Dimension and Extensions

DIMENSION AND EXTENSIONS

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. Kempennan, H.A. Lauwerier, W.A.J. Luxemburg, L. Nachbin, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 48

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

Dimension and hxtensions

J.M. AARTS Faculty of Technical Mathematics and Informatics Deljit University of Technology Delft, The Netherlands

T. NISHIURA Department of Mathematics Wayne State University Detroit, MI, USA

1993 NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

L i b r a r y of Congress C a t a l o g i n g - i n - P u b l t c a t i o n D a t a

Aarts, J. M. Dimension and extensions / J.M. Aarts. T. Nishiura. p. cm. -- (North-Holland mathematical library ; v . 48) Includes bibliographical r e f e r e n c e s and index. I S B N 0-444-89740-2 talk. paper) 1. Dimension theory (Topology) 2. M a p p i n g s (Mathematics) 3. Compactifications. I. Nishiura. T. 11. Title. 111. Series. PA611.3.A27 1993 514’.32--d~20 92-44402 CIP

ISBN: 0 444 89740 2

0 1993 ELSEVIER SCIENCE PUBLISHERS B.V. 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., Copyright & Permissions Department, P.O. Box 521, 1000 AM 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 and/or 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. This book is printed on acid-free paper Printed in The Netherlands

Dedicated to the memory of Johannes de Groot

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PREFACE

Two types of seemingly unrelated extension problems are discussed in this book. Their common focus is a long standing problem proposed by Johannes de Groot, the main conjecture of which was recently resolved. As is true of many important conjectures, a wide range of mathematical investigations had developed. These investigations have been grouped into the two extension problems under discussion. The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early successes in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, two extension problems were spawned. The first extension problem concerns the extending of spaces. Among the various extensions studied here, the important ones are compactifications and metrizable completions. Also, a-compact extensions are discussed. The natural problems are the construction of dimension preserving extensions satisfying various extra conditions and the construction of extensions possessing adjoined subsets satisfying certain restrictions. The second extension problem concerns extending the theory of dimension by replacing the empty space with other spaces. The compactness degree of de Groot was defined by the replacement of the empty space with compact spaces in the initial step of the definition of the small inductive dimension. Such replacements lead t o two ki& of investigations. The first is the development of a generalized vii

Viii

PREFACE

dimension theory. The other kind is the search for a dimensionlike invariant which would internally characterize the first extension problem. It took almost fifty years to settle the original problem of de Groot. Surprisingly, the analogous problem for metric completions turned out to be much more manageable. It also became apparent in many other analogous problems that “excision” was a more natural concept than that of extension. In 1980 Pol produced an example to show that the compactness degree was not the right candidate for characterizing the compactness deficiency. Then Kimura showed in 1988 that the compactness dimension function that was introduced by Sklyarenko in 1964 characterized the compactness deficiency. Also, from 1942 to the present, a substantial theory of the extension of dimension theory t o dimension-like functions evolved. This book is a presentation of the current status of the two extension problems.

The organization of the book The material has been arranged in six chapters with the following themes: history, mappings into spheres, inductive invariants, covering dimension, basic dimension and compactifications. The first chapter is a historical introduction, dealing mainly with separable metrizable spaces. In it we discuss prototypes of the results which will be obtained in the subsequent chapters. We have also included a short introduction into dimension theory, making the book essentially self-contained. In Chapter TI the theory of dimension and mappings into spheres is generalized to dimension-like functions. A wealth of examples is presented, including those related t o the Bore1 classes. Throughout the book, use is made of the material in Chapters I and 11. Chapters I11 and IV deal with functions of inductive dimensional type and with functions of covering dimensional type. In Chapter V we return t o the class of metrizable spaces t o discuss the basic dimension functions, that is, functions that depend upon the existence of special bases for the open sets of a space. These functions are used to relate the many dimension functions that were developed in the earlier chapters. In the final chapter the various compactifications are discussed in a unified way. Included in this discussion are the compactification of de Groot [1942] which was the origin of the theory presented in the book and the compactification of Kimura [1988] which finally resolved the compactification problem.

PREFACE

ix

What is new? Much of the material is taken from the literature. Some of the new material appearing in the book is summarized below.

