduction
The theory of dynamics begins with Newton who formulated the law of motion and who stated the universal law of gravitation. In Newton’s theory the motion of a dynamical system is governed by a system of differential equations. The problem of applying the theory to planetary motion was challenging, and people might even succeed in explicitly integrating the differential equations, as Newton did in the case of the two-body problem. However, their hope did not materialize. In the late nineteenth century, Poincari created a new branch of mathematics by publishing his famous memoir (“Methodes nouvelles de la mkcanique celeste”) on the qualitative theory of differential equations. The major ingredient of the change was to relate the geometry of the phase space with
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the analysis. As a result of PoincarC's qualitative approach, the focal point in the dynamical theory shifted away from the differential equations. In Birkhoff's treatise on dynamical systems, he discussed many dynamical phenomena in the context of transformation groups acting on general metric spaces. Since then, many mathematicians have considered this theory in view of the geometrical theory. The survey presented in this chapter is written in view of the topological theory of dynamical systems, and is meant to be introductory and, therefore, to give the reader an understanding of some of the basic ideas involved in geometrical (differential) dynamics studied by Smale, Bowen and many other mathematicians. The only requisite for reading this chapter is a basic course on General Topology, but the relevant results on differential manifolds are mentioned briefly. The central theme of the chapter is the theory of dynamics in General Topology. Section 1 explains dynamical phenomena known as minimality, topological transitivity and distal property of homeomorphisms of compact metric spaces. In Section 2, we state the existence of spaces admitting expansive homeomorphisms and of metrics taking hold of expansive behaviours, the so-called hyperbolic metrics. In Section 3, we discuss that every compact metric space admitting expansive homeomorphisms is finite(-topological)-dimensional and every minimal subset of the space is zero-dimensional. Section 4 contains fundamental properties of the pseudo-orbit-tracing property. Here we follow the rule that the distal property rejects the pseudoorbit-tracing property whenever the space is connected, but such a phenomenon can not happen for totally disconnected spaces. By making use of local (unstable) stable sets and (unstable) stable sets, we introduce in Section 5 the notions of canonical coordinate systems, hyperbolic coordinate systems and local product structures which play an important role in the discussion of dynamical phenomena of continuous maps. We follow here the relation between hyperbolic coordinate systems and topological stability. Section 6 deals with the fact that every continuous map with hyperbolic coordinate systems of tori is topologically conjugate to a toral endomorphism. This topic has recently been proved for homeomorphisms, but we do not touch upon this theme here. In Section 7, we investigate the relation between recurrent sets, chainrecurrent sets and nonwandering sets, and state that the Smale decomposition
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621
theorem (spectral decomposition theorem) and the Bowen decomposition theorem are established for the case when a hyperbolic coordinate system holds on the chain-recurrent set. In Section 8 we discuss Markov partitions, which play an important role in the study of the theory of dynamical systems. A necessary and sufficient condition for expansive homeomorphisms to have Markov partitions and the fact that there are no new expansive homeomorphisms except pseudo-Anosov homeomorphisms of compact surfaces are briefly mentioned. Section 9 introduces topological entropy and briefly states some related topics. The author is most grateful to Professor K. Shiraiwa who encouraged him to present these introductory ideas on topological dynamics.
1. Orbit structures Let X be a compact metric space and f:X morphism.
-+
X be a (surjective) homeo-
1.1. Definition. A homeomorphism f is minimal if for every x E X , Of(x) = {f”(x):n E Z} is dense in X. Of(x) is called theforbit of x. 1.2. Remark. f i s minimal if and only iff(E) = E and E is closed, then E = 8 or X . Proof. Suppose f is minimal and suppose E is closed and f(E) = E # 8. Choose x E E. Then Of(x) c E since f ( E ) = E, and X = O,(x), hence I!? = X , i.e., E = X. Conversely, for every x E X , o/ois a closedfinvariant nonempty set and hence O,(x) = X. 1.3. Definition. A closed subset E of X which is $invariant is called a minimal set with respect to f : X -+ X iff;, is minimal. 1.4. Remark, Every homeomorphismf: X
-+
X has a minimal set.
Proof. Let 0 denote the collection of all closed nonemptyfinvariant subsets of X. Clearly 0 # 8 since X E 0. 0 is a partially ordered set under inclusion. Every linearly ordered subset of 0 has a least element. By Zorn’s
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lemma 0 has a minimum element. This element is a minimal set with respect to f.
1.5. Remark. Let f : X + X be a minimal homeomorphism. Then (1) f has no nonconstant invariant continuous functions; (2) if X is not finite thenfhas no periodic points. 1.6. Definition. A homeomorphism f : X + X is topologically transitive if there is xoE X such that O,(x,) is dense in X. It is clear that iff is minimal then f is topologically transitive.
1.7. Theorem. The following assertions are equivalent: (1) f is topologically transitive. ( 2 ) If U, V are nonempty open sets then there is an n E h such that f " ( U ) n V z 8. Proof. (1)*(2): Suppose O,(xo) = X and U, V are non-empty open sets. Then there are n, m E E such that f " ( x o )E U and f " ( x o ) E V , and so f " - " ( u ) n V # 8. (2)*(1): Let { V,: j 2 I } be a countable base for X. For x E X , the following areequivalent: (a) O,b) z X , (b) O,(x) n U, = 8 for some n, (c) f "(x) E X\ U, for every m E Z, (d) x E f"(X\U,) for some n, (e) x E un"=,nnmm=-mfv\u,). Since -,f"(U,)is dense by (2), (7,"- -,f"(X\U,,)is nowhere dense for every n I . Hence { x E X : O,(x) # X} is a set of first category since so is a compact metric space. 0
nz=-,
u,"=
1.8. Definition. Let f : X --* X , g : Y + Y be homeomorphisms of compact metric spaces. Thenfis said to be topologically conjugate to g if there exists a (surjective) homeomorphism cp : X + Y such that cp 0f = go cp. If, in particular, cp is not 1-1, then f is said to be topologically semi-conjugate to g ; in other words, g is called afactor off. 1.9. Remark. ( I ) Topological conjugacy is an equivalence relation. (2) Iff and g are topologically conjugate then f is minimal if and only if g is minimal, and f is topologically transitive if and only if g is so.
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1.10. Theorem (Halmos and von Newman [1942]). Let f : X --* X be a homeomorphism of a compact metric space. I f f is topologically transitive and an isometry under some metric for X , then f is topologically conjugate to a minimal rotation on a compact abelian metric group.
Proof. Let d be an isometry metric for X. Suppose O f ( x o )= X for some xo E X . Define a multiplication * in O f ( x o )byf"(xo) * f " ( x o ) = f"+"(xo). Then d(f"(x0) *f"(xo),f'(xo) * f 9 ( x o ) ) = d ( f " + " ( x 0 ) , f p + 9 ( ~ o ) )
< d(f"+"(xo),f '+"(xo))+ d(f'+"(xo),f p + 9 ( ~ 0 ) ) = d ( f " ( x ~ ~) ,' ( X O ) + ) d ( f ' " ( ~ f0 ' )( ~~ 0 ) ) ~ from which the map * : O f ( x o )x Of(xo)+ Of(xo)is uniformly continuous and can therefore be extended uniquely to a continuous map * : X x X + X . Since d ( f -"(xd,f-"(x0)) = d ( f " ( x o ) , f " ( x o ) )the , map: Of(xo)-, Of(xo) (inverse) is uniformly continuous and can be uniquely extended to a continuous map of X . Thus Xis a topological group and is abelian since { f " ( x o:)n E Z} is a dense abelian group of X . Since f ( f " x o )= f " + ' ( x 0 = ) f ( x o )* f " ( x o ) , we have f ( x ) = f ( x o )* x and so f is the rotation by f (xo). 0 The book of Furstenberg [ 198I] systematically investigates the combinatorial number theory on a situation of dynamical systems. As an application of minimality, we have the following theorem. 1.11. Theorem (Shur and Brauer). Let N be the set of natural numbers. For any finite partition N = So v S, v * . * v Sk-,and n >, 1 there exist Si, a 2 0 and b > 0 such that {a, a b, . . . , a nb} c Si.
+
+
We give a proof due to Furstenberg and Weiss [1978]. The theorem follows from the following proposition. 1.12. Proposition. Let X be a compact metric space and let n 2 1. Suppose + X (1 < i < n) are homeomorphisms with J Q ~ =;4.f; for I < i, j < n. Then there are for every E > 0, x E X and b 2 1 such that d ( J b ( x ) x, ) < E for 1 < i < n.
I;: X
For the moment, we will assume this proposition to be true and give the proof later. Once Proposition I . 12 is established, Theorem 1.1 1 can be obtained after the following preparation.
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630
1.13. Definition. Let k be a fixed natural number and let Y, = (0, 1, . . . , - l}. Put the discrete topology on Y, and construct the product space Y," = X Tm Y, equipped with the product topology, and the shift (homeomorphism) a: Y t -+ Y t is defined by (a(x)), = xi+, where x = (xi)"".
k
Let k be the number of elements of the partition {Si} into N and let Y, = (0, I , . . . , k - l}.Givenm > 1 weputx,,, = j i f m ~qandx,,, = 0 if m < 0. Then xo = (. . . , x - ~ x, o , x I , . . .) E .Y: Since Y," is compact, X = n ~ = l { a ' ( x o ) :2 j m} is nonempty and a-invariant. Write ai= ai for 1 < i < n and use Proposition 1.12 for ai (1 < i < n). Then for E > 0 there are y E X and b 2 1 with d(a:(y), y ) -= E (1 < i < n), and hence Yib = aib(y)o = yo for 1 < i < n. Since y E X,we can choose a 2 0 such that the distance d(a"(xo), y) is small enough, from which
X:+nb = (a"Xo)nb = ynb,
Since yo = yb = * = ynb, we have {a, a + 6, . . . , a + nb} t Syo. For the proof of Proposition 1.12 we need the notion of minimality for a group of homeomorphisms. 1
1.14. Definition. Let G be a group consisting of homeomorphisms of a compact metric space X.The group G is said to be minimal if no proper closed subset Y c X is left invariant by all the homeomorphisms of G (i.e., if g(Y) c Y for all g E G then Y = X or Y = 8). 1.15. Remark. If G is minimal then each G-orbit {g(x) :g E G } is dense in
X,and conversely, this condition implies minimality for G. 1.16. Definition. Letf: X + X be a homeomorphism of a compact metric space and G as in Definition 1.14. A closed subset Y is said to be homogeneous withrespecttofifg(Y) = Y f o r a l l g ~ G ( G Y= Y)andgof = fogforall g E G and Y is minimal for G. 1.17. Lemma. Let Y be a closed subset of X and let GY = Y. If Y is minimal for G, there exists for every E > 0 a Jinite set of homeomorphisms g,, . . . , g, E G such that for x , y E Y, min{d(gi(x), y): 1 < i < N } < E.
Proof. If Vis any nonempty open subset of Y, then U{g-'( V ):gE G } is an open G-invariant subset of Y, which by minimality is all of Y. Since Y is
63 1
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compact, there are g , , . . . ,g , E G such that U L l g ; ' ( V ) = Y. Now let { & } be a finite open cover of Y with diameter < E . We see that UiN_'lg;l(K)= Y for some N' 2 1. Therefore, for x , y E Y we can find k, such that y E V, ,and 1 < i < N' such that x E g ; l ( K ) , and hence y, g i ( x ) E K. 0 1.18. Lemma. Let Y be a homogeneous closed subset of X.Suppose that for every E > 0 there is an n 3 1 with d ( f " ( Y ) , Y ) < E. Then for every E > 0 there exist n 2 1 and x E Y such that d ( f " ( x ) , x ) < E.
Proof. We first prove that for each E > 0 and y E Y there are n 2 1 and x E Y with d ( f " ( x ) ,y ) < E . Since Y is minimal for G, by Lemma 1.17 there are g , , . . . , g , E G such that min{d(g,(x), y ) : 1 < i < N} < + E for x, y E Y . Now choose x,, yo and no such that d(f"O(x0),yo) < 6 where 6 is so small that d(x, x') c 6 implies d(gi(x),g,(x')) < $ E for 1 < i < N. Then for any y E Y we can find 1 < i < N such that d(f"Oogi(x0),y ) < 0 d(f"oog;(xo), g;(yo)) + d(g;(yo),Y ) < t E + $ 8 = E. To obtain the conclusion of Lemma 1.18, let E > 0. From the above it , < E. follows that for zo E Y there are z , E Y and n, 3 1 with d ( f " ' ( z , )zo) Put rll = E - d ( f " ' ( z , ) 20) , and take E~ > 0 with
E~
+ qI <
4x9 Y ) <
E2
(x, y
E
E
such that
Y ) =-d ( f " ' ( x ) , f " ' ( y ) < ) 'I,.
Then for zI E Y we can find z2 E Y and n2 2 1 with d ( f n 2 ( z 2z) ,I ) < .z2 (use the above fact), and then d( f "I +"2 @ 2 ) , 20) < d(fn'+nz(Z2),f"'(ZI)) + d(f"'(z,),20) < E . Next we put ?2
+
- [ d ( f " ' + " Z ( z 2 ) , f " ' ( z , )d(f"'(z,), ) Zo)l,
= E2
and take
E~
> 0 with
E~
-
z,)} such that
d(fn2(Z2)9
+ q2 < E
N.Aoki
632
and d(fn2+"'(Z3),ZI)
< d(f"2+n3(Z3),f"2(Z*))+ d(f"*(Z,), ZI) < E .
Continue inductively in this fashion, then infinite sequence {zo,zI, . . .} of points of Y and { n l ,n,, . . .} of positive integers are chosen such that d(f"(zj), zi) < E where n = ni+l + . * + nj for 0 < i < j . On the other hand, since Y is compact, we have d(z,, z j ) < E for some i, j. Therefore, 4%Zj) < 28. d(f"(Zj), Zj) < d(f"(Zj), Zi)
+
Proof of Proposition 1.12. Let G be a group generated by a finite set of homeomorphismsf,, . . . ,f, of X.Since Xis compact, we can find, by using Zorn's Lemma, a minimal subset X' for G. Hereafter we may suppose that Xis minimal for G. If m = 1 then G = (fi":n E Z} and Xis minimal for G. For this case we have X = {f;(x): n E Z} for x E X, and hence d(f,*(x), x) < E for some b 2 1. Let A be the diagonal set of X" and define (1 < j < m) by 3 = f, x x fm and homeomorphisms 3 and = f ; x . x f ; . Obviously3(A) = A andx(A) = A for 1 < j < m. Let c" be a group generated by the set {J,. . . ,Tm}.It is easily checked then that (A, G") is homeomorphic to (X,G). Thus A is minimal for .?t Since f ; o f = y0f;for 1 < i < m, A is homogeneous with respect toJ Let hi = f;' for 1 < i < m - 1. By our inductive hypothesis we have that there are x E X and b 2 1 such that
x
x
-
-
d(hP(x), x) = d(Xbofib(x),x) < For X
=
(xl, . . . , x,,,), J = ( y I , . . . , y , )
d ( ~J ,)
=
max{d(x,, y i ) :1
E
E,
1
- 1.
X" put
< i < m}.
Then d is a metric for X" and d(Tb(2), 5 ) < E for 5 = (x, . . . , x), P = (f;*(x), . . . ,fl*(x)) E A. Henced(p(A), A) < E. Using Lemma 1.18 we see that there are n 2 1 and X E A with d(p(X), 3 ) < E, and therefore d(f;"(x), x) < E for 1 < i < m. 0 1.19. Definition. Let f :X -, X be a homeomorphism of a compact metric space. We say that f is distal if inf{d(f"(x),f"( y ) ) :n E Z} = 0 then x = y . Let A be a subset of the integer group Z. Then A is syndetic if there is a finite set K of Z with Z = K A. Let x E X.Then x is an almost periodic point if {n E Z :f"(x) E U } is a syndetic set for all neighborhood U of x. We say that f is uniformly almost periodic if for every E 0 there is a syndetic set A such thatf"(x) E U,(x) for all x E Xand n E A . Here U,(x) denotes an open neighborhood with radius E of x.
+
=-
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1.20. Remark. Products and factors of distal homeomorphisms are distal. Every equi-continuous homeomorphism is distal (see the proof of Theorem 1.21 below). But the converse does not hold. For example, let T2 be the 2-torus and define a homeomorphism f:T2+ T2 by f(x,, x2) = (a x,, nxl x2) ((xl, x2) E T2)where ct E TI and 0 # n E Z. It is easily checked that f is distal, but {f":n E Z} is not equi-continuous.
+
+
1.21. Theorem.
Zf f:X
.+
X is uniformly almost periodic, then f is distal.
Proof. We first prove that iffis uniformly almost periodic, then { f" : n E Z} is equi-continuous. By the assumption, for every E > 0 there is a syndetic set A c Z such thatf"(x) E U,,,(x) for all x E X and n E A. Since A is syndetic, we can find a finite set K = {nl, . . . , n I } with Z = A + K. Take 0 < 6 < $ E such that d(x, y) < 6
* max{d(f"'(x),f"'(y)):
1
< i < I}
<
$E.
If d(x, y) < 6, for each n E A and n, E K we have y ) ) < d(f" of"'(x), f"'(x)) d(f"+"'(x), f"+"'(
+ d ( f " ' ( f" A
9
<$&+$&+$&
of"'(
=
+ d(f"'(x), f"'(y ) ) Y)) &
and so {f" :n E Z} is equicontinuous. Suppose inf{d(f"(x),f"( y)) :n E Z} = 0. For j > 0 there is n, E Z with d(f"J(x),f"'( y)) < l/j. Thus to each v > 0 there are p > 0 and J > 0 such thatj 2 Jimplies d(f"(x), fnJ(y)) < p and d(x, y) < p implies sup{d(f"(x),f"(y)):n E Z} < v, and hence d(f"'of''(x), f - " ' o f " ' ( y ) ) = d(x, y) < v. Since v is arbitrary, we have d(x, y) = 0 and then x = y. 0
1.22. Definition. Let g : X + X be a continuous map of a compact metric space. Then (g(x):x E X) will be considered to be a point of the compact Hausdorff space X x . An Ellis semigroup E ( X ) of a homeomorphism f:X + X is the closure of { f" :n E Z} in X x . A nonempty subset I of E ( X ) is called a right ideal if g 0 h E I for every g E E ( X ) and h E I. 1.23. Theorem. point.
Zff:
X
.+
X is distal, then each x
E
X is an almost periodic
Proof. We prove the following three claims to obtain the conclusion. Claim 1. Let I c E ( X ) and I # 8.If Z is a right ideal of E ( X ) , then there is u E I such that u2 = u.
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N. Aoki
Proof. Define a map R, : E ( X ) -+ E ( X ) by R,(h) = h 0 g for g, h E E ( X ) . Then R, is continuous. For, if a directed sequence {h,} of E ( X ) converges to h E E ( X ) , then h,(gx) -+ h(gx) for every x E X . Hence &(ha) -+ RJh). Let A denote the family of closed non-empty subsets S c Z with So S c S. Then A is the partially ordered set under the inclusion and is inductive. Zorn's lemma guarantees the existence of a minimal element S of A. If g is in S, then Sog = R,(S) # 0 and (Sog)o(Sog) c S o S o S o g c Sog. Since S is minimal, we have Sog = S and then there is an h E S with h o g = g. Obviously R ; ' ( S ) n S # 0. Since J o g = h o g = g for J , fi E R;'(g) n S, we have (fi o h ) 0 g = fi o g = g and soh oh E R; '(g) n S. Therefore (R;'(g) n S)o(R;'(g) n S ) c R;'(g) n S, and by minimality R;'(g) n S = S. This implies g2 = g. Claim 2. Let M be a closed subset of E ( X ) . Iff" 0 g E M for g E M and n E Z, then M = E ( X ) . Proof. Since E ( X ) 0 M c M, by Claim 1 there is a g E M with g2 = g. Put y = g ( x ) . Then g(x) = g(y). Since g E E ( X ) , we have f"'-+ g for some { f"}and
Since g(x) = g(y), obviously d ( f " ' ( x ) , f " ' ( y ) ) 0 as i 00, and hence inf{d(f"(x),f"(y)):m E Z} = 0. Sincef:X + X i s distal, we have x = y and then g is the identity since x is arbitrary: i.e., f" 0 id = f" E M for all n E Z. Therefore M = E ( X ) . Claim 3. From now on the theorem is proved. Note that the identity id is contained in E ( X ) . Let U be an open neighborhood of id in E ( X ) . We first show that there is a syndetic set A c Z such thatf" E U for all n E A. Fix h E E(X) and define a map Lh :E ( X ) -+ E ( X ) by Lh(g)= h o g (g E E(X)). It is easily checked that if h is continuous then so is Lh. Hence L,.. is continuous since so is f" for n E Z, and L, 0 L, = Lr- Lr = id. Hence L,.is injective. Since Lr is continuous, Lr ( U ) = f" 0 U is open in E ( X ) and so W = E(X)\(U?mf"o U ) is closed in E ( X ) . Suppose W # 0. Then we have f"0g E W for g E W and n E Z. Hence W = E ( X ) by Claim 2, thus contradicting W # E ( X ) . Therefore W = 0, from which E ( X ) c UTm,f"o U.Since E ( X ) is compact, we have E ( X ) c U ~ J " o U for some finite set K = {nl, . . . , n,}. Let A = {n E Z : f " E U}. Then it follows that there is n, E K withf" E f " i o U for all n E Z. Hencef"-"' E U : i.e., n - niE A and hence n E A -I- K. Since n is arbitrary, we have Z = K A and then A is the syndetic set. We next prove that each x E X is an almost periodic point. To do this, -+
-+
0
+
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635
define a continuous map Ox:E ( X ) + X by OJg) = g(x) (g E E ( X ) ) . Let V be a neighborhood of x in X. Then there is a neighborhood U of id in E ( X ) such that O x ( U ) c V. Replacing this U by the above U , for n E A f"(x)
= f"oU,(id)
= U,(f"oid)
= U,(f")
E
U,(U) c V
and so x is the almost periodic point off.
1.24. Theorem. Iff: X set for f.
0 -
+
X i s distal, then for each x E X , Of(x) is a minimal
Proof. By Theorem 1.23 each x E Xis an almost periodic point for$ Hence, for U an open neighborhood of x , letting A = {n E Z : f " ( x ) E U}, Z is expressed as Z = A + K for some finite set K = { n , , . . . , n k } . Hence
u {f " + " ' ( x ) : n k
O,<X) =
E
A } u { f " ( x ) : nE A }
i= I
-
from which we have 0,( y) n 0 # 0 for every y E O,(x). This follows from the fact that f y) E { f " ( x ) :n E A } c 0 since y E { f "+"'(x):nE A } for n,. Since U is arbitrary, we have x E Of(y) and so O f ( ~= ) Of( y), i.e., some 0, ( x ) is a minimal set for f. 0 -"I(
1.25. Definition. Letf: X + X be a homeomorphism of a compact metric space. We say that f is semisimple if there exists a sequence {E,) of closed subsets such that f(E,)= E,, X = uBEaand each E, is a minimal set forf. Since each E, is minimal, it is clear that E, n E, = 8 if ct # B. 1.26. Theorem. I f f : X Proof.
--*
X
is distal, then f is semisimple.
-
{o/o: x E X } is a cover of X and each O,(x) is minimal by Theorem 0
1.24.
2. Expansive behaviours Let X be a metric space with metric d and f :X homeomorphism.
+
X be a (surjective)
2.1. Definition. A homeomorphism f is said to be expansive if there exist a metric d for X (d means compatible with the topology of X ) and a number
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636
c > 0 such that x # y implies the existence of an integer n such that d ( f " ( x ) ,f"(y ) ) > c. The number c is called an expansive constant for$ This property (although not c ) is independent of the choice of metrics for X when X is compact, but not so for noncompact spaces.
2.2. Definition. Let X be a compact metric space. A finite open cover a of X is a generator (weak generator) for f if for every bisequence {A,} of f -"(A,) is at most one point (0;- -f, -"(A,) is at most members of a, one point). These concepts are due to Keynes and Robertson [1969].
n;=-,
2.3. Definition. If a, /?are open covers of X , then their join a v /3 is given by a v B = { A n B : A E a, B E S}. An open cover fi is a refinement of an open cover a (written a < B ) if every member of B is a subset of a member ofa.Itisclearthata < a v BandB < a v /?.Iff:X+Xisacontinuous map then f - ' ( a ) = { f - ' ( A ) : A E a } is an open cover of X . It is easily checked that f - ' ( a v /?) = f -'(a) v f -'(fl)and f - ' ( a ) < f - ' ( p ) ifa < /?. 2.4. Theorem. Let f :X + X be a homeomorphism of a compact metric space. Then the following are equivalent: (1) f is expansive, (2) f has a generator, (3) f has a weak generator. Proof. (2)=(3) is clear. (3)*(2): Let /? = {Bl, . . . , B,} be a weak generator for f and let 6 > 0 be a Lebesgue number for b. Let a be a finite open cover by sets A, with diam(A,) < 6. If A,"is a bisequence in a, then for every n there isj, such that A, c Bjn,and so 0: f -"(A,,) c f -"(Bjn).Therefore a is a generator. (1)*(2): Let 6 > 0 be an expansive constant for f and a be a finite cover by open balls of radius $ 6 . Suppose x, y E n'?,,f-"(A,) where A, E a. Then d ( f " ( x ) , f " ( y ) )< 6 for every n , and by assumption x = y. (3)=4 1): Suppose a is a weak generator. Let 6 > 0 be a Lebesgue number for a. If d(f"(x),f"( y)) c 6 for all n, then for every n there is A , E a such thatf"(x),f"( y) E A, and so x, y E f -"(A,) which is at most one point. Therefore x = y. 0
n?m
n?,
2.5. Remark. Let f : X + X be a homeomorphism of a compact metric space. f is expansive if and only if f k (k # 0) is expansive.
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Topological Dynamics
Proof. If a is a generator for f then a v f -'(a) v . . . v f -(,,-')(a) is a generator for f".If a is a generator for f " then u is also a generator forf.
0 2.6. Remark. (1) Iff : X + Xis expansive and Y is a closed subset of X with f (Y) = Y , then is expansive. (2) If fi:: X, + X, (i = 1, 2) are expansive then so is fi x fi :XI x X , + XI x X,. Every finite directed product of expansive homeomorphisms is expansive, but an infinite product is not.
2.7. Remark. Let k 2 2. Then the shift a : Y,"
+
Y," is expansive.
Proof. Let Ai = {(x,,)'?, :xo = i } for 0 < i < k - 1. Then A, u . . . u A".-' = Y," and each A i is open. a = { A o , . . . , A " - ' } is a generator for IT . . since if x E n'?,,o-"(Ain) where A, E a then x = (. . . , iL2, i - , , l o , i I r . . .).
0 2.8. Theorem. Let f : X + X be an expansive homeomorphism of a compact metric space. Then there exists an integer k > 0 and a closed subset R of Y f such that a(R) = R (a is the shijt of Y,")anda continuous surjection K :R + X such that n o 6 = f o x .
Proof. Let 26 > 0 be an expansive constant forf. Choose a cover c1 = { A o , . . . , A " - , } by closed sets such that diam(Ai) c 6 for 0 < i < k - 1, and such that Ai intersect only in their boundaries. Let 8 denote the union of the boundaries of A i . Then 8, = U'?J'(8) is a set of first category and so X\8, is denselin X. For every x E X\8,, we can assign uniquely a member of Y f by x (a,)'?wwheneverf.(x) E Aan.Let A denote the collection of all sequences arising in this way. Then cp 0 a( y ) = f 0 cp( y) for y E A. If we hold that cp is uniformly continuous, then cp can be uniquely extended to a continuous map n : A + X such that n 0 IT(y ) = f n( y) for y E A. Let E > 0 be given. By the following lemma we can choose N > 0 such that each member of V -NNf"(a)has diameter less than E since we can enlarge each Ai to an open set to obtain a generator (remembering diam(Ai) < 6). If (a,,),(b,) E A and a,, = b, for I n I < N then cp(a,), cp(b,,) are in the same member of V NNf"(a)and so d(cp(a,), cp(b,)) < E . Therefore cp is uniformly continuous. 0 0
2.9. Lemma.
Let a be a generator for$ Then the following hold.
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( I ) For every E > 0 there is N > 0 such that each set in V !N f -"(a) has diameter < E . ( 2 ) For every N > 0 there is E > 0 such that d ( x , y ) < E implies x , y E n f N f-"(A,) for some A-N, . . . , AN E a. Proof. Suppose (1) does not hold, i.e., there is E > 0 such that for every
j > 0 there are xi, yj E X with d(xj, yi) > E and Aj,i E a ( - j < i < j ) with xi, y j E - j f -i(Aj,i). Since X is compact, we can suppose that xi + x , yj --* y, and hence x # y . Consider the sets Then infinitely many of them coincide since a is finite. Thus xi, y j E A , for infinitely manyj and hence x , y E Ao. Similarly, for each n, infinitely many Aj,, coincide and we obtain
n/=
n?,
A, E a withx,y E f -"(A,).Thereforex,y E f -"(A,), thuscontradicting the fact that a is a generator. (2): Let N > 0 be given. Let 6 > 0 be a Lebesgue number for a. Choose E > 0 such that d ( x , y ) < E implies d ( f ' ( x ) , f ' ( y ) ) < 6 for - N < i < N . If d ( x , y ) < E and I i 1 < N then f '(x)$( y ) E A i for some Ai E a. Therefore O x, y E n ! N f -yAi).
2.10. Definition. I f f : X + X is a homeomorphism of a compact metric space and x is a point of X , the a-limit set of x denoted by a ( x ) consists of those points y of X such that y = limj+, f "'(x) for some strictly decreasing sequence of integers nj. The w-limit set of x denoted by w ( x ) is similarly defined for strictly increasing sequence. If, for some point x , a ( x ) and w ( x ) each consists of a single point, we say that x has converging semiorbits under f. 2.11. Theorem (Reddy [1965]). Iff is an expansive homeomorphism, the set of points having converging semiorbits under f is a countable set. Proof. Let c > 0 be an expansive constant forf and let d ( x , y ) < +c. Since f is expansive,fhas at most finitely many fixed points, say q , , . . . , qk. Let A denote the set of points having converging semiorbits under$ Suppose A is uncountable. If x E A, both a ( x ) and w(x) are fixed points. Let A(i,j ) be the set of points such that a ( x ) = qi and w ( x ) = qj. Then A is the union of the finitely many sets A(i,j ) , so that one of these, say B, is uncountable. For each positive integer N , let B ( N ) be the collection of points x such that, for n 2 N , f " ( x ) is close to o(x) and f -"(x) is close to a ( x ) . Since B is the union of the sets B ( N ) , one of them (say B ( M ) )must be infinite. Since Xis compact, there exist distinct points y and z of B ( M ) with d( y , z) < ) c so small that f " ( z ) is close to f"( y ) if In 1 2 M . Thus, from the definition of B ( M ) , we
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conclude that d ( f ” (y), f ” ( z ) ) < c for all integers n, thus contradicting the choice of c. Therefore A is countable. 0
2.12. Theorem (Jacobsen and Utz [ 19601). There exists no expansive homeomorphism of a closed arc.
Proof. I f f is a homeomorphism of [0, I] onto [0, I], then f has either f(0) = 0 and f(1) = 1, or f(0) = 1 and f(1) = 0. In both cases f 2 induces a homeomorphism of [0, I]. Put Fix( f 2, = { x E [0, 13 : f 2 ( x )= x}. Then Fix( f ’) is closed. If Fix( f 2 , = [0, I] then all points in [0, I] have converging semiorbits under f2,and so f 2 is not expansive. If Fix(f2) # [0, I], then U = [0, ]]\Fix( f ’) is a nonempty open set. Thus U is the union of the countable intervals 4 where 4’s are mutually disjoint. For x E U , there is an 4 such that x E 4. Then the extremal points of 1;. are fixed points off Thus we have that for x E 4, x > f 2 ( x ) > f 4 ( x ) > * . , or x < f 2 ( x ) < f4(x) < * * . In any case { f ”(x) :j 2 0} converges to a fixed point off *.Since 4 is uncountable, f is not expansive. 0
’.
3
2.13. Theorem. There exist no expansive homeomorphism of the unit circle S‘ . Proof. Supposef : S’ + S’is expansive. By Theorem 2.12,f has no periodic points in S ’ . Let M be a minimal set off in S ’ . Sincef : M + M is expansive and minimal, we see that M is totally disconnected (see Theorem 3.8 for the proof). Note that M is an infinite set. We claim that isolated points are not = { x } for some in M. Indeed, if x E M is an isolated point, then U,(x) n M 6 > 0 and so { x } is open in M. Since M is minimal, we have O f ( x ) = M and hencef”(x) E U,(x) n M = { x } for some n E Z,a contradiction. Since S’\M is open, S’\M is expressed as a countable disjoint union S’\ M = Uj4 of open arcs in S ’ . Thus diam(4) + 0 a s j + co and for fixed j,f”(4) # 4 for all n # 0. Thus we have diam(f”(4)) + 0 as In I + 00. For a, b E 4, let f”’(z) be the arc with the longest length in :n E Z}. Choose E > 0 and take a, b E 4 such that the length between f “‘(a)and f “‘(b) is less than E . Then diam(f”(z)) < E for all n E Z and hence f :S ’ -,S’ is not expansive. This is inconsistent with assumption. 0
{f”(z)
Recently the following has been proved by Hiraide [1988].
2.14. Theorem. There exists no expansive homeomorphism of the 2-sphere S2.
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By Theorem 2.13 the 1-torus does not admit expansive homeomorphisms, but any torus of dimension greater than one admits such homeomorphisms (see Theorem 2.18 below). 2.15. Proposition. Let f be a linear map of the euclidean space R" (n > 1) and d be the usual metric for R". Then f is expansive under d if and only iff has no eigenvalues of modulus 1.
Proof. Suppose all the eigenvalues off are off the unit circle. Then (w" splits into the direct sum R" = Es @ E" of subspaces E" and E" such that f(Es) = E" and f(E") = E", and such that there are c > 1 , 0 < 1 < 1 so that
IIf
-"(v) II
< c1" II v 11,
v E E",
n 2 0.
Thus f is expansive under d. Conversely, iff has an eigenvalue of modulus 1, then R" splits into the direct sum of subspaces as follows: (i) R" = E' @ E" 0 E", (ii) f(E") = E" (a = c, s, u), (iii) there are c > 0 and 0 < 1 < 1 such that (*) holds, and (iv) J E c is topologically conjugate to a linear map corresponding to Jordan form J (in real field)
where either f l
1
0
J. =
or 0
for 1
0
4
=
1
Z
fl
R(ej)
< j < k. Here I is a 2 x 2 identity matrix and R(ej)
cos 0,
-sin 0,
sin 0,
cos 0,
=
(0 <
e,
<
-.I.
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641
By (iv) we can find a subspace F of E' such that g l , is an isometry under d. Thus g is not expansive.
a
2.16. Definition. Let 8 and X' be metric spaces with metrics and d respectively, and A :8 + X be a continuous surjection. Then A is called a locally isometric covering map if for each x E X there exists a neighborhood U ( x )of x such that z - ' ( V ( x ) )= U, U, (a # a' * U, n U,. = 0)where U, is open and TI,, : U, + V ( x )is an isometry. 2.17. Proposition. With the notations of Definition 2.16, let f : 8 + 8 and g : X + X be homeomorphisms. If A 0f = g 0 A and X is compact, and i f there exists 6, > 0 such that to each x and 0 < 6 < 6, there is a connected open neighborhood U,(x) of x with radius 6 such that A : U,(x) + U,(AX)is an isometry, then g : X + X is expansive if and only i f f :8 + 8 is expansive under a.
Proof. From the conditions given for X and A, it follows that for eachp E 8 and 0 < 6 < 6, there is a connected open neighborhood U,(p) with radius 6 such that A ~ ~ , (U,(p) ~ , : + U,(np) is an isometry. We first show that f is uniformly continuous. It is enough to see that for E with 0 < E < 6, there is 6 > 0 such that
4 P , 4) < 6
(PI
4 E 2)* J ( f ( P ) , f ( d ) <
E.