- The discussion of the basic and the order dimension leading t o the equality of the generalized covering and inductive dimensions in many cases. - The characterization of the generalized covering dimension by means of mappings into spheres, resulting in the proof of the equality of strong inductive compactness dimension and covering compactness dimension. - The introduction of the class of spaces called the Dowker universe which is also of interest in dimension theory proper. - The line-up of the various compactness dimension functions. - The discussion of recent results in the axiomatics of the dimension functions, with emphasis on the class of metrizable spaces, including the result of Hayashi. - The integrated discussion of compactifications constructed by Zippin, Freudenthal, de Groot, de Vries and Kimura as Wallman compactifications.

What is next? The future is not ours to see, of course, b t sure1 there are ma Y possible investigations yet to be pursued. Some unsolved problems have been posed at the end of each chapter. The listing is by no means exhaustive. We have tried to indicate what we think are the more interesting open problems. This book has not t o u c h 4 upon the transfinite dimension and its possible generalizations. The 1987 paper by Pol may serve as an introduction into this problem area.

Acknowledgements We are indebted to K. P. Hart for valuable advice on the use of and A M S - W . To Eva Coplakova go our thanks for reading various versions of the book and for providing us with the English translations of the Russian papers cited in the Bibliography whose translations were not available in the literature. Our thanks is due to Jan van Mill for helpful comments and continuous encouragement.

X

PREFACE

Finally, we would like to thank both the Faculty of Technical Mathematics and Informatics of the Delft University of Technology and the Department of Mathematics of Wayne State University. Their generous support enabled us to work together on the book. The book was typeset by dn,lS-m.

J. A. and T. N.

CONTENTS

Preface Chapter I. The separable case in historical perspective 1.1. A compactification problem 1.2. Dimensionsgrad 1.3. The small inductive dimension ind 1.4. The large inductive dimension Ind 1.5. The compactness degree; de Groot’s problem 1.6. Splitting the compactification problem 1.7. The completeness degree 1.8. The covering dimension dim 1.9. The covering completeness degree 1.10. The a-compactness degree I. 11. Pol’s example 1.12. Kimura’s theorem 1.13. Guide to dimension theory 1.14. Historical comments and unsolved problems Chapter 11. Mappings into spheres 11.1, Classes and universe 11.2. Inductive dimension modulo a class P 11.3. Kernels and surplus 11.4. P-Ind and mappings into spheres 11.5. Covering dimensions modulo a class P 11.6. P-dim and mappings into spheres 11.7. Comparison of P-Ind and P-dim 11.8. Hulls and deficiency 11.9. Absolute Borel classes in metric spaces 11.10. Dimension modulo Borel classes xi

vii

14 20 28 41 48

53 59 66 68 69 73 74 76 85 87

93 100 106 109 112 119

xii

CONTENTS

11.1 1. Historical comments and unsolved problems Chapter 111. Functions of inductive dimensional type 111.1. Additivity 111.2. Normal families 111.3. Optimal universe 111.4. Embedding theorems 111.5. Axioms for the dimension function 111.6. Historical comments and unsolved problems Chapter IV. Functions of covering dimensional type IV.l. Finite unions IV.2. Normal families IV.3. The Dowker universe D ’ IV.4. Dimension and mappings IV.5. Historical comments and unsolved problems Chapter V. Functions of basic dimensional type V.1. The basic inductive dimension V.2. Excision and extension V.3. The order dimension V.4. The mixed inductive dimension V.5. Historical comments and unsolved problems Chapter VI. Compactifications V I . l . Wallman compactifications VI.2. Dimension preserving compactifications VI.3. The Fkeudenthal compactification VI.4. The inequality IC-Ind 2 K-Def V1.5. Kimura’s characterization of K-def VI.6. The inequality K-dim 2 K-def VI.7. Historical comments and unsolved problems Chart 1. The absolute Bore1 classes Chart 2. Compactness dimension functions Bibliography List of symbols Index

126 129 130 139 146 160 171 181 185 186 192 197 204 215 217 217 223 234 241 244 247 248 258 266 282 287 300 312 72 246 315 327 329

Sculpture "NeedleTower 11" by Kenneth Snelson Photo: Jan Aarts (reprinted with permission of K. Snelson)