Take 6, > 0 such that d ( x , y ) < 6 , ( x , y E X ) implies d(g(x), g ( y ) ) < E and let 6 = min{b,, S,}. Now if d ( p , q) < 6 ( p , q E 8), then d ( x ( p ) , = &P, 4 ) < 6, and s o d ( a o f ( p ) , n o f ( q ) ) = d ( g 0 x ( p ) ,g o m ) < E . Since E < do, we have n o f ( q ) E UJ0(goa ( p ) ) and hence f ( U , , ( p ) ) c n--'U,,(gon(p)). Since f ( U , , ( p ) ) is connected, we let 0 be the connected component of f ( p ) in n-'U,,(gon(p)). Then 0 3 f ( U , , ( p ) ) and A : 0 + U,&g 0 ~ ( p )is)an isometry. Sincef (q) E 0,it follows that f (p),f ( q ) ) = d(a 0f ( p ) , A 0f (4)) < E , which implies that f is uniformly continuous. Next we show that there is > 0 such that for p , q E 8 ( p # q )
m)
a(
4 P ) = n(q) * a ( P I 4 ) 2
El.
Let 6, > 0 be as above. If the statement does not hold, then for n there are pn, q. E 8 such that Pn
+ qn,
~ ( p n )= n(qn) and
J(Pn,
qn) < ~ / n .
For no > 0 with l/n, < do, it follows that a(pno,qno) < 6,. Let x = ~ ( p , , , ) , then A : Ubo(pn0) + Uao(x)is an isometry. Since qnoE U,,,(p,,J and n(qno)= ~ ( p , , , ,= ) x, we have qn0 = p,, thus contradicting.
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We prove that f : 8 + 8 i s expansive whenever g :X + Xis so. Let E,, > 0 be an expansive constant for g and So, E , be as above. Put ~2 = min(S,, E , , ~ g } . Then c2 is an expansive constant for $ For, let p , q E 8 ( p # q) and put n ( p ) = x , n(q) = y . I f x = y then J ( p , q) 2 el 2 c2 as we saw above. When x # y , by expansivity of g there is k e Z such that d(gk(x),d ( Y N 2 t o . I f 4 f k ( P ) , f k ( q N < 60 then
aifk(P),fk(d)
= d ( z o f k ( p ) ,n o f k ( q ) ) = d(gk(x),g k ( y ) )2 Eo
2 E2 and thereforef is expansive. Conversely, let c3 > 0 be an expansive constant for$ Then we prove that E~ = min{s,, E ~ So} , is an expansive constant for g. Let d(gk(x),gk( y ) ) < E~ for all k E Z. Fix p E 8 with x = ~ ( p and ) choose q E U,,(p) with y = n(q). Then it can be easily proved that f k ( q )E Ubo(f k p ) for all k E Z. Since R is a local isometry, a i f k ( P ) , f k ( q ) ) = d(gk(x),sk(Y))<
64,
k E Z.
Since f is expansive, we have p = q and so x = x ( p ) = x(q) implies that g is expansive.
=
y , which
0
Combining Propositions 2.15 and 2.17, we have the following theorem.
2.18. Theorem. Let f be a toral automorphism of the n-torus. Then f is expansive if and only i f f has no eigenvalues of modulus 1. 2.19. Definition. Let X and Y be topological spaces. A map cp :X + Y is open if cp(U) is open in Y whenever U is open in X . The branch set B, of cp is the set of points at which cp is not a local homeomorphism. Obviously B, is closed in X and the compactness of Ximplies the compactness of B,, cp(B,) and cp-'(cpB,). 2.20. Definition. A map cp :X + Y is called a pseudo-covering map if cp is an open map which is a k-to-1 continuous surjection on the set X\rp-'(cpB,) for k > 0. In particular when B, is empty, cp is called a covering map. Let A be a subset. Then we designate the &-neighborhoodof A by N ( A , E ) .
2.21. Theorem (Hemmingsen and Reddy [1968]). Let X and Y be compact metric spaces andf :X -+ X be a homeomorphism. I f g : Y + Y is an expansive homeomorphism and rp is an open continuous map of X onto Y with cp f = g rp, and i f f is expansive on cp-'(cpB,) and there is q > 0 such that O,(y ) N(cp(B,), q) for all y E Y\rp(B,), then f is expansive. 0
+
0
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Proof. Let dand e be the metrics for X and Y respectively. Supposefis not expansive. Then for every E > 0 there corresponds a pair of points {a, b} with 0 < d(f"(a),f"(b)) < E for all integers n. Two cases will be considered: (a) For all sufficiently small E and all corresponding pairs {a, b},
cp(4 z
cpm.
(b) No matter how small E is, there is a corresponding pair {a, b} with cp(4 = db). Case (a): Let c > 0 be an expansive constant for g. Choose a finite open cover { Q } of Xsuch that diam(cpU,) < t c for all i. Let E~ > 0 be a Lebesgue number of the cover {U,}. Then for each n there is i such that {f"(a),f"(b)} c U, where {a, b} is the pair corresponding to E ~ Hence . for all n, e(cp o f " ( a ) , cp of"@)) < $c. Since cp of" = g" 0 cp, it follows that for all n, e(g"cp(a),g"cp(b)) < t c , which contradicts the fact that c is an expansive constant for g. Therefore, go is an expansive constant for pairs of points with distinct cp images. Case (b): Since cp(a) = cp(b),a and b E cp-'(cpE,,,)or a, b E X\cp-'(cpEq). If a, b E cp-'(cpE,,,) thenf is expansive by hypothesis. If a, b E X\cp-'(cpE,,,), then attention will be given to pairs of points lying outside a certain neighborhood of E,,,. Let V, be a neighborhood of the branch point p so small that diam(cp(5))< q. Denote by V the union of all the sets V, for p E E,. If e( y, cp(E,)) > q, then y 4: cp(V). Now we prove that if x and x' are distinct points of the compact set X\cp-'(rpV) such that cp(x) = cp(x'), then there is /3 > 0 with d(x, x ' ) > /3. For otherwise, there is a sequence of pairs {xnrx ; } c X\cp-'(cpV) for which 0 < d(x,, xi) < l/nandcp(x,) = cp(x;)forn 2 1. Withoutlossofgenerality it may be assumed that the sequences { x , } and { x : } converge to the common limit point 2. Then f E X\cp-'(cpV) for which there is no neighborhood on which cp is injective, i.e., 2 E E,,, ,thus contradicting the choice of V . Choose c2 with 0 < E~ < /3. If d(x, x ' ) < t 2 and cp(x) = cp(x'). then { x , x ' } c cp-'(cpV). Next consider the pair {a, b} for which cp(a) = cp(b), i.e., a, b 4 cp-'(cpV) and d(f"(a),f"(b))c c2 for all m. By hypothesis there is n with e(g"0 cp(a), cp(E,)) > q, and so g" 0 cp(a) 4: cp(V). Since ~ ( a =) cp(b), it is clear that g" 0 cp(a) = g" 0 cp(b) and then g" cp(a) = cp o f " ( a ) = cp of"(b) since g o cp = cp of. This implies thatf"(a),f"(b) 4 cp-'(cpV) for this particular n. Therefore d(f"(a),f"(b))> /3 > c2, which contradicts the choice of a and b. Therefore e2 is an expansive constant for pairs of points in X\cp-'(cpE,) that have the same cp image. It follows that E = min{Eo, E ' , E ~ } is an expansive constant for f. 0 0
The following is a corollary of Theorem 2.21.
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2.22. Corollary. Let X and Y be compact metric spaces and cp : X + Y be a covering map. I f g : Y + Y is an expansive homeomorphism andf :X + X is a homeomorphism that covers g (i.e., cp 0f = g 0 cp), then f is expansive. As a result related to Corollary 2.22, we have the following theorem.
2.23. Theorem. Let f : X + X be an expansive homeomorphism and cp : X + Y be a covering map for which there is a homeomorphism g : Y Y with g 0 cp = cp of. Then g is expansive.
Proof. Let c > 0 be an expansive constant for$ From the compactness and the fact that B, = 0,it follows that there is q > 0 such that d(x,, x2) > q whenever rp(xl) = cp(x2)and xI # x2.Since Xis compact and cp is a covering map, there is a k > 0 such that cp is k-to-1 at all points of X. To each sufficiently small open neighborhood V, of y E Y there is a collection of open sets { Uy,i: 1 < i < k} such that cp-'( V,) = Uf=,Uy,iand cp, uv, is a homeomorphism of Uy,ionto V,,. The set V, can be chosen so that the diameter of the sets Uy,iare less than some preassigned number by picking sufficiently small open sets U,', . . . , UL, one around each of the points of q - ' ( y ) , and by taking
instead of with Uy,iand V,. Let the diameter of Uy,ibe so small that IY~,~ and f(Uy,i)andf -'(Uy,i)have diameter less thanmin{c, t q } for ally and i. Choose a finite cover { y : 1 < j < r} of Y from { V,, :y E Y}, and let the corresponding U's be called {q,i: 1 < j < r, 1 < i < k}. Let jl > 0 be a Lebesgue number for each of the covers {Y}, {g(c)} and {g-'(V;)}. Note that ~of(U,,= ~ )g(y)andcpof-'(V,,,) = g-'(y),andthat U,,iand ycouldhave been chosen so small that cp is a homeomorphism of f(U,,i) and f -'(V,,i). If g is not expansive, for every E > 0 there corresponds a pair of distinct points {a, b} c Y with .p(g"(a), g"(b)) < E for all n. Let E c jl. Then for each n, there i s j = j(n) such that {g"(a), g"(b)} c y(,,).Hence for a given n, such that images there is to each i (1 < i < k) a unique pair of points in q(n),i of cp are g"(a) and g"(b). Choose i and denote by p and q the points of q(o),i that map onto a and b respectively: i.e. {p, q } c q(o),i and cp(p) = a, q(q) = b. Since f has the expansive constant c, there is m # 0, least in absolute value, such that f " ( p ) and f "(4) fail to belong to the same element of { U,,i}.The remainder of the proof is carried out under the assumption that rn is positive. Since there is a pair of integers {s, t} such that {f"-'(p), f"-'(q)} c Us,,,
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it follows that { f " ( p ) , f " ( q ) } is inf(U,,,) and d ( f " ( p ) , f " ( q ) ) < ) q . Since e(g"(a), g"(b)) < fl, it follows that there is an index u such that {g"(a), g"(b)} c V, (1 < u < r). Then we choose U,,wE {Uu,i:1 < i < k } such that f"(p) E Uu,wand denote the unique representative of c p - l o cp o f m ( q ) = cp--'og"o(b)in Uu,wby 4. Then d ( f " ( p ) , 4) < t q . From the choice of m it follows that 4 # f"(q). Hence d(cl,f"(q)) < d ( f " ( p ) , f " ( d )
+ d(f"(p), 1) < ?.
But cp o f " ( q ) = g" 0 q ( q ) = g"(b) and this implies that two distinct points of cp-'(g"(b))are closer than q . This contradicts the choice of q and proves that g is expansive. 0 2.24. Remark. Let p : E + B be a continuous map. A vector bundle chart on ( p , E, B ) with domain U and dimension n is a homeomorphism cp:p-I(U) + U x R" where U c B is open, such that the diagram
commutes: here n I ( x ,y ) = x. For each x E U we define the homeomorphism cpx to be the composition cpx : p - ' ( x ) 5 x x R" + R". Thus ify E p - ' ( x ) we have the formula cp( y ) = (x, cp,( y)). A vector bundle atlas @ on ( p , E, B ) is a family of vector bundle charts on ( p , E, B ) with values in the same R", whose domain cover B, and such that whenever (cp, 17) and (9,V ) are in @ and x E U n V , the homeomorphism 9, cp'; : R" + R" is linear. The map U n V + GL(n), x H 9, 0 cp'; is required to be continuous: it is called the transition function of the pair of charts (cp, U ) , (9,V ) . If @ = {(cp,, U , ) : i E A} we obtain a family {g,,} of traaition functions g , : U, n V, + GL(n). These maps satisfy the identities 0
gt,(x)og,k(X) = gik(X),
X E
uk,
g,,(x) = 1 E GL(n).
The family {g,,} is also called the cocycle of the vector bundle atlas @. A maximal vector bundle atlas @ is a vector bundle structure on ( p , E, B ) . We call 5 = ( p , E, B, @) a vector bundle having dimension n, projection p , total space E and base space B. Let V be a real finite-dimensional vector space of dimension n > 0. Two bases (el, . . . , en), (kl, . . . , k,,) of V are equivalent if the automorphism A : V + V such that Ae, = k,has positive determinant. An orientation of V
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646
is an equivalence class [ e , , . . . , en]of bases. If dim( V ) > 0 then there are just two orientations. If one of them is denoted by w , then - w denotes the other one. An oriented vector space is a pair (V, w ) where w is an orientation of V.' Given (V, w ) and (V', w ' ) an isomorphism L: V + V' is called orientation preserving if L(w) = w', otherwise L is orientation reversing. Now let = ( p , E, E ) be a vector bundle. An orientation for is a family w = { w x :x E E } where w, is an orientation of the fiber Ex such that 5 has an atlas @ with the following property: if cp:<,. + R" is in @ then rp, : ( E x ,w,) + (R", 0,) is orientation preserving. We call w a coherent family of orientation of the fibers. The atlas @ is an orientated atlas belonging to <. If has an orientation w, then is called orientable and the pair (<, w ) is an oriented vector bundle. Let M be a manifold. M is called orientable if TM is an orientable vector bundle. An orientation of M means an orientation of TM, an oriented manifold is a pair (M, w ) where w is an orientation of M . We define - w to be the orientation of M such that ( - w ) ~= -0, for all x E M. If M is connected and orientable then it has exactly two orientations w and -a. Every simply connected manifold is orientable. An alternative definition of orientable manifold is that M is orientable if it has an atlas whose coordinate changes have positive Jacobian determinants at all points. Let (M, w ) and (N, a) be oriented manifolds. A diffeomorphismf : M + N is called orientation preserving if Tf :(TM, w ) + (TN, a) preserves Orientation. Every connected 2-manifold without boundary is called a surface.
<
<
<
By lifting the expansive homeomorphism of the 2-torus given by
through the psuedo covering map, the following is proved. 2.25. Theorem. The compact orientable surface of genus 2 admits an expansive homeomorphism. See O'Brien [ 19701 for the proof (such a homeomorphism is called pseudoAnosov maps (see Definition 8.51)).
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2.26. Definition. Letf : X + X be a continuous surjection of a metric space. The map f is called a positively expansive map if there exist a metric d (compatible with the topology of X) and c > 0 (called an expansive constant) such that x # y implies d ( f " ( x ) ,f"( y ) ) > c for some n > 0 (compare with Definition 2.1). For compact spaces this concept is independent of the compatible metric used. 2.27. Remark. Let f be a toral endomorphism of the n-torus. Then f is positively expansive if and only iffhas only the eigenvalues of modulus > 1 (compare with Theorem 2.18). 2.28. Remark. We can construct an example of positively expansive map which is not open. The following is due to Rosenholtz [1976]. Consider the subset X in the plane defined by
Given X the arclength metric. A map is defined as follows: stretch each of the small circles onto the big circle, stretch each of the upper and lower semicircles of the big circle first around a smaller circle, and then across the other semicircle and finally around the other smaller circle. More precisely we describe the map f by
2(2
z3
+ +)
if Re(z)
if
<
- 1,
-3 < Re(z) < 4,
2.29. Definition. Let X be a compact metric space. Forf :X ous surjection, we let
x,
= {( x , , K : x , E X , f ( X n + I )= x,, n
3((xfl)0") = (f(xn))om,
X((Xfl)gm)
+
X a continu-
2 01,
= xo-
03
T h e n j : X, +&is called the shift map determined byfand a = f a holds. The mapyis the homeomorphism and3-l((xn)F) = ( x , , + ~ )for ; (x,,); E 4. We say that (x/,3)is the inverse limit of ( X , f ) . 0
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648
2.30. Remark. Let f :X + X be a continuous surjection of a compact metric space and let ( X , , f ) be the inverse limit of ( X , f ) . Define a metric a for X, by a(2, J ) = C;d(x,, yi)/2’for 2 = (xi):, J = (y,): E X,. I f f : X + X is positively expansive, thenf: X, + X, is expansive. Proof. Let 6 > 0 be an expansive constant f o r 5 Then there is n 2 0 such that x # y implies d(f”(x),f”(y)) > 6. Suppose 2 # 9, then x,,, # ym for some m 2 0 and by positive expansivity d(f”(x,,,),f”(y,,,)) > 6 for some n2O.Ifk=n-m>Othen
J < f k < 2 > , r k ( J= ) ) J((fk(X0),fk(XI), . . .), ( f k C Y o ) , f k ( Y , ) ,* .))
> d ( f k ( x 0 ) , f k ( ~ o )= ) d(fkf”(xm),fkfm(ym)) = d(f”(xrn),f”(ym)) >
6,
and if k < 0 then
a(fk(2),rk(J))
= d((x-k,
>
x-k+l,.
.
(Y-kr
y-k+l,.
*
.))
4 x 4 , y-k)
= d(fm+k(xm), frn+&(
Ym))
= d(f”(xrn),f”(ym))
> 6.
2.31. Remark. (a) The closed interval I = [0, 11 does not admit positively expansive maps. Proof. Suppose g :I + I is a positively expansive map. Let I , ( & ) denote the closed subinterval [ x , x E ] for x E [0, 1 - E ] . Then there is E > 0 such that is injective. The point a E lo(+)which gives the for each x E [0, 1 - E ] , g,,x(E) greatest value of g,,0(J2)is one of the end points of Z o ( + ~ ) : i.e., a = 0 or a = + ~ / 2 . For, if a is an interior point of I&&), then we have g(0) < g(a) - v < g(a) and g(+E) < g(a) - v < g(a) for some v > 0, so that there are x I , x2 E I&) with x I # x2 such that g ( x , ) = g ( x 2 )(by the mean value theorem). This cannot happen since g is a local homeomorphism. If a = + E gives the greatest value of g on I,(+&),then it is easily checked that the point E in I must be a point of the greatest value of g on ZE/2($9. Continuing this argument, it is obtained that g : I + I is a homeomorphism, contradicting Theorem 2.38 below. 0
+
2.31. Remark. (b) The inverse limit of every continuous surjection of a closed interval is not expansive (this was asserted in Jacobsen and Utz [1960]). Proof. We give here a simple proof due to Dateyama. Let f :I + I be a continuous surjection of a closed interval I = [0, 11 and ( I , , f ) denote the
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649
inverse limit of ( I , f ) . If in particular f is a homeomorphism then f is topologically conjugate tof It is clear that (+, f') is topologically conjugate to (I,,?). Sincefis surjective, there are a, b E I with 0 < a < b < 1 such that either (l)f(a) = 0 andf(b) = 1, or ( 2 ) f ( a ) = 1 andf(b) = 0. For the case (2) we see thatf2(a') = 0 andf2(b') = 1 for some a', b' E I with 0 < a' < 6' < 1. Therefore it is enough to give a proof for the case ( I ) . Sincef: I + I satisfies (l), there are two points u, v E I (u < v ) such that f ( u ) = u , f ( o ) = v andf(x) # x for u < x < v. For, it is clear thatfhas at least two fixed points in I. Sincefis continuous, the set of fixed points is not dense in I. Thus f has the following cases: (3)f(x) > x (u < x < v ) and (4)f(x) < x (u < x < v ) . We first give a proof for case (3). Note thatf;,,,, cannot be bijective. Then we can find two points X I and x2 such that u < X I < x' < ZI and f(x') = f(x2). By using the point x2, we define yof(xo) = max{f(x): u < x < x'}. Then XI < xo < x2 < yo. Since the graph offon [u, v ] is over the diagonal set of I x I, there is a sequence x: > xy > * * * in [u, v ] with x: = xo andf(xp) = for i 2 1. For every 0 < E < x2 - xo, we have + E for some N > 0. Take 6 > 0 so small that Ix - y l < 26 xi < u (x, y E I) implies If"(x) - f"(y)I < +e for 0 < n < N. Write c' = x i - 6 and c2 = xt 6. Then f o r j = 1, 2
+
+
If"(C')
- f"(Xt>l
=
If"@')
- X0I
< +E,
and hencef"(c') < xo + E < x' since x' - xo > E . By the definition of yo we havef'"+I(d) < yo f o r j = 1,2. IffN+'(c') = yo for somej, then we can find a sequence {c:: 0 < i} c [u, v ] such that c; = cJ andf(cj) = for i 2 1. Note that {c:: 0 < i } is strictly decreasing. Define a sequence {b,} in [u, v ] by b, = f"-'(c') (0 < i < N ) and bN+,= c: (i 2 0). Since Id - xi1 < 26 f o r j = 1, 2, we have)'-@"fI -f"(xi)I = 1bN-"<E > . * * > u. for 0 < n < N. As we saw above, we obtain u + +E > x i > Since x i E [u, v ] , u E > cJ = c; > c: > * . > u and hence I b, - xp I < E for i 2 N. Therefore I b, - XPI < E for all i 2 0. Sincef"+I(d) = f(xo) = yo, we have f ' ( b o ) = fl(x,") for i > 0. Thus for i > 0, ith components of f " ( b o , b,, . . .) andf"(x,O, xy, . . .) are closer than E for all n E Z. Let x3 = max{f"+I(c'):j = I , 2). If x3 < yo, then there are c3, c4 with c' < c3 < xN < c4 < c2such thatfN+'(c3)= fN+l(c4) = c4. By (3) wecan find sequence {c::O < i} ( j = 3, 4) in [u, v ] such that cd = c' andf(cj) = c:- for i 2 1. They are strictly decreasing sequences. By a similar argument
+
,
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650
mentioned above we have that for i 2 0, ith components of p(f "(c;), f " - ' ( c i ) , . . . , c,,3 c:, . . .) and J"( f "(c,'),f "-'(c,"),. . . , c,', c f , . . .) are closer than E for all n E E . Thereforefis not expansive. The case (4) can be proved as above. 0
2.32. Theorem (Reddy [1982]). I f f : X + X i s apositively expansive map of a compact metric space, then there exist a compatible metric D , E > 0 and I > 1 such that
w,y ) G
(x, y
E
X)
* D ( f ( x ) , f ( y ) )2
m x , y).
For the proof we need the following proposition.
2.33. Proposition (Frink [1937]). Let V , be a decreasing sequence of open symmetric neighborhoods of the diagonal set A of X x X such that = XxX,npV, = AandV,+,~V,+,~ c ~V,foraNn +, 2 0.Thenthere is a metric D compatible with the topology of X such that for n 2 I , K = { ( x , y ) : D ( x , y )< 1/2"} = K-I.
Proof. Define a function ( : X x X -, R by
and for ( x , y ) E X x X we denote by c(x, y) a finite sequence x = x,, x I , . . . , x,, xn+' = y . We write U ( x , y ) a collection of all such sequences c(x, y). Let us put D ( x , y ) = inf{Ci<(xi, xi+1 )
C(X,
V ) E %(x, Y>}*
It is easily checked that (i) H X , y ) 2 D(X9 y>, ( x , y ) E X x X , (ii) D ( x , y ) < D ( x , z ) D(z, x ) , x , y, z E X , (iii) D ( x , y ) = D( y, x ) . Since D ( x , y ) < ((x, y ) < 1/2" for ( x , y ) E V., we have V. c { ( x , y ) : D ( x , y ) < 1/2"}. If we hold that for n 2 1 (iv) { ( x , y ) : D ( x ,y ) < 1/2"} c K-1, then x = y whenever D ( x , y ) = 0 since ( x , y ) E nV, = A. From (ii) and (iii) together with this fact, it follows that D is a metric compatible with the topology of X. If we have the following: (v) for each finite sequence x = xo,x , , . . . , x,,+~= y
+
n
((x,Y)
< 2 1 ( ( x i , xi+\), i=O
Topological Dynamics
65 1
then (iv) is obtained. For, if D ( x , y) < 1/2", by (v) we have
< 2 . 112" = 1/2n-1 and so (x, y ) E V,- I . The remainder of the proof is only to prove that (v) is true. If n = 0 then t(xo,xI) < 2t(x0, xI). Suppose (v) holds for 0 < k < n - 1, i.e., t(xo, xk) < 2Zf&'t(xi, x i + l ) . Letting a = ZI:i((x,, xi+l), we have that either ( 4 a x 0 3 XI) d la, or (vii) t(xo, xI) > +a. Case (vi): Let k 2 1 be the greatest integer among {k'} such that XfLilt(xi, xi+,) < ;a. Then we have Zi!it(xi, xi+l) d $a and by hypothesis it- I
5(Xk+l,
x,) Q a,
t(&,& + I )
< a. 1/2" < a
For this a there is m > 0 such that < 1/2"-' and (xo, xk), (xk, (xk+l, x,) E Vm-l, from which (xo, x n ) E Vm-2 (we consider Vm-2= K, if m - 2 < 0) and hence n- I
Case (vii): Since t(xo, xI) > +a, we have XYi;t(xi, x i + l ) < $a and by hypothesis of induction n- I
< 2 1 < ( x i , xi+l) < a, E Vm-2.Therefore t(xo, x,) < 1/2"-' xn)
i=O
and so (xo,x,) In any case (v) holds.
< 2a = 2ZYzit(xi, xi+l).
Proof of Theorem 2.32. Let c > 0 be an expansive constant for J Let Wo = X x X and for n 2 1 let
w,
= {(x,
y):d(f'(x),f'(y))
< c, 0 < j
I}.
Then it is easily checked that { W,} is a decreasing sequence of open symmetric neighborhoods of the diagonal set A such that 0 : W, = A and letting g =fxf:XxX+XxX,g(W,) = W,-l n g ( W , ) f o r n 2 I.SinceW,is open, there is 6 > 0 such that Nd(A,d) = {(x, y ) :d(x, y ) < S} c W,.
N.Aoki
652
Since X is compact and 0;Wn = A, there is N 2 1 such that W 1 + ,c N 6 / 3 ( 4 d ) . Hence W + N O K+NoW , + N c W,. Now define a new sequence { V , } as = W, and V , = for n 2 1 . By Proposition 2.33 there is a metric D for X such that V , c {(x, y):D(x, y) < 1/2”} c K-, (n 2 1). Indeed, the sequence { V , } of neighborhoods of A satisfies all the assumptions of Proposition 2.33 except the relation V,+, V,+,0 V,+, c V , for n 2 0. The relation holds for n = 0, 1 . To prove that the relation is true for all n 2 2, let (x, y) E V,+,o V,+,o V,+I for n 2 2. Then there are a, b E Xsuch that (x, a), (a, b), (b, y ) E V,+,and s o f o r 0 < j < (n - l ) N
w+(n-l)N
0
gJ(x9 Y ) = ( f W
c g’(
f’W) (f’(4,f’(b)) O
O
(f’(b), fJ<
V,+I ) “I?’(V,+I ) og’(K+ I )
w+nN-jo w+nN-jo w + n N - j
c
& o b o &
c
&.
Therefore d(f’(x), f J ( y ) ) < c for 0 w+(n-I)N
=
K.
< j < (n -
1) N and hence (x, y) E
A.
Suppose 0 < D(x, y) < Then there is n 2 - 1 such that (x, y) E V,+l\K+2and so I / P 3 < D(x, y) < min{&, 1/2“+’}.Hence n 2 3. Since (x, y ) -5 K+i\K+2 = w + n N \ w + ( n + l ) N , there i s j such that
nN < j
< (n +
I ) N and d(f’(x),f’(y))
> c.
Let (2, w ) = ( f ” ( ~ ) , f ~ ~ ( y )Then ) . 0 < (n - 3 ) N < J - 3 N < (n - 2 ) N and d(f’-3N(z),f’-3N(~)) = d(f’(x),f’(y)) > c. Hence (z, w ) # W , + ( n - Z ) N and D(f3N(~),f3N(y)) 2 1/2“ > 2 D ( x , y). Put y = 2-1/3N, L = y and define a metric D‘ by
c YiD(Y(4,fi(Y)).
3N- I
D’(X9
Y) =
i=O
Then D‘ has the desired property.
2.34. Remark. Letf: X + X be a positively expansive map of a compact connected metric space. Iff is an open map, then f has fixed points in X. Proof. Sincefis positively expansive and open,fis a local homeomorphism. This follows from the fact that there are no pairs (x, y) of distinct points of X arbitrarily closed to x withf(x) = f(y) by positive expansivity. Since X is compact and connected, it is checked thatf: X + Xis a k-to-1 map with k > 1 (such a map is called a covering map). In fact,f : X + X is not bijective by Theorem 2.35. Hence there is q > 0 such that for x, y E X
Topological Dynamics
653
with x # y,f(x) = f(y ) implies d(x, y ) > 4q.If this is false then for every n > 0 there are x,, y, E X (x, # y,) such that f(x,) = f ( y , ) and d(x,, y,) < l/n. Since d(x,, y,) + 0 as n + co, we have that x, + x and y , + x as n + co. Sincefis a local homeomorphism,f: U(x) + f ( U ( x ) )is bijective for some neighborhood U(x) of x. This cannot happen. Let Y, = ( y E X:card(f-'x) = k} fork 2 1. Then each Y, is open and closed, and X = Y, u * * u Y, for some 1 2 1. To see this, for each x E X let U(x) be an open neighborhood of x with diam(U(x)) < 4q where q is as above. Since X is compact, there is no > 0 with u:Il U(xi) = X. Assuming I = 00, we can find Y, # 0 such that for y E Y,, there is 1 < i < no so that A = f-'(y) n U(xi) contains at most two points. Let x, x' E A with x # x'. Then 0 < d(x, x') < 4q and f(x) = f ( x ' ) , thus contradicting a local homeomorphism. Therefore Xis covered by the union of finite sets Y,, . . . , Y,. To show openness of Y,, let y E Y,. Then d(xi, xi) > 4q for x i , xj ~ f - l y() (i # j ) . Sincefis a local homeomorphism, there are open neighborhoods U(xi) of xi such that U(xi) n U(xj) = 0 and diam(U(xi)) < 4q for 1 < i < k. Note that nf=,f(U(xi)) is an open neighborhood o f y . To obtain the conclusion, it is enough to see that n;=,f(U(x,)) c Y,. Since ( ) f = z l f ( U ( x ic) ) Y, u Y,,, u * . . u Y,, we prove that n f = , f ( U ( x i )n ) (Y,, u . . . u Y,) = 0.If this is false, then there is for each 1 < i < k, an open neighborhood U,(x,) c V ( x i ) with small diameter A such that u . u Y,) # 0, from which we take a point nf=,f(Ui<xi>> n y,. Then there is XI") E U,(xi) with f(x,'L')= y,. If x,'L) xi as % 0, then lim,y, = lim,f(x,!')) = y. But y, E Y,,, u * * u Y, and s o f - ' ( y l ) \ uf=,UA(xi)# 0. Hence we can find x,' 4 u~=,U,(xi)with f(x;) = y,. If xi -+ x' as % + 0, then x' # U i ( x i ) and f(x') = y. This contradicts f - ' ( y ) c Uf=,U,(x,). Since each Y, is open and closed and X is connected, X is k-to-1. we have X = Y, for some 1 < k 6 I, i.e., f :X By Theorem 2.32, there are a compatible metric d, 6 > 0 and A > 1 such that d(x, y ) < 6 implies d(f(x),f( y)) 2 M(x, y). Since f :X -, X is a covering map, if V is an open set of diameter less than 6 and if q E f - I ( V) then there is a unique open set containing q of diameter less than (I/%) 6 which maps homeomorphically onto V . This condition enables us to lift chains of small open sets. Let f V,: 1 < i < n} be a fixed open cover of X so that the diameter of each is less than 6. Let xo E X withf(x,) # xo.Since X is connected, there is a chain of open sets chosen from among { V,} fromf(xo) to xo. Lift this chain to a chain of open sets from xo to x, such thatf(x,) = x,. Further, lift this chain to a chain of open sets from x , to x2 such thatf(x,) = x,, and continue this inductively. Since the original chain had length at most n6, the length of 1
,
u;=l
N.Aoki
654
the first lifted chain is at most ( l / I ) n 6 and in general the length of the kth lifted chain is at most (l/A)kn6. If j < k then d(xj, x k ) < {(l/I)j+’+ . . * ( 1 / I ) k } n 6 c (l/A)j(nd/(A- 1)). Thus {xi} is a Cauchy sequence, which therefore converges to y E X . Since f ( x i ) = xi-‘, we have f ( y ) = f(limjxj) = limjf(xj) = limjxj-, = y. 0
+
2.35. Theorem. I f f : X + X is a positively expansive homeomorphism of a compact metric space, then X is ajinite set. Proof. Sincefis positively expansive, by Theorem 2.32 there exist a compatible metric D, 6 > 0 and 0 < I < 1 such that D(x, y) < 6 ( x , y E X ) implies D(f-’(x),f-’( y)) < I D ( x , y). Let 0 < E < 6. Then we can find 6’ > 0 such that for all i 2 0 D(x, y ) < 6’ ( x , y
E
X ) => D(f-i(x),f-i(Y)) <
8,
from which the set @- = { f - i :i >, 0 } is equi-continuous. Write @+ = { f i :i > 0 } and put 0 = @- u @+. By a metric e for @ defined by
e ( f ,g )
= m a x W f ( x ) , g ( 4 ): x E
x>,
is totally bounded, i.e., for B > 0 there is in with U i = I U , ( f - i n )3 @-. Defining a map G : 0- + @+ by G(f-’)= f i for i 2 0, we claim that G is an isometry. Therefore @+ = G ( W ) c U k = l G ( U , ( f - i n ) c ) U,k_,U.(f”)and so @+ is totally bounded. Since Xis compact, 0’ is uniformly equicontinuous. Let c > 0 be an expansive constant forf. By equicontinuity there is v > 0 such that D(x, y ) < v implies D(f’(x),f’(y)) < c for all i 2 0. But x = y by positive expansivity, which implies that each x is an isolated point. Therefore Xis finite. 0 2.36. Definition. Letf: X + X be a homeomorphism of a compact metric space. Let x E X and E > 0. We define the local stable and local unstable set of x by K ( x , d ) = { y E X : d ( f ” ( x ) , f ” ( y ) )< E for n > O } , e ( x ,d )
=
{y
E
X : d ( f - ” ( x ) , f - “ ( y ) )<
E
for n 2 O}.
2.37. Proposition. f is expansive with expansive constant c ifand only if there is c > 0 such that for all r > 0 there exists N > 0 such that for all x E X and all n 2 N
Topological Dynamics
655
Proof. If this is false then we can find sequences x,, yn E X , m, > 0 such that y , E v ( x , , , d), lim m, = + 00 and d ( f m " ( x , ) , f m " ( y ,> ) ) r. Since Y , E ~(x,,d),wehaved(f"of""(x,),f"~f""(y,)) < cforall -m, < n.If f m n ( x n+ ) x and f " " ( y , ) + y as n + co, then d ( f " ( x ) , f " ( y ) )< c for all n E H. Furthermore, d(x, y) = lim d(f""(x,),f " " ( y , ) ) 2 r, thus contradicting expansivity. Conversely if d ( f " ( x ) , f " ( y ) )< c for all n E Z, for any r > 0 we have y E w(x)by the first relation of (*) and hence y = x. 2.38. Definition. For x
E
X the stable and unstable sets are defined by
W s( x,d ) = { y E X : lirn d ( f " ( x ) , f " ( y ) )= 0}, n-m
W u ( x ,d ) = { y
E
X :n-m lim d( f
- " ( x ) , f -"( y ) ) = 01.
2.39. Proposition. Let E > 0 be the number less than expansive constant for f . I f f is an expansive homeomorphism, then
wyx, d )
=
u u f"
f -"WE"(fnx, d ) ,
n2O
W U ( xd, ) =
w,u(f-"x, d ) .
n bO
, then , there is n 2 0 with f " ( y ) E Proof. If y E Una,f - n V ( f d")~ v ( f " x , d). By Proposition 2.37, for r > 0 there is N 2 0 such that for all m 2 N f " + " ( y )E f " K ( f " x , d ) c
W(f"+,x, d ) ,
and so d(f " + " ( y ) f"+"(x)) , < r (m 2 N ) , and therefore y E W s ( x ,d). Conversely, if z E W s ( x ,d ) and E > 0 then we can find N 2 0 such that d ( f " ( x ) , f " ( y ) )< E for n 2 N. Hence d(f'of"(x), f'of"(y)) < E for all j 2 0 i.e., f " ( z ) E K(f Nx,d ) and so zEf-"K(f"x,d) c
u
f-"K(f"x,d).
0
n,O
2.40. Theorem (Reddy [1983]). Let f :X + X be a homeomorphism of a compact metric space. I f f is expansive, then there exist a compatible metric D , y > 0, 0 > 1 c 1 and a 2 1 such that y
E
y < x ,D )
=>
D ( f " ( x ) , f " ( y ) )< a l " D ( x ,Y ) ,
y
E
y ( x ,D)
3
D ( f - " ( x ) , f - n ( y )< ) apD(x, Y )
for all n 2 0.
N . Aoki
656
2.41. Definition. The metrics D stated in Theorems 2.32 and 2.40 are called hyperbolic metrics forf, and the numbers I are called a skewness off. Proof of Theorem 2.40. As before let W, = X x X and for n 2 1
w,
= {(x, y):d(fJ(x),f’(y)) < c,
l j l < n}
where c is an expansive constant forf. Then { W,} is a decreasing sequence W,, if (x, y) 4 A, of open symmetric neighborhoods of A. Since A c Un2, then there is n 2 0 with (x, y) 4 W,. This implies W, = A. Hence there are E > 0 and N 2 1 with P t ( + N c NE,3(A,d ) c NE(A,d ) c W,. We now define a new sequence of open symmetric neighborhoods of A as V, = W, and V , = P t ( + ( n - I ) N (n 2 1). Obviously 0; V , = A. To use Proposition 2.33, we prove that V,,, V,+, V,,, c V , (n 2 0), which is one of the conditions of Proposition 2.33. Obviously V, V, V, c V, and by the choice of E and N, V , o V , o c V,. The proof is done by induction on n. Suppose (x, y) E V,+, V,,, 0 V,+, for k > 1. Then there are z , w E X with (x, w), ( w , z), ( 2 , y) EG+,. If (p, q) is any of these three pairs then d ( f J ( p ) , f ’ ( q ) )< c for l j l < 1 kN. If Iil < 1 + (k - l ) N and Ikl < 1 + Nthen Ii + kl < 1 + k N a n d s o
n,
0
0
0
0
0
+
4fkOf’(P),
fkof’(q))
=
d ( f k ” ( p ) , f k + ’ ( q )< ) c.
Therefore ( f ’ ( p ) ,f i ( q ) )E V, and hence ( f i ( x ) , f i ( y ) )V~, o V , o V , c V,
Iil < 1
+ ( k - 1)N,
from which d ( f ‘ ( x ) , f i ( y ) ) < c for l i l < 1 + (k - 1)N. Therefore V,andso v k + l o v k + l o v k + l c V,. By Proposition 2.33 there is a metric D such that V, c {(x, y ) :D ( x , y) < 1/2“} c V,-, for all n 2 1. Let A c X x X and denote by x A the subset {y~X:(x,y)~A}ofX.Thenwehavefo2 rn1
(X,Y)E
f(Wc(X,
4 n xw,)
=
W f x , 4 n f ( x ) K+l,
f-’(w(x, d ) n x W , ) = T ( f - ’ x , d ) n f - ’ ( x ) W,,,. Indeed, the first equality is computed as f(Wc(X, 4 n xw,)
=
f{ y : d ( f ’ ( x ) , f ‘ ( y ) )<
= {z:d(fkf(x),fk(z)) =
<
Wc(fx3 4 n f ( x )% + I .
c, i c,
2 -n}
k 2 -(n
+ l)}
Topological Dynamics
657
Let N be as above. By induction we have for all n 2 1 fN(K(X,
4
nXK)
=
w,s(f"x,
fpN(K"(x, d ) nX K ) = ~ " ( f
4 n f N W K+l, - ~ dx ), n f - " ( x ) K+,.
From now on we show that D is a hyperbolic metric for$ Since there is > 0 such that D ( x , y) < q implies d ( x , y) < c, we have W,"(x, D ) c W , ( x , d ) for x E X. Put v = min{r], i} and lety E F ( x , D).We first prove that D ( f ' " ( ~ ) , f ' ~ ( y ) )< +D(x, A. D),we have in+*< D ( x , y ) < and so Since (x, y) 4 V,+, 3 N1,2n+2(A, n 2 1. Since y E W , ( x , d ) n x V , , we have r]
a
f3"(
.Y)E
W.'(f3"x3
4
nf3'"(x)
K+3
c f 3 " ( x ) N1p+3(A3D ) ,
i.e., ~ ( f ' " ( x ) , f ~ ~ (
< (1/2>kD(X,Y).
We choose 0 < A < 1 with A - L
=
2 and show that if y E v ( x , D ) then
D ( f " ( x ) , f r n ( y ) ) G 8A"D(x, y ) , m 2 0.
+
Put m = kL j for k 2 0 and 0 V,\ V,+I for some n 2 1, we have
<j
< L. Since ( f " ( x ) , f k L ( y ) ) E
f k " ( y ) E W , ( f k ' x , d ) n f k L ( x )W + ( n - I ) N ,
and therefore
f from which (
ukL
(x) Since and hence
1
Since m =
1
1 .
- _ .
.
- .
By replacing f by f-',we can establish the required condition for y E wv"(x, D ) . 0
658
N.Aoki
3. Expansivity and dimension In this section we shall try to characterize topologically compact metric spaces which admit expansive homeomorphisms. 3.1. Definition. Let X be a compact metric space. The dimension of the space Xis said to be less than n if for all r > 0 there exists a cover a of X by 1 open sets with diameter c r such that each point belongs to at most n sets of a. (If the topological dimension is of interest, the reader should see Hurwicz and Wallman [1948]). It is known that Xis 0-dimensional if and only if it is totally disconnected, i.e. if the connected component of each point is a single point.
+
3.2. Theorem (Maiii [ 19791). I f f : X + X is an expansive homeomorphism of a compact metric space, then the dimension of X is finite.
Hereafter let c > 0 be an expansive constant for f and fix 0 c
E
< +c.
3.3. Lemma. There is a 6 > 0 such that if d(x, y ) < 6 ( x , y E X ) and E < sup{d(f J ( x ) f, ' ( y ) ) : O < j < n} < 2.3 for some n 2 0, then d ( f " ( x ) , f " ( Y ) ) 2 6.
Proof. If this is false, then there are sequences x,,, y,, E X , m,, > I, > 0 such that d(xn, yn) + 0, d(Smn(Xn),fmn(yn)) 0, W " ( X n ) , f C ( Y n ) ) 3 E and sup{d(f"(x,,),f"(y,,)):O < m < m,,} < 2.5. Since X i s compact, we suppose that f " ( x , ) + x and f ' " ( y , ) + y as n + 00. Then d(x, y) 2 E and d ( f " ( x ) f, " ( y)) < 2.5 for all n E Z. 0 +
3.4. Lemma. For all @ > 0 there exists N = N ( q ) > 0 such that E < sup{d(f"(x),f"(y)):InJ < N} whenever d(x, y ) 2 q . Proof. If this is false, then there exist sequences x,, y , E X with d(x,, y,) 2 e such that sup{d(f J ( x , ) ,f'( y,)): l j I < n } < E . If x, + x and y,, + y as n + 00, then d(x, y ) 2 q and d ( f " ( x ) ,f"( y)) < E for all n E Z.
0 Proof of Theorem 3.2. Let 6 be as in Lemma 3.3 and choose a cover { V ,: 1 < i < I } of X by open sets with diameter < 6. To obtain the conclusion of the theorem, we show that dim(X) < 1' - 1.
659
Topological Dynamics
For each n 2 0, choose 6, > 0 such that d(x, y)
f'( y)) c E for all ( j I < n. We write
-
qj =
< 6,
implies d(f'(x),
(*I
f " ( Q ) nf-"(U,)
Define x y for x, y E vi:,if there exists a sequence x = x,, x I , . . . ,xp = y such that d(x,, x,+,) c 6, for all 0 < r < p - 1 and x, E Qyj for all 0 < r < p. Denote by U$ (1 < k < k ( i , j , n)) the d,-components of qj, i.e., the equivalence classes of y j under the relation x y. it is easily checked that each Q:jk is open and they cover X. We claim that
-
(**I
lim (sup diam(U,4k)) = 0. k.i.j
Indeed, if this is false then we could find e > 0 and a large number n, say n > 2 N ( e ) , N ( e ) given in Lemma 3.4, such that in some Y';."there are x, y with d(x, y) > e. Let x = x,, x l r. . . ,xp = y be a sequence in such that d(x,, x,+,) < 6, for all 0 < r < p - 1. Let us put
vj
< n}.
s, = sup{d(f"(x,),f"(x,)):Iml
Then sp >
by Lemma 3.4. From the choice of 6,, it follows that sI < E and for 1 < r c p. Choose r such that s,. < E if r' < r and s, > E . Then s, < 28 and therefore d(f-"(x,),f-"(x,)) < 6 sincef-"(x,), f-"(x,) E Q..Sincef"(x,),f"(x,) E U,, we haved(f"(x,),f"(x,)) < 6, and so - s,l
E
<E
s,
=
sup{d(f"(x,),f"(x,)):Iml
< n} < 2.5.
Letting xi = f-"(xo) and x; = f-"(x,), we have sup(d(f"(x~),f"(x~)):O< m
< 2n} < 2.5.
Since d(x& xi) c 6, by Lemma 3.3 we have d(f2"(x& f2"(x;)) = d(f"(x,), f"(x,)) > 6, thus contradicting d(f"(x,), f"(x,)) < 6. Thus (**) was proved. It only remains to be shown that for each n, every point of X belongs to at most 1' sets of the cover {Uilt;.k: 1 < i , j < I, 1 < k < k ( i , j ) } . Suppose
n {Q;?:I
s
m
< s} z 8.
n.k,
If (im,j,,,)= (i*, j f i ) , then we have Qh,jm = Q:: since they are both 6,-components of U,"', and have nonempty intersection. This implies that to different values of m correspond different values of the couple (im,j m ) . Therefore, s < 1'. 0 3.5. Remark. (a) Letf: X -+ X be a positively expansive map of a compact connected metric space. Iffis an open map, then the dimension of X is finite.
660
N. Aoki
Proof. We consider the inverse limit (X,,3 ) of (X, f ). Sincef is k-to-1, there is a neighborhood base for %each of whose members is the directed product of a basic neighborhood in X with the Cantor set. Sincef : X + Xis positively expansive,f: X, + X, is expansive by Remark 2.31. By Theorem 3.2, X,is finite-dimensional, and hence the dimension of X is finite by the above fact. 0
3.5. Remark. (b) Iff :X + Xis a positively expansive map (need not open) of a compact metric space (need not connected), then Xis finite-dimensional. This is easily checked by taking v,:i = V, nf ( q ) instead of (*) in the proof of Theorem 3.2. 3.6. Theorem (MaiiC [1979]). Iff: X -, X is an expansive homeomorphism of a compact metric space, then every minimal set for f is zero-dimensional. For the proof we need the deep structure theorems due to MaiiC for expansive homeomorphisms. Before starting with the proof, we shall give the same for positively expansive maps to clarify the structure of the behaviour of expansivity. 3.7. Theorem. Ifa continuous surjectionf : X expansive, then X is zero-dimensional.
+
X is minimal andpositively
3.8. Lemma. Let X be a compact connected metric space and A be a proper closed subset of X . If x E A and C is the connected component of x in A , then C n a A # 8 where a A denotes the boundary of A . Proof. Let G$ be the collection of all open and closed subsets under the relative topology of A such that C c K c A . Since X is compact we have C = n { K : K E W } . Suppose C n a A = 8. Then U { X \ K : K E G $ } =) a A . Since d A is compact and X\ K is open and closed, a A is covered by the finite union of X\ Ki. But n i K i = K is open and closed, K E G$ and K n a A = 8. Since K is open in A , there is an open subset U of X such that U n A = K . Since A = a A u int(A), we have K = U n A = U n ( a A u int(A)) = U n int(A), i.e., K is open in X . Since K is closed in X , we have X = K (i.e., a A = 8) and therefore A = X , a contradiction. 0 Let y > 0. For x E X let X,(x) be the connected component of x in B,(x) where B,(x) = { y E X : d ( x ,y ) < y } , and put aBy(x) = { y E X : d ( x ,y ) = y } .
Topological Dynamics
66 1
Proof of Theorem 3.7. Sincefis positively expansive, by Theorem 2.32 there exists a compatible metric d, to > 0 and M 2 1 such that d(x, y) < E, (x, y E X ) implies d(f"'(x),f "( y)) 2 3d(x, y). Sincef: X -,Xis minimal, we have X = o/o = u & ' O f M ( f k ~for ) x E X . Write Ek = n,"=,f'"Of,(fkx) for 0 G k 4 M - 1. Since f M O f M ( f k x )c Of,(fkx), f M ( E k )= Ek is nonempty and ur=i'Ek= X . It is easily checked that each Ek is a minimal set forf"'. Henceforth, we replace Ek by X and f"' by f and give the proof of the theorem. Suppose dim(X) > 0. By Lemma 3.8 there are x, E X and t j > 0 such that & , ( x o )n aB,,(xo)# 0. If 0 < 6 < t j it follows that Xa(x,) n aB,(x,) # 0. Let S = min{+c, +q,}where c is an expansive constant for fi and fix O < E < S . The proof is done by running on the three claims. Claim I. Let x E X and A be the connected component of x in B,(x). If A n B,(x) # 0 for some 0 < 6 < E , then there exist a, /IE ~ ( A and ) compact connected sets A,, A, such that (a) a E Am = Ba(a), B E Ap = &(B), (b) A, n aB,(a) f 0,A, n a&(b) # 0, (c) inf{d(z, w):z E &(a), w E B,(b)} > 6. Proof. Take y E A n aB,(x), then
Thus there are a, b E f ( A ) with inf{d(z, w):z
E
Ba(a),w
E
B,(b)} > 6 ,
and so &(a), B&?) satisfy (c). Let A,, Ap be the connected components of a, B inf(A) n B 6 ( a ) , f ( A )n B,(/3) respectively. Sincef (A) is connected and a 4 Ba(fi)and f l $ &(a), it follows that A, n dB,(a) # 0 and Ap n all&?) # 0, and so Am,Ap satisfy (b). Obviously a and fl satisfy (a). Claim 2. There exist compact connected sets A,,, x,, E A,,, an open set U and t j > 0 such that for 0 < 6 < t j and all n (a)' A, n a&&,,) # 0, (b)'f(An) 3 An+l, (c)' U n A, = 0. Proof: As we saw above, we have Ea(xo)n aB,(x,) # 0.Write A,, = Ca(xo). Since diam(A,) < 26 and diam(f(l2,)) 2 36, we can find a nonempty open set U such that diam(U) < +6 and U n A,, = 0.Suppose that we constructed
N.Aoki
662
Ak and xk E Ak (0 < k < n - 1) satisfying (a)', (b)' and (c)'. By (a)' and Claim 1 there are c1, B E f (A,-l) and compact connected set Au c f(A,,-l) n &(a) and A, c f(A,,-l) n B,(B) which satisfy (a), (b) and (c) of Claim 1. Since d(Aa,A,) > 6, we have U n A, = 8 if U n Au # 8. Thus, letting A, = A, and x,, = B, (a)', (b)' and (c)' would be satisfied. Claim 3. Now we are ready to derive a contradiction. By Claim 2 there is a collection of compact connected sets Afl(n 2 0) such that n:=of-"(A,) # 8 and U n f -"(A,)) = 0.Ifp E n;f-"(A,,) then we havef"(p) 4 Ufor all n 2 0. This cannot happen sincefis minimal.
(or=,
We shall prepare some propositions to be used in proving Theorem 3.7. 3.9. Proposition. Iffor some x E X and m 2 O , f m W ( x ,d ) n W ( x ,d ) # then X contains a periodic point.
8,
Proof. Take y E W s ( x ,d ) nf"W s ( x ,d ) and put z = f - " ( y ) . Then f"(z) E W s ( x ,d ) = Ws(z, d ) . Therefore lim,,+,d(f"f"(z),f"(z)) = 0. If f m n ( z )-P w as n + co then d(w, f " ( w ) ) = lim d(f"f""(z), fmn(z)) = lim,d(fmnfm(z),fmn(z))= 0. 0
Define Xi(x) and X:(x) as the connected components of x in y ( x , d ) n Bs(x)and w ( x , d ) n B,(x) respectively. Let c > 0 be an expansive constant forf. Henceforth, fix 0 < E < ic. 3.10. Proposition. Ifdim(X) > 0, then there exist 0 < r < E such that for 3 < 6 < r there is a E X such that Xi(a) n aB,(a) # 8 or X:(a) n aB,(a) # 8. Proof. Let Xr(x)be the connected component ofx in B,(x).Since dim(X) > 0, we can find x E X and r > 0 such that X,(x) n ~ B , ( X#) 8 (by Lemma 3.8), and so &(x) n aB,(x) # 8 for 0 < 6 < r. Suppose there is 0 < 6 < r such that C,U(y)n aB,(y) = 8 for all y . We now prove that there exists a E X such that Xi(a) n dB,(a) # 8. To find a E X we construct a collection of compact connected sets A, (n 2 0) and a sequence of points x, E A, such that for some sequence of mn > 0 the following conditions hold: (i) f - " " ( X n ) = xn+1, (ii) f-mn(An)3 A n + I , (iii) A,, n aB,(x,) # 8, (iv) f'(A,) c B,(fjx,) (0 < j < m , - l ) .
If we established the above conditions, then there is a subsequence { x , , }
Topological Dynamics
of {x,} such that xni -, a as i
+
663
co. Fix { x n i }and define
It is clear that a E A and A is compact. We show that A is connected. Indeed, suppose A is not connected. Then there are nonempty closed sets Fi ,Fz such that Fl n F2 = 8 and A = Fi v F,. If a E F,, then for b E F, we can find b,,., E A,,,, such that b,,,, + b ( j + 00). Since {x,,,} is a subsequence of {x,,}, we have xn,,, + a ( j -, 00). Since Fl n F2 = 0, there is a neighborhood U(FJ of F, such that U(F,) n U(F,) = 8. But each A,,., is connected and we can find { yn,,,} such that Y n , , , 4 u ( F i ) n u(F2) and ~ n , , ,E An,,,. If ~ n , , , Y ( j + 00) then y E A, but y 4 Fi u Fz. This is a contradiction. By (iv) we have +
0 <j
< m,,-l =>f’(A,) 0 < j < mn *f’(An+I)
c B,(fx,), c B&(f’xn+i)*
Since x, = f ’ ” ” ( ~ , +by~ (i), ) by (iv) we have (v) f J ( A n ) c B & ( f m n + ’ X n + i ) , 0 < j < m,-i, i) and by (ii), A, 3 Jm”(An+1). Hence by (v>,fm.+’(~,+I ) c f’W c &(f””+’Xn+ (0 < j < m f l - l ) Hence . we have (vi) f’(An+l) c B&(fJxn+i), mn < j < “ I n - 1 + mn, andinductivelyforn Z l , f J ( A n )c B,(f’xn), 0 < j < mo ml .* * m f l P If i . n + 00 thenfJ(A) c B , ( f a ) , 0 < j < co,and hence A c y ( u , d ) . Since x,,., + a ( j + a),there is y , , E A,,., n aB,(x,,,,) such that 6 = lim,,,d(yn ,.,, xn,,,) = d ( y , a), i.e. A n all,@) # 0 and by Lemma 3.8, c;(a)n # 0. The remainder of the proof is only to construct the sequence { A n } of compact connected subsets satisfying (i), (ii), (iii) and (iv). Let A, = C6(x). Then l\o n aB,(x) # 8.Note that Cg( y) n aB,( y ) = 0 for y E A’. Suppose are constructed. Then $ W(X,-,,d ) . For, if An-l c &, A1, . . . , V(X,-~, d) then 2 : ( ~ , , -13~An-l ) since C,”(x,_,) is the connected component n W ~ ( X ,d). - ~Hence , of x,-~ in B,(X,-~)
+
C:(X,-~) n aB,(xn-,) = AnP1n a B , ( x , - , ) #
thus contradicting. Hence An-i\ W ” ( x , - , ,d) # point y. Then for some m > 0 we have sup{d(f-“(z),
fPm(xn-l
0, from
+
8, which choose a
1) :z E An-I} Z d ( f - ” ( ~ )f,- “ ( x n - i >
(vii) sup{d(f-’(~),f-~(x,-~)):z E A,-,, 0
<j
E,
< m}
+
< E.
))
664
N. Aoki
Put m,- I = m and x, = f - " " - ' ( ~ , - ~and ) , let A,, be the connected component ). (ii) holds and by A, n dE,(x,) # 8, of x , in E,(x,) n f - m n - ' ( A n - l Then (iii) holds. From (vii) we obtain (iv). 0 3.11. Proposition. For 0 < 6 c E there exists N > 0 such that for all x E X andy E W , ( x , d ) with d( y , x ) = 6 there is 0 < n < N such that d ( f - " (y ) , f - " ( x ) ) > E. Proof. If this is false, then there exist sequences x, E X, y , E W s ( x ,, d ) such that d(x,, y , ) = 6 and d ( f - j ( x , ) , f - J (y,)) < E for all 0 < j < n. If x, + x and y , + y , n + co,it is easy to check that x # y and d ( f " ( x ) , f " ( y ) )< E for all n E Z. 0
=-
3.12. Proposition. There exists 6, 0 such that v ( x , d ) n & ( x ) = W i ( x , d ) n B,(x) for all x E X and 0 < 6 < 6,. Proof. If the proposition is false then there exist sequences x,, yn E X such that d( x,, y,) + 0, and y, E W;3E(xn,d ) . Hence for some m, > 0 we have d(fn,(xn),fm,(yn)) > E and m, + co.We also have d(f"f""(x,),f"P(y,)) < 2.5 for all -m, < n since y, E W;e(x,, d ) . Iff""(x,) + x andf""(y,) + y (as n + co), we conclude that d(x, y) 2 E and d ( f " ( x ) , f " ( y ) ) < 2.5 for all m E Z,thus contradicting the expansivity off. 0
3.13. Proposition. Let E = inf{d(x, y ) : d ( f - ' ( x ) , f - ' ( y ) )> E } and 6, > 0 be as in Proposition 3.12. For all 0 c 6 c min{6,, +E}, there are N = N(6) > 0 such that i f x E X and A c W:(x, d ) is a compact connected set containing x and A n E,(x) # 8, then there exist 0 c m c N , a, B E f -"(A) and compact connected sets A#, A fl such that (a) a E A,, fl E Afl,a E W,S,(B,4, (b) Amn a W 4 # 8, Afl n a&,(B) f 8, (c) inf{d(z, w):z E &(a), w E E,(B)} > 6 , ( 4 Am = w E s ( 4 d ) n B&), A, c K(B,d ) n &(B). Proof. Let N = N(6) > 0 be as in Proposition 3.11. Since A n aE,(x) # 8 and x E A c r ( x , d ) , for y E W:(x, d ) with d(x, y ) = 6 we can find 0 < m < N - 1 such that (e) sup{d(f-"+''(z),f-'"+l)(x)) :z E A} 2 d(f-("+l'(y ) , f - ( " + ' ) ( x ) )> E , (f) sup(d(f-j(z), f - j ( x ) ) :z E A, 0 < j < m} < E . By (f) and the fact that A c W:(x, d ) , (g)f-"(A) = WE((f-"(x),d ) and clearly f - " ( x ) E ~ - " ( A ) .Hence we have
Topological Dynamics
665
(h)f-"(x) E W , ( w , d), w E ~ - " ( A ) . For, by (f) and the fact that f"(w) E A,
< E,
d(f-'f"(w),f-'(x))
s j < m.
0
Letting i = - j + m, (i) d(f'(w),f'f-"(x)) < E , 0 < i < m. Sincefm(w) E A c W,(x, d), for all k 2 0 we have d(f"f"(w),fk(x)) < E . Letting i = k + m, for all i 2 m (j) d(f'(w),fff-"(x)) < E and hencef-"(x) E W , ( w , d) (by (i) and (j)). From (g) and (h) it follows that for all w ~ f - " ( l \ ) (k)f-"(N c W2sE(w, d ) . By (e), d ( f - ' ~ f - " ( z ) , f - 'of-"(x)) > E for some z E A, and so d ( f - " ( z ) , f-"(x)) 2 8. Since z, x E A, we have diam(f-") 2 8. Since 36 < 8, we can find a, /3 e f - " ( A ) such that d(a, /3) > 36. Obviously a # &(/I) and /3 $ &(a). Hence (c) holds. Let Aa, Ab be the connected component of a and /3 in f - " ( h )n &(a), f-"(A) n &(/I) respectively. Then (b) holds by Lemma 3.8. From (g) it follows that (a) holds. Since A, c f-"(A) and A, c f-"(A), we have A. c Wi(a,d) and Ab c W2E(/3,d) by (k), and hence Aa c Bjr
Ve(a, d ) n Ba(a),
A, c Wi(P,d ) n Ba(B)*
Proposition 3.12, (d) holds.
0
Proof of Theorem 3.2. It is enough to verify that iff: X + Xis minimal and expansive then dim(X) = 0. Let r > 0 be as in Proposition 3.10, N = N(6) be as in Proposition 3.1 I and 6, > 0 be as in Proposition 3.12. Let 8 > 0 be as in Proposition 3.13. For 0 < 6 < min{b,, $S, r} we define rl
=
inf{d(f'(x),f'(y)): y 0
E
W&(x,d ) , d(x, y ) 2 6,
< i, j < N ( 6 ) ) .
If rl = 0 then it is checked that X is finite. Indeed, for all n 2 0 there exist x,, y, E X and 0 < i n , j , < N ( 6 ) such that y,
E
W,",(xn,4,
d(f'"(x,),
fJ7y,))
We can suppose that x, n 2 0. Hence y
E WAX,
+
4,
d(x,, y n ) 2 --*
x, y ,
0, n +y
+
as n
6 9
a. + 00
4x9 Y ) 2 6 ,
and thatj, f'(4 =
= j, in = i
f'(v)
for all
N.Aoki
666
and so i # j. Suppose i - j > 0, then we have f J - j ( x ) = y E W&(x,d ) c W s ( x ,d) and furthermore f j - i W s ( x ,d) n W s ( x ,d) # 8 since f J P i ( x )E fj-j W s ( x ,d). By Proposition 3.9, X contains a periodic point x, and hence O,(x,) = X (sincef : X + X is minimal). It remains to check the case rl > 0. We shall derive a contradiction by showing the existence of an open set U and a point p such thatf"(p) $ U for all n 2 0. We first construct a family of compact connected sets A, (n 2 0), a sequence of points x, E A, and an open set U c X such that (a) An n aEa(xn) # 8, (b) An c wEs(xn, d ) , ( C ) ~ - ~ " ( A3, ) A,,+, for some 0 < m, < N(6), (d)fJ(An) n U = 8 for all 0 < j < m,. Since 6 < r, there is a E X with Z;(a) n aEa((a)# 8 by Proposition 3.10. Now put X,
= a,
A0
= Z;(X~,),
and we show the existence of U # 8 with diam(U) < t r , such that U n A, # 8. Suppose U n A,, # 8 for all open sets U . Then A, is dense in Xand so A,, = X (since A, is closed). Since diam(A,,) < 26 andfis expansive, X consists of one point set, thus contradicting dim(X) > 0. If m, = 0 then (a), (b) and (d) hold for Ao. From now on we construct A, satisfying (c). Since fb n aE,(xo) # 8 and A, c W,(x,, d), by using Proposition 3.13, there exists 0 < m, c N(6), a, fl ES-""(A,) and compact connected subsets Aa, A, such that (a), (b), (c) and (d) of Proposition 3.13 hold. We can prove that (el U n {Ujmlof'(Aa)} # 8 * U n {UJ20f'(AS)} = 8. Indeed, if y E U n {uYL,f'(A,)} then y e f J ( A 8 ) for some 0 < j < m,. Hence there is z E A, such that fJ(z) = y. By hypothesis of (e), for x E U nT.(Aa)there is w E A, such that f'(w) = x . Since x , y E U , (f) d(T(w),f'(z)) = 4 x 9 Y ) < t r , , and since w E A, and z E A,
d(w,z) > 6 (by (c) and (d)). E K(a,d) and z E w,S(fl, d), and by (a), a E W i ( f l ,d). Hence W&(z, d). But d(f'(w),f'(z)) = d(x, y) 2 rl by the definition, thus
By (d), w
w
E
contradicting (f). Let A, be the connected component of B inf-"(A,) n Ea(B).For the case when (c) holds, we put x , = fl and A, = A,. Then the following is obtained: (a)' A, W ( X 1 ) # 8 (by (b)' A, c K(xI,d ) (by (d) in Proposition 3.13),
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667
(c)' c f-"'(Ao), ( 4 ' {U,20f'(Al)} n U = 0 (by (el). Replace A,, by A1 and repeat the above argument, then there exist a compact connected subset A2 and m, > 0 such that A2 c f-"'(A,), and (a)', (b)', (c)' and (d)' hold. Construct inductively such compact connected subsets {A,,}. Then we have
I\,
f"I(A,)
2fml+m2(1\2)
...
f"'l+'''+"n
(A,,)
3
* *
.
Since X is compact, we have r)n,Ifm'+'"+mn(A,,) # 8, from which take a point p. Thenf-"l(p) E A,. Sincef-"'+'(p) 4 U (0 < j < m l ) by (d)', we havef-'(p) 4 Ufor 0 < i < m , . By (d)'and the fact thatf-'"'+"2'(p) E A2, we have f -(ml+m2)+i(p) 4 U for 0 < j < m, and so f -i(p) 4 U for 0 < i < m , + m,. Consequently, we obtainf-"(p) 4 U for all n 0. 0
4. Pseudo-orbit-tracing property 4.1. Definition. Letf: X + X be a (surjective) homeomorphism of a metric space and d be a compatible metric for X. A sequence of points { x i:i E (a, b)} (- 00 < a < b < co) is called a 6-pseudo-orbit for f if d ( f ( x i ) ,x i + l ) < 6 for i E (a,b - I). A sequence { x i } is called to be c-traced by x E X if d ( f i ( x ) ,x i ) < E holds for i E (a, b). A homeomorphism is said to have the pseudo-orbit-tracingproperty if, for every E > 0 there is 6 > 0 such that each 6-pseudo-orbit for f i s &-tracedby some point of X. We abbreviate pseudoorbit-tracing property to POTP. Note that the notion of POTP is defined for continuous maps. 4.2. Remark.
Let d' be a metric for X that is uniformly equivalent to the metric d. Iff : X + X has the POTP under d, then f has the same property under d'. For some time the fundamental properties of the POTP will be investigated.
4.3. Theorem. Iff: X for k > 0.
+
X has POTP under some metric d, then so does f k
Proof. For every E > 0 choose 6 > 0 such that each 8-pseudo-orbit { x i } is &-tracedby some point of X. We prove that each b-pseudo-orbit { yi} forfk is &-tracedby a point. For 0 < j < k and n E Z put xnk+' = f'( yn). Then {xi}is a b-pseudoorbit forfand hence { x i } is c-traced by y E X, i.e., d ( f i (y), x i ) < E for all i E Z. In particular, for i = kn we have d ( ( f k ) " ( y )y,) , = d ( P k (y), xnk) < E (n E Z) and so y is a tracing point of { y i } for f k . 0
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4.4. Lemma. I f f :X + X is uniformly continuous under d and N is a natural number, thenfor every E > 0 there is 6 > 0 such that each &finite pseudo-orbit {x,:O < i < N } satisfies d ( f ’ ( x o ) ,x i ) < E for 0 < i < N. Proof. Suppose the lemma holds for N - 1. Sincef is uniformly continuous, for every E > 0 there is 0 < el < E such that d ( x , y) < E , implies d( f ( x ) , f ( y ) ) < ) E (x, y E X). By the assumption there is 0 < 6, < E such that each 6,-pseudo-orbit { x i :0 < i < N - l } is &,-tracedby xo E X . We show that each $d,-pseudo-orbit { x i :0 < i < N } for f is &-tracedby xo E X. Since the +a,-pseudo-orbit { x i :0 < i < N - l} is a 6,-pseudo-orbit, the pseudo-orbit is &,-tracedby the point x,. Hence d( f i(xo),x i ) c E , < E , 0 < i < N - 1 (especially d( f N-’(xo),x N - , ) < E , ) , and so d( f 0f N-I(xo), f(XN-I)) < + E . Since {x,:O < i < N } is a )6,-pseudo-orbit, we have d( f(XN-I)r X N ) < i d l < $ 8 and therefore d( f N ( x O x) ,N ) < E. 0
4.5. Theorem. Let f : X 4 X be a uniformly continuous homeomorphism of a metric space with metric d. I f f k has POTP under d for k > 0, then so doesf.
Proof. (1) Let E > 0. Then there is an E > E , > 0 such that (i) each c,-pseudo-orbit {xi:O < i < k} satisfies d(f’(xo),x i ) < + E for 0 < i < k (by Lemma 4.4), and (ii) d(x, y) < el implies m a x { d ( f ’ ( x ) , f ’ ( y ) ) : O < i < k } < +&. (2) Let E , be as above. Then there is a 6, > 0 such that each 6,-pseudoorbit for f is &,-tracedby some point. (3) Let E , and 6, be as above. Then there is a 6 > 0 such that each 6-finite pseudo-orbit { z i : 0 < i < k } is 6,-traced by z, E X (by Lemma 4.4). With these properties we prove that each 6-pseudo-orbit { y i :i E Z } for f is &-traced by some point. Write xi = yki for i E Z and fix i E Z . Since { y k i + j0: < j < k) is a 6-finite pseudo-orbit for f , by (3) we have d ( f J (yki), yki+,) < 6 , (0 < j < k) and especially if j = k then d( f k ( yki), yki+j) = d( f k ( x i ) ,xi+,) < 6 , , i.e., { x i }is a 6,-pseudo-orbit for f Hence by (2) there is y E X with d( f ” ( y ) , xi) < E , ( i E Z ) , i.e., d( f k i ( y ) ,y k i ) < E , (i E Z). On the other hand, since { yki+j:0 < j < k} is a 6,-pseudo-orbit forf, from (l)(i) it follows that d(f J (yki),yki+j)< ) E (0 < j < k), and from (l)(ii), d(fki+’( y), f’(yki)) < +& (0 < j < k). Therefore, d( fki+’(y), yki+,) < E (0 < j < k). Since i is arbitrary, we have d(f”( y), y,) < E (n E Z ) and hence the point y &-tracesthe 6-pseudo-orbit { y,}. 0
’.
4.6. Theorem. With the notations and the assumptions of Theorem 4.5, i f f has the POTP then so does f - ’ .
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Proof. For every E > 0 there is 6 > 0 such that each 6-pseudo-orbit { x i } is &-tracedby a point y E X. Choose 6' > 0 such that d ( x , y) < 6' implies d ( f ( x ) ,f ( y ) ) < 6 , and put g = f - ' . It is enough to verify that each 6'-pseudo-orbit { y i } for g is &-tracedby some point. Since d(g( y i ) , y i + ' ) < 6' for all i E Z,we have d( yi, f ( yi+')) < 6 ( i E Z). Letting xi = y-i for i E Z,{ x i }is a 6-pseudo-orbit for f and hence there is y E Xwith d( f i ( y), xi) < E for all i E Z. This implies d( f - j ( y ) , x - ~ )< E for i E Z and therefore d ( g ' ( y ) , y i ) < E for all i E Z. 0 4.7. Theorem. Let X and X' be metric spaces and X x X' be the product space with metric d"((x, x'), ( y , y ' ) ) = max{d(x, y ) , d'(x', y ' ) } where dand d' are, respectively, metrics for X and X'. Lei f ; X -+ X and g : X' + X' be homeomorphisms and f x g be the homeomorphism defined by f x g ( x , x') = ( f (x),g ( x ' ) ) ( x E X , x' E X ' ) . Thenf x g has POTP ifand only ifboth f and g have the POTP. Proof. Supposef x g has the POTP. For every E > 0 let { x i }and { X I } be 6-pseudo-orbits for f and g, respectively. Obviously, { ( x i ,x,)} is a b-pseudoorbit for f x g, and hence there is ( y , y') in X x X' with d"((f x g)'( y , y'), ( x i , xlr)) < E for all i E Z. Then d ( f ' ( y), xi) < E and d'(gi( y'), X I ) < E for i E Z. The converse is proved by the same technique. 0 4.8. Theorem. Lei f and h be (surjective) homeomorphisms of a compact metric space X . Suppose h and h-' be continuous andput g = h f h-'. Then f has the POTP if and only if so does g. 0
0
Proof. For every E > 0 there is E' > 0 such that d ( x , y) < E' ( x , y E X ) implies d(h(x), h ( y ) ) < E . I f f has the POTP, then there is 6' 0 such that each d'-pseudo-orbit { x i }forf is &'-tracedby some point. Choose 6 > 0 such that d ( x , y) < 6 ( x , y E X ) implies d(h-'(x), h - ' ( y ) ) < 6'. We prove that each 6-pseudo-orbit { y i } for g is &-tracedby some point. To do this, put xi = h - ' ( y i ) for i E Z. Since d ( g ( y i ) ,y i + ' ) < 6 for i E Z, we have d(h-'g( yi), h-'( yi+')) c a', i.e., d( f (xi),xi+')< 6' for i E Z. Thus { x i }is a 6'-pseudo-orbit for f and so d( f i ( y ) , xi) < E' (i E Z) for some y E X. Thus d ( h o f ' ( y ) ,h ( x i ) ) < E and d(g'oh(y), y i ) < E ( i E Z). Therefore h( y) is an &-tracingpoint of { y i } for y. The converse is clear.