CHAPTER I

THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE This introductory chapter presents an exposition of the history of a compactification problem in dimension theory and the dimensionlike functions that have grown out of it. At the same time, it presents the plan of the book. By no means is the chapter just a collection of historical facts, rather it is a mathematical text in which some emphasis has been placed on the history of the compactification problem. For the sake of simplicity, many of the results will not be presented in full generality; the later chapters will make up for this. 1. A compactification problem

An appropriate subtitle for the book would be “A compactification problem in dimension theory.” The theory presented in this book originated from a compactification problem posed by Johannes de Groot [1942] in his thesis. This problem was posed in a form that made it appear to be a natural generalization of the small inductive dimension. It was finally solved in the negative by Pol [1982]. Meanwhile a substantial theory of dimension-like functions had been developed in the various attempts to solve the problem. It is this theory that is the core of our book. Since the emphasis in this chapter is being placed on the historical perspective of the compactification problem and the theory of dimension-like functions, it is proper that the chapter begin with a few remarks about the origins of dimension theory itself and the most important results of dimension theory. The dimension-like functions and the compactification problem will then appear in their natural historical settings. Though the chapter has been made as self-contained as possible, there are details of two major points which have been delayed t o later

2

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

chapters. The first of these points is the sum and decomposition theorems whose details will be discussed in Chapter 111, and the second is the constructions of various compactifications whose details will be given in Chapter VI. 2. Dimensionsgrad

The need for a dimension theory came at the end of the 19th century when examples of “dimension-raising” maps had been discovered. In 1878 Cantor showed that the line and the plane have the same number of points by constructing a bijective function from the real line R onto the plane W2. Then in 1890 Peano constructed a continuous mapping of the interval [0,1] onto its square and thereby exhibited a continuous parametrization of the square by means of an interval. It was the first space-filling curve. Apparently a naive concept of dimension was no longer adequate. In the paper [1912], published in a philosophical journal, Poincard pointed out that dimension can be defined in an inductive way by using the notion of separation of a space. In the next year Brouwer [1913] gave the first definition of a topological dimension function, the Dimensionsgrad, and showed that the Dimensionsgrad of the n-cube I ” is equal to n. The definition of the Dimensionsgrad is an inductive one based on a notion of separation. But Brouwer’s notion of separation is somewhat different from the one which is used nowadays in dimension theory. Because of the inductive nature of the definition of dimension, the value of the dimension function depends in a sensitive way on the start of the definition, that is, on the definition of the zero-dimensional sets. In this respect, Brouwer’s definition was not the best possible choice. In Chapter I11 we shall study the inductive dimensions and their generalizations and reveal the dependence of the value of the dimension-like functions upon the start of the definitions. After the structure of zero-dimensional spaces had been explored by various mathematicians, Urysohn in [1922] and Menger in [1923] finally and independently defined the dimension function which eventually became known as the small inductive dimension. The fine point of the definition is the step before the first step, namely the assigning of the dimension number minus one to the empty space. In this way the correct choice of the collection of zero-dimensional spaces was made. But one should not be deceived, the step from 0

3. THE SMALL INDUCTIVE DIMENSION ind

3

to 1 still contains many delicate points as the general theory of dimension-like functions will show. 3. The small inductive dimension ind

We shall now give a precise formulation of the definition of the small inductive dimension and state the most important theorems. The following notion of separation will be used. 3.1. Definition. Let X be a topological space and let F and G be disjoint sets in X . A subset S of X is called a partition between F and G if X\S can be written as a union of disjoint open sets U and V with one containing F and the other containing G, that is,

X\S=UUV,

UnV=0,

FCU,

GCV.

If F consists of only one point, F = { p } , we say that S is a partition between p and G. Notice that a partition is always a closed set. 3.2. Definition. To every topological space X one assigns the small inductive dimension, denoted by ind X , as follows.

(i) i n d X = -1 if and only if X = 0. (ij) For each natural number n, i n d X 5 n if for each point p in X and for each closed set G of X with p 4 G there is a partition S between p and G such that ind S 5 n - 1. (iij) ind X = n if ind X 5 n and ind X $ n - 1. (iv) ind X = 00 if the inequality ind X 5 n does not hold for any natural number n. The reader may have noticed that we have adopted the convention that the natural numbers start with 0. We shall denote the set of natural numbers by N. It is immediately clear that ind is a topological invariant, i.e., i n d X = indY whenever the topological spaces X and Y are homeomorphic. A somewhat closer look at the definition of ind reveals that ind X = 00 for each nonregular topological space X. Here are a few examples to illustrate the definition.