=-
0 4.9. Definition. Let A be a bounded subset of a metric space. Then A is said to be relatively compact if the closure of A is compact. Every bounded subset of the euclidean space is relatively compact.
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4.10. Proposition. Let X be a metric space such that each bounded subset is relatively compact. If to every E > 0 there is 6 = 6 ( ~ >) 0 such that f o r each k > 0, each 6-pseudo-orbit { x i :0 < i < k } for f is c-traced by some point,
then f has the POTP.
Proof. Let { x i: i E Z} be a 6-pseudo-orbit for$ Then it is enough to see that { x i } is &-traced by some point of X. Fix k > 0 and put zi = x i - k for 0 < i < 2k. Obviously {zi:O < i < 2 k } is a 6-pseudo-orbit for $ Thus there is z = z ( k ) E X with d( f ’(z),z i ) c E for 0 < i < 2k. If y ( k ) = f k ( z ) , then d ( f ’ ( y ( k ) ) ,x i ) < E for I il < k . Since { y ( k ) } is bounded, there is a subsequence { y(k,)} such that y(k,,) + x as n + a.For every i E Z we have k , > I i I ( n 2 no) for some no, and so d(f’(y(k,)), x i ) < E. Therefore, d(f’(x),x i )
< E for all i.
0
4.11. Theorem. Under the notations and the assumptions of Proposition 2.17,
g :X
+
X has the POTP ifand only i f f : 8+ 8 has the POTP.
Proof. It follows from the proof of Proposition 2.17 that f : 8 + 8 is uniformly continuous. We first show that iff : 8 + 8 has the POTP then so does g : X + X . Let do > 0 be as in Proposition 2.17. For every E > 0 there is do > 6 > 0 such that each 6-pseudo-orbit { p i } for f is c-traced by some point of 8.From now on we prove that each 6-pseudo-orbit { y i ) for g is &-tracedby some point of X. To do this, choose po E 8 with z ( p o ) = yo and fix po, and choose inductively pi E 8 where .(pi) = yi (i 2 1). Since f ( p , ) E 8, we have IL 0f ( p i ) = g 0 n ( p , ) = g( y i ) . On the other hand, since ~ ; +E i uh(g(yi)), there is auniquepi+i E Ua(f(Pi))s ~ ~ h t h a t n ( p i += i )Y i + i By the choice of { p i: i 3 0 } , { p i } is a 6-pseudo-orbit for f and hence there is an &-tracingpointy of { p,}, i.e., & f ’ ( y > , p i ) < E for i 2 0. Let ~ ( y =) z. Then for i 2 0
d(g‘(Z),yi) = d(g’on(y), y i ) =
WC Y),
=
d ( n o f ’ ( y ) , n(pi))
Pi)*
By Proposition 4.10, g has the POTP. To prove the converse, let do > 0 be as above. For every E > 0 we can find 0 c 6 c +do such that each 6-pseudo-orbit { y i } for g is &-tracedby some point of X. We may suppose that &p, q) < E ( p , q E 8 )implies f ( p ) , f (4)) c $6,. Let { p i :i 2 0 } be a 6-pseudo-orbit forfand let yi = n ( p i )for i >, 0. Obviously { y i } is a 6-pseudo-orbit for g. For, since f ( p i ) , p i + l )< 6 c 6, for all i 2 0,
a(
a(
d(g(yi)yi+l) = d ( a o f ( p i ) , n(Pi+l)) = J ( f ( P i ) , Pi+l) < 6 .
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Hence there is z E X with d(g'(z), y i ) < E for all i 2 0. In particular, since d(z, y o ) < E , there is q E U,(po)with n(q) = z. We show that c?(f'(q),p i ) < E for all i 2 0. Note that d(xofi(qI9 x ( p i ) )
=
d(giox(q), yi) = d(gi(z),yi) <
6.
It is clear that c?(q,p o ) < E when i = 0. If c?(f'-'(q),p i - l ) < E for i 2 1, then c?v'( pq i)) <, E. For, since c?Cr.fl-l(q),f(pi-I)) < 46, and c?(.f(pi-l),pi) < 6 < +So, we have c?(fi(q),p i ) < do. Since x : U,,(p) --+ U,,(n(p)) is an isometry, = d(gi(z), yi) <
aifi(q),pi) = d(noY(q), .(pi))
E,
and hence q is an &-tracingpoint of the 6-pseudo-orbit { p i } .Using Proposition 4.10, the conclusion is obtained. 0 4.12. Remark. Let X be a compact metric space and denote the product space by 2 = X where each X, is a copy of X . A metric c? for 8 is defined by J ( x , y ) = max{d(xi,yi)/2li1:i
E
Z}
( x = (xi), y = ( y , ) E 8 ) .
Obviously 8 is a compact metric space. The shift map a : 8 --+ 8, defined usually by a ( x i ) = ( y i ) , y i = x i + l , i E Z,is a homeomorphism. Then a : 8 --+ 8has the POTP.
Proof. For every E > 0, choose 6 with 0 < 26 < b-pseudo-orbit for a. Then for i E Z J(o(x'), x i + I ) 2 d(a(x'),, x/y)/21kl
for all k
E
x i ' , xi, xi,
E.
Let { x i :i E Z} be a
= d(x;+l, xp)/2lkl
Z, and hence d ( ~ : + x~?," ) < 2Ik16(i, k E Z). Write x = (. . . , . . .). Then x E 8 and (dx)k = x F k (i, k E Z). If k 2 0 then k- I
d(XL, XA+k)
< 1 d(XLt5, X L t T ; )
< 2k+16
j=O
and if k < 0 then d(xL, x:') for i E Z.
< 2Ik1+'6.Therefore &xi,
ai(x))
< 26
<
E
0
4.13. Remark. The shift a : Y f + Yf has POTP (by Remark 4.12) and is expansive (by Remark 2.7). 4.14. Definition. A new metric d for Y : is defined by d(x, y) = 2-" if m is the largest natural number with x, = y . for all I n I < m,and d(x, y) = 1 if
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xo # y o . Such a metric is uniformly equivalent to the metric defined in
Remark 4.12. If S is a closed subset of Yf with o ( S ) = S then cIs:S -, S is called a subshift. We sometimes write oISas B : S + S. A subshift B : S -, S is said to be of finite type if there exists some natural number N and a subset C c X :Yk with the property that x = ( x i ) E S if and only if each block ( x i , . . . , x i + N )in x of length N 1 is one of the prescribed blocks. The smallest such natural number N is called the order of the subshift of finite type.
+
4.15. Theorem. Every subshift offinite type is topologically conjugate to one of order 1.
Proof. We just take a new symbolic space Y consisting of the allowable blocks of length N, and then the conjugacy is given by
Obviously the shift 6 : Y E -, Yz is of order 1 and cp
0
B
= 6 0 cp holds.
0
4.16. Definition. Let A be a k x k matrix of 0’s and l’s, called a transition matrix. Associated to A is the directed graph G ( A ) with vertices { 1, . . . , k } and an edge j -+ i if and only if A , = 1. For example,
G(A) =
c@-@ t-2 0
A is called irreducible if to every pair of vertices i, j there is a path in G ( A ) beginning at j and ending at i. We define the compact set X A = { ( x i ) : A , + , ,= 1, i E Z} c Yf for k 2 1. The point ( x i )E Z A called a k-symbol sequence. These concerned to two sided infinite paths in G ( A ) with both points of Y,. The shift B : X A -, X A is given as usual. Such a shift is called the Markov subshift. Every subshift of finite type with order 1 is a Markov subshift (see Definition 4.14).
4.17. Theorem (Walters [1979]). Let B : S -, S be a subshift. Then B :S has the POTP if and only if. is a Markov subshift.
-+
S
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Proof. (e): Note that a is of order 1. Let E > 0 and take m > 0 such that 2--"' < E . Then xi = yi for I i I < m when d(x, y ) < 2-('"+'). Let { x i :i E Z} be a 2-("+')-pseudo-orbit for qS,i.e., { x i } c S a n d d(o(x'), x i + ' ) < 2-(m+') for all i E Z. Then X I = xF' for all i E Z. Now put x,, = x; for n E Z and x = (x,,). Then (x,,, % + I ) = (x;,
Xgn+l)
= (x;, x39
n
E
Z,
and hence x E S. Since ( 0 ' ~=) ~xi+j = xi for l j I < m and i, we have d ( a i ( x ) ,x i ) < 2-" < E for i E Z, and therefore x is the &-tracingpoint of {Xi}.
(a): Let E > 0 be an expansive constant for alsand let 0 < 6 < there is N > 0 such that
=. d ( x , y ) < 6. Denote by B the set of all blocks ( x i - N , . . . , x i , . . . , x i = yi (I il
2N
+E.
Then
- 1)
+ 1 for all x E S and write
in x with length
S ( B ) = { x E Y;:(xi-,,,, . . . , x i , . . . , xi+,,,)E B, i E Z}. Then S ( B ) 3 Sand a(S(B)) = S ( B ) . It is easily checked that S ( B ) is closed and ols(B)is of finite type. Since ( yi-,,,, . . . , yi+,,,) E B for each y E S ( B ) and i E Z, there is x i E S with yi+j = xi for 1 j I < N , and hence d(oi(y), x i ) < 6 (i E Z). Hence { x i } is a b-pseudo-orbit for oIS.Since a : S -, S has the POTP by assumption, there is z E S with d(a'(z), x i ) < + E for all i E Z, and so d(oi(z), a i ( y ) ) < d(oi(z), x i ) d(x', a'( y ) ) < E (i E Z). Since z, y E Sand a : S + S is expansive, we have z = y and so y E S: i.e., S ( B ) = S. 0
+
4.18. Remark. Let X be a compact totally disconnected metric space. If
f :X + X is an expansive homeomorphism with the POTP, thenf is topologically conjugate to a subshift of finite type (use Theorems 2.8 and 4.17 for the proof). 4.19. Definition. Letf: X -, X be a (surjective) homeomorphism of a compact metric space. A point p E Xis nonwundering forfif, for any neighborhood U of p and any integer no > 0, there exists n > 0 such that In I > no and f " ( U ) n U # 8. 4.20. Remark. The set n(f)of nonwandering points is closed andfinvariant (f(Q(f))= Q ( f ) ) .The limit sets w(q) and a(q), for q E X , are contained in Q(f).In particular, every fixed or periodic point off belongs to n(f).Such a set Q(f)is called the nonwundering set.
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A point x E Xis said to be recurrent if, for any neighborhood U of x, there exist infinitely many n with f " ( x ) E U (in other words, x E a(x) u w ( x ) ) . Every recurrent point belongs to f2(f ). Let c( f ) denote the set of all recurrent points off. The closure of c( f ) is called the set of central motion (Birkhoff [1972]). We can check that if c( f ) is dense in Q( f ) then c( f ) is a Baire set in f2( f ). In fact, let F(x) = inf{d(f"(x), x ) : n > O}. Then ccf) c {x E nCf):F(x) = O}. Since F(x) is upper semi-continuous, { x E f2( f ) : F ( x ) < l/k} (k 3 1) is open dense in f2( f ), and consequently c( f ) is a Barie set in f2( f ). 4.21. Lemma.
Let f : X + X be a (surjective)homeomorphism of a compact metric space. I f f has the POTP, for every E > 0 and x E Q( f ) there exist y E X and k = k ( x , E ) > 0 with O,k(y) c U,(x).
Proof. Let +E > 0 and 0 < 6 < E be the number with the property of the POTP. Since x E f2( f ) , there are z E X and k > 0 such that z, f k ( z )E VaiZ(x). Write znk+i= f ( z ) for n E E and 0 < i < k. Then { z i :i E E } = {. . . , z, f (z), . . . ,f'-'(z), z , . . .} is a periodic b-pseudo-orbit, and hence there is y E X with d ( f ' ( y), z ) < $ E for i E Z. Since d ( y k (y), z ) < $6, we have for n E Z and so Of,(y ) c U,(x). 0 d ( f " ' ( y ) , x) < 4.22. Theorem. Let f :X + X be a (surjective)homeomorphism of a compact metric space. Suppose X is connected and not one point. I f f : X + X is minimal, then f does not satisfy the POTP.
Proof. Let 1 > 0 be the diameter of X and put E = $l. Supposef has POTP. is y E X and k > 0 such that Let x E f2( f ) . By Lemma 4.21 there O,,( y ) c U,(x), and obviously X = O,,( y ) u O,,( f y ) u * u O,,(f-'y). By connectednessand minimality we have O,,( y ) = Xand therefore 1 < 2 ~ . This is a contradiction.
-
4.23. Convention. Let
<
a
X be a compact metric space and for n > 0 let
l}beafinitesubsetsuchthatpi#pj(i#j). Choose E with 0 < 2.5 < min{d(p,, pi) : i # j } . Supposef : X + Xis a continuous surjection with the POTP. By definition there is a > 0 such that each a-pseudo-orbit is &-tracedby some point. Let m > 0 and define an n x n matrix A(m) by 1 if there is an a-pseudo-orbit from pi to pj of length A 0, . = = { p , ~ X : 0 i < n -
0 otherwise.
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Then the two sided Markov subshift c : XA(,,,) + XA(,,,) with transition matrix A(m) is defined. 4.24. Theorem. Let f : X + X be a homeomorphism of a compact metric space. I f f has the POTP, for m > 0 there exist a closed subset Y and a continuous surjection 7~ : Y + XA(,,,) such that f "( Y ) = Y and the diagram
n
i/"f X:A(m)
commutes.
7X A ( m )
Proof. The proof is due to (Shimomura [1988]). For n > 0 let Y, and A(m) be as in Convention 4.23. We define
Xi(,,,)= { ( x - , , . . . , x,)
E
Y?+' :Akl = 1 if xi = pk and xi+, = p, for - a f i < a }
for a > 0, and B,(x) = { y E X : d ( x , y) f E } where E is as in Convention 4.23. It follows then that for x = ( x - , , . . . , x,) E Zf;,), a
P,(x) =
i=
-*
f-'"(Be(xi))
is nonempty. Indeed, put yj = x - . + ~for 0 < j f 2a. Then for 0 < j < 2a there is an a-pseudo-orbit Oj from y j to yj+I with length m + 1. By combining {Oj:O f j < 2a} we can construct an a-pseudo-orbit {z,,, . . . , z2-} with zjm = y j for 0 f j f 2a. Since f has the POTP, there is y E X with d(f k ( y), z k ) < E for 0 f k f 2am. This implies that d( fj"( y), y j ) f E for 0 f j < 2 4 and hencef"(y) E P,(x) # 0 as desired. Since d ( p , , p,) > 2.5 for k # I, it follows that P,(x) n P,(x') = 0 for x , x' E X",,, with x # x'. Write K, = U { P , ( x ) : x E X'&")} and K = ll,"lK,. Sincef " ( P a + l ( x ) )c P,(x) for a > 0 and x E Xcf;?;), we have f " ( K , + l ) c K, for a > 0 and so f"(K) c K . For the case when the set XA(,,,) = { ( x i ) E Y,' :Akl = 1 xi = pk and xi+l = p , for i
E
Z}.
is not empty, it is enough to verify the theorem. Note that K 3 U { P ( x ) : x E XA(,,,)} where P ( x ) = f -'"(BE(xi)).It is easily checked then that P ( x ) n P ( x ' ) = 0 for x , x' E XA(") with x # x'. On the other hand, since P,(x) n P,(x') = 0 for a > 0, we see that K c U { P ( x ): x E XA(,,,)}.Therefore K = U { P ( x ) : xE XA(,,,)}.
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Define a surjection n : K + ZA(")by n ( P ( x ) ) = x for x E CA(").Then it is checked that n is continuous. Indeed, fix a > 0 and put /3 = min{d(p,,p,):O
< k, I < n
- 1, k # l } - 2.5 > 0.
By continuity we can find 6 > 0 such that d ( y , y') < 6 implies d ( f ' " ( y ) , f " ( y ' ) ) < /3 for l i l < a. If xi # x,! for some I i l < a, then 2.5 p < d(x,!, x i ) and so d ( x i ,xl) < d ( x i , f ' " ( y ) ) d ( f i " ( y ) , f ' " ( y ' ) ) d ( f ' " ( y ' ) , x,:) < 2.5 + @. This is a contradiction which shows that n is continuous. Since n - ' ( x ) 3 f ' ( n - ' u - ' ( x ) ) for x E ZA(") and i 2 0, we have n - l ( x ) 3 n E o f i m ( n - ' u - ' ( x ) )# 8 and so n-'(x) n Y # 8 where Y = n E o f " ' ( K ) . Therefore K ( Y ) = ZA("). It is easily checked that a n = n of" where a is the shift of CA("). 0
+
+ +
0
4.25. Remark. Let (X,,?) denote the inverse limit of a continuous surjection f of a compact metric space X and be a metric for X, defined as in Remark 2.30. 1ff:X + X has the POTP then so doesf: X, + X,.
a
Proof. Let a = diam(X) and choose N > 0 with ~ r / 2 ~ < - ' E for E > 0. Then there is E , > 0 such that d(x, y ) < E , implies max{d(f'(x),fJ(y)): 0 < j < N} < Let 6, > 0 be a number such that each d,-pseudo-orbit in X is &,-traced.Choose and fix 6 with 0 < 2"'6 < 6,. Let k > 0 and {?: 0 < i < k ) be a b-finite pseudo-orbit in X,, i.e.,
a&.
c d ( f ( x j ) ,xj+I)/2j 2 d ( f ( x k ) ,x?')/2N rn
6 > 2(3(xi), x i + , ) =
j=O
for 0 < i < k - 1. Then d ( f ( x h ) ,x c ' ) < 2N6 < 6, for 0 < i < k - 1. Sincef: X 4 X has the POTP, there is yN E X with d ( f ' ( yN), xk) < E , for 0 < i < k. Let xi = f N - ' ( y N ) for 0 < j < Nand xi+, = f ( x j ) f o r j 2 N, and put 2 = (xo,x,, . . . , x N , x N + , , . . .). Then 2 E X, and m
for 0
N
< i < k. This implies that 3:X, + X, has POTP.
0
4.26. Remark. Let X be a compact metric space and 93 be the family of Borel sets of X, i.e., the smallest a-algebra containing all open subsets of X. By a Borel probability measure p on X we shall understand a countably additive nonnegative set function p on 93 with the property that p ( X ) = 1.
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Let C ( X ) be the Banach space of continuous real valued functions on X with the sup-norm. Every measure on X induces a nonnegative linear functional on C ( X )by the map 5 H 5 dp. Conversely, by the Riesz representation theorem, to any nonnegative linear functional M on C ( X ) with M ( 1) = 1, there corresponds a unique measure p on X such that M ( 5 ) = J 5 dp. Let f:X + X be a homeomorphism. A measure p on X is said to be finvariant if p ( f - I B ) = p ( B ) = p(fB)for all B E 98 or, which is the same, if 5 ofdp = J 5 d p for all 5 E C ( X ) .Since Xis compact, there exists always finvariant measures on X (Krylov and Bogolioubov's theorem). Let Af (X) denote the space of allf-invariant measures on X. J f l ( X ) is obviously a convex set. With the weak topology for . I t / ( X ) : a topology given by taking as a base of open neighborhoods for p E A f ( X ) the set
G(r1,
* *
. r k : & l ,. . 3
*
9
Ek)
= ( v E A f ( X ) : l f c j d p - ftjdv1
<
cj,
1< j
< k}
with ej > 0 and 4; E C ( X ) , A,(X) becomes a compact metric space. 4.27. Remark. Let c ( f ) be the set of all recurrent points offand A,(X) be as in Remark 4.26. Then c ( f ) is a Borel set and p ( c ( f ) ) = 1 for all c1 E -,e,(X).
Proof. Let A be a Borel set of X. We write m
An = f " ( A ) ~ Ak.n
=
An,
Ak,m
= Ak\
u
Ak,n
n=k+l
for n, k 2 0. It is checked then that P ( A , , ~ = ) 0 for all p E A,(X).For, sincef(A,,,) = f ( A , n A , ) = A , n A n + Ifor n > I , it follows that
and by induction f ( A k , m )= A k + l , mfor k 2 1. Since Ak,, n A k + l , m= 0 for all k 2 0, p(UkAk,m) = p ( U k f k ( A , , ) ) = &p(AO,m). Therefore P(AO,,) = 0. Let { U @ ) }be a countable base of the topology of X , and define U$'L as above for U'"). Obviously for all p E A,(X), p ( H - ) = 0 where H - = UnUtL. To see x E a(x) when x E X\H-. Let x E X\ H-. Fix U$) with x E Ur'. If x 4 U:) for all m 2 1, x E UP)\ U:=lU:) = U$'L c H-,thus contradicting x E X \ H - . Hence x E @') = f ' " ( U 2 ) ) for some m 2 1, and so f - " ( x ) E U$", i.e., x E a(x). In the similar argument, letting
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678
H + = U;==,U&,, we have p(H+)= 0 for all p E Af(X)and x E o(x) whenever x E X \ H + . Therefore X\(H- u H+)c c( f ). If we established that X\(H- u H + )= c( f ) , it is concluded then that c( f ) is a Bore1 set and p(c(f )) = 1 for all p E &,(X). Let x E c ( f ) , then there is U(') with x E W ) .Since f " ' ( x ) E U(J)for an increasing sequence ni, x E f -"'(U(')) for all i 2 0 and so x E U, f -"(U(')). Hence x $ U(')\U, f -'(U(')) = Ui;:. This is true for all j with x E U"). Therefore x $ U7Ut: = H - , and in the similar way, x $ H+. We have x E X\(H- u H'). 4.28. Theorem (Aoki [ 19821). Let f :X + X be a homeomorphism of a compact metric space. If X is connected and not one point, and i f f is distal, then f does not satisfy the POTP.
Proof. Supposef has POTP and let diam(X) = 1. Let E = $1. By Lemma 4.21, for yo E Q ( f ) there are y E X and k > 0 with Ofk(y) c U,( yo). We write g = f for simplicity. Note that g :X + X has the POTP and is distal. Let 0 < 6 < E be a number with the property of the POTP for g. Since Xis compact and connected, we can find a finite sequence p I = y , p 2 , . . . ,p N such that d(pi, p i + l ) < +S (1 < i < N - 1) and X = UEl U,(pi). Since each x E X is almost periodic (by Theorem 1.21), there are c ( i ) > 0 with d(pi, gC(")(pi)) < +S, and so we define a sequence { x i :i E Z} by putting xi = g i ( y ) ,
xi
< 0, 0 < i < c(1) - 1, 0 < i < 4 2 ) - 1, i
= gi(pl),
xc(l)+i= g i ( p 2 ) ,
= g'(Y), i
2 0. Obviously {xi}is a b-pseudo-orbit and is 6-dense in X . Since g has the POTP, there is z E X with d(g'(z), x i ) < E for all i E Z, and so for w E X there is w' E O,(z) with d(w, w') < 2.5. Write Xc(l)+2(c(2)+~~~+c(N-l)}+c(N)+i
N- I
c = c(1)
+ 2 1 c(i) + c(N). i=2
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Then we have d(gi(z), gi(pl)) = d(g'(z), g'(y))
E,
d(g'+'(z), g'(pl)) = d(g'+'(z), gi(y)) c
i E,
< ~ ( 1 )- 1, i 2 0.
and hence gi(z) E U,(giy)
=
gi+'(z) E U,(g'y)
U{U,(h):h E O,) = U,(O,(Y)),
=
i
< 0,
U,(O,(y)), i 2 0.
Letting O,-(z) = {gi(z):i < 0} and O,'(z) = {gi(z):i2 0}, we have do,- (4) = U,(Og(YN and g'(O,+ (z)) = U,(O,( Y)). Since O,(z) = 0,(4 0;(z), byBaire theorem one of O,-(z) and 0 ; (z) has an interior point in O,(z). Sinceg : OJz) + O,(z) is a homeomorphism, there -is a g-invariant (probability) measure p on O,(z) (Remark 4.26). Since O,(z) is a minimal set for g by Theorem 1.24, it follows that p( U ) # 0 for each non-empty open set U. This follows from the facts that UTagi(U) = o,(z)and p ( m ) < XTap(U). If O,-(z) contains interior points in O,(z), then O,-(z) = g(0,-(z)). Suppose O;(z). If V = nkrIg-k(O,-(z)) contains interiorpoints in g-'(O;(z)) Og(z),then there i s j 2 0 with g-J(z) E V, and since g(V) c V, O,(z)c V. Henceg-'(O,(z)) = 0; (z), a contradiction. If Vcontains no interior points in OJz), then ~(0,(z)\ V ) > 0 since 0,-(z)\ V contains their points in O,(z). Since O,-(z) = Ukzog-k(W)u V, we have Put W = O;(z)\g-'(O;(z)). p( W) > 0, thus contradicting p(OE-(z)) <1. If O2fz) contains interior Doints inOAz). then can Drove that 6 -we g(O: (z)) = 0 : (z). In any case we have O,(z) = 0,(z) or O,(z) = 0 : (z). Hence O,(z) = U,(O,(y)) and since O,(y) = V,(yo), & \
"
O,(z)
=
U,(O,(YN
=
U2,(Yo).
Since {xi} is b-dense in X and d(g'(z), xi) c
-
max{d(O,(z), x ) : x E X }
-
,I
E
(i E Z), we have
< 26
and hence X = U2,(0,(z)) = U,( yo), i.e., diam(X) = 1 < 8.2, thus contradicting that E = 61. 0 4.29. Remark. Let X be a compact totally disconnected metric group and f :X + X be a group automorphism. Supposefhas zero topological entropy
(see Section 9 for the definition). It is known (cf. see Aoki [1984]) then that Xcontains a sequence X = Xo 3 XI 3 * . of normal subgroups such that f ( X , ) = X, (n 2 0), nnXnis trivial and for n 2 0, the factor group X,/X,+, is finite. Then f : X + X is distal. For, to x , y E X (x # y ) there is n > 0
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680
such that xy-’ 4 X , . Since f J ( X , ) = X, for all j E Z, we have fJ(xy-’) # X, ( j E Z), which implies that d(f’(x), f’(y ) ) 2 d ( f ’ ( x )X , , f’(y ) X , ) > 0 (the metric d is translation invariant). Since X / X , is finite, we have inf{d(fJ(x),f’(y)) : j E Z} > 0. This implies thatfis distal. It is checked in (Aoki [1984]), however, that every group automorphism of compact totally disconnected metric groups has the POTP. Therefore distal homeomorphisms with the POTP exist in general on totally disconnected spaces. 4.30. Remark. Let X be a compact metric space and denote by id the identity map of X , Then id :X -+ X has the POTP if and only if X is totally disconnected (this was pointed out by Fuji). Proof. If Xis totally disconnected, then to every E > 0 there is a finite open cover% = {U,, . . . , U n } o f X s u c h t h a t U i n U, = 0 f o r i Z j . C h o o s e 6 with 0 < 6 < min{d(U,, q):i # j } and fix a 6-orbit {xk:k E Z} for the identity map. Then there is V, such that { x k } c U,.and so we can find in U, a &-tracingpoint for { xk},i.e., id : X + X has the POTP. Conversely, suppose dim(X) # 0. Then there is a closed connected subset F such that diam(F) > 0. Since F is compact, diam(F) = d(x,, y o ) = c0 for some x,, yo E F. Let E , = 3cO. Since F is connected, for any 6 > 0 we can construct a &orbit from xoto y o in F which is not &,-tracedin X . This is a contradiction.
0 4.31. Remark. (1) The set 9 of all homeomorphisms with the POTP is dense in the space of all the homeomorphisms S of the Cantor set (Dateyama [19831). (2) 9 is a Baire set in S of the circle (Yano [1987]).
5. Coordinate systems and topological stability Expansivity and the POTP are used to give some coordinate systems in a compact metric space. 5.1. Definition. Let f:X -+ X be a homeomorphism of a compact metric space. Thenfis said to have a canonical coordinate if for each 6 > 0 there is = ~ ( 8 )> 0 such that d(x, y ) < E implies W;(x, d ) n W;(y, d ) # 8. 5.2. Remark. Iff: X
+
X has the POTP, thenfhas canonical coordinates.
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Proof. For E > 0 let 6 > 0 be a number with the property of the POTP. For x, y E X we suppose d ( x , y) < 6 and define a sequence { x i }of X by xi = f ' ( x )for i 2 0 and xi = f '( y ) for i < 0. Then { x i }is a b-pseudo-orbit for$ Hence there is z E Xwith d ( f ' ( z ) , x i ) < E for i E Z. Then z E v ( x , d ) and z E w,"(y, d ) . 0 5.3. Definition. A surjective homeomorphism f : X + X is said to have a hyperbolic coordinate iff has a canonical coordinate and there are a compatible metric d, y > 0, 0 < 1 < 1 and a a 1 such that for each x E X
y E q < x ,d )
-
d ( f " ( x ) , f " ( y ) ) < a W x , y).
y E wy"(x,d ) * d( f -"(x),f - " ( y ) ) < al"d(x, y), n 2 0.
5.4. Theorem. A homeomorphism f : X + X i s expansive and has the POTP i f and only i f f has hyperbolic coordinates.
Proof. The implication (=+)follows from Remark 5.2 and Theorem 2.40. To obtain the converse we first show that f : X + X is expansive. Let d, a > 0, y > 0 and 0 < 1 < 1 be as in Definition 5.3. If y E wyS(x,d ) n v ( x , d ) then we have d ( x , y) < (aX')2d(x,y ) since d ( f " ( x ) , f " ( y ) ) < y for all n E Z. This implies x = y and hence W;(x, d ) n W y ( x , d ) consists of one point, i.e., f : X + X is expansive. Let E > 0 be a number to be determined later and 6 > 0 such that V ( X ,
d)n
V ( Y 9
d)
z0
if d ( x , y ) < 6. Choose N so large that aLNE < $5 and then a > 0 such that if { yi: 0 < i G N } isana-pseudo-orbit, then d( f '( y,,),y,) < + b (0 6 i < N ) by Lemma 4.4. Let r > 0 and consider an a-pseudo-orbit {xi:0 < i < r N } . Put xh = x, and x;i+l)N E
for 0
W(fNXh, d ) n K(x(i+l)N, d)
< i < N. Then we have d(f"(xb),f N ( x i N < ) ) adN&< 46, d(fN(XiN),~ ( i + l ) N ) < 46
(by the choice of a). Thus d( f N(x,!N), x('+,)< ~ )6 and so {x,;: 0 < i G N} is a 6-pseudo-orbit. Now put x = f +'"x:N) and for 0 < i < rN choose s such
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682
that sN f i
< (s +
1)N. Then we have r
d(f'(x),f'-N(x:,))
< 1
d(fi-lN(X;N),fi-rN+N(X;l-I)N))
I=J+l
r
< 1
< asA/(l
- A)
r=s+1
since x;,,,E w,"(f'"~~,-~),,,, d ) . On the other hand, since xi,,,E e ( x , , , d ) , we have d ( f i - " ( ~ ~ N ) , f i - " N ( < ~ sE, ,and , ) ) by the choice of a, d(f-"(x,,,,), x i ) i +6. Thus
d ( f ' ( x ) ,X i )
< UEA/(l
-
A)
+ + $5. E
This is less than /3 if E is small. 5.5. Definition. Letf :X + X be a (surjective) homeomorphism of a compact metric space. Let A(&) = { ( x , y ) : d(x, y ) < E } for E > 0. Thenfis said to have local product structure if: (A) there are 6, > 0 and a continuous function [ , ] :A(6,) --* X such that for x , y , z E X
1x9
XI
=
"x, yl, ZI = [ x , 4,
x,
[ x , [ Y , zll = [ x , 21,
f [ x , YI = [ f ( x ) , f ( y ) l
when the two sides of these relations are defined; (B) there exist 0 < dl < $6, and 0 < e < 6, such that for each x letting %(XI = { y E WO(X, d ) : d ( x ,y ) < dl},
vb,(x) = { y
E
E
X
w g x , d ) : d ( x , y ) < SI>,
N, = [%<x>, v,s,(x)l, the following conditions hold: (a) N, is an open set of X and diam(N,) < do, (b) [ , ] : q ( x ) x G, ( x ) + N, is a homeomorphism, (c) N, 3 B,(x) where B,(x) = { y E X : d ( x , y ) f e}. If f : X + X has the POTP, for sufficiently small 6 > 0 we have W;(x, d) n W;(y, d) # 8 i f x is neary (by Remark 5.2) and it is one point set iffis expansive. In the case write [ x , y ] = Wi ( x , d ) n W;( y, d). Then we check in the following theorem that [ , ] satisfies all conditions of Definition 5.5.
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5.6. Theorem. Let f : X -, X be a (surjective) homeomorphism of a compact metric space. I f f is expansive and has the POTP, then f has a local product structure. Proof. Let c > 0 be an expansive constant forf and fix Q, = ac. Then there is 0 < 6, < E , such that W;(x, d ) n W,"(y,d ) = {one point} for x, y E X with d(x, y) < 6, (by Remark 5.2 and expansivity). We define a map [ , ] :A@,) + X by assigning [x, y ] = q ( x , d ) n %(y , d ) to (x, y ) E A(6,). The map [ , ] is continuous. For, suppose a sequence {(x,, y,)} of A(6,) converges to (x, y) E A(6,). Put z , = [x,, y , ] . Since Xis compact, there is a subsequence {zn,}of { z , } that converges to z E X. Since z,, E W;(x,,, d ) , we have d(f'(x,,),f ' ( z , ) ) < E, (i 2 0, nj) and so d ( f ' ( x ) ,f i ( z ) ) < Q,, and z E W;(x, d). Similarly, z E W,"(y,d ) and z = [x, y ] . This shows that { z , } converges to [x, y ] . It is clear that [x, x ] = x for all x E X. Since [x, y ] E y o ( x , d ) we have [[x, y ] , z] E W&(x, d ) n W;(z, d ) and by expansivity then [[x, y ] , z] = [x, 21. Similarly, [x, [ y, z ] ] = [x, z ] . It is easily checked that f [ x , y ] = [f (x), f(y ) ] by uniform continuity. To obtain the conclusion of the theorem, we must prove (a), (b) and (c) of Definition 5.5. Define a map g, :X x A(6,) -, R by g,(x, ( y , z)) = d(x, [ y , z ] ) for x E Xand ( y , z ) E A(6,). Then g, is continuous and g,(x, (x, x)) = 0. By uniform continuity of g, we can find 0 < 6, < $5, such that diam{x, y , z} < 26,impliesd(x, [y, z ] ) < fd,.If(y, z ) E G ( x ) x GI(x)thend(x,[y, z ] ) < 46, and hence diam(N,) < 6,. To show openness of N,, let w E N,. Then there are y E G(x) and z E G,(x) with w = [y, z ] . Since d ( x , w ) < $ao, we can define maps P,,: Bs,,(w) -, W,U(x,d ) and P,:Bb013(w)-, W;(x, d ) by P,(4 = [v, XI and PS(V) = [x, 4, .u E 4 0 / 3 ( w ) . They are clearly continuous. Since w = [ y, 21, we have P,,(w) = [ y , x] = y and Ps(w)= z . Hence there is a neighborhood U c B,,,(w) of w in X such that P , ( U ) c q ( x ) and P , ( U ) c GI(x). If w E U then w E N, since w = [[w, x], [x, v ] ]by expansivity. This implies that N, is open in X . Hence (a) was proved. (b) is easily checked as follows: Define a map h :N, -+ (x) x G, (x) by h(w) = ([w,x], [x, w ] )for w E N,. Obviously, h is continuous and h itself is the inverse map of [ , 1. Put g,(x, y) = diam{x, [y, X I , [x, y ] } for (x, y ) E A(6,). Then g, :A(6,) + R is a continuous map, and hence there is 0 < e < 6, such that d(x, y) < e implies g,(x, y ) < 6,. This shows that [y, x] E G(x) and [x, y ] E GI(x) and therefore y = [[ y , x ] , [x, y ] ] E N,. (c) was proved.
G
N . Aoki
684
5.7. Definition. Let f : X + X be a (surjective) homeomorphism of a metric space. Then f is said to be topologically stable if, for every E > 0, there is a 6 > 0 such that ifg :X -+ Xis any homeomorphism with d( f (x),g ( x ) ) < 6 for all x E X , then there is a continuous map (need not onto) h : X + X with hog = f'o h and d(h(x), x ) < E for all x E X . 5.8. Theorem. Let f : X + X be a homeomorphism of a compact metric space. I f f has a hyperbolic coordinate, then f is topologically stable. Proof. By Theorem 5.4, f : X + X is expansive and has the POTP. Let c > 0 be an expansive constant for f and 0 < E < tic be a number with the property of the POTP for$ By expansivity it is checked that there is a unique x E X which &-tracesa given 6-pseudo-orbit. Indeed, let { x i }be a &pseudoorbit. Suppose d( f i ( y ) , x i ) < E (i E Z) for some y E X. Then we have d(f i ( y ) , f ' ( x ) ) < d ( f ' ( y ) , x i ) d ( x i ,f i ( x ) ) < 28 < c for all i E Z,and hence x = y . Let g : X + X be a (surjective) homeomorphism with d(g(x), f ( x ) ) < S for all x E X . Let x E X. Since d ( f o g " ( x ) , g"+'(x)) < 6 for n, { g n ( x ) }is a 6-pseudo-orbit for$ Hence there is a unique point h ( x ) E X whosef-orbit &-traces{g"(x)}.This defines a map h : X + X with d ( f " h ( x ) , g"(x)) < E for all n E Z and x E X. We have d(h(x), x ) < E ( x E X ) by putting n = 0. S i n c e d ( f " o h o g ( x ) , g " + l ( x ) )< ~ f o r Zna n~d d ( f " o f o h ( x ) , g " + ' ( x ) )= , d ( f " + l o h ( x ) ,g"+l(x)) < E (n E Z), we have h o g ( x ) = f o h ( x ) for x E X. Finally, we show that h is continuous. Let 1 > 0. Using Lemma 2.9, we can choose N > 0 such that d ( f " ( x ) , f " ( y ) ) < c for In1 < N implies d(x, y ) < 1.Choose q > 0 such that d(x, y ) < q implies d(g"(x),g"( y ) ) < tic for In 1 < N. If d ( x , y ) < q then
+
d ( f " o h ( x ) , f " o h ( y ) )= d ( h o g " ( x ) ,h 0 g " ( y ) )
< d(hog"(x),g"(x)) + d(g"(x),g " ( y ) ) + d(g"(y), h o g " ( y ) )
< E + c/3 + E
<
c,
In1
< N.
Therefore d ( x , y ) < q implies d(h(x), h( y ) ) < 1and the continuity of h is proved. 0 5.9. Remark. If X is a compact topological manifold and E > 0 is sufficiently small, then d(h(x), x ) < E for all x E X implies that h : X + X is surjective if h is continuous (Munkress [ 19631). However, this needs not be as in general spaces. We give an example due to Walters.
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Consider the shift a : Yf -+ Yf where Y, = (0, l}. Then a is expansive (by Remark 2.7) and has the POTP (by Remark 4.12). Let m > 0 and define g : GZ + YT by (g(x),) = xi,
i < -m,
(g(x)), =
-m
i > m,
< m,
(g(x))m = x-m*
Then d(g(x), a ( x ) ) < 1/2" where d is the metric for Y," given in Definition 4.14. If rn is sufficiently large then we have h o a = g o h for a continuous map h with d(h(x),x) < 1/2" ( x E Y f ) . Since g2"+'(x)= x for all x E Yf,we have h ( x ) = hog2"+I(x)= a2"+'oh(x) for x E Y,". Therefore h is not surjective. 5.10. Remark. Let f : X + X be an expansive homeomorphism with the POTP. Thenf is topologically stable by Theorem 5.8, i.e., to a homeomorphism g with d(g(x),f ( x ) ) < 6 for all x there is a continuous map h with hog = f o h and d(h(x),x ) < E for x E X (note that 0 < E < 3 c where c is an expansive constant for f ) . Suppose g is expansive with an expansive constant c' 2 c. Then h is injective.
5.11. Remark (Yano [1980a]). There exists a homeomorphism of the circle which has the POTP but not topologically stable. 5.12. Theorem (Morimoto [1979]). Let f : X -+ X be a homeomorphism of a compact topological manifold. I f f is topologically stable, then the set of all periodic points o f f , per( f ), is dense in Q(f ).
Proof. Let E > 0 and 0 .c 6 < E be a number with the property of topological stability. Since Xis compact, a(f ) is compact. Take and fix x E Q( f ). Let U be a coordinate neighborhood of x with U c I!I~,~(X). Since x E Q( f ), there is k > 0 with f k ( U )n U # 8. We may suppose that f i ( U ) n U = 8 for 0 < i < k. Choose x' E U with f k ( x ' )E U . Since U is connected, we can
686
N.Aoki
.construct a homeomorphism q : X + X such that q 0f k(x’) = x‘, q ( U ) = U and q( y ) = y ( y 4 V ) . Obviously, d(q(x),x ) < 6 for all x E X . Letting l = q of, d ( l ( x ) ,f ( x ) ) < 6 for x E X . Hence we have h 0 = f 0 h for a continuous map h with d(h(x),x ) < E for x E X. I t is easily checked that tk(x’) = x’. Since h 0 tk = f k 0 h, we have h(x’) = f k 0 h(x’) and so h(x’) E per(f ) . On the other hand, since d(x, h(x’)) < d(x, x ’ ) + d(x’, h(x’)) < 2c, we have h(x’) E V,(x). Since E is arbitrary, per(f ) is dense in a(f). 0
<
5.13. Theorem (Walters [1978]). Let f : X + X be a topologically stable homeomorphism of a compact manifold. If dim(X) > 2, then f : X --+ X has the POTP. For the proof we need the following two lemmas. 5.14. Lemma. Let f : X + X be a (surjective)homeomorphism of a compact metric space. Let k > 0 be an integer and T > 0, q > 0 be given. Then for any sequence {xo, . . . , x k }with d ( f ( x i ) ,x i + l )< T (0 < i < k - l), there exists a sequence { x i , . . . , x i } such that (a) d(xi, x,:) < q, 0 < i < k, < 22, 0 < i < k - 1, (b) d(f ( x ! ) , (c) x,! # xi, i # j , 0 < i , j < k. Roof. The statement is true for k = 0. Suppose the lemma is true for k - 1. Let T > 0 and q > 0 be given. We can suppose q < 2. Choose 1 > 0 such that d(x, y ) < l implies d ( f ( x ) , f ( y ) )< T and 1 < q. Let { x o ,. . . ,x k } be given such that d( f ( x , ) ,x , + ~<) T (0 < i < k - 1). By hypothesis there is { x i , . . . , x i - I } such that d(x,, x,’) < 1 (0 < i < k - l), d(f(x,’),x,’+~) < 22 (0 < i < k - 2) and x,‘ # XI if i # j (i < k - 1 , j < k - 1). Since d ( f ( x ; - I ) ,X k ) < d ( f ( X i - I ) , f ( X k - , ) ) d(f(Xk-,), X k ) < 22, wechoosex; such that x; # XJ’ i f j < k - 1, d(x;, x k ) < q and d(f(xL-,), x;) < 22.
+
0 5.15. Lemma (Nitecki and Shub [1976]). Let X be a compact manifold of dimension 2 2. Then there is 1 > 0 such that fi { ( p i ,qi) E X x X: 1 d i < k} is a finite set satisfying (a) for each 1 < i < k, d(pi, qi) < 1, (b) i f i # j then pi # pj and qi # qj, then there is a homeomorphism cp : X --+ X such that cp(pi) = qi for 1 < i < k, and d(cp(x), x ) < 2nl for x E X .
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Proof of Theorem 5.13. For E > 0 let 6 > 0 be as in the definition of topological stability. Suppose { x o , . . . , x k } be chosen such that d ( f ( x , ) , x i + l ) < t6/11for 0 < i < k - 1. By Lemma 5.14 there is { x i , . . . , x ; } such that d ( x i , xl) < E (0 < i < k ) , d ( f ( x ; ) , x ; + , ) < ad/. (0 < i < k - l), xl! # xi if i # j (i < k , j < k), andf(x() # f ( x J if i # j (i < k, j < k). By Lemma 5.15 there is a homeomorphism cp: X + X with d(cp(x), x ) .c 6 ( x E X ) and cp of(x,!) = (0 < i < k - 1). Put g = cp 0s. Then d ( g ( x ) , f ( x ) ) < 6 ( x E X ) and g ( x l f )= xi;, (0 < i < k - I). By topological stability there is a continuous map h : X + X such that d(h(x), x ) < E ( x E X ) and hog = f o h . Then d(f'oh(&), x i ) = d(hog'(xA), x i ) = d(h(x,!),x i )
<
E + E
=
< d(h(xl), x l f )+ d ( x l , x i )
2.5, O < i < k .
Hence for each { x o , . . . , x k } with d ( f ( x i ) ,x i + , ) < :6/n (0 < i < k - 1) there is y E Xsuch that d(f'( y), x i ) < 2~ (0 < i < k - 1). Thereforefhas the POTP. 17
6. Representations of maps with hyperbolic coordinates
We had seen in Sections 2 and 3 the properties of expansive homeomorphisms and positively expansive maps of compact metric spaces. In this section we shall characterize homeomorphisms with hyperbolic coordinates and in particular continuous surjections that are called expanding maps. 6.1. Definition. A C'-mapf : X + Xof a compact manifold without boundary is called an expanding diyerentiable map if there is a compatible Rieman-
nian structure, with induced norm 11 (1, and a constant 1 > 1 such that )I Df"(w)I( > I" 11 w 11 for n > 1 and all tangent vectors w . 6.2. Remark. Every positively expansive toral endomorphism of the n-torus is an expanding differentiable map (see Remark 2.27). 6.3. Remark. The existence of a positively expansive map of the circle which is not an expanding differentiable map is described in Coven and Reddy [1980] as follows: Let g : R + R be a C1-mapwith the following properties: (1) g(0) = 0, g(1) = 2,
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(2) g'(0) = g'(1) = 1 and g'(x) > 1 whenever 0 < x < 1, ( 3 ) g ( x ) = 2n g ( x - n) whenever n < x < n 1 (n E Z). In fact, the map g ( x ) = x x2(x - 2)' of [0, 11 satisfies (1) and (2), and is extended g to all of R by (3). We denote byf: S' + S' the C'-map induced by g. Since Dfn(v) = o for all n 2 1 and all tangent vectors o at z = 1,fis not expanding. Since the projection P : R + S' is uniformly continuous, we first show that g : R -+ R is positively expansive, and then f : S' --* S' is positively expansive by an argument similar to that in Proposition 2.17. Choose 0 < 6 < 3 such that I x - yl c 26 implies g'(x) - g ( y ) l < 5 and 1 > 1 such that I x - n I 2 6 (n E Z) implies g ' ( x ) 2 1.Then we obtain the following: If g'(x) rj Z for allj 2 0 then there exist infinitely many i 2 0 such that I g ' ( x ) - n I 2 26 for all n E Z. Indeed, if this is false and k is the largest integer with I g k ( x ) - nl 2 26 for all n E Z,then there exists n E Z such that either n < g k + ' ( x ) < n + 26 or n - 26 < gk+'(x) < n. We suppose the former without loss of generality. Let y = g k + ' ( x ) - n. By induction it is easily checked that 0 < g ' ( y ) < 26 for all i 2 0. By the mean value theorem, g ' + ' ( y ) = g ' ( t l ) g ' ( y ) where 0 < t, < g ' ( y ) . Since g ' ( y ) < 1, we have g'(t,) > 1 . Therefore { g ' ( y)} is increasing. Since it is bounded, g ' ( y ) + z < 1 and then g(z) > z. But g(z) = g'(t)z where 0 < t < 1 , and so g(z) # z. This is a contradiction. We now prove that 6 is an expansive constant for g. Let x > y. If g ' ( x ) , g ' ( y ) E Z for some i, j 2 0, then g'+'(x), g'+'(y) E Z and so Jg'+'(x) - g'+'(y)l 2 6. Suppose g'(x) rj Z for all j 2 0. The sequence {Ig'(x) - g ' ( y ) l } is nondecreasing. This follows from the fact that l g ' + ' ( x ) - g ' + ' ( Y ) l = I g ' W - g'(y)llg'(t,)lforsomeg'(y) < t, < &). Since g ' ( x ) .$ Z for all j 2 0, there are infinitely many i > 0 such that I g ' ( x ) - n 1 2 26 for all n E Z.For any such i and any n E Z we have that
+
lg'(x)
+
+
- g'(y)l < 6
* It, - n l 2
Ig'W -
nl -
I g ' W - g ' ( y ) l > 6,
and hence g'(t,) 2 1.If I g ' ( x ) - g'( y ) I < 6 for all i 2 0, then I g ' + ' ( x ) g"'(y) I 2 1 lg'(x) - g'( y ) 1 for infinitely many i 2 0. Thus ( g ' ( x ) - g'( y)} is unbounded, a contradiction. Therefore g and hencef is positively expansive.
0 6.4. Definition. A continuous surjection of a compact metric space is called an expanding map if it is positively expansive and open. 6.5. Remark. Let f :X + X be a positively expansive map of a compact
metric space. For an open set U with small diameter,f:U + f(U ) is homeomorphic. Thus if in particular X is a compact topological manifold without
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boundary thenfis a local homemorphism by the Brouwer Theorem (Spanier [1966, p. 1991). 6.6. Remark. If X is a compact manifold with metric d, its universal covering space is denoted by 8 with covering map P : 8 + X . Then there exist a metric 2 for 2 and q > 0 such that 0) if x , y E 2,&x, Y ) < q then d ( P ( x ) , P( y ) ) = z ( x , y ) , (ii) the covering transformations are isometries, (iii) d is complete, (iv) if A c 8 and the diameter of A is less than q, then the diameter of P ( A ) is less than q. We give a proof due to Duvall and Husch [1972]. Let 42 = { U,}be a finite open cover of X such that each U, is evenly covered by P . Let a be the Lebesgue number of 42. If A c X with diam(A) < a, then there exists a continuous map cp : A + 2 such that Po cp(x) = x for each x E A . Note that if A is connected and if cpI and cp2 are two liftings of A such that cp,(x) = cp2(x)for some x then cp, = cp2. Let V = { K} be a finite open cover of X by connected sets such that the diameter of each K is less than $a. Let /? be the Lebesgue number of V .For x , y E 8, define
Jcx, y ) =
i
min{+/?,d(P(x), P ( y ) ) } , if x , y lie in the same component of P - ' ( K ) for some i, )/3,
otherwise.
a
The difficulty in showing that is a metric is the triangle inequality. To see this, let x , y , z E 8. If either J ( x , y ) or J ( y , z ) is +/? then obviously Jky ) J ( Y , z ) 3 six, 4. y ) and y , z ) are both less than f/?.Then x , y E and y, Suppose z E y for some components and of P-'(K) and P - ' ( y ) respectively. Note that the diameter of { P ( x ) , P( y), P(z)}is less than 8. Then P ( x ) , P( y ) , P ( z ) E V, for some k . Let be the component of P-'(V,) which contains x . Then &' n # 0 and therefore V, u K has diameter less than a. Hence there is a unique lift cp: V, u V, + 8 such that K' u = cp(& u K). Since x , y , z lie in the same component of P - ' ( &) and d'(a, b) = min{+/?, d(a, b)} is a metric, we have J(x, y ) J ( y , z ) 3 J(x, z). Let q = +/?, then (iv) is obtained. The remainder is easily checked. 0
+
a(,,
a(
r
r
c
r
+
6.7. Remark. Let 2 be the universal covering space of a compact connected topological manifold X andf : X + X be a continuous surjection. A lifting
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690
3 o f f is a continuous surjection f : 8 -+ 8 such that
p
i
'
'
f '
1.
commutes.
X
Since 8 is simply connected,ris a homeomorphism ifJis a local homeomorphism. Let r be the group of covering transformations of (8,P). For y E r, letf*(y) = 30y of-'. Then3* :r -+ r is an injective group homomorphism. 6.8. Proposition. Let f :X + X be a positively expansive map of a compact topological manifold without boundary. Then its lijiingr: 8 + 8 is positively expansive.
This follows from the proof of Proposition 2.17.
6.9. Proposition. Under the notations and assumptions of Proposition 6.8, its lifting7 :8 -+ 8 is a homeomorphism and bothf a n d y - ' are uniformly continuOMS under
a.
The proof is obvious. 6.10. Proposition. Let (8,P ) be the universal covering space of a compact connected topological manifold X and f:8 -+ 8 be a lifting of a continuous surjectionf : X + X . SupposeTis a positively expansive map. Then there exist a compatible metric 6, > 0 and 1, > 1 such that
a,,
ab(K 9 ) < 6, (2,9 E 8 )* 12,Jo<% 9 ) < 4(3(2>, f(9)). In particular, 2, is uniformly equivalent to 2. The proof is similar to that of Theorem 2.32, in which compactness is used only to obtain the following through Proposition 6.8: for S > 0 there is no > 0 such that max{d(p(Z),p(J)):O < j < no} < c implies d ( 2 , J )< 6 . Here c is an expansive constant forJ 6.11. Proposition. Under the notations and the assumptions of Proposition 6.10, i f in addition X is connected, then there is a compatible metric for 8 such that for all 2,9 E 2,
a,
Ji,J-'(.?>> < (l/no)Ji<% 9).
69 1
Topological Dynamics
Proof. Let 6, be as in Proposition 6.10. Sincey-' is uniformly continuous under &, there is 0 < 6, < 6, such that ao(y(2),y(jj)) c 6, implies a,(% j j ) < 6,. Then ao(2,j j ) < 6,/A0 by Proposition 6.10. Let 2, j j E 2 and { Z i : 1 < i < 1 + l} be a finite sequence such that 2 , = 2, j j = RI+, and a,(:,, Zi+,)c 6, for 1 < i 6 1. Define a metric a, for .f by
where the infimum is taken over all finite sequences from 2 to J . Then a,) < (1/$)2,(2, 5). For, let {2,, . . . , 2,,,} be a finite sequence from 3(2)to 3(jj)with ao(Zi, Zi+,)c 6, for 0 < i < m. Then { 7 - ' ( 2 , ) ,. . . ,7-'(2,,,)}is a finite sequence from 2 to j j such that a0(3-1(2i),3-1(2i+l)) < 6, (0 < i < m - 1). Hence by Proposition 6.10 &(2i, 2i+l) =
a~<~-',3-'(~i+
and therefore X&(Zi, Z i + , ) 2 $al(2, j j ) , from which
t?,(f(.f),f(J))
2
noad:, 7).
Since the set of (2, j j ) for which there is a finite sequence {gi:1 < i < I + l} zi+,) c 6, (1 < i < I ) is open and closed, all pairs from Z to J with (2, j j ) of 2 have such finite sequence since X is connected. If ao(2,7) c 6, then a,(2, j j ) = 4(2,j j ) and so & is uniformly equivalent to El
a,.
6.12. Theorem. Every positively expansive map of the n-torus is topologically conjugate to an expanding toral endomorphism. 6.13. Remark. Theorem 6.12 was proved in Shub [1969] for expanding differentiable maps of tori. After that, it was shown in Manning [1974] that every Anosov diffeomorphism of tori is topologically conjugate to a hyperbolic toral automorphism. Recently, Manning's result has been extended in Hiraide [1988] to homeomorphisms with hyperbolic coordinates of tori. It seems likely that every Anosov differentiable map (not 1-1) of tori could not be topologically conjugate to a hyperbolic toral endomorphism, but its inverse limit is topologically conjugate to a hyperbolic group automorphism of a solenoidal group (for the definition of solenoidal groups, see Aoki, Dateyama and Komuro [1982]. Theorem 6.12 has been recently extended to positively expansive maps of solenoidal groups (Aoki [1988]). However it is yet unknown whether the same result is true for homeomorphisms with hyperbolic coordinates of solenoidal groups.
Proof of Theorem 6.12. We denote by 8" = R"/E" the n-torus and by P :R" + T" the natural projection. Let r be the group of covering
692
N. Aoki
transformations for P.Then r may be considered as the subgroup Z"of R". Letf: 8"+ T" be a continuous surjection and?: R" + R" be a lifting off. Then there is an injective group homomorphism 3 * : Z" + Z" such that 3(z + v) = 3*(z) + 3(v) for all z E H" and v E W. Since 3 * induces the linear map y : R" + R" such that ylZn = J(*, y projects to a toral endomorphism yo : T" + T". Then y is a lifting of yo. Hereafter let f:T" + T" be a positively expansive map. By Proposition 6.8,3: R" + R" is a positively expansive homeomorphism. We first prove that there is a continuous surjection F: R" + R" such that 3 0 h" = L o y and ii: Z" + Z"is the identity map. If this has been established, we have f o h = hoy, on T". To get the conclusion, we must prove that h is bijective. To establish the former, let V be the set of continuous maps B: R" + R" which are liftings of continuous maps from T" onto itself such that BIZ.= id. If A, B E V then D ( A , B ) = sup{d(A(v), B(v)):v E W"},where d denotes the euclidean metric, is a complete metric for V. We can check this as follows: (a) D ( A , B) < CO: let C be a compact subset such that Z" C = R" (C is called a covering domain). Then for any x E R" there is m E Z"with m + x E C . Since A* = B* = id,
+
d(A(x),B(x)) = d(A(x = d(A(x
+ m) - n, B ( x + m) - m)
+ m),B ( x + m)).
Hence sup{d(A(v), B(v)):vE R"} = sup{d(A(x), B ( x ) ) : xE C } . Since Cis compact, we have D ( A , B) < co. (b) D is complete: if Ai is a Cauchy sequence in V then Ai converges to a continuous map A : 88" + R" and A i ( v m) + A ( v + m) under D for any m E Z". But A i ( v + m) = m + A i ( v )and soA(v m) = m + A ( v ) . Then A E V. For A E V define T ( A ) = 3-Io A o y . Then we have T ( V ) c V. For, let A E V and m E Z". Then ?-IoAoy(v + m) =3-'oAoy(v) + m and so T ( A ) E V. Sinceyis positively expansive, by Proposition 6.1 1 there exist a compatible , for all x , metric dl and 1 > 1 such that d1(3-i(x),3-i(y)) < ( l / l ) i d l ( xy) y E R" and i > 0. Since d, is uniformly equivalent to d, for every E > 0 there is 6 > 0 such that d(v, u) < 6 (v, u E R") implies d,(w,u ) < E . Let a = { U, : I < i < k} be a finite open cover of a covering domain C such that the diameter of Ui E c( is less than 6. Then for a, b E C there is a finite sequence { X I , . . . , xk} Of c such that a = X I , b = xk and d(Xi, X i + l ) < 6 ( 1 < i < k - 1 ) . Since C is compact, length k of such sequences is independent
+
+
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693
of the choice of points a, b of C. For A, B E V, we have R" = A ( C ) + Z" = B ( C ) H" and so choose C, such that C, =) A ( C ) u B ( C ) as a covering domain. Then for v E R", A(w) and B ( v ) are represented as A ( v ) = a + 1 and B(w) = b 1 for some a, b E C, and some 1 E E". Since d is translate invariant, we have
+
+
+ I, b + I ) < 1 dl(xi + 1, xi+l + I)
dl(A(TJ),B(v)) = d,(a k
c ke
i= I
where { x i :1 < i < k} is a finite sequence as above. Therefore D'(A, B) = sup{d,(A(v), B(w)):v E W} < 00, i.e., D' is a metric for I/. It is easily checked that V is complete under D'. Since y : R" -+ R" is surjective, we have D'(A 0 y, B o y ) = D'(A, B) and hence
D'(T(A), T(B)) = sup d l ( J - l o ~ O y ( v ) , r - ' O ~ O y ( ~ ) ) vsuan
<
(l/A)D'(A, B ) .
Therefore T: V -+ V is a contraction under D'. Using the metric D', T has the unique fixed point Kin V. Thus 60y = 70fi on R" and so h o y , = f o h on 8". The remainder of the proof is to show that h is bijective. If we established that all the eigenvalues of y : R" -+ R" had modulus greater than one, then there is A' =- 1 such that A'd(v, u) 2 d(y(o), y(u)) for all TJ,u E R". Put T'(A) = y-' A orfor A E V. It follows then that T'( V) c Vand under the metric D constructed above, T'is a contraction. Thus there is a unique 6' E V with T'(6') = 6'. Since f i o y = 30k and / ? o r = y o R , we have L o y o / ? = 606'03and 6 ' o J b h " = 6 ' o L . y and hence~o(fio6')= ( f i 0 6 ' ) o r and y 0 (6'o i;> = (R o i;> o y. Since f i o R and 6' fi are in V, by the uniqueness of the existence we have f i o /? = 6' fi = id and consequently h : T" --* T"is bijective. It only remains to show that y is expanding. As usual, R" splits into the directed sum R" = E" 0 Es 0 ECwhere E", E s and E" are subspaces corresponding to the eigenvalues of y with modulus greater than one, less than one and equal to one. Now ylEUis essentially an expansion. Thus there is a norm on E" relative to which y,€$ is actually a contraction. With these norms, there is 0 c A' < 1 such that 0
0
0
II Y(v)IIu 2 A'-' II TJ IIuy
v E E",
II Y(TJ)IIs < A' II v [Is,
v
E
Es.
By using Jordan's normal form in the real field for y I E E ,E" splits into a finite directed sum E' = E'' 8 . . * 0 ECkof subspaces satisfying (iv) in the proof of Proposition 2.15, i.e.,
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694
(c) for 0
< i < k the dimension of E" is 1 or 2,
Here each p, : El' + E" is an isometry under some norm 11 I1 of E" and each 4 : El' + EL$-'is either a zero map or a map corresponding to the identity I
matrix. Let v, w
E
R" and split v, w into the sum
+ v s + v o + + vk, w" + w s + w o + . . . + wk,
v = v' w =
* * .
where v", w" E E", vs,w sE E s and v', w' E E" (0 a metric $ for R" by d(v, w)
=
II us - w sIIs,
max{II 0" - wuIIu, llv' - w' 11, (0
Then
induces a compatible metric
< i < k). Now we define
< i < k)}.
e for T"by
e(P(v),P(w)) = inf{G(v
+ z, w + z ' ) : z , z'
E
Z"}
such that e(P(v),P(w)) = G(v, w)whenever the distance from v to w is small enough. Let 6, > 0 be an expansive constant for f under e. Since h :T" + T" is continuous, we can choose E , > 0 such that e ( x , y ) < E , implies e(h(x), h ( y ) ) < 6 , . Put U(v, ) E , ) = (w E E'O:B(v, w) < +el} for v E E'O. Then there is a countable sequence {v, : J > 0 } such that UJU(v,, + E , ) = E'O. Since y(E'O) = Elo and Po = ylEcois a &isometry, for v, w E U(v,, + E , ) e(P(v), P ( W = 6(v, w) =
G(Y'(~)~ ~ ' ( 4 )
e(rP(v), Y;fYw)),
i 2 0, and hence e ( f ' 0h P(v),f'o h P(w)) < 6, for i > 0. By positive expansivity we have h P(v) = h P ( w ) for o, w E U(vJ,+ e l ) , and hence h-'(h P (v )) 3 P(E'O). Denote by M the closure of P(Eco).Then M is a torus subgroup of T". Let A be the subspace of R" corresponding to M . Then Z" n I@ # (0)because if not P , M : A + M is injective. Since h P(O) = h P(A)= P O &A),we =
0
0
0
0
0
0
0
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+
have L ( k ) c l ( 0 ) Z" and hence &A?) is a set consisting of one point. Since L(z + v) = L*(z) + &v), we have L*(z) = 0 for all z E Z" n k. However this cannot happen since h" is the identity map. Thus E'O = {0}and by induction E' = (0). By running on the above argument we have E s = (0). Therefore R" = E" from which y : R" + R" is expanding. 0
7. Chain components and decompositions Letf: X + X be a (surjective) homeomorphism of a compact metric space. The following concept was introduced by Conely [1976] in his study of flows on manifolds. 7.1. Definition. Let d be a compatible metric for X. A sequence of points { x i :a < i < b} ( - 00 < a < b < 0 0 ) is called a 6-pseudo-orbit of f if d(f(xi), x i f l ) < 6 for a < i < b - 1. Let x , y E Xand let a > 0, then x is a-related by y (written x 5 y) if there are a-pseudo-orbits off such that x,, = x, xI, . . . , xk = y and yo = y , y , , . . . , y, = x. If x k y for every a > 0, then x is related to y (written x y). We say x is chain-recurrent if x x. The set of all chain-recurrent points is called the chain-recurrent set and denoted by R(f).
-
-
7.2. Remark. (a) R(f) = R(fk)for every integer k # 0. (b) If x E R(f)thenfk(x) E R(f)for every k # 0.
Proof. This is immediate when x is a periodic point off. Suppose x is not a periodic point. Let k > 0 and 6, be the distance from x to the set {f(x),f'(x), . . . ,fk+"(x)}.Since f i is uniformly continuous, for E > o there is 6 > 0 such that d ( y , z ) < 6 ( y , z E X) implies d(fi(y),f"(z)) < min{s/(k l), 6 , / ( k 1)) for 0 < i < k 1. Let {xo,x I , . . . , x,} be a 6-pseudo-orbit from x to x. Then for 1 < i < min{k + 1, n }
+
+
d(fi(x), Xi)
+
< d(f'(xo),f'-'(x,N + . . . + d(f(x,-,),
+ . . . + d(f(Xi< 6 , / ( k + 1) + - + 6 , / ( k + 1) < 6 , .
=
d ( f ' - If(XO),f'-'(x1 )) *
Xi) I),
Xi)
'
Since d(f'(x), x) 2 6,, we see that xi # x for 1 < i < min{k + I , n } . Hence every 6-pseudo-orbit from x to x have more than k + 2 elements. A similar argument shows that d(fk'I(x), x ~ + <~ E) and so {f"(x), Xk+l, . . . , x,} is an E-pseudo-orbit from f k ( x ) to x. 0
N.Aoki
696
7.2. Remark. (c) R(f ) is closed.
Proof. Let a > 0. Since X is compact, f : X -, Xis uniformly continuous. Thus there is 0 c 6 < f a such that d(x, y) c 6 implies d( f ( x ) ,f ( y ) ) c fa. } a sequence in R(f ) such that x(’) --+ x as k -, m. Then Let { x ( ~ )be d(x(k),x ) c 6 for some k > 0. Thus there are $a-pseudo-orbit { x i } and n > 0 such that x, = x , = x(’). Put y i = xi ( i # 0 mod n) and yo = x (i = 0). Then { yi} is an a-pseudo-orbit through the point x , and therefore x E R ( f ). 0
n;==,{uk,,
7.3. Definition. For Y a subset let o ( Y ) = f k ( Y ) } .If y E Y then o({y}) is the w-limit set of y. A subset K of X is called an attractor if there is an open set U containing K with K = w ( U ) . 7.4. Theorem (Block and Franke [1983]). Let x E X . Then the following are equivalent: (1) x
B Wf).
(2) There is an open set U such that x 4 0,f ( x ) E U and f ( 0 )c U . ( 3 ) There is an open set U such that x 4 U , f ( x ) E U and f ( 0 ) c U . (4) There is an open set U such that x 4 U, f ( 0 ) c U and f ‘ ( x ) E U for some k > 0. (5) There is an attractor K such that x 4 K and ~ ( { x } )c K. The implications (2)*(3), (3)*(4) and (4)=45) are immediate. To show (1)=-(2) and (5)*(1), we need some lemmas.
7.5. Definition. We say that a subset Y of X is positively chain-invariant if for every y E Y and x E X\ Y , y is not related to x . 7.6. Lemma. Let x E X and E > 0. Let R,(x) denote the set of y E X such that there is an E-pseudo-orbit from x to y. Then R,(x) is an open positively chain-recurrent set such that f ( R , ( x ) ) c R,(x).
Proof. It follows easily from the definition that R,(x) is open and positively chain-invariant. To showf(R,(x)) t R,(x), let y E R,(x).There is 0 < 6 c E such that d( y , z) c 6 implies d( f ( y), f (z)) c E . Since y E R,(x), there is z E R,(x) with d( y , z) c 6. Let ( x , , . , . , x , } be an E-pseudo-orbitfrom x to z . Then { x , , . . . , x -’ f ( y ) } is an E-pseudo-orbit from x to f ( y ) and so f ( Y ) E R&(Y)* Thusf(R,(xN = R,(x).
691
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7.7. Lemma. Let Y be a positively chain-invariant subset of X . I f x fk(x) E Yfor k > 0 , then x # ~ ( f ) .
E
Y and
Proof. The proof follows directly from Remark 7.2(b) and the definition of positively chain-invariant subset. 0 7.8. Lemma. Let K c X. K is an attractor ifand only i f there is an open set U =) K such that f( 8 )c U and K = n,"3J"(8). Proof. Suppose K is an attractor. Then there is an open set Y 3 Ksuch that K = o(Y). If for each i > O , f i ( P) contains points outside of Y, there is a point of o(Y) outside of Y , thus contradicting w(Y) c Y. Thusfk(E) c Y for some k > 0. Let Yk-l be an open set such that Yk-1 3 f k - ' ( 8 ) , and let Y k - 2 , .. . , & and Yl be open sets such that 3 f ( Y ) andf( FJ c &+, for 1 < i < k - 2. Put u = Y, u & u * u Yk-1 u Y. Then f ( 8 ) c u and K = w(Y) = w ( U ) = The converse is obtained by taking Y = u. 0
n:=o,f.(o).
Proof of Theorem 7.4. It remains only to prove implications (1)-(2) and (5)=4). (1)*(2): Let x # R,(x) for some E > 0, and put W = R,(x). By Lemma 7.6, Wis open and c W. Suppose x E Sincef(w) c U c c W for some open set U, we have x 4 8,f ( x ) E U and f ( 8 ) c U . (5)*(1): By Lemma 7.8 there is an open set U such t h a t f ( 8 ) c U and K = n;=os"(u). Since x # K , there is m > 0 with x # f " ( 8 Let ) . W be an Sincef"(w) c W , open subset of X such thatf"(0) c W c U and x 4 it follows that W is positively chain-invariant underf". On the other hand, as ~ ( { x } )c K we have f ' ( x ) E U for some j > 0. Thusf(x) E for all i 2 j + m andfmk(x)E Wfor some k > 0. By Lemma 7.7, x # R(f") and 0 since R(f) = R(f'"), x # R(f).
f(w)
u
w.
w.
In order to see the interesting property of the chain-recurrent set, we prepare the following.
7.9. Definition. Let X be a topological space and 2' be the family of all nonempty closed subsets of X . The exponential topology of 2' is defined by assuming that the family of all sets B ( G ) = { F E 2': F c G } and C ( H ) = { F E 2': F n H # S} is an open subbase of 2' provided G and H are open subsets of X. It is easily checked that the family of all sets &Go, GI, . . . , Gn) = { F E 2': F c Go, F n Gi# 8, l < i < n } is a base of 2'.
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7.10. Proposition. I f X is compact, so is 2'. Proof. It is enough to prove that every cover of 2'whose sets belong to an open subbase of 2' contains a finite subcover. Suppose 2' = U,B(G,) u U,C(H,) where G, and H, are open in X . Put F, = X\U,H,. Then for each s we have F, n H, = 8, i.e., Fo 4 C ( H , ) and hence F, E B(G,) for some to, i.e., F, c G,, and hence X\G,, c X\F, = USA!,.Since X\C, is compact, there are si (1 < i < n) such that X\G, c H,, u . . . u Hsn. Let F E 2'. Then there are the following two cases: (i) F c GI,. Then F E B(G,,). (ii) F Q G,o,i.e., F n (X\G,,) # 8. Then there i s j such that F n Hs, # 8, i.e., F E C(HJ. Thus in both cases F E B(G,) u C(H,,) u . . . u C(H,"). 0 7.11. Proposition. Suppose X is Hausdor- If2' is compact, so is X . Proof. Let X = U,G,where each G, is open. It follows that 2' = U,C(G,). Since 2' is compact, we have 2' = C(G,,) u * . u C(G,,) for some n > 0. Let x, E X . Then { x , } E C(G,,) for somej < n. This implies that xo E G,, and therefore X = G,, u . . . u G,n.
n
7.12. Definition. Let X be a bounded metric space. A metric @ ( A ,B)
where d ( A , b)
=
= SUP
LB SUP
d ( A , b),
SUP
e defined by
I
d ( ~B) , , A , B E 2',
OEA
inf(d(a, b ) : a E A}, is called the Hausdorflmetric for 2'.