4

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

3.3. Examples. The sets of real, rational and irrational numbers will be denoted by R, Q and B respectively. a. For a and b in P with a < b the set ( a , b ) n Q is closed as well as open in Q.It follows that for p in ( a , b ) n Q the empty set is a partition between p and Q \ ( a , b) in Q. Consequently, ind Q = 0. A similar argument will show that i n d P = 0.

b. As W is connected, indW > 0. For E > 0 and for p E R the two-point set { p - E , p E } is a partition between p and the closed set ( - o o , ~ - ~ E ] ~ [ ~ + ~ E , + o o ) . It easily follows that indR = 1. Notice that the 1-dimensional space R can be covered by two 0-dimensional spaces since R = P U Q.

-+

c . The n-dimensional sphere S n is the subset of Rn+' defined by

8" = { 2 : x E R "+I,

IIXlI

= 1}

where n is in N and 11 denotes the Euclidean norm. Denoting the spherical neighborhood of the point p in 8" with radius E by S , ( p ) , we have that its boundary B ( S c ( p ) ) is homeomorphic with Sn-l when 0 < E < 2 and n 2 1. It readily follows that indS" 5 n. The inequality ind S" 2 n (whence ind 8" = n) will be established in Theorem 4.8. c, namely ind $" = n, we shall prove ind W " = n. Recall that $" is homeomorphic with the one-point compactification of R". Note that S" is homogeneous, that is, for any two points 2 and y of 8" there is a homeomorphism of S" onto itself sending x t o y . Thus indR" = n.

d. With the aid of

Therefore the small inductive dimension distinguishes R " from R for n # m and thus satisfies a basic requirement of a dimension function. The theory of the small inductive dimension was developed between 1920 and 1940. A beautiful account can be found in the book by Hurewicz and Wallman [1941]. See also the more recent book by Engelking [1978], in particular Chapter 1. This theory of ind has become the paradigm of the theories about inductive invariants. The subspace theorem, the sum theorem and the decomposition theorem, to be discussed below, are essential features of it. As these

3. THE SMALL INDUCTIVE DIMENSION ind

5

types of theorems make their appearances in the book they will be so identified. 3.4. Theorem (Subspace theorem). For every subspace Y of a space X, ind Y 5 ind X .

Proof. The proof is a very nice example of the type of inductive proofs found in dimension theory. The statement in the theorem holds trivially for ind X = 00. So, without loss of generality, we may assume that i n d X = n < 00. The statement is obvious if X = 0 (equivalently, i n d X = -1, the first step of the inductive proof). Let us assume for a natural number n that the statement has been proved for all spaces 2 with ind 2 5 n - 1 and let X be a space with i n d X = n. Suppose that p is a point in a subspace Y of X and that G is a closed subset of Y with p 4 G. Then clX(G) is a closed subset of X with p I$ clx(G). Because ind X = n, there is a partition S in X between p and clx(G) such that ind S _< 7t - 1. It follows directly from the definition of a partition that 5’ n Y is a partition in Y between p and G. By the induction hypothesis we have ind (Sn Y ) 5 ind S 5 n - 1. Thus we have ind Y 5 n. Thereby the theorem is proved. The following two propositions suggest other ways of defining the small inductive dimension. For regular spaces these alternative definitions result in the same dimension function. The straightforward proofs will be left to the reader. 3.5. Proposition. Let X be a regular space. For each natural number n, i n d X 5 n if and only if each point p of X has a neighborhood base B ( p ) such that ind B ( U ) n - 1 for every U in f ? ( p ) .

<

In the statement of the proposition, the notation B (U)has been used t o denote the boundary of the set U. The content of the proposition is sometimes stated as follows: ind X 5 n if and only if each point p of X has arbitrarily small neighborhoods U with ind B (U)5 n - 1. 3.6. Proposition. Suppose that X is a regular space. For each natural number n, ind X _< n if and only if there exists a base f? for the open sets of X such that ind B (U)5 n - 1 for every U in f?.