7.13. Proposition. I f a metric space X is compact, so is 2'. Proof. We denote by CO(2') and 0(2'), the exponential topology of 2' and the topology induced by e respectively. We first show that 0(2'), c 0(2'), i.e., each open ball with center A, R = ( F E 2' : @ ( A F , ) < E } , is open in 2'. Since A is compact in X , there is for each k 2 1, a:, . . . , a:k E A such that for each x E A and k we have d(a,k, x ) < l/k for some i. Let us put (I) (2)
G: G
= =
{ x E x : d ( x , a;) < ( x E X : d ( x ,A ) <
E
- l/k},
E}.
To obtain the conclusion, it is enough to prove that m
(3)
R
=
U k=
I
B(G, G:, . . . , G:k)
m
I
= U { F E ~ ' : F C G , F ~ G / <# i~< ,n ~k } . k=l
Topological Dynamics
If F E R, then p ( A , F ) < (4)
x
E
A
(5)
y
E
F
E
* d(x, F )
699
- l/k for some k. It follows that <
E
- l/k,
d(y,A) <
E
-
l/k.
(2) and ( 5 ) imply that F c G. On the other hand, letting x = a:, by (4) there is y E F such that d ( y , a!) < E - l/k. Thus F n Gf # 0 for 1 < i < nk (compare with (l)), and so F belongs to the right member of (3)). Suppose conversely that F c G and F n Gf # 0 for some k and all i with 1 < i < nk. Then we have by (2) (6)
y E F
=- d( y, A )
<
E.
On the other hand, let x E A and let i be such that d(x, a:) < I / k . Since F n Gf # 0, there is y E F such that d ( y , a:) < E - I/k (compare with (1)). It follows that d( y, x ) < E and hence d(x, F ) < E. Thus if x E A then d(x, F) < E . This fact and (6) yield F E R . Next we show that O(2') c 0(2'),. We can of course, restrict ourselves to the case when a subbase of 2' is contained in 0(2'),. Let A = {FE 2' : F c G } where G is open in X and G # X. Then A is a member of the subbase of 2'. If A is a compact proper subset of G then d ( A , X\G) = E > 0. Thus we have that F c G when F E A. For, suppose p E F\G. Then d ( p , A) 2 E and hence e(F, A ) 2 E , so F # R . Therefore R c A. LetA= U{F~2~:FnG#0}andAbecompactandAnG #@.For a E A n G put d(a, X \ G ) = E . Then we have F n G # @ for F E R, i.e., R c A. This completes the proof of the proposition. 0 7.14. Theorem (Robinson [1977]). r f f : X compact metric space, then R(f;R) = R( f ).
+
X is a homeomorphism of a
Proof. Let x E R ( f ) and C, = {xy)} be a periodic l/n-pseudo-orbit through x. Then C, is a finite set. In the Hausdorff metric, there is a subsequence Cnkthat converges to some compact set C c X. If we established that for every y E C and E > 0 there are a periodic c-pseudo-orbit {zi}through y with ziE C, then it follows that y E R ( f ; , ) c R(f).Since y E C,we have C c R ( f ) and x E C c R(f;,) c R(f;R).This is true for all x E R ( f ) and thus R ( f ) R ( f ; R ) . It only remains to show that there is a periodic c-pseudo-orbit through y with ziE C. Sincef: X + Xis uniformly continuous, there is 6 = 6 ( f e ) > 0 such that d(a, b) < 6 implies d ( f ( a ) ,f ( b ) ) < E . Since C,, converges to C , we
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can find n = nk such that ljn < $ 8 and the distance from C, to C in the Hausdorff metric is less than 6. Suppose {xi'.)}has a periodj and x z j = xi'.' for all i . For each x(")take zi E C with d(xj?),z i ) < 6, zi+j = zi for all i, and zi = y for some i. Then d(f(zi),zi+,) < d(f (zi), f ( x y ' ) ) d(f(xy'), x@,) d ( x c l ,z , + ~ )< E . Therefore { z i } is a periodic E-pseudo-orbit in C through y .
+
+
0 7.15. Definition. A subset Y of R ( f )is called chain-transitive if to any pair of points p , q E Y and any a > 0 there is a periodic a-pseudo-orbit { x i } through p and q. Y is a chain component off if Y is chain-transitive and each set that contains Y as a proper subset is not chain-transitive. Equivalently, a q. chain component is an equivalence class in R(f)under the relation p
-
7.16. Remark. (1) Every chain component off is closed and finvariant. (2) A chain component is a maximal chain-transitive set. 7.17. Theorem. Let A be a chain component of$ Zfg ;X + X i s a homeomarphism and h : X + X is a continuous surjection with h 0 g = f 0 h, then the following holds: (a) h(R(g)) = W f ) . (b) h k l ( A )contains every chain component of g that it meets. Proof. Since h is uniformly continuous, for each a > 0 there is fl > 0 such that if { x i } is a fl-pseudo-orbit for g then { h ( x i ) }is an a-pseudo-orbit for$ If C is a chain-transitive set for g , then h ( C ) is chain-transitive for S. Therefore (a) and (b) hold. 0 7.18. Theorem. I f f : X
+
X has the POTP, then R ( f ) = Q ( f ) .
Proof. It is enough to verify that R ( f ) c Q ( f ) . If x E R(f),then for every a > 0 there is a pseudo-orbit (xi}such that xo = x , x I ,. . . ,xk = x, and d ( f i ( y ) ,x i ) < a (0 < i < k ) for some y E X . Therefore, U,(x) n f-'(Ua(x)) f 8 where V a ( x )= { y E X : d ( x , y ) < a}, and hence x E Q ( f ) .
0 Take and fix E > 0. Let 6 = 6 ( ~ )> 0 be the number with the property of POTP and split Q ( f ) into a union Q( f) = U A , of equivalence classes Ai under the &relation ( x A y). Each Ai is clearly closed andfinvariant. From the following proposition we see that { A , } is finite, i.e., Q ( f ) = U f A , .
Topological Dynamics
7.19. Proposition.
Iff: X
-+
70 I
X has the POTP, then each A , is open in Q( f).
Proof. Let x E A i . For every y E Al there is in Q(f)a b-pseudo-orbit . . , x p = y.Choose0 c y c ibsuchthatf(U,(x,)) c V , ( x , ) . Then for every xi E U y ( x o )n Q(f),{xi, x,, . . . , x,} is a b-pseudo-orbit in Q ( f ) .On the other hand, let { yo = y , y , , . . . ,y , = x} = 0 be a b-pseudoorbit in Wf). Iff(Y,-l) E V,(xo) n Q(f)then @ \ { Y / } ) u {xi} = { y o , y , , . . . , y'-,, xi} is a b-pseudo-orbit since d ( f ( y,- ,), xi) < 2y c 6 , and hence y d xi. Supposef( y,- ,) # U,(xo) n Q ( f ) .Then there is z E V y ( x o n ) Q(f) with d ( f ( A- I ), U y ( x o )n Q(fN = d ( f ( y,- I 1, 4 c 6 and so d(x& z) < 27. Since z E Q(f)= R(f),we have z z, i.e., there is a periodic y-pseudoorbit {zo = z, z,, . . . , zb, z } in Q ( f ) . Since d(f(Zb), xi) < d(f(Zb), z) + d(z, xi) < 3y c 6 , the sequence (0\{ y , } ) u {zo, . . . , zb, xi} = { y o ,. . . , y , - , , z,, . . . , xi} is a b-pseudo-orbit from yo to xi. Therefore xi E A, and so 0 U,(xo) (7 Q(f)= A i . xo = x,xI,.
-
-
7.20. Theorem (Aoki [1983]). Let f:X + X be a homeomorphism of a compact metric space. I f f has the POTP, then so does .
Proof. Let E and 6 be as above. Since each Ai is open and closed, we have d(Ai, Aj) = inf{d(a, b ) : aE Ai, b E A j } > 0 if i # j . Put 6 , = min{d(Ai, A j ) : i # j } . Let 0 c a c min{b, S,} and { x i } be an a-pseudo-orbit in Q ( f ) . It is only to show that an &-tracingpoint of { x i } is chosen in Q ( f ) . It is easily checked that { x i } is contained in some A j . Choose x,, xb E { x i } and let a > b. Then we have x, A x b , so that there are k , , k, > 0 and a (k, + k,)-periodic b-pseudo-orbit { z i } such that x, = z(k,+k2)iand xb = z ~ , + ( ~ , +for ~ , )all ~ i. Put k = k, + k2. Sincef:X -+ X has the POTP, there is yo,$E X such that d(f'( y,,b), zi) c E for all i, and so d(fki+j(y,,b), zj) c E (i E Z, 0 < j < k). If D = { f k i ( y c r , b ) :Ei Z} is discrete, then there is 1 > 0 such that f'(y,,b) = y,b. Hence Y,,b E Q ( f ) . If D is not discrete, then we can find a subsequence {fk"(y,,b)} with f k i ' (ya,b) + y:,b E X as i' -+ co. Obviously, d( yi.6, x,) < E and d ( f J (y,,b), zj) < E for allj. To see yA.6 E Q ( f ) ,for B > 0 take J > 0 such that d(fki'( Ya,b), yi.6) c B and d(fki'+'(Ya,b),f( yi.6)) c B for i' > J. Then we have yi,6 i!,y& for /? > 0, and hence Y i + bE R(f) = Q ( f ) . If a subsequence of { & } converges to y as a -P - 00 and b -P 00, then y E Q(f)and d ( f ' ( y ) ,x i ) < E for all i, i.e.,An(f) has the POTP. 0 Let X be a compact metric space and f :X -+ X be a (surjective) homeomorphism. Consider the chain-recurrent set R(f) which is closed and
N. Aoki
702
-
finvariant. Then R ( f ) splits into a union R ( f ) = U R , of the equivalence classes R, under the relation x y. Each R, is closed andfinvariant. 7.21. Theorem (Smale [1967]). Iff: R ( f ) --* R ( f ) is expansive and has the POTP, then R( f)is represented as a j n i t e union R(f) = R, off-invariant closed subsets Ri such that J R , is topologically transitive.
uf=,
Proof. Note that the set per(f) of all periodic points is dense in Q(f).Let 6 > 0 be the number with the property of the POTP and let U,(R,) = { y E R ( f ) : d (y , R , ) < 6 ) . For p E U,(R,) n per(f) there is y E Ri with d( y , p ) < 6. Since A,(,) has the POTP, it follows that W " ( p , d ) n Ws ( ( y,d ) # 8 and W ' ( p , d ) n W " ( y ,d ) # 8. Thus there is yo E R1 with yo y , i.e., p E R,, and so R , 3 U,(R,) n per(f) 3 U,(R,) n per(f) = U,(R,). Therefore, each R, is open and closed. By compactness R ( f ) is expressed as a union of a finite set of {R,}, i.e., R ( f ) = UfR,. We next show thatf;,, is topologically transitive. Since each Ri is open and closed,f;,, has the POTP. Let U and V be nonempty open sets in R,. Since x y for x E U and y E V , we can always find in Ri a tracing point for a pseudo-orbit from x to y. Thus U nf'(V) # 0 for some I > 0. 0
-
-
7.22. Definition. The sets Ri are called basic sets forf. 7.23. Definition. A homeomorphism f :X + Xis topologically mixing if for nonempty open sets U and V , there is no > 0 such that U n f " ( V ) # 0 for all n > no. It follows from these definitions that topological mixing implies topological transitivity. 7.24. Theorem (Bowen [1975]). Under the assumptions of Theorem 7.21, there exists a subset E of a basic set R and a > 0 such that f " ( E ) = E, E n f ' ( E ) = 8 (0 < j < a),f;., is topologically mixing and R = U:-lfi(E). Proof. Define E = W " ( p , d ) n R for p E R n per(f), then it is checked that E is open in R. Indeed, let m be a period of p ( f " ( p ) = p ) and 6 > 0 be the number with the property of the POTP for J R . For q E U,(E) n per(f) ( f " ( q ) = q for some n > 0), there is x E W u ( p ,d ) n R with d(q, x ) < 6. Thus there is y E Ws(q,d ) n W u ( x ,d ) n R by the POTP for AR.Since W ( x ,d ) = W ( p ,d ) , we havefkm(y ) ~ f ~ u / u (d p) =, W ( p ,d ) for every k > 0, and so d ( f k m (y ) , f k m ( q ) ) = d(fkm"(y ) , q) + 0 as k + 00. Therefore q E W " ( p , d ) n R = E, i.e., U,(E) = E.
Topological Dynamics
703
Since f ( E ) = W " ( f p ,d) n R, obviously f'"(E) = E. We can find 0 < a < m such thatf"(E) = E. Since R is a chain component set, we have R = E uf(E) u . . . uP-'(E). We next prove that if F = W"(q,d ) n R for q E E n per(f) then F = E. Since U,(E) = E, we have W t ( q , d) c E. Let n > 0 be the period of q. Then W"(q,d ) = U,,Ofnm'W t ( q , d) (see Proposition 2.39) and so F c E. To obtain the conclusion, suppose p $ F. Then 0 < d(F, K )where K = E\ F. Since q E E = W " ( p ,d) n R, there is z E W " ( p ,d) n R with d(z, p ) < d(K, F ) . Thus z E F. On the other hand, d(f-"'"'(z), p ) = d(f-""'(z), f-"'"'(p))-+ 0 as j -+ 00. Thus f-""'(z) # F for sufficiently large j , and so z $f"'"'(F) = F, which is a contradiction. Finally, we prove that f Eis topologically mixing. Let U , V be nonempty open sets of E. Since E = W"(q,d) n R for q E V n per(f), we have U n W " ( f " q ,d) # 0 for j~ Z. Let n > 0 be the period of q, for 0 < j < n - 1 there isz, E U n W " ( f " q .d)such thatf-""'(z,) -f"(q)as t -+ co.Obviouslyf"'(q) ~f"(V). Fix 0 < j < n - 1. Then there is N, > 0 such that for every t > N,,f-@"'(z,) EP"'(V). Put N = max{N,:O < j < n - I } . For every t 2 N we have t = sn + j for some s > 0 and 0 < j < n - 1. If s 2 N then f - " ' ( z j ) = f-""""'(z,) E V . Since zj E U , f " ' ( V ) n U # 0 for t 2 nN. 0 7.25. Remark. Supposef: R(f) -+ R(f)is expansive and has the POTP. If, in additioqf: R(f) + R(f)is topologically transitive and R(f)contains a fixed point ofS, thenf: R(f)-, R(f)is topologically mixing. 7.26. Theorem. Suppose f:X -+ X is expansive and has the POTP. Iff;Rcr, is topologically transitive, then X = R(f).
Proof. Suppose X # R(f)and then we derive a contradiction by showing ( X \ R ( f ) ) n per(f) # 8.Since o ( x o )and a ( x o )are contained in R(f)for xo E X \ R ( f ) , we have 0; (x,,) n R(f) # 0 and 0; (xo) n R(f) # 0. Let 0 < E < d(xo, R(f))and 6 > 0 be the number with the property of the POTP f o r 5 Then d ( R ( f ) , f " ( x , ) )< 6 and d ( R ( f ) , f - " ( x , ) )< 6 for large n, and hence d ( x , + , , f " ( x o ) < ) 6 and d ( f ( x - , - , ) , f - " ( x o ) < ) 6 for some &+I, x-,-I E W). Now construct a 6-pseudo-orbit { x - , - ,,f - " ( x o ) f, - " + ' ( x 0 ) ., . . , f " - ' ( x o ) ,x,,+'} from x - , - ' to x n + , .Since J R ( / ) is topologically transitive, we can find in R(f) a 6-pseudo-orbit {x,,+',2,,+2r. . . , K,,,x - , - I > from xn+, to x - , - , . By combining the above two 6-pseudo-orbits, we construct a periodic 6-pseudo-orbit { x - , - ,,f - " ( x 0 ) ,. . . , xo, f ( x o ) ,. . . ,
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704
f"-'(x,,), x,,+~, 5,,+*, . . . ,Z-,,, x - , - ~ } .Sincef has the POTP, there is y E X &-tracingthe periodic 6-pseudo-orbit.Sincef is expansive,the pointy is periodic and since d(x,, Rcf)) > E , we have y $ R ( f ) , thus contradicting. 0
7.27. Remark. There exists an example of an expansive homeomorphism having the POTP of a compact metric space such that X # R( f ) . For example, let Y: = X i c z Y2and a : Y: -+ Y: be as before. Let S = { ( x i )E &':(xi, x i + l E) C, i E E } where C = ((0, 0), (0, l), (1, I)}. Then a : S + S is a Markov subshift. Since ols is expansive and has the POTP, per(al,) is dense in R(a,,). But S contains only two periodic points x = (. . . ,0, 0, . . .) a n d y = ( . . . , 1,1, . . .).Ontheotherhand,thepointz=( . . . , 0,0,1,1, ...) is in S and not periodic. Therefore z 4 per(a,,) = R(al,). 7.28. Theorem. I f f : X + X is expansive and f : R c f ) + R( f ) has the POTP, then there exists an open set U 3 R( f ) such that n'?,f"(U) = R( f ).
Proof. Let&> ObeanexpansiveconstantforJForO < B < $eleta > 0 be the number with the property of the POTP. By uniform continuity off there is 0 < y < min{ta, + E } such that d(x, y ) < y ( x , y E X ) implies 4 f ( x ) , f ( y ) ) < .4. Now let U = { y E X : d ( y , R ( f ) ) < y } and take y E n Y m f " ( U ) . Since fi( y ) E U for all i, there is xi E R(f ) with d ( f ' ( y), x i ) < y for all i, and hence d ( f ( x i ) ,xi+l)
d ( f ( x i ) , f ' + ' ( y ) )+ d ( f ' + ' ( ~xi+l) ), < +a + y < a
for i E Z. Since { x i }is an a-pseudo-orbit in R( f ), there is a B-tracing point x E R( f ) and for all i d ( f ' ( y ) , f ' ( x ) )G d ( f ' ( y ) ,xi) Thus x = y and y E R( f ) .
+ d(xi,f'(x))< Y + P
<
E.
0
7.29. Convention. Let f :X X be a homeomorphism of a compact metric space and A be a subset of X. Then we write Ws((A) = W"(A) =
U{W S ( x d, ) : x E A } , U{W " ( X ,d ) : x E A }
and for E > 0 v(A) = W,"(A) =
U{W ( Xd, ) : x E A } , U{W ( X , d ) : x E A } .
7.30. Theorem. I f f : X -+ X is expansive and has the POTP, and if R( f ) splits into the finite union R( f ) = RI v v R, of basic sets Ri,
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705
then Ws(Rj) = Wu(Rj) =
{ {
xE
X :lim d ( f " ( x ) ,Rj)
xE
X :lim d ( f - " ( x ) ,Rj) n-m
and for each E > 0 there exists an open set V, k>O
1
=0
n-m
3
= 0
Rj such thaf
0f'(V,) c
f - ' ( ~ , )c W,(Rj),
,
k>O
W,"(Rj).
Proof. Since each Rj is open and closed in R(f),JR, has the POTP. Thus to every E > 0 there is a > 0 such that every a-pseudo-orbit of Rj is ~/Ztraced by some point in Ri. Since f:X -P X is uniformly continuous, there is 0 c y c min{+a, + E } such that d ( x , y) < y implies d ( f ( x ) , f ( y ) )c +a. Suppose lim,+,d(f"(y), Rj) = 0 for y E X. Then we can find N > 0 with d ( f " ( y ) , Rj) c y for n 2 N,and so there is x,, E Rj such that d(x,, f"( y)) c y for all n 2 N. Letting x,, = y v N ( x N )for n < N , { x , , : nE E } is an a-pseudo-orbit in Rj. For, if n 2 N then d(f(xn), xn+l) G d(f(xn),f"+'(y)) + d(f"+'(yL x,,+,) G +a + y c a. When n c N we have d ( f ( x , ) , x,,+') = d(f(xn),f(xn)) = 0 a. Let x E Rj be a +tracing point of {x,,}. Then
d(f'of"(~),fi~f~(~))
G
xi+N)
d ( f i + N ( ~ ) q
< ~ + ) E < E ,
+ d(xi+N,f'+N(x))
i20.
HencefN(y) E w ( f " ' x ,d) c W s ( f " x ,d), and so y ~ f - ~ W ( f "dx) ,= W s ( x ,d). Therefore, y
E
W S ( x d, ) c
u{W ' ( X ,d): x
E
Rj} = Ws(Rj),
which implies
{
y
E
X :lim d(f"(y), Rj) n-m
1
= 0
c Ws(Rj).
nk,of-k(V,).
Next let V, = { y E X : d ( y , Rj) c y } and take z E Since d(f'(z), Rj) c y fork 2 0, thereisy, E Rjwithd(f'(z), yk) c yfor k 2 0. By putting y - i = f - i ( y o ) for i > 0 we obtain an a-pseudo-orbit { yk : k E E} of Rj. If x E Rj is an +&-tracingpoint of { y k } , then we have for k 2 0 4fk(X),fk(ZN
and hence z
E
G d ( f k ( X ) ,Y k )
+ d(Y,,f'(Z)) G
)E
w(x,d ) c V(Rj), i.e. nk20f-k(L$) c W,(Rj).
+Y
E,
0
N.Aoki
706
7.31. Theorem. Under the assumptions of Theorem 7.30, for y exists a unique Ri ( R j )such that
E
X there
lim d ( f - " ( y ) ,R j ) = 0
lim d ( f " ( y ) , R i ) = 0
(n-m
n-m
and furthermore,
X =
I
I
i= I
j=l
U W s ( R j ) = IJ
W"(Rj).
Ifi # jthen w s ( R i )n W s ( R j ) =
0,
W u ( R l )n W"(R,) =
0.
Proof. Letx E X. SinceXiscompact, we havew(x) c R U ) = R, u . . . u R,. We show the existence of Rj with o ( x ) c R j . To do this, it is enough to see that there is a unique Ri with Ri n w ( x ) # 0. Suppose that
0,w ( x ) n Rj, # 8 ( j # j ' ) , w(x) n Ri = 8 (i # j , i # j ' ) . Since R, n Rb = 0 for a # b, we can find open neighborhoods U, (1 < a < I ) with U, n ub = 0 (a # b). Sincef(R,) = R, for all a, we can (1)
w ( x ) n Ri #
take an open neighborhood V , such that (2)
f ( K ) c U , , f - ' ( K ) c U, (0
< a < I).
By assumption choose subsequences {f"'(x):i 2 l } and {f"'(x):i 3 1) such that
(3)
limf"'(x)
E
Rj, limfmg(x)E Rj.
1-00
1-w
We may assume that the subsequences { xn,}and { m , } are chosen such that (4)
n, < m , < n2 < m2 <
*
- . *
From (3) it follows that for a large Z (5)
i 2
=> fn8(x)E
y,
f""(x)
E
v,.
From (2), (4) and (5) together with the fact that U, n U, = follows that there is such that i 2
z => n, < 1, < m,,
f"(x)
4 y u y,.
0 (a #
b), it
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Topological Dynamics
Since X \ ( y u I$)is closed and { f " ( x ) : i 2 I } c X\( t: u I$),there is a subsequence of { f ' ( x ) } for which the limit point z is contained in X\( u I$). Thus z 4 R, u R,,. On the other hand, since z E w ( x ) , there is Rksuch that z E Rk for k # j , j ' . This contradicts (1). Finally we prove that lim,,,d(f"(x), R,)= 0 whenever w ( x ) c R,.If this is false then there are E,, > 0 and an infinite sequence {n,} such that d(f"l(x),R , ) 2 e0 for a l l j 2 0. Iff"'(x) + z ( j + 0 0 ) then d(z, R , ) b eO. But, since z E w ( x ) , we have z E R, which contradicts. Therefore lim,,,d(f"(x), R,)= 0. The proof of the rest of the statement is easy and therefore omitted. 0
8. Markov partitions and subshifts Markov partitions for Anosov diffeomorphisms were first constructed by (Sinai [1968]). After that, the same partitions for basic sets of Axiom A diffeomorphisms were founded by Bowen [ 197Obl. Ruelle [ 19781constructed Markov partitions for homeomorphisms satisfying some conditions that imply hyperbolic coordinates. Let f : X + X be a homeomorphism of a compact metric space. Suppose f has a hyperbolic coordinate. Then f has a local product structure by Theorem 5.6. Thus I$(x), v,S(x) and N, are defined as in Definition 5.5. 8.1. Remark. Let x E X. Then [ y, z] E N, for y, z
E
N,.
v(x)and wl,w 2
Proof. By the definition of N, there are u , , u2 E such that y = [uI,w , ]and z = [u,, w,]. Thus [y, z] [u,
9
2121 E
Nx.
E
vS;(x)
= [ [ u , , q ] , [uz, vJ] =
[7
8.2. Convention. We write D:,, = I$(x) n N, and D:,, = v,S,(x) n N, for x , y E X. 8.3. Lemma. Let Q be as in Dejinition 5.5 and let x, y E X with d(x, y ) < e. Then 0x4, is an open neighborhood of x in G(x) ( 0 = u, s), and the maps D;*, are homeomorphisms. [ , y] : D;, + DYU,, and [ y, ] : D:,, Proof. By definition N, is open in X. Thus 0x4, is open in V,q(x). Since E B , ( y ) c N,, we have x E 0x4, and hence 0x4, is an open neighborhood
x
N. Aoki
708
of x in G(x). Let z E OX",,.Since z E N,, we have [z, y] E v ( y ) and since z E y ( x ) c N, and y E B,(x) c N,, [z, y] E N'. Thus [z, y] E Dlx. Similarly we see that if z E D,!" then [z, x] E D:,. By using the properties of [ , ] it follows that [DXU,,,y] = Dix.By Definition 5.5(B)(b) the map [ ,y] :DX",,+ 0 ; ' is a homeomorphism and [ , x] :0;' + DXU,, is its inverse map. It is easily checked that the same is true for r~ = s.
Proof. Let w E ~ ( N ,n ) V,,(fx). Thenf-'(w) E N, and sof-'(w) = [y, z] for some y E q (x) and z E GI(x). Since % (2) c Wi(z, d) by definition, we have f - ' ( w ) E W i ( z , d). On the other hand, since w E V,:(fx), we have w E v,S,(fx) c W&(fz, d). By expansivity we have f-'(w) = z and so w = f(z) EYE,(x). Therefore,f(N,) n V,q(fx) = fv,S,(x) n v,S,(fx). Since N, is open in X , we have (a) and similarly (b). 8.5. Definition. A subset R of X is called a rectangle if diam(R) [x, y ] E R for x, y E R.
< e and
Hereafter, let R be a rectangle of X. 8.6. Remark. Since e is chosen such that e < d,, we have [x, y] = V,(x) n q ( y ) for x , y E R. It is clear that the closure of R, Cl(R), is a rectangle.
8.7. Convention. Let x E R. We write V(x, R) = q ( x ) n R, and
GI( x ) n R and V(x, R)
=
aVs(x, R) = V s ( x ,R)\Int Vs(x, R), aV"(x, R)
=
V"(x, R)\Int V"(x, R),
where Int V'(x, R) denotes the interior of V'(x, R) in Y ( x )
(0
= s, u).
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Topological Dynamics
Proof. Since R is a rectangle. Obviously [V"(x,R ) , V s ( x ,R)] c R . Let E R, then [z, x] E R n v ( x ) = V " ( x ,R ) and [ x , z] E V s ( x ,R). Hence z = [[z,XI, [ x , z ] ]E [V"(x,R ) , V s ( x ,R)].(a) was proved. Since diam(R) < e and y E R, we have R c B , ( y ) c Ny and so V"(x,R ) c D&. Similarly V"(y,R ) c DY,. Since [V"(y,R), y] = V"(y, R), by Lemma 8.3 we have [aV"(y,R ) , x ] = aV"(x, R ) and hence
z
[ a v " ( x ,R ) , v s ( x ,R)1 = [[aV"(y,R), X I , [x, V'((Y,WII = [aV"(y,R ) , V S ( y R)1 ,
(b) was proved. (c) is shown in the same way.
0
8.9. Convention. We write a'R = [aV"(x,R), V ( x ,R)]and a"R = [V"(x,R), aVs(x,R)].Note that these sets do not depend on x E R (by Lemma 8.8). Since R is a rectangle, obviously PR, a"R c R. The interior of R in X is denoted by Int R, and write aR = R\Int R.
8.10. Lemma. (a) Int R = [Int V u ( x ,R ) , Int V'(x, R)]forx E R, (b) aR = asR v a"R.
Proof. Since R c N, and N, is open in X , the interior of R in N, coincides with Int R. By Lemma 8.8(a), (a) follows from Definition 5.5(B)(b), and hence aR = R\Int R = [V"(x,R), V s ( x ,R)]\[Int V u ( x ,R ) , Int V s ( x ,R)]
0
= asR v PR.
8.11. Remark. Int R is also a rectangle by Lemma 8.10(a).
8.12. Lemma. Let x E Int R. Then Int V"(x,R ) = V"(x,Int R)for (7 = u, s. Proof. Since Int R is open in X , V s ( x ,Int R ) is open in c , ( x ) . Hence x E Int V s ( x ,R ) since x E Int R. Let z E Int V ( x , R), then z = [x, z] and by Lemma 8.10(a), z E Int R. Therefore z E Y s ( x ,Int R). It is easily checked that the same result is true for o = u. 8.13. Lemma. Let x E R. C1 V " ( x ,R ) = V ( x , Cl(R))for(7
=
u, s.
Proof. Since Cl(R)is a rectangle, we have [x, Cl(R)] c Cl(R) n GI( x ) = V ( x ,Cl(R)).Let z E V s ( x ,Cl(R)),then z E v l ( x )and z = [x,z] E [x, Cl(R)].
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710
Hence Vs(x, CI(R)) = [x, CI(R)].Since [x, CI(R)] is closed in X,we have CI Vs(x, R ) c Vs(x, CI(R)). Similarly C1 V u ( x ,R ) c Vu(x, CI(R)). By Lemma 8.8(a), R = [ V " ( x ,R ) , V'(x, R)] and hence R c [CI V"(x, R ) , C1 V ( x , R)] which is closed in X.Hence CI(R) c [Cl V(x, R), CI V ( x , R)] c [v"(x, CI(R)),V(x, Cl(R))] = Cl(R), i.e., Cl(R) = [Cl V"(x, R), C1 V(x, R)]. Therefore, V'(X, CI(R)) = [x, CI(R)] = [x, [CI V"(X, R ) , CI V ' ( X , R ) ] ] = [x, C1 V'(x, R)] = CI V ' ( x , R ) .
0
8.14. Definition. A rectangle R is said to be proper if R = CI Int R.
8.15. Definition. A Markov partition forfis a finite cover { R,,. . . , R,} of X such that (a) each Ri is a proper rectangle, (b) Int Ri n Int Rj = 8 for i # j , (c) let x E Int Ri n f - ' I n t Rj, then fV'(x, R i ) c V'(f(x), R,) and fV"(x, Ri) 3 V"(f(x),Rj). Construction of Markov partitions. From now on we shall construct a Markov partition for f by the method in Bowen [ 19751. Let c be an expansive constant for f and 6,, e be as in Definition 5.5. Choose 0 < b < min{)e, c } such that d(x, y ) < b implies max{d(f(x), f( y)), d(f-'(x),f-'(y))} < 6,. Sincefhas the POTP, let 0 < a < tfl be a number such that any a-pseudo-orbit is tp-traced by some point of X. We choose 0 < y < :a such that d(x, y ) < y implies d ( f ( x ) , f ( y ) )< $a. Let P = { p , , . . . ,p,} be a y-dense finite subset of X and define
Z(P)
= {(q,) E
P z : d(f(qj),q j + l )
a, i E
z>*
Then to each q E X ( P ) there is a unique Q(q) E X which +b-traces q. This implies that for any x E X there exists q E Z ( P ) with x = O(q). Thus 0 : Z ( P ) + X is surjective and the diagram
X(P) ( I X(P)
x
f
I
o
commutes.
+x
Here CJ denotes the shift defined as usual. We now define
T,
= (0(q):q E
UP),qo = p,}, 1 ,< s
< r.
Topological Dynamics
71 1
Then diam(Ts) 6 fl and T = {TI,. . . , T } is a cover of X . Fix s with 1 6 s 6 r . If x , y E T, then there exist q, q' E C ( P ) with x = O(q) and y = O(q') and qo = qi = p s . Put q: = q, f o r j 2 0, q: = q,' f o r j 6 0 and write q* = qq' E C ( P ) . By the definition of 0 we have
D d(fJ(B(q*)),fJ(O(q'))) < B d ( f J ( w l * ) )f'(O(q))) , <
f o r j 2 0, f o r j 6 0.
Since [ x , yl = [O(q), O(q')l E w,S(O(q), d ) n w;(O(q'), d ) and D < c, we have [ x , y ] = O(q*) which contains in T,. Therefore T, is a rectangle. We next prove that 0 : X ( P ) + X is continuous. If this is false then there is 1 > 0 so that for every N we can find qN, 4" E X(P)with 4;" = 4;" for l j l 6 N such that d(O(qN),O(q"')) > A. Let x , = O(qN) and y , = O(q"), then d ( f J ( x N ) , f y,)) J ( < 28 for I j 1 6 N . By taking a subsequence we have x , --* x and y , + y as N co. Then d ( f J ( x ) , f J ( y )6) 28 for all j and d(x, y ) 2 1, thus contradicting expansivity. By continuity of O it follows that T, is closed in X . Let x E X and define T ( x ) = {T, E T : x E T,}, T*(x) = {T, E T : T, n T, #
z
=
\
0 for some
E
T(x)},
x naq. ,:I
Obviously, Z is open dense in X . Furthermore, define
z*
= {X
E
x:G,(X)n asTk = 0, q ( x ) n auTk =
0 for all
TkE T * ( x ) } .
8.16. Lemma. Z * is dense in X .
Proof. For x E X put a; = U{PT, : T, E T * ( x ) } and a: = U{auTk:Tk E T * ( x ) } . Since diam(T,) 6 p, we have u{Tk:Tk E T * ( x ) } c B,(x) c N, and hence
where yk E Tk. Since V u ( y , , T,) c D,",,,, by Lemma 8.3 we have [aV'( y,, T,), x] c DXU,yk.But [a?'"( y,, Tk), x] is nowhere dense in D;yk and so is in q ( x ) . Thus [a;, x ] is nowhere dense in q ( x ) . Let x E Z , then there exists an open neighborhood U, c iV, of x in X such that for any y E U,, T ( x ) = T( y ) , from which 8; = 8; for 0 = u, s. Let us Put U,' = u, n [Vg:(X>\[as,,X I , vQ,(x)\[x, a:]].
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712
By Definition 5.5(B)(b) it follows that U,' is dense in U,. For y E U,', y = [ y , , y,] for somey, E q(x)\[a:, x] and somey, E vI(x)\[x, $3. Thus vi(y) n 8 : = 8. For, if z E vB,(y) n 8; then y , = [ y , XI= [ [ y ,z], XI= [z, x] E [a;, x], a contradiction. Similarly q ( y ) n 8; = 8. Therefore y E Z* and so CJ; c Z* # 8 for x E Z. Since U, is an open neighborhood of x, it follows that Z * is dense in 2. Since 2 is dense in X , Z * is dense in X. 0
8.18. Remark. It is clear that
T,!k
=
T, n Tkand T, = u{Tk:1 < n G 4).
8.19. Remark. Each y k is a rectangle. It is clear that q ! k is a rectangle. To see that q ? k is a rectangle, let x , y E ??k. Then z E & ( y ) n Tk z 8 and e i ( x ) n Tk = 8. Since z = [z, y ] = [z, [x, y ] ] and d(z, [x, y ] ) < 28 < 6 , , we have z E q ( [ x , y ] ) n Tk # 0. If z' E Gl([x, y ] ) n Tk # 8 then z' = [ [ x , y], 2'1 = [x, 2'3. Since d(z', x) < 28 < a,, we have z' E vi(x) n Tk # 8, a contradiction. Hence GI([x, y ] ) n Tk = 8 and [x, y ] E q?k. This implies that q:k is a rectangle. It is shown by the similar way that l$ and c k are rectangles.
Proof. Let x E Int yk and z E V,l(x)n asT, # 8. By the definition of ZT,, z = [z,, z,] for some z, E aV"(x, q) and some z, E V s ( x ,T,). Since x = [z,, z,] E Vs(x, TJ n dV'(x, T,), we have x = [[z,,z,], [z,, z l ] ] = z, E asT,. This contradicts x E Int T, by Lemma 8.10(b). If z E v,S,(x) n PTk then z = [z,, z z ] for some zI E dV"(z, Tk)and some E vs(Z,Tk).Thus Z = "21, Z,], [Z,, z l ] ] = ZI and SO Z E av"(Z, Tk).Since d(x, z) < e, by Lemma 8.3 there is a homeomorphism [ , z] : D& + D& and
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Topological Dynamics
then [x, z] = z since z E GI(x).Note that OX",;n y k is a neighborhood of x in Dt:,and that Vu(z, T,) has no interior points in T,. Then we can find v E 0;: n y k with [v, z] 4 V"(z, T,). For this v , V,l(v) n T, = 8 holds. n T, # 0 then [v, z] = This follows from the fact that if w E G,(v) [[w, v ] ,z] E V"(z, T,). But [v, 23 4: V"(z, T,). This cannot happen. Therefore v,S,(x) n ( 8 T , u dsT,) = 8 when x E Int y,. By the similar way we can prove that x E Int y k implies V{(x) n ( d " ~ ,u duTk)= 0. Suppose that X E a " q k and Gl(X) n (a"T, U dsTk)= 0.Then x E dv"(x, q,). If x E aV"(x, T,) then x E [dV"(x, q), Vs(x, T,)] = d 5 q and so v,Sl(x) n a"T, # 8, thus contradicting. Therefore x E Int V"(x, T,). Since d ( x , y,) < e for yk E Tk,by Lemma 8.3 there is a homeomorphism [ , y , ] : D t , , 4 D ~ k , , . S i n C e y , " , ( X ) n a s=@anddV"(y,,Tk) T~ C dsTk,We have ~ l ( xn) aV"(y,, Tk) = 8 and hence [x, y,] 4 dV"(yk, T k ) . Since Tkis closed, V (yk, T,) is closed in X by Lemma 8.13. Since V "( y , , T,) c D,",,,, it follows that d V u (yk, T,) is the boundary of V"(yk, T,) in DL,r.Thus there is a neighborhood U t c Vu(x, T,) of x in D;,,such that
[ u , ?yk1
V " ( y k , Tk)
[u:,ykl
Or
DL.r\VU(Yk,
Tk),
namely GI(v) n Tk # 0 for all v E U t or GI( v ) n Tk = 0 for all v E U,U.On the other hand, since U,!' c V"(x, T,), we have that q ( v ) n T, # 8 for all v E u,"or q ( v ) n Tk = 0 for all v E U,!'.Therefore u,!'c q k which contradicts x E dV"(x, T,lfc). It was proved that for x E q k G,(x) f?(a"qu dsTk) =
8*x 4 a"qk.
Similarly we can prove that Q(x) n (d"T, u PT,) = 8 implies x 4 8"yk. Therefore the conclusion is obtained by Lemma 8.10(b). 0
x
Let x E Z * . If T, n T, # 0 for T, E T(x) and T, Int y k by Lemma 8.20.
E
T, then there is n with
E
8.21. Convention. Let x
E
R(x) = n{Int
Z y and write q k :
q n T, and
8.22. Remark. T(x) = T(Y).
0 for r j E T(x) Tk E T and x E Int Tk}. #
R(x) is an open rectangle. For y
E
R(x), R ( x ) = R( y ) and
Proof. The former is clear. If T, E T( y ) and T, E T(x), then y E T, n T, # 0 and hence y E R ( x ) c T,!, and Tk E T(x). Therefore T( y ) c T(x).Obviously
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714
T ( x ) c T( y) and so T(x) = T( y). By Lemma 8.21 we have R(x) = R( y) for y E R(x). 0 Since T is finite, so is {R(x):x E Z * } . Thus there exist x I , . . . , x,,, E Z * such that Z* = R(x,) u . . * u R(x,,,)is a disjoint union.
8. Then
8.23. Lemma. Let x E R(x,) n f-'R(x,) # ( a > f J " ( ~R(xi)) , c Vs(f(x), R(xj)), (b)fV"(x, R(xi)) 3 v"(f(x), R(xj))*
Proof. Fix x E R(xi) n f - I R ( x j ) and let v E X. Then there is q E Z(P)with v = O(q). Suppose qo = p , and q1 = p , for p , , p , E P = { p l , . . . ,p l } . If w E Vs(v,T,) then there is q' E Z ( P ) such that w = O(q') and qk = p , . Hence w = [v,w] = [O(q), O(q')] = O(qq') so that f ( w ) = f o O ( q q ' ) = 8 0 o(qq') E T, (qq' = (. . . , q t 2 ,qLl, qo, q l , q2, . . .) as defined above). Since w = [v, w ] E v,S,(
f W , T,) =
V"(s(v), Tt),
and in a similar way, (2)
fV"(V9
T,)
= V"(f(v),T,).
Let y E V ( x ,R(xi)). Then y E V;,(x) and R ( x ) = R( y) = R(xi).We first prove that T(fx)= T ( f y ) . Iff(x) E Ti andf(x) = O o a ( q ) where q, = pi and qo = p , , then x = O(q) E T,. By (1) we have f ( y ) efVS(x, T,) c Vs(f(x), q ) and hencef(y) E q. Similarlyf(x) E T, wheneverf( y ) E T,. Next we show thatf(x),f(y) E yk if T, E T ( f x ) = T(fy) and T, n Tk # 8 for Tk E T. Sincef( y ) E V(f(x), T,),f(x) andf( y) belong to q ! k u q:k or q:ku Suppose q ( f y ) n Tk = 8 and V ( f x ) n Tk # 8.This assumption is equivalent to
ck.
(3)
V " ( f ( y ) , q ) n Tk
=
8,
V"(f(x),
Tk #
8.
Takef(z) E V"(f(x), T,)n Tk.Letf(x) = O a(q) for q E C ( P )with q, = pj and qo = p , . By using (2) we have 0
f(z> E V"(f(4, T,) c f V " ( X 9 Ts). Let f(z) = 0 a(@) for q' E C ( P ) with q; = pk and qi = p t . Then z E T, and z E T, n T, # 8. Since x E T,, we have T, E T(x) = T( y). Since z E V " ( x , T,) n Tt and x, y belong to the same set T,lt, it follows that V"(y , T , ) n Tt # 8. Hence there is z' E V"(y , T,) n T, and then 0
' 2
=
[z, ~1
=
[z, z'l
E
Vs(z, Td
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Topological Dynamics
and by (l),f(z")
E
V s ( f ( z ) ,Tk).Sincef(z), f(y ) E T,, we have
f ( z " ) = [f(zh f(Y)l E V " ( f (Y ) , T,), which contradicts (3). Now define R(fx)'
=
n{Tk: T, n Tk # 0 for T, E T(fx) and Tk E T and f(x) E
yk}.
Then we have Int R(fx)' = R(fx) = R(x,) since f(x) E R(x,). Since diam(R(x,)) < fi, we havejV(x, R(x,)) t l $ ~ x )and hencefV"(x, &xi)) c Vs(f(x), R(fx)'). By Lemma 8.4(a),fc, (x) n GI(fx) is open in GI(fx) and R(xi) is also open in X . ThusfVs(x, R(xi)) is open in Cl(fx). Therefore by Lemma 8.12
With the above preparations we have the following theorem. 8.24. Theorem. Iff: X + X i s a (surjective)homeomorphism with hyperbolic coordinate, then there exist in X Markov partitions with arbitrary small diameter.
Proof. Let R, = CI R(x,) for 1 < j < m. Then W = { R l , . . . , R,} is a Markov partition of X. W is a cover of X , each R, is a proper rectangle with diam(R,) < fi and Int R, n Int R, = 0 for i # j . The remainder of the proof is to check Definition 8.15(c). If Int R, nf-'(Int R,) # 0, then there is x E R(x,) n f-'(R(x,)), and thenfVs(x, R(x,)) c V'(f(x), R(x,)) by Lemma 8.23. By Lemma 8.13 we havefVs(x, R,) c VS((f(x), R,) and then for y E R, n f - ' ( R , ) , fV"Y,
Rl)
= f [ y , VS(X,R,)1 = [f(Y)?fV"X, RJ1
= [f(Y)l
V"f(x),
411
= V " ( f ( Y ) , R,).
The proof of one half of Definition 8.15(c) has been given. The proof of the other half is similar and is therefore omitted. 0 8.25. Remark. Letf: X + X be a (surjective) homeomorphism of a compact
metric space and Q ( f ) be the nonwandering set o f f in X. Suppose f is
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expansive and has the POTP. ThenJ;,,,, has the POTP by Theorem 7.20 and it is expansive. Thus R( f ) has Markov partitions. Furthermore, R( f ) is expressed as the finite union R( f ) = U= ! ,R, of basic sets R, which are open and closed in R( f ) . Thus the restriction o f f on R, has the POTP and is expansive, and so R, has Markov partitions. Symbolic dynamics. Let R = (&, . . . , Rm-,} be a Markov partition of a basic set R, and define the transition matrix A = A ( W ) by 1
if Int Ri n f -'(Int R,) #
8,
0 otherwise. 8.26. Lemma. Suppose x E Ri n f -'(R,) and A , = 1. Then f V s ( x , R i ) c V s ( f(x), Rj) andf Vu(x, Ri) 3 V " ( f ( x ) ,Rj). Proof. This is exactly the same as the last part of the proof of Theorem 8.24.
0 8.27. Lemma. Let D c V,:(x) n R, and C c q ( x ) n 0,. Then the rectangle [C,D] is proper if and only if D = CI Int D and C = CI Int C as subsets of v,S,(x)n R, and q ( x ) n R,, respectively. Proof. This follows from Definition 5.5(b).
0
8.28. Definition. Let R, S be two rectangles in 0,. Then S is called a
u-subrectangle o f R if (a) S # 8, S c R and S is proper, (b) Vu( y, S) = V"(y , R ) for y E S. 8.29. Lemma. Suppose S is a u-subrectangle of Ri and A , = 1. Then f ( S ) n R, is a u-subrectangle of R,.
Proof. Choose x E R, nf -'(R,) and put D = V s ( x ,R , ) n S. Since S is a u-subrectangle, we have S = U{ Vu(y , R , ) :y E D } = [ V " ( x ,R,), D ] . Since S is proper and nonempty, by Lemma 8.27 we have 8 # D = CI Int D and f ( S ) n R, = U{f V " ( y , R , ) n R , : y E D } . It follows that f ( y ) E R, for y E D and y E R, nf - ' ( R , ) . Hence f V " ( y , R , ) n Rj = V " ( f ( y ) ,R,) by using lemma 8.26 and so
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Since Rj = [V"(f(x), Rj), Vs(f(x), Rj)]by Lemma 8.8 and Rj is proper, so is V"(f(x), Rj). Sincefmaps V(x, R i ) homeomorphically onto a neighborhood of V(f(x), R,), we have f ( D ) = CI Intf(D) and so f(S)n Rj is proper by Lemma 8.27. Since f ( D ) # 8, obviously f(S)n R, # 0. If y" E ~ ( Sn ) Rj then y" E vll( y', Rj) for some y' E ~ ( Dand ) hence V"(y", Rj) = V"(y', R,) c f ( S ) n Rj. Therefore V"(y", Rj) = V"(y " , f ( S ) n R,) for y" E ~ ( Sn)Rj. This implies thatf(S) n Rj is a u-subrectangle of R,. 0
8.30. Theorem. Let C, be the compact subset of Y', defined by C A = {x = (xi): A , , + , = 1 for i E h } and c : C A+ C A be the shift defined by a(xi) = (xifl) as usual. For each u E C,, the set n { f - ' ( R , , ) : j E Z}consists of a single point which is denoted by n(u). The map n : CA + R, is a continuous surjection with n o a = f o n , and n is injective on the Baire set Y = Q \ U { f ' ( W : i E hl. Proof. If A , , , + , = 1 for 1 we see that
- 1, by using inductively Lemma 8.29
is a u-subrectangle of Ran. Hence we obtain that &(a) = n { f - , ( R a J ) : - n < j < n } is nonempty. Since K J a ) 2 K,+,(a)2 . . , we have
n s-J(R,)
,= - m
n K , ( ~ )z 0. m
iD
K ( ~ )=
=
n= I
If X, y E K(a) thenfJ(x),fJ( y ) E R, are close for all j E Z and so x = y by expansivity. Since K(aa) = n , f - ' ( R , + , ) = f ( ) J f - J ( R , ) = f K ( a ) , we have n a = fo n. Continuity of n is easily checked. Since dB is nowhere dense, Y is a Baire set. For x E Y take a, withfJ(x) E R,. Since x E Y,fJ(x) E Int R, and SO Aa,4+, = I. Thus a = (a,) E C, and x = n(a). If x = n(b) then fJ(x) E Rb, and b, = u, sincefJ(x) 4 89. Hence n is injective on Y. Since n(C,) is a compact subset of Rscontaining Y, we have n ( C A ) = R,. 0 0
8.31. Proposition. The shift CT : C, + C, is topologically transitive. topologically mixing, SO is 0 : CA + C, .
vAnsis
Proof. Let U and V be nonempty open subsets of X A . Then there are a, b E C, and N > 0 such that
U
2
U, =
V
3
V, = { X ECA:xi = bi, I i l
{XE
CA:xi= ai, Iil
< N},
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and N
8#
Int K N ( a ) =
8#
Int &(b)
n f-J(Int Raj) =
U2,
j= -N
N
f-'(Int Rbj) = b.
= j= -N
If a(x) E U2 then f ' o a(x) E R , and f J o a ( x ) E Int(R,,) implies xi = aj for ljl < N, and hence a - ' ( U 2 )c U I . Similarly n-'(V,) c 5 . Since is topologically transitive, we havef"(U2) n V, # 8 for some n. Then
An,
8 # a-'(fn(U2)n b)
= a - l ( f n ~n ~ a-'(&) ) c f " ( ~ )n V .
By the same argument we can prove that a,2,is topologically mixing iff;% is.
8.32. Theorem (Bowen [197Oc]). There is an integer d such that card(n-'(x)) < d for all x E R,, i.e., K : CA + Q is a bounded-to-one map. 8.33. Remark. X A 3 x is a periodic point of a if and only if a ( x ) is. Proof. Since f o a = R O O , we see easily that a(x) is periodic if x is. If f " ( y ) = y where y = a(a), then n - ' ( y ) 3 {a, a"(a), a2"(a),. . .}. Since a-'(y) is finite, a is a periodic point of a. 0 Before starting with the proof of Theorem 8.32, we shall prepare some notations and some lemmas. As before let W be a Markov partition for f : R, + R, where Q is a basic set for a homeomorphism f with a hyperbolic coordinate. For R i , Rj E W we define 1 if f(1nt Ri) n Int Rj # 8, t(Ri, Rj) = 0 otherwise, and let C = {(Rn,)Ym: R ,E
W and t(R,,, R,,+ I ) = 1, i E H}.
Then Z is a compact metric space and the shift a : C + C defined by a ( R , ) , = R , + ,is a homeomorphism. Hereafter we write x = (xi) an element of X. Let (Z,,, a) be defined by W as in Theorem 8.30. Obviously ( Z A ,a) is isomorphic to (C, a).
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8.34. Convention. For x E 0, let Set(x) = { R E W : x E R } .
8.35. Lemma. (a) Set(x) = { R E 9: R = xofor some ( x i ) E ~ ‘ ( x ) } , (b) Set([x, y ] ) 3 Set(x) n Set(y). Proof. (a): If x = n((x,)) then clearly x,, = R E Set(x). That there is such an ( x i ) for any member of Set(x) was proved in Theorem 8.30. (b): [ x , y ] is defined only for x and y near each other. If x, y E R, then [ x , y ] E R because R is a rectangle. 0 8.36. Lemma. Let z be near enough x so that [x, z] is deJned. Zfz 4 PW,then to every 6 > 0 there is z’ E c(z)\U{f”(d’W):n E Z} with Set(z’) = Set(z) and [x, z’] 4 8.9. Zf z 4 39, then for every 6 > 0 there is z‘ E l$”(z)\ U { f ” ( a s W ) n: E Z} with Set(z’) = Set(z) and [z’, x ] 4 8%. Proof. Remember that a‘B = U{a”R: R E W} where d”R = [ V ” ( x ,R), a V s ( x ,R)] and a V s ( x , R ) = V s ( x ,R)\Int V s ( x ,R ) . Since a V s ( x , R ) is a closed nowhere dense subset of e ( x ) , we see that whenever z 4 a”&?,
v
=
Int R~Set(z)
vS(z,R )
\
R#!d(z)
is an open neighborhood of z in c(z). Obviously Set(z’) = Set(z) for all z’ E V and V n 8 ’ 9 = 8. Since PW n G(x) is contained in a finite union of nowhere dense sets of the form 8V(y , R ) , d”W n G(x) is nowhere dense in v(x). Since eachf-“(V,”(z)) can be covered by finitely many G(x), we have that
f”(d”W) n G(z) c f ” ( U x G ( x )n 8%’) is nowhere dense in G(z). Hence by Baire’s theorem U{f”(i?”&?) :n E Z} n V(z) is nowhere dense in G(z).By Lemma 8.3, [x, 1: G(z) n N, + v(z) n N, is a homeomorphism. It follows that [x, z’] 8 9 for some z’ E ~(z)\U{f”(PW): n E Z}.
+
8.37. Lemma. Let R,, R 2 , R3 E W.Suppose that R , # R 2 , x f - ’ ( R , ) and A13 = 1, A23 = 1. Then x E a”&?.
E
R , n R2 n
Proof. Lemma 8.26 gives V = V’(x, R , ) n V’(x, R,)
3
f-’(V”(f(x),R 3 ) ) .
Since V ” ( f ( x ) ,R , ) contains an open subset of every neighborhood off(x) in q (fx), V contains an open subset of every neighborhood of x in q (x).
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720
As we saw in the proof of Lemma 8.36, we can find x' E V with x' 4 as%?. Since R, # R,, we have V c R, n R, c aR,. SinceaR, = aURlv aSR,,we havex' E P R , and then Vu(x', R,) c P R , c 8.9. Thereforex E 8.9. 0 8.38. Lemma. (a) Suppose z 4 8.9. Then to every Ri E Set(fz) there is a unique Rj = T,(R,) E Set(z) such that A, = 1. The map T,: Set(fz) + Set(z) is surjective. r f y E I$(z)\au9 and Set( y) = Set(z), then Set(fy) = Set(fz) and T, = T,. (b) Suppose z 4 8'9. Then to each Ri E Set(f-'z) there is a unique Rj = Tz'(Ri) E Set(z) such that A, = 1. The map T:: Set(f-'z) + Set(z) is surjective. r f y E q(z)\asW and Set(y) = Set(z), then Set(f-'y) = Set(f-lz) and T"' = Tz'.
Proof. By Lemma 8.35, for Ri E Set(fz) there is (wi) E R - ' ( ~ zwith ) wo E Ri and so r.-'(wi) = n-'(z). Hence z E R,,,-,. Since (wi) E C,, Aw-,w, = Aw,-,i= 1. By Lemma 8.37, w-, is a unique since z 4 8.9. Hence T, is well defined. If Ri E Set(z), then there is (zi) E n-'(z) with zo E R;. Then R,, E Set(fz) and A,,,, = 1. Hence T,(R,,) = R,, and T, is surjective. Consider y E va",(z)\auWwith Set(y) = Set(z). Then y E Vs(z, Ri) for every Ri E Set(z). If Ri E Set(fz) and Ri = Tz(Rj) then Lemma 8.26 shows f ( y ) ~ f V ~ ( Ri) z , c Vs(f(z),Rj) c Rj.
Thus Set(fz) c Set(fy). Symmetrically Set(fy) c Set(fz). Thus Set(fy) = Set(fz). Since the definition of T, depends only on the sets Set(z) and Set(fz) (not z itself), we see that T, = T,. (b) is shown in a similar way. 0 8.39. Convention. Let 6, > 0 be as in Definition 5.5(B). For x E R, define Js(x) = {9 c 9 : 6 1 > V6 2 0, 32 E v ( x ) \ a ' 9 with Set(z) = 9 } , J"(X)
= {9 c
9 : d 1 > V6 2 0,
32 E e ( x ) \ a s W with Set(z) = 9 } . Remark that we can choose 6 = 6(x) so small that Set(z) E J,(x) whenever z E G ( x ) \ a U 9and Set(z) E J,(x) whenever z E ~ ( x ) \ 8 " W . 8.40. Definition. Let N > 0 and KN be the set of all n-tuples (a,, . . . , a,) of integers with 1 < n < N , 1 < ai < N a n d a, 2 a, 2 3 a,. Define a partial ordering G on KN by (a,, . . . , an) > (b,, . . . , b,) if either n > m or n = m and bi 2 ai for all 1 < i < m (note that this ordering is not the natural condition ai 2 bi). We write (al, . . . , an) > (bl, . . . , b,) if (a19 . . . an) 2 (bl, . * * b m ) but ( a , , . . . a n ) # (bl, * * . b m ) . 9
9
9
9
72 I
Topological Dynamics
8.41. Definition. Let E denote the cardinality of E and B ( B ) denote the set of subsets of B. Fix N 2 2". For L E B ( B ( W ) ) we define L* E KN to be (al, . . . , a,,) where n = and a,, . . . , a, denote the cardinalities of the n sets in L . 8.42. Lemma. Js(fx)*
< Js(x)* and J,,(f-'x)* < Ju(x)*
Proof. Define the map Ri : Js(x) + J,(fx) by Ri(9) = Set(fz) for 9 E J,(x). Lemma 8.38(a) ensures that Ri is well defined. We can check that R: is surjective. For, if d E Js(fx), then by Lemma 8.36, for arbitrarily small 6 > 0 we can find w E v,"(fx)\d"W with Set(w) = 8'and w # f ( d " B ) . Then d = Ri(Set(f-'w)). Let Js(x)* = ( a i , . . . , a,,) where ai = gi and J s ( x ) = {g1, . . . , 9,}, and let J,(fx)* = ( b l , . . . , b,) where bi = 8iand Js(fx) = {g1,. . . , &}. Then Ri induces a surjection g : [ l , n] + [l, m] by RT;(Qi) = dg)B(i). We can consider the cases: either n > m and (al, . . . , a,) > ( b l ,. . . , b,) or n = m and g is bijective. Considering the second case. For certain z we have the surjection T,: dgg(i) -, Qjand thus bg(i)2 ai. For any i there i s j E [l, i ] with g ( j ) 2 i. Otherwise g([l, i ] ) c [ l , i - 11 which contradicts g one-to-one. Then bi 2 bg(,)2 a, 2 ai. Here we use the fact that all ai and b, are arranged in 0 descending order. The second statement is proved in the same way. 8.43. Definition. We say x E R, is an s-branch (u-branch) point if there are (xi), ( y i )E c l ( x ) with xo = yo but x I # yI (but x - ~# Y - ~ ) . 8.44. Lemma. If x is an s-branch point, then Js(x)* > Js(fx)*. u-branch point, then J,,(x)* > J,(f-'x)*.
If x
is a
Proof. Let x be an s-branch point. It is enough to show that one of T, with z E V&)(x)\C?"' is not bijective. Indeed, if (al, . . . , a,) = (bl, . . . , b,) then g is a bijection and so
2 bi
i= I
n
=
C ai 6
,=I
i= I
bg)B(i) =
CI bi.
i=
Here we use ai < bg(i). To have the above equality we need a, = bg,i,for all . . I , i.e., T, is always a bijection. Since x is an s-branch point, we can find Ri E Set(x) and R,, Rk E Set(fx) with R, # Rk and A , = Alk = 1. Using Lemma 8.36, we can choose
z
E
{ V(x, Ri) n v,"(,,(x)}\duW.
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By Lemma 8.26 f V s ( X , Ri)
Vs(f(x),Rj) n V'(f(x), Rk)*
C
Thus Rj, Rk E Set(fz) and T,(R,) = T Z ( R k= ) R, and so T, is not bijective.
0 Proof of Theorem 8.32. Let c = & and e be the length of the longest chain in the partially ordered set (K,,,, <). Let 0 < n, < n2 < * . < nk be all the nonnegative integers such that f"'(x) is an s-branch point. By Lemmas 8.42 and 8.44 J s ( f " ' x ) *>
JS(f"2X)*
> ... >
JS(f"kX)*,
which shows k < e. Let A, be the set of all sequences (Rko,. . . , Rk,)of elements of W such that there is ( x i ) E n-'(x) with xi E Rk, for all 0 < i < n. By the definition of s-branch points fi,,,, = A,, unless f"(x) is an s-branch point. Hence Art+,< cA,, for every n. Since Jo < c, we have that A, < ck+l < ce+' for all n 2 0. From this it follows that there are at most c'+' possibilities for (xi): with ( x i ) E n - ' ( x ) . Similarly there are at most c'+' possibilities for (xi)O, with ( x i ) E n - ' ( x ) .
-<
Therefore n - l ( x )
0
c*('+').
8.45. Definition. Letf : X -,X be a homeomorphism of a compact metric space. Let 9 be a partition of X , i.e., for A, B E 9 , A n B = 8 and u { D : D E 9 } = X. A sequence { x i }of points of X is said to be an a-pseudoorbit with respect to 9 if d ( f ( x i ) ,x i + ' ) < a andf(xi) x i + , for all i where x y implies that x and y are in the same element of 9. The following concept was introduced by Y. Takahashi. A (surjective) homeomorphism f is said to have the special POTP if there exists a finite partition 9 such that to every p > 0 there is a > 0 such that each a-pseudo-orbit with respect to 9 is P-traced. If in particular 9 = {X}, the special POTP implies the POTP. 8.46. Convention. Let
E
> 0 and write
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Obviously, y E Z,(x) if and only if x Z & ( X ) c Y&(X>.
E
Z,( y ) , and r ( x , d ) u w,"(x, d ) c
8.47. Theorem (Dateyama [ 19881). Let f : X + X be an expansive homeomorphism of a compact metric space and c > 0 be an expansive constant for f. Then the following are equivalent: (a) f : X + X has a Markov partition, (b) f has the special POTP, (c) for each x E X there exists 6 = 6 ( x ) > 0 such that {ZC,,(y ) n B s ( x ) : y E B,(x)} isfnite, (d) for each x E X there exists q = q(x) > 0 such that { Y,,,( y ) n B,,(x) : y E B,,(x)} isfinite. 8.48. Definition. Let k 2 2 and a: Y t + YE be the shift. A subshift a : S + S is said to be sofc if there exists a subshift a : S' 4 S' of finite type such that there is a continuous surjection cp : S' + S with a l s o cp = cp o aIs,. 8.49. Remark. A sofic subshift does not have the POTP unless it is of finite type (see Theorems 4.15 and 4.17).
8.50. Remark. A subshift a : C + C has the special POTP if and only if it is sofic (compare with Theorem 4.17).
Proof. If the subshift alxhas the special POTP, then it has Markov partitions by Theorem 8.47. Thus aIxis a factor of a subshift of finite type and therefore alzis sofic. Conversely, suppose a : C + C be sofic. Let W be the set of blocks which occur in C and let w E W. Put F(w) = {w' E W :ww' E W } .Since alr is sofic, we claim that {F(w): w E W }is finite. Indeed, since a : C + C is sofic, is a factor of some subshift a : S + S of finite type of order L, i.e., there is a continuous surjection h : S + X such that hoa = aoh. Note that S c YE for some k > 0. Let w = ( w l , . . . , w L )be a block which occur in 72 and for n E Z define Cyl,[w] = { x E C : x , =
WI,
x,,, =
w2,.
. . , X,+L-l
= WL}.
For s E Yk,Cyb[s] is open and closed and so is h-'(Cyb[s]). Let W, be the set of all blocks with length p which occur in S. Since Y, is finite, there is N > L such that for s E Y, there is C, c W,,+, so that h-I(Cyl,[s]) = U{CylL.[b] : b E C,}. For w = ( w l , . . . , w L ) E W , write C, = {b E C,,: h-I(CyILL+I[~]) n CylL,[b] #
s}.
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Since c : S + S is of finite type and L < 2N + I , we have that u E F(w) if and only if Cyl_,[b] n h-'(cyl,[u]) # 8 for some b E C,. If C,, = C,, for w', w2 E W , then we have F ( d ) = F(wz). Since W,,,, is finite, so is {CH': w E W} and therefore { F ( w ) :w E W } is finite. For x = (x,) E Z, write Fn = F ( x - , , . . . , xo). Then { F , : n 2 0} is a decreasing sequence of W. Since {F(w): w E W } is finite, there is N > 0 such that FN = F, for all n 2 N. For simplicity we write &x) = FN. Then { $ ( x ) : x ~ Z }c { F ( w ) : w ~ W } . L e t G ( x )= ( y ~ C : ( y , , . . . , y , ) ~ ~ ( x ) for all n 2 I } for x E C.For x, y E C we define a point z = (z,) by z, = x, (i < 0) and z, = y , (i 2 1). Then it follows that z E C if and only if y E G(x). It is clear that {G(x) : x E Z} is finite. Let a metric d for C be defined by d(x, y) = max{6,,,,/21'1:i E Z} (x = (x,), y = ( y,) E C) where ~5,,~, denotes the Kronecker delta. If c = $ then 2c is an expansive constant for oIzunder d. Let x ( x ) be as in Convention 8.46. Then we have Y,(x) = { y E C : y E G(x), yo = xo and y - , = x-, } for x E C.Since (G(x) : x E C} is finite, so is { (x) : x E C}.Thus the condition (d) of Theorem 8.47 holds and therefore qZ has the special POTP. 0
x
8.51. Definition. The natural numbers, the real numbers and the complex numbers will be denoted by N,R and C respectively. For p E N, let np:C -, C be the map which sends z to zp. We define the domains gP(1 < p < a)of C by
g2 = { z E C : I R e z l < 1 , I I m z l < l}, 9, = n2(gZ)and = nil@,). It is easily checked that n,:Qp -, 9, is a p-fold branched cover for every p E N.Denote by X ; and V2the horizontal and vertical foliations on QZ respectively. We write XI(resp. V , )the decomposition of 9, as the projection of X2(resp. V 2 by ) n 2 : g 2-, 9,, and write Xp(resp. Vp)the decomposition of gPas the lifting of S,(resp. <) by np:Qp+ 9,. A decomposition 9of a compact surface M without boundary is called a Co-singular-$oliation if every L E 9 is arcwised connected and if to every x E M there are p ( x ) E N and a Cochart cp,: U, + C around x such that (1) cpx(x) = 0, (2) rP,(C) = qlp(.,, (3) cp, sends each connected component of U, n L onto some element of SP(,, unless Uyn L = 8 for L E 9. Let 9be a Co-singular-foliation on M. Each element of 9is called a leaf and equipped with the leaf topology. The number p(x) is called the number of separatrices at x. We say that x is a regular point if p(x) = 2, and that
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x is a singular point with p(x)-separatrices if p ( x ) # 2. Since M is compact, 3 9 the obviously the set S of all singular points is finite. We denote by 9 Co-foliation on M\ S obtained by taking singular points away from each leaf of 9. For materials of Co-foliations on surfaces, the reader may refer to 3 9is dense in M , then 9is called Hector and Hirsch [1981]. If every leaf of 9 minimal. We say that 9 is orientable (resp. transversally orientable). A subset A of M is an arc (resp. open arc) if there is a Co-embedding h from a compact (resp. open) interval I of R to M such that h ( I ) = A . Let 9 and S be as above. An arc A is called a transversal of 9 if the interior of A is contained in M\ S and if to every x E A \S there is a Co-chart cpx : U, 4 C around x satisfying (I), ( 2 ) and (3) such that P,0 cp., is injective on U, n A where P, denotes the projection from C onto the imaginary axis. Let A, and A be transversals of 9. We say A, 2: A if there is a continuous x~,l m a p H : [0, I] x [0, 11 + Msuch that H , = H l ~ o , I laxn~doHl , = H l ~ o , , lare homeomorphisms from [0, 11 x {0} onto A, and from [0, 11 x { I } onto A , respectively, and such that if L E 9 then H - ' ( L ) = B x [0, 11 for some B c [0, I]. Let h:[O, 11 x { 0 } + [0, 11 x (1) be the homeomorphism which sends (t, 0) to (t, 1). If A, N A , , the homeomorphism H, 0 h 0 H;' : A, + A , is called a projection along the leaves. A transverse invariant measure p for 9is a collection { pA:A is a transversal} of finite Bore1 measures on all transversals of 9 such that p A I A .= pA, if A' c A and such that p A , h = pAOif h : A, -+ A , is a projection along the leaves; p ( A ) # 0 for some transversal A. A measured Co-foliation(9, p ) is a Co-singular-foliation 9 equipped with a transverse invariant measure p. We denote by A(9)the set of all transverse invariant measures for 9. For { P . ~ } {, v A }E A(9)and a > 0, we write { p A } { v A } = { p A v A } and a { p A }= { a p A } .Obviously { p A } { v A } ,a { p A }E A(9).Let f:M -, M be a homeomorphism. Then f sends 9 to a Co-singular-foliation 9'. If A is a transversal of 9'then f - ' ( A ) is a transversal of 9. Hence we can define the map f*:A ( 9 )+ A(9')by f*({pA}) = { p Aof-' }. Obviously f*(ap bv) = af,(p) bf*(v) for p, v E A(9) and a, b 2 0 with a+b>O. I f f ( 9 ) = 9' andf,(p) = p', we writef(9, p ) = (9', p'). Let 9and 9' be Co-singular-foliations on M. We say that 9' is transverse to 9 if 9 and 9' have the same number p ( x ) of separatrices at all x E M and if every x E M has a Co-chart cpx: U, + C such that (4) % ( X ) = 0, ( 5 ) cpAU,) = gP)p(,)r (6) cpx sends each connected component of U, n L onto some element of Xp(.,unless U, n L = 0 for L E 9,
,
,
0
+
+
+
+
+
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N. Aoki
(7) cpx sends each connected component of U, n L' onto some element of Vpcx, unless U, n L' = 8 for L' E 9'. be transverse C'-singular-foliations on M and S be the set Let 9 and 9' and the interior of all singular points. If A is an arc in a leaf of 9(resp. 9') of A is contained in M\ S,then it is easily checked that A is a transversal of 9' (resp. 9). A homeomorphism f of M is called pseudo-Anosov if there are a constant 1 > 1 and a pair (SS, p') and (F", p") of transverse measured CO-foliations with the number of separatrices at each singular point greater than 2 and with every finite Bore1 measure of p' and of p" nonatomic and positive on all nonempty open sets such that
f ( F ' ,$1
=
f P "p") ,
( 9 ' 9
= (9", Ap")
(this means that f preserves the transverse Co-singular-foliations 9' and 9", i.e., it contracts all arcs in the leaves of 9' by 1-'and it expands all arcs in the leaves of 9" by A). Let f be a homeomorphism of a compact metric space ( X , d ) . For x E X we define the stable set W'(x) and the unstable set W u ( x )as before, and put
9 ; = { W " ( x ) : xE X } ,
0
= s, u.
Then 9 ;is a decomposition of X and f (9;= )9;. If Xis a compact surface and f is pseudo-Anosov, then it is easily checked that every leaf L of the coincides with W " ( x )for all x E L, that associated Co-singular-foliation 9" is 9"= 9;.
8.52. Remark. By Theorem 8.47, every pseudo-Anosov map has Markov partitions. Such maps have the special POTP, but not the POTP because they have singular points. 8.53. Theorem (Hiraide [1988]). Let f : M compact surface without boundary. I f f : M pseudo-Anosov map.
+ +
M be a homeomorphism of a M is expansive, then it is a
For the proof we need the following proposition.
8.54. Proposition. Let f : M + M be an expansive homeomorphism. Then 9;(a = s, u) have the following properties: (i) 9 ;is a CO-singular-foliation, (ii) every leaf W"(x)E 9 ;is homeomorphic to Lp = { z E C : Im(zJ"*)= 0} for some p 2 2,
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121
(iii) 9 :is transverse to 9,, (iv) 9 ;is minimal. If Proposition 8.54 has been established, then the transverse invariant ;(a = s, u) and the stretching factor 1 > 1 o f f are measures pa for 9 obtained from the following proposition.
8.55. Proposition. Let f : M --t M be a homeomorphism and Ss and 9" be transverse C'-singuIar-foliations on M . If 9" is minimal and f(9'= )sa for a = s, u, then there are constants I > 0 and transverse invariant measures pa for 9 ;(a = s, u) such that (i) pa is nonatomic and positive on all nonempty open sets (ii) f*(ps) = l - ' p sandf,(p") = 1p". Propositions 8.54 and 8.55 are obtained by investigating deeply the orbit structures of homeomorphisms of compact surfaces. If the proof is of interest to the reader, he should see the preprint by Hiraide [1988].