6

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

We shall now formulate the countable sum theorem and the decomposition theorem. The proofs of both of them will be presented in Chapter I11 in a much more general setting, see Theorem 111.2.10 in particular. Up t o the discussion in Section 111.2 we shall freely use both theorems. The reader should not worry too much about this because the sum and decomposition theorems can be regarded as axioms of small inductive dimension until Chapter 111. Though this arrangement of material is not the most logical one, it will enable us to present some motivating examples first. Indeed, the arrangement is quite natural for the thematic approach in this chapter. 3.7. Theorem (Countable sum theorem). Let { Fi : i E N} be a countable, closed cover o f a separable metrizable space X . Then

ind X = sup { ind Fi : i E N}. Recall that a subset of a space X is said t o be an Fu-set if it can be represented as a countable union of closed subsets of X . Note that an open subset of a metrizable space is an Fu-set of that space. 3.8. Theorem (Decomposition theorem). Let X be a separable metrizable space. If ind X 5 n, then X can be partitioned into n -t 1 disjoint subsets X i with ind X i 5 0, i = 0,1,. . .,n. Moreover, one of the Xi’s may be assumed to be an F,-set. Conversely, if X = U{ X i : i = 0 , 1 , . . ., n } is such that ind X i 5 0 for every i , then ind X 5 n.

The sum theorem and the decomposition theorem are very powerful. Their power will be best illustrated by presenting some applications. Just as in the statements of the previous two theorems we shall sometimes assign a type t o a theorem for easy reference. 3.9. Theorem (Addition theorem). Let X be a separable metrizable space. I f X = Y U 2, then

ind X

5 ind Y -t ind Z

+ 1.

Proof. We may assume that both Y and 2 are not empty. Let ind Y = n and ind 2 = m. Then we have the two decompositions

Y = u { Y , : i = O , l , ..., n } , i n d Y , = O , Z = U{ Zj : j = 0, 1,. . .,m }, ind Zj = 0,

..., n, = 0, 1, . . .,m.

i=O,l, j

3. THE SMALL INDUCTIVE DIMENSION ind

7

Because of the equality

X = ( u { ~ , : i = o,..., , i n } ) ( ~U { ~ j : j = 0 ,,..., 1 m}), we have by the decomposition theorem that ind X 5 ( n

+ 1) + ( m t 1) - 1.

The theorem now follows. By applying the decomposition theorem to P,Q and R of Examples 3.3.a and b, we can prove i n d R 1 in a sophisticated way.

<

In the sequel we shall frequently use the product theorem for the small inductive dimension. 3.10. Theorem (Product theorem). Let X x Y be the topological product of two separable metrizable spaces X and Y a t least one of which is not empty. Then

ind (X x Y) 5 ind X

+ indY

Proof. The proof is by way of a double induction on ind X = n and indY = m. The statement of the theorem is evident if either i n d X = -1 or i n d Y = -1. Let i n d X = n and i n d Y = m and suppose that the statement has been proved correct for the cases (1) i n d X 5 n and i n d Y 5 m - 1 , (2) i n d X 5 n - 1 and indY 5 m. Each point (z,y) in X x Y has arbitrarily small neighborhoods of the form U x V where U and V are open neighborhoods of x and y in X and Y respectively with ind B x ( U ) 5 n - 1 and ind By(V) 5 m - 1. We have BXxY(U x V ) = (ClX(U) x BY(V)) u ( B x ( U ) x ClY(V)). As both sets in the union of the right-hand side are closed, we have ind B x x y ( U x V)

5 ntm - 1

by the subspace and sum theorems and the induction hypotheses (1) and (2); thereby the induction is completed. Another consequence of the sum theorem is the point addition theorem: The dimension of a nonempty separable metrizable space cannot be raised by the adjunction of a single point.

8

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

3.11. Theorem (Point addition theorem). Suppose that X is a separable metrizable space consisting of more than one point. For each point p in X , i n d X = ind ( X

\ {p } ) .