9. Topological entropy In Adler, Konheim and McAndrew [ 19651 they introduced topological entropy as an invariant of topological conjugacy for compact spaces. The definition of their entropy is given by open covers. We first give a definition of topological entropy by open covers. All logarithms have base e. Let X be a compact topological space. Open covers of X are denoted by a, /?,. . . . 9.1. Definition. Thejointa v Bisgivenbya v B = { A n B : A E a , B E B } . /? is a refinement of a (written a < j)if every member of /? is a subset of a member of a. Thus a < a v p and j < a v j.Iff: X + Xis continuous, then f - ' ( a ) = { f - ' ( A ) :A E a } is an open cover of X. It is clear that
f -'(a v
8)
= f -'(a) v
f -'(/I)andf-'(a) < f-'(/I) if a < B.
9.2. Definition. Let N(a) be the number of sets in a finite subcover of a with smallest cardinality. We define the entropy of a by H(a) = log N(a). 9.3. Remark. (a) H(a) 2 0. (b) H(a) = 0 o N(a) = 1 o ct = {X}. (c) a < B H(a) < H(Bh
728
N. Aoki
( 4 H(a v P ) < H ( 4 + W P ) . (e) Iff: X + Xis a continuous map then H ( f - ' a ) < H(a). Iff is bijective then H( f - ' a ) = H(a).
9.4. Lemma. If{a,,},,21 satisfies a,, 2 0, a,,+,,,< a,, lim a,,/n exists and equals Inf a,,/n.
+ a,,,for n, m
2 1, then
Proof. Fixm > O . F o r j > O w e w r i t e j = k m + n w h e r e O < n < m . T h e n ajlj = a,,+k,,,/(n km) < a,,/km ak,,,/km < a,/km a,,,/m, and then lim aj/j < a,,,/m. Therefore lim ajlj < Inf a,,,lm. But Inf a,/m < !ig aj/j. Hence lim ajlj exists and equals Inf uj/j. 0
+
+
+
9.5. Theorem. I f f : X
-, X i s
continuous, then lim( l / n )H( V : - ' f -ia) exists.
Proof. We set a,, = H ( V ; - ' f - i a ) . Then by Lemma 9.4 it is enough to show that a,, 2 0 and a,,+,,,< a,, + a,,,. It is clear that a,, 2 0, and a,,,,,, = H( V$+m-lf-ia)= H( V:-lf-ia v f - n V,!,'-Ypia) < H ( V:-lf-ia) + H ( V 7 - l f - i a ) = a,, a,,,. 0
+
9.6. Definition. I f f : X -, X is a continuous map, then the entropy o f f relative to a is given by h ( f , a) = lim( l / n )H ( V [ - ' f p i a ) .
9.8. Definition. I f f : X + X is continuous, then the topological entropy of f is given by h( f ) = sup,h(f, a ) where a ranges over all open covers of X. 9.9. Remark. (a) h(f) 2 0. (b) h(id) = 0 where id denotes the identity map of X.
9.10. Theorem. Let XI and X2 be compact spaces. I f J :X, -, X, are continuous ( i = 1, 2), and are topologically conjugate, then h ( f , ) = h(f2).
Proof. Let cp be a homeomorphism such that cp of, = f2 cp. Then hV2,a) = lim(l/n)H(V $ - ' j - ' a ) = lim(l/n)H(cp-l(V$-lf,-ia)) = lim(l/n) H( V0n-l f,-'cp-'(a))= h ( f , , cp-'(a)). By taking suprema the result follows. 0 0
9.11. Theorem. I f f : X
+
X is a homeomorphism, then h( f ) = h( f
-I).
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Proof. h(f, a) = lim(l/n)H(V,"-'j-'a) = lim(l/n)HCf-~+'(V,"-lf'a)) = lim( I In) H ( V,"- " a ) . 0 We give the new definition of topological entropy by Bowen [1970d]. Let (X, d) be a metric space and let x E X. B , ( x ) denotes the open ball centered at x and of radius E . Letf: X + X be a uniformly continuous map, and let n E Z (n > 0) and E > 0. If K c X, then a subset F of X is said to (n,&)-span K with respect t o f i f for x E K there is y E F such that max{d(f'x,f'y): O < i < n - I } < c . 9.12. Remark. For K compact, let r,,(&,K) be the smallest cardinality of any (n,&)-spanningset for K with respect tof. Then r,,(&,K ) < 00.
Proof. Since f is uniformly continuous, there is a 6 > 0 such that d(x, y ) < 6 implies max{d(f'(x),f'(y)):O < i < n - l} < E. Then rn(&,K) is less than the number of b-balls needed to cover K and hence is finite. 0 9.13. Definition. We write
?/(E,
K)
=
G ( l / n ) log r,,(&,K).
9.14. Definition. A set E c Xis (n,&)-separatedwith respect to f if x , y E E and x # y then max{d(f'(x),f'(y)):O < i < n - I } > E . 9.15. Remark. For Kcompact, let s,,(E, K) denote the largest cardinality of any (n, &)-separatedsubset of K with respect tof. By the following theorem, s,,(E, K) is finite. 9.16. Definition. We set Sf(&, K ) = G ( l / n ) log s,,(E, K). If h(f, K ) = lim,:+,Tf(&, K) = lim,&(&, K) then we define hd(f) = sup{h(f, K): K compact}.
Proof. (i): That K) < 00 holds is proved in Remark 9.12. We prove K) < r , , ( + ~K). , Suppose E c K is an (n,&)-separatedset and that F (n,+&)-spansK. Define cp: E -+ F by choosing for each x E E some point q ( x ) E Fwith max{d(f'ocp(x),f'(x)): 0 < i < n - I } < + E . If cp(x) = cp(y) then max{d(f'(x),f'(y)):O < i < n - l} < + E + + E = E and so x = y. I+,,(+&,
s,,(E,
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Hence cp is one-to-one and the cardinality of E is less than or equal to the cardinality of F. Therefore s,,(E, K) < r,,(+&,K). We next prove r,,(&,K) < s,(E, K). Let E be an (n, &)-separatedsubset of K maximum cardinality. Then E (n, &)-spansK. For, if not, there is x E K such that max{d(fj(x), f’(y ) ) : 0 < j < n - 1) > E for y E E. Then E u { x j is an (n, &)-separatedsubset of K, contradicting the choice of E. (ii) is clear from the definition of ?(,&, K ) and S,(E, K). 0 9.18. Remark. (a) h d ( f ) depends on d. (b) If K c K, u . * . u K,,, are all compact then h ( f , K ) 1 < i G m}.
< max{h(f, K i ) :
Proof. It is checked that s,,(&, K) < s,(E, K l ) + . . . + s,@, K,,,). Fix > 0. For n > 0 choose Kjn(&)such that s,(E, Kin(&))= max{s,(e, 4 ) : j } . Then s,,(E, K ) < ms,,(&,Ki,(&))and so log s,,(E, K) < log m + log s,,(E, Kin(&)). Choose nj + co such that ( l / n j ) log s,,(E, K ) + G ( l / n ) log s,,(E, K) and so E
Kjnj(&) does not depend onj. Hence 3,(q K) < S(&,, K(E)). Choose 8. -, 0 such that K(cJ is constant (say, K(E,) = Kh). Hence h ( f , K ) < h ( f , Ki,) < max{h(f, 4 ) : j } . 0 9.18. Remark. (c) For any 6 > 0, h d ( f ) = sup{h(f, K): diam(K) compact}.
< 6, K
Proof. Let a be an open cover of Ksuch that diam(U) c 6 for U E a. Then K = U.0 and by (2), h ( f , K) < max{h(f, 0 ) :U E E}. I7 9.18. Remark. (d) If X is compact then h d ( f ) = h ( f , X ) . 9.19. Theorem. Let d and d’ be uniformly equivalent and f is uniformly continuous. Then h d ( f ) = h d ‘ ( f ) .
Proof. Let > 0. Choose Q > 0 such that d’(x,y ) < c2 implies d(x, y ) c and choose c3 > 0 such that d ( x , y ) < c3 implies d’(x, y ) c c2. Let K be compact. Then rn(El, K, d) < r , , ( ~K, ~ ,d’) and r,,(c2, K, d’) < r,,(E3, K, d ) . Hence ?,(el, K , d) < ?,(e2, K , d’) < ? / ( E ~ , K , d). If cl + 0, then e2 + 0 and c3 -+ 0. Therefore h d ( f ,K ) = h d , ( f ,K ) . 0 9.20. Theorem. Let X be compact and h * ( f ) denote the entropy akjined by open covers. Then h ( f ) = h * ( f ) .
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= { A l , . . . , A,,} be an open cover of X. We first show h ( f , a) < h ( f ) . Let 6 be a Lebesgue number for a and F be an (n, ;@-spanning set for X of minimum cardinality. For z E F choose Aio(z), . . . , Ai,,-l(z)E a such that Bai2cfk(z)) c Aik(z).Write C(z) = Ah@)nf-I(Ail(z)) n . . . nf -(n-I) (Aia-l(z)),which is a member of V:-lf-i(a). If x E Xthen there is z E F such that max{d(f'(x),f'(z)):O < i < n - l } < i d . Hence X = UZcFC(z) and x ~ f - ~ ( B ~ , ~ ( f 'c( zf k) ()A) i k ( z ) ) for 0 < k < n - 1. So we have x E C(z). Hence N(V:-'f-'(a)) < IF1 = r"(t.8, X) and h*(f, a) < X ) < h ( f ,X ) = N f ) . To prove the converse let S > 0. Choose a finite open cover a of X such that diam(A) < 6 for A E a. Let E be an (n, &separated subset of X with maximum cardinality. If x , y E n ; - ' f - j ( A , . )and x , y E E, then max{d(f'(x), f'(y)):O < j < n - 1) < 6 and so x = y . So N(V:-'f-'(a)) 2 [ E l = s,(S, X). Therefore h * ( f ) 2 h*(f, a) 2 $(a, X ) .
Proof. Let a
9.21. Theorem. ( 1 ) mh(f )*
Iff
is uniformly continuous and m > 0, then h(f "') =
(2) LetA be uniformly continuous (i = 1, 2) and dejine a metricfor XI x X, by d((xl, x2)t (yl, y2)) = max{dl(x,, yl), d2(x2, yZ>}* Then h d ( f i x f z ) < h d l ( f i ) + hd2(.h)* (3) I f XI and X , are compact, then hd(fi x f 2 ) = h d l ( f i ) hdz(f2).
+
Proof. (1): For K compact and E > 0 it is checked that rn(&,K , f " ) < r,,,,,(E, K , f ) . Hence ( l / n )log r,,(&,K , f " ' ) < (m/mn)log r,,,,,(E, K , f ) and therefore h ( f " ) < mh(f ). Since f is uniformly continuous, there is 6 > 0 such that d(x, y) < 6 implies max{d(f'(x),f'(y)): 0 < j < n - l } < E. Every (n, &spanning set for Kwith respect tof" is also an (nm, &)-spanningset for K with respect to$ Hence rn(6,K , f " ' ) 2 rnm(&,K , f ) and so m?/(&,K) < i$6, K). Therefore mhdu,K) < hd(fm,K ) . (2): Let Ki c X, be compact (i = 1, 2). If 6. is an (n, &)-spanningset for Ki with respect tax, then Fl x F, is an (n, &)-spanningset for Kl x K2 with x h ) < r,(&, Kl,fi)r,(&,K 2 , h ) respect to fi x f 2 . Hence r,,(&,KI x and thereforehd(fi x f z , Kl x K 2 ) < h d l ( f i K , l ) hd2(f2,K2).Let n i : X l x X2 -,X, (i = 1, 2) be the projection map. If K c XI x X, is compact, Kl = n , ( K ) and K2 = n 2 ( K ) are compact and K c KI x K,. Hence hd(fi f i g K ) < h d ( f i f2, Kl K 2 ) and so
+
hd(fi X f z ) = sup{hd(fi X f 2 , K ) : K c KI X K2 compact}
< sup{hd(fi x f 2 , Kl x K 2 ) :K , , K2 compact} < h d l ( f i ) + hd2(fz)*
N. Aoki
732
(3): Suppose XI and X , are compact. Let ai be an open cover of X, and have Lebesgue numbers di ( i = 1, 2). If Siis a maximal (r, +di)-separated set for X, with respect toJ;, then S, x S, is an (n, 6)-separated set for XI x X , with respect tofi x f i where 6 = min{+6,, $5,). Therefore sn(6, XI x
Xz) 2
sn(+sl,
X l ) s n ( ! i S ~ ,X,)
N( V :plfi--i(al))N( v :-1fi-i(a2))
and so
+ lim -n1 log N( v :-'f2-j(a2)) = h(A, El)
+ N f i , a,).
Since a1 and a, are arbitrary, we have h(fi x f i ) 2 h ( f , ) + h ( f i ) .
0
9.22. Theorem. Let f : X + X be an expansive homeomorphism with expansive constant c. Then for any 0 < 6 < +c, 1 h ( f ) = lim - log rn(6, X), n-a, n
h ( f ) = lim n-co
1
-n log sn(6, X).
Proof. Since X is compact, there are x I , . . . , xk E X such that X = UfZlUc,z-6(xi).Hence a = { UC,,(xi): 1 < i < k} is an open cover of X and 26 is a Lebesgue number of a. From the first part of the proof of Theorem 9.19 we have that for n > 0, N ( V : - l f - i ( a ) ) < rn(6,X). By expansivity h( f) < lim(l/n) log rn(6, X) and from the second part of the proof of Theorem9.19, &(l/n) log rn(S/2,X ) < h ( f ) . By Theorem 9.17 we have 1
lim - log rn(6,X) n
I < !& -n1 log sn(6, X) < -lim - log s,,(6, X) n 1 < -lim - log rn(6/2,X) n
and so h( f) = lim(l/n) log ~ ~ (X). 6 ,
733
Topological Dynamics
From now one we consider some connections between the entropy and periodic points. We first prove some results about periodic points for homeomorphisms. + X be a homeomorphism of a compact metric space. We say that f satisfies specijication if for E > 0 there is M ( E )= M > 0 such that for any finite sequence x I , x2, . . . , xk E X , any integers a, < 6 , < a2 < 6, < . . . < uk d b,witha, - b,-I 2- M ( 2 < j < k)and any p > M (bk - a,), there is x E X such that fp(x) = x and d(f’(x),f’(x,)) < E (a, f i d b,, 1 d j f k ) .
9.23. Definition. Let f : X
+
9.24. Remark. If f : X mixing.
.+
X has specification, then it is topologically
Proof. Let U , V be nonempty open subsets of X . For any x E U and y E V choose E > 0 such that U&(x)c U and V,(y) c V. J2et M ( E )= M > 0 be a number with property in the definition of specification. For n 2- M there is z E X such that d(z, x) < E and d ( f ” ( z ) , y) < E . Hence z ~ f - “ ( U , ( y ) ) andf-“(U,(y)) n UAx) # 0. 0 9.25. Theorem. Let f : X + X be a homeomorphism of a compact metric space. rff is topologically mixing, expansive and has the POTP, then f satisfies specification.
Proof. Let c be an expansive constant for$ For 0 < E < +c let 6 be a number with property in the definition of the POTP. Choose a finite open cover a of 6-open balls. Sincef is topologically mixing, for U , , V, E a there is MI, > 0 such thatf”(U,) n V, # 8 for n 2 MI,.Write M = max{M,,}. To get the conclusion, take any finite sequence x l r. . . , xk E X , and choose an integer sequence a, < 6 , < * . . < ak d bk with a, - 6,- 2 M (2 f j d k) andp > M (bk - a,)SuCh thata,,, = bk+l = p -I- a,,xk+, =f4’-akt’(X~). We write U(z) if z E U E a for z E A’. Then for 1 ,< j f k there is E U(f””’(x,,,)). From this it follows that y, E U(f4(x,)) such thatful+’-bl(yJ) for 1 < J d k , U(faJ+‘(x,+,))nfa”’-bU(fbl(x,)) # 8 since a,,, - b, 2 M. Let {z,} be a sequence such that z, = f’(x,) (a, f i < b,), z, = f’-’/(y,) (b, d i d a,,,) and z , + ~= z, (i E Z).Then {z,} is a 6-pseudo-orbit for f. Since f has the POTP, there is x E X such that d(f ‘(x), z,) < E for i E Z. Since z,+,, = z, ( i E Z), we have d ( f ’ + P ( ~z,) ) , < E (iE Z) and so d(flfp(x),f’(x)) < 2.5 -= c (i E Z). By expansiveness, f p ( ( x ) = x and therefore f satisfies the specification. 0
+
,
N. Aoki
734
9.26. Remark. I f f : X
+
X has specification, then h(f) > 0.
Proof. For x , y E X ( x # y ) choose E > 0 such that d(x, y) > 3.5. Let M be a number with property in the definition of specification. Take and fix n > 0. Then for ( z , , . . . ,z,,) with zi E { x , y } (1 Q i Q n) we have d ( Y M ( z )zi) , Q E (1 Q i Q n) for some z E X by specification. It is easily checked that for distinct n-tuples there correspond different z. Hence there are at least 2" points which are (nM, &)-separated.This implies h(f) 2 lim(l/nM) log 2" = 1/M > 0. 0 9.27. Theorem (Bowen [1974]). Iff: X + X is expansive and satisfies specification, then h ( f ) = lim(l/n) log # p , ( f ) wherep,( f)denotes the set of all n-periodic points and # E denotes the cardinality of E. This is obtained from the following five lemmas.
0
9.28. Lemma. Zfe > 0 and 6 > 0 are small enough, then there is C,,E> 0 such that ~ " ( 6X, ) < C6,e~,,(~, X ) f o r n 2 0. Proof. Let E > t 6 . By expansiveness there is N = N(6) such that d ( f ' ( x ) , f ' ( y ) )Q 2.5 for l i l Q N implies d(x, y) < 6. Choose fl > 0 so small that d(x, y) Q fl implies d ( f ' ( x ) ,f'( y)) Q 6 for I i I < N. Let K be the maximal number of (1, @separated points in X x X (with the maximal metric). Let F c X be maximal (n, &)-separated and E c X maximal (n, 6)separated. For x E E there is z ( x ) E F such that d ( f ( x ) , f ' ( z ( x ) ) )< E for 0 < i < n. If not F u { x }is (n, &)-separated,a contradiction to the maximality of F. For z E F write E, = { x E E : z ( x ) = z } . Then we have # E, Q K. For, if x , y E E, then d ( f ' ( x ) ,f ' ( y ) ) < 2~ for 0 d j < n and hence d(f'(x), f ' ( y ) ) < 6 for N Q j < n - N. But { x , y} is (n, &separated. Hence we or d ( f " ( x ) , f " ( y ) ) > fl. Thus { ( x , f " ( x ) ) : xE E,} c have d(x, y) > X x Xis (1, fl)-separated and so s,,(6, X) = # E Q K # F = Kin(&,X ) . 0 9.29. Lemma. If E > 0 is small enough, then there is D, > 0 such that s,, + ... + n k ( ~ , X ) Q llf-, D&s,,(E,X ) whenever n, , . . . , nk 2 1. Proof. Let E c X be (n, + - . . + nk, &)-separatedand 1;1. c X be maximal (ni, ~/2)-separatedfor 1 < i < k. For x E E choose z ( x ) = (z,(x),. . . , zk(x)) E Fl X * . . X Fk such that d ( f " ' f " ' + " i - i +(x),f'(zi(x)) J < $ E for 0 Q j < n (zi(x)exists by the maximality of 6). The map z( ) : E + F, x
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Topological Dynamics
+
+
* x F, is injective since E is (n, * * n,, &)-separated.Hence # E < llf=, #t;l = llf=,sn,(+e, X). By Lemma 9.27 we haves,,(& X ) < C , ~ 2 , , ~X, ,)( ~ , and hence the lemma follows from D, = C,/2,E. 0 1
9.30. Lemma. If E > 0 is small enough, then there is E, such that S n , + . . . + n , ( G X ) 2 nf=lE,sni(&,X ) .
Proof. Choose a maximal (n,, 3&)-separatedset El c X for 1 M be as in the definition of specification and write
< i < k. Let
a,, = 0,
+ . . . + n,-, + ( i - 1)M, n, + + nk + ( k - 1)M.
a, = n, m =
2
< i < k,
*
By specification property, for each z = ( z , , . . . , z,) E El x * x Ek there is x = x(z) with d(f”’+’(x), f’(z,)) < E for 0 < j < n,, 1 < i < k. Obviously E = { x ( z ) :z E El x * x E,} is (m, +separated. Hence ll~=,s,,(3~, X) < s,,,(E, X). On the other hand, Lemma 9.29 implies that 1
~ ( 6 X,
)
< D,ksn,+
+nk(&,
x ) ( ~ w ( & X)Y-’ ,
and Lemma 9.28 ensures that CETJ:Sn,(&,
X)
< ~n,(3&9 X).
Letting E, = {C,,3rDc~Y(~, X ) } - ’ , we have the result.
9.31. Lemma. If& > 0 is small, then D;leh n 2 1 . Here h = h ( f ) .
< s,,(E, X) < E;lehn
0 for all
Proof. Suppose s,(E, X ) < D;’e” for some n. By Lemma 9.29 we have X ) < {D,s,,(E, X)},and henceskn(&, X) < em.Thereforeh = lim(l/kn) log s,,,(E, X) and h < ( l / n ) log ehn= h, a contradiction. The proof goes similarly for the other inequality. 0
s&,
9.32. Lemma. There are 0 < D < E such that f o r a large enough, De”” < #p,(f) < Eeh”. Proof. If #pn(f)
E
> 0 is small enough, then p,( f) is (n, &)-separated. Hence therefore #p,( f) < Eeh” where E = EL1 by
< s,,(E, X) and
Lemma 9.3 1 . Let M be as in the definition of specification property and E c X be (n - M y3~)-separatedset (n > M ) . Then for each z E E there is x(z) E p,( f) such that d( f’(z), f’(x(z)) < E for 0 < j < n - M . Clearly z # z’
736
implies x ( z ) # x(z'). Hence where D = {D,,ehM}-'.
N. Aoki
##p,(f)
2
s,_,,,(~E, X ) 2
DjE'eh(n-M)= Deh"
0
Theorem 9.27 follows immediately from Lemma 9.32.
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W. The existence of expansive homeomorphisms of manifolds, Duke Marh. J. 32, 627-632. Expanding maps on compact metric spaces, Topology Appl. 13 327-334. Expansive canonical coordinates are hyperbolic, Topofogy Appl. 15, 205-2 10.
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K. Morita, J. Nagata, Eds., Topics in General Topology
0Elsevier Science Publishers B.V. (1989)
SUBJECT INDEX &-filter 465 d-ultrafilter 465 AE (absolute extensor) 50 ANE (absolute neighborhood extensor)
branch 721 branch set 642
50
ANR (absolute neighborhood retract) AR (absolute retract) 50 absorbent 438 accumulation degree 95 additive annihilator 248 K-space 316 KO-barrelled 439 KO-bounded 446 KO-space 316,425 almost disjoint 358 almost locally finite 377 almost periodic 632 almost a-compact 419 a-i.p. 471 a-limit set 638 a-related 695 a,X 462 annihilator 248, 305 Arzela-Ascoli's Theorem 42 I Ascoli Theorem 35 attractor 696 Bake space 427 balanced category 570 barrel 438 barrelled 439 basic cover 122 basic semi-cover 143 basic set 702 BCO 327 p-space 306 pL.X 462 bimorphism 570 Bing's example G 191 Bing-type example 112 bireflective subcategory 589 Bore1 probability measure 676 bounded 439 boundedly point-finite 88
50
c.u.b. set 476 C-embedded 51, 237 C-scattered space 530 c-space 278 c, 501 cO(A)
501
canonical coordinate 680 canonical homomorphism 489 canonical operator SO8 Cartesian closed category 621 category 205, 565 Cauchy filter I I Cauchy stack 14 CCR-operator 265 CCR-space 265 cellularity 428 central motion 674 chain 252, 327 chain component 700 chain recurrent 695 chain recurrent set 695 chain transitive 700 character 86 X(P. X ) 86 circled 438 clopen ultrafilter 465 closed under extremal epimorphism 597 closed under extremal monomorphism 594 closed under K-colimit 597 closed under K-limit 592 closure preserving 163, 368 co-(inverse image) 578 co-topological functor 616 co-well-powered category 575 cocomplete category 579 coequalizer 578 coextension (morphism) 590 cointersection 578 collectionwise 6-normal 196 compact-open topology 412 complete 15
742
Subject Index
complete category 579 completely separated 5 I complete topological linear space 437 complete valued field 498 completion 18, 20 concrete category 618 condence 419 connecting morphism 576 continuous annihilator 248 continuously completely regular space 265 continuously perfectly normal space 265 continuous pseudo-metric 57 contravariant functor 568 convergence semi-orbit 638 convex 438 coproduct 578 coreflection 590 coreflective subcategory 590 coreflector 590 coretraction 572 coseparator 590 cosmic space 425 countable intersection property (= c.i.p.) 465
countable order 327 countably complete 465 countably paracompact 82 covariant functor 205 cover for compact subsets 422 covering dimension 234, 553 covering map 642 cozero-set rectangle 208 cozero set 84 CP-expandable 382 CPN-operator 265 CPN-space 265 cs-network 316 cs*-network 323 C*-embedded 51, 206 cushioned refinement 167 D-product 550 decreasing 94 6-metrizable space 305 &normal 145 density 432 developable space 300 diagonal map 461 diagonal morphism 583, 599 diagram 575 dim 399 dimension 658
directed collection 163 direct limit of Banach spaces 502 direct limit of groups 484 direct product of Banach spaces 501 direct product of groups 483 direct sum of Banach spaces 501 direct sum of groups 483 direct system of Banach spaces 501 direct system of groups 484 discrete category 566 discrete collection 163 dissectable 331 dissection 331 dominate 377 Dowker space 188 dual category 567 duality principle for categories 573 dual of a group 485 dual of a homomorphism 485 dual of a linear operator 503 dual of a normed space 500 (E, M)-category 599 (E, M)-diagonalization property 599 (E, M)-factorizable 599 (E, M)-factorization 599 E-co-well-powered category 601 E-compactification 463 E-compact space 461, 597 E-reflective hull 601 E-reflective subcategory 589 E-regular space 461, 597 Eberlein compact 434 ECP 386 Ellis semi group 633 EM, 380 embedding (morphism) 618 empty category 566 encircling net 380 (Epi, Ex mono)-diagonarization property 584 (Epi, Ex mono)-factorizable 584 (Epi, Ex mono)-factorization 584 epi-sink 615 epimorphism 569 epireflective hull 596 epireflective subcategory 589 &-support 505 equalizer 573 equicontinuous 35, 414 equivalence natural relation 606 evenly continuous 35
Subject Index
(Ex epi, Mono)-diagonarization property 583
(Exepi, Mono)-factorizable 583 (Ex epi, Mono)-factorization 583 (Ex epi)-co-well-powered category 575 (Ex epi)-reflective category 589 (Ex mono)-well-powered category 574 expanding differentiable map 687 expanding map 688 expansive 635 expansive constant 636 exponential topology 697 extension (morphism) 588 extremal epimorphism 571 extremal monomorphism 571 8-compact space 108 F-final lift 616 F-initial lift 615 F-limit point 108, 108 F-sink 615 F-source 614 faithful functor 616 finally in B 597 finitely closure preserving (FCP) 99, 173, 377
finitely cocomplete category 579 finitely complete category 579 finite partition 722 finite topology 265 finite type 672 fit 94 fit family 94 fit part 94 fixed I-filter 465 foliation 725 forgetful functor 568 Fpp-space 259 free I-filter 465 free base 490 free group 490 free L-space 400 free sequence 91 front closed embedding 572 Frichet mace 276 F,-metrizable 155, 380 F,-screenable 184 full subcategory 205, 566 fully normal 171 functionally complete 434 functor 567
743
g-function 245 g-function (n.n.0.) 245 y-space 306 Ga-closed 464 G,-closure 464 G,-diagonal 294 G,+-diagonal 334 generalized ultraproduct 479 generalized uniform space 4 generalized uniformity (-base) 3, 4 generator 636
A@ (the class)
380
H-closed 209 Hausdorff metric 698 hemi-compact 417 HEP (homotopy extension property) hereditarily closure-preserving 250 hereditarily closure-preserving family hereditarily M , 380 hereditarily metacornpact 530 homogeneous 630 hyperbolic coordinate 68 I hyperbolic metric 656 hyperconnected space 610 i-weight 433 idempotent 491 identity (morphism) 565 identity functor 568 inclusion functor 568 increasing 94 Ind 399 independent 252 initially in B 593 initial map (in Top) 600 initial object 578 interior-preserving 163 intersection 577 invariant 677 inverse image 578 inverse limit 647 inverse limit of Banach spaces 501 inverse limit of groups 484 inverse system of Banach spaces 501 inverse system of groups 484 irreducible 400, 672 isomorphic 500 isomorphism 500, 569 isomorphism-closed subcategory 589 k-continuous 438, 474 k-network 287, 316,425
75 285
744 X-network 346 K-scattered space 529 k-space 36,216, 277,475 K-collectionwise normal 88 K-Dowker space 98, 188, 194 ~-metrizablespace 305 K-paracompact 82 k,-leader 474 k,-space 474 k,X 474 k,-space 438 /-equivalent 447 /-separated space 604 A (the space A) 68 I-shrinking 97 I-ultraparacompact 104 large inductive dimension 555 Lahev space 140, 281 length 450 limit 576 Lindelof number 428 linear functional 500 linear isometry 500 linear topological space 47 I" 501 P ( A ) 501 local W-refinement 17I local W-refining sequence 175 local isometric covering map 641 locally convex 47, 439 locally countable 163 locally small category 574 locally starring 172 local product structure 682 local stable set 654 local star-refinement 3, 17I local unstable set 654 lower limit operator 603 m-collectionwise normal 59 M-embedded 62 m-map 341 m-separable pseudo-metric 57 M-space 75, 130, 298 M,-space 399 M ,-space 172, 369 M,-space 369 M,-space 75, 369 M-structure 380 Markov partition 710
Subject Index
Markov subshift 672 maximal E-compactification 463 maximal natural relation 608 measurable cardinal 467 metacompact 179 Michael line 74, 121, 130 minimal 627, 630, 725 minimal set 627 M" -embedded 62 (mod /+network 346 mono-coreflective hull 597 mono-coreflective subcategory 590 mono-source 614 monolithic 434 monomorphism 569 monoreflective subcategory 589 monotone annihilator 248 monotone decreasing 94 monotone increasing 94. 163 monotone normality operator 369 monotonically normal space 307, 369 Morita-Rudin-Starbird Theorem 125 morphism 565 mosaic 373 mosaical 373 mosaic space 373 M*-space 341 p-space 380 multiple coequalizer 578 multiple equalizer 578 multiple pullback 578 multiple pushout 578 .N (class N ) M ( X ) 102
136, 155
N-compact space 459 natural relation in A 608 natural relation in Top 606 natural sink 576 natural source 575 natural transformation 608 neighborhood retract 50 net-weight 433 network 369, 425 Niemytzki space 67 non-Archimedean Banach space 499 non-Archimedean norm 499 non-Archimedean normed space 499 non-Archimedean valuation 497 non-Archimedean valued field 497 nonwandering 673
Subject Index
nonwandering set 673 normal cover 43 normal sequence of open covers norm topology 412
43
object 565 o-limit set 638 open map 642 operation norm 499 opposite category 567 orbit 627 order 531, 672 orientable 646, 725 orientation 645, 646 orthogonal 491 orthonormal 509 orthonormal base 509 outer base 369 outer quasi-base 368
9 (class 9) 136, 155, 380 P(m)-space 128 p-adic valuation 497 P-embedded 55 P-space 92, 11 I , 128, 437, 517, 537 p-space 130 9-space associated to E 441 para-Lindelof 190 paracompact 82 partial refinement 162 partition 373 perfect 371 perfect m-map 341 play 524 Pm-embedded 55 point-countable type 41I point-finite rank 252 point-finite 163, 375 pointwise-countable type 304 pointwise W-refinement 171 pointwise W-refining sequence I75 pointwise star-refinement 171 pointwise starring 185 positively chain invariant 696 positively expansive 647 primarily fit 95 product 577 product of categories 566 proper 710 proper basic cover 124 proper basic semi-cover 146
745
property (6) 145 property (*) 145 property (**) 145 property D 97, 187 property 9 ? ( ~ ) 97 property D 97, 187 property 9 ( ~ )97 property g ( Z ) 96 property g ( Z , 2) 96 pseudo-character 432 pseudo-m-compact 469 pseudo-Anosov map 726 pseudo-covering map 642 pseudometric associated with a cover 82 pseudo-orbit 667, 695, 722 pseudo-orbit-tracing property 667 pullback 577 pushout 578 q-space 417 quasi-perfect rn-map 34 I quasi base 368 quasi-k-space 310 quotient category 567 quotient map (morphism) 618 rank 252 realcomponent space 459 rectangle 543, 708 rectangular product 234, 558 recurrence 674 refinement 162, 636 reflection 588 reflective subcategory 588 reflector 205, 588 reflexive group 489 reflexive normed space 508 regular G,-set 145 regular cardinal 238 regular epimorphim 573 regularly stratifiable 380 regular monomorphism 573 regular ordinal 238 relatively compact 669 representative set of epimorphisms from X 575 representative set of monomorphisms into X 574 retract 50 retraction 569 e-connected object 609
746
Subject Index
p-separated object 608 p-separated space 607 right ideal 633 rimcompact 271 ring homomorphism 479 S, (Arens’ space) 276 scattered set 534 section 569 semi-open cover 180 semi-stratifiable 369 semi-stratification 369 semi-uniformity (-base) 3, 4 semi-uniform space 3 semi-simple 635 separate 251, 415 separated 369 separate points 88 separating cover 294 separator 590 sequentially complete topological linear space 437 sequential space 277 shift 630 shift map 647 shrinkable 186 shrinking 97, 186 shrunk 97 a-locally compact 130 a-locally finite 163 I55 a-metric Z-network 345 a-paracompact 184 Z-product 187 Z-space 153, 345 a-space 151, 184, 369 simple accumulation point 86 simply generated (simple) epireflective 61 1 singular foliation 724 sink 615 skewness 656 slender group 489 slice 82 small category 566 sofic 723 S, (sequential fan) 276 source 614 Souslin number 428 space determined by (B 284 space dominated by (B 285 special outer functional 445 special refinement 122
special refinement relative to 144 specification 733 spherically complete metric space 498 spherically complete valued field 498 stable 655 stable set 655 star-filter 13 star-finite 375 stationary set 476 stationary winning strategy 525 Stone-Weierstrass Theorem 41 7 stratifiable 369 stratification 369 strict extension of a space 20 strictly Cauchy filter 11 strong C-space 187 strongly Frkhet space 276 strongly closed under Kcolimit 597 strongly closed under K-limit 593 strongly point finite 375 strongly 0-dimensional 470 strong Z-network 345 strong C-space 345 subcategory 566 submetacompact 175 submetrizable 262, 419, 469 subparacompact space 184, 294 subshift 672 summand 490 support for a homomorphism 487 support for a linear operator 513 surface 646 symbolic sequence 672 syndetic 632 r-closed 205 r-closure 209 r - l ~ ~ lfinite ly 206 r-open 205 r-paracompact I63 terminal object 577 test space 103 0-base 330 O-refinable space 175, 299 0-refinement 175, 329 0-sequence 175 tightness 88, 422 top 105 topological category 618 topological entropy 728 topological functor 615 topological game 524
T
topologically conjugate 628 topologically mixing 702 topologically stable 684 topologically transitive 628 topology of I convergence 412 topology of pointwise convergence torsion free 491 totally normal space 555 trace 667 transition matrix 672 trivial valuation 497 Tychonoff functor 205 Tychonoff space 204
v ( Y ) 88 valuation 497 valued field 497 Vietori’s topology 412
88 Ulam measure 467 ultraconnected space 610 ultraproduct of Banach spaces 507 ultraproduct of groups 488 uniform Eberlein compact space 88 uniformity of a topological linear space 436 uniformly almost periodic 632 uniformly locally finite 68 uniquely (E, M)-factorizable 599 uniquely (Epi, Ex mono)-factorizable u(C)
747
265
w-compact 209 w-normal 230 w-paracompact 224 WCP-expandable 382 wcs-network 316 wA-condition 259 wA-space 300 weak 93-property 187 weak base 294 weak generator 636 weakly Cauchy 224 weakly Lindelof space 267 weakly [-equivalent 452 weakly paracompact 179 weakly regular 8 weakly O-refinable 178 weak star-filter 13 weak topological basis 452 well-powered category 574 winning strategy 525 9-product 382 Y-countably compact
341
584
uniquely (Ex epi. Mono)-factorizable 583 universal space 351 unstable set 655 upper intersection 93 upper limit operator 603
z-embedded 65, 206 Z-map 213 0-dimensional 390, 463 0-dimensionally a-compact zero-filter 467 zero object 579
471
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