Proof. By the subspace theorem we have ind (X \ { p } ) 5 ind X . For each n in N define Fn = { z E X : 5 d ( s , p ) } . As X is the union { p } U (U{ F, : n E N}), the countable sum theorem gives ind X = max { 0, sup { ind F, : n E N } }. Because Fn is not empty for some n, we have i n d X 5 sup{indF, : n E N} = i n d ( X \ { p } ) . Henceforth in our discussions involving the small inductive dimension we shall always assume that our spaces are regular. This is done so as to avoid some pathological situations. As an illustration, consider the following example. 3.12. Example. Let the space Y be the set of real numbers R with the topology determined by the subbase consisting of the usual open sets of W and the set Q of the rational numbers. We note that Y is the standard example of a Hausdorff space which is not regular; there are no partitions between the closed set P of irrational numbers and any rational point q. By the observation made after Definition 3.2 we have indY = 00. The collection f? of all sets of the forms ( a , b ) and (a,b ) n Q,a < b, is a base for the open sets of Y . Also, a set of the form ( a ,b ) n Q has ([a,b]n P) U { a , b } as its boundary. Now note that the subspace topology of P induced by Y coincides with the one induced by the usual topology of R. From this we find ind B ( U ) = 0 for each member U off?. Let us define a dimension function ind* for (not necessarily regular) topological spaces in the way suggested by Proposition 3.6, namely ind* 0 = -1 and ind* X 5 n if there exists a base B for the open sets of X such that ind* B ( U ) 6 n - 1 for every U in B. For this dimension function we find ind* Y = 1 as opposed to ind Y = 00. In this way one can define a dimension function for which some nonregular spaces will have finite dimensional values. However, such a function will have very limited consequences for the theory that will evolve. As we have seen in Proposition 3.6, the gap between the dimension functions ind and ind* will disappear in the realm of regular spaces.

4. T H E LARGE INDUCTIVE DIMENSION Ind

9

4. The large inductive dimension Ind In [1931] cech introduced the large inductive dimension Ind which is closely related to the Dimensionsgrad of Brouwer. We shall present the definition of Ind and deduce some basic results. The definition of Ind is similar t o that of ind. It is based on the separation of disjoint closed sets rather than the separation of points and closed sets. 4.1. Definition. To every topological space X one assigns the large inductive dimension, denoted by Ind X , as follows. (i) Ind X = -1 if and only if X = 0. (ij) For each natural number n , I n d X 5 n if for each pair of disjoint closed sets F and G there is a partition S between F and G such that Ind S 5 n - 1. (iij) Ind X = n if Ind X 5 n and Ind X f n - 1. (iv) Ind X = 00 if the inequality Ind X 5 n does not hold for any natural number n.

It is immediately clear that Ind is a topological invariant. If the topological space X is not a normal space, then I n d X = 00. The following results can be established by easy inductive arguments. 4.2. Proposition. For every TI-space X ,

ind X

5 Ind X.

4.3. Proposition. For every closed subspace F of a space X,

Ind F 5 IndX. For many spaces X the equality ind X = I n d X holds. More specifically, ind and Ind coincide for every separable metrizable space as we shall prove shortly. But, as a matter of fact, it was not until [1962] that an example was constructed by Roy of a metrizable space A with ind A = 0 and Ind A = 1. Other examples of such metrizable spaces have been presented in Kulesza 119901 and Ostaszewski [1990]. The space A will appear later in various examples. A detailed exposition of the space A can be found in Pears [1975]. The coincidence of ind and Ind will be discussed further in Chapters I11 and V.

10

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

4.4. Theorem (Coincidence theorem). Suppose that X is a separable metrizable space. Then

ind X = Ind X .

Proof. In view of Proposition 4.2, only Ind X 5 ind X needs to be proved. We may assume that i n d X < 00. The proof is by induction on the value of i n d X . The inequality obviously holds for X = 8. We assume that the inequality has been proved for all separable metrizable spaces 2 with ind 2 5 n - l. Let X be a separable metrizable space with i n d X = n. Suppose that F and G are disjoint closed sets of X . By proposition 3.6 there exists a base B for the open sets such that ind B ( U ) 5 n - 1 for every U in B. As X is a second countable space, we may assume that B is countable as well (see Engelking [1977], Theorem 1.1.15). Consider the collection { ( C j , D j ) : j E N } of all pairs of elements of B with cl(Cj) C Dj such that cl (Dj)fl G = 0 or Dj n F = 0. For each k in N define

The open collection and

V = { vk : k

B (Vk) C B (Co)U

E

* * *

N} is a locally finite cover of X ,

U B (ck-1) U B (Dk)

u{

for each k. Now let W be the open set vk : cl(Dh) fl G = @ } . Clearly F is a subset of W . As V is locally finite, cl ( W )fl G = 8 and B ( W ) c U{ B (V,) : cl (Dh)n G = 8 } hold. Subsequently we have ind (U{ B (V,) : cl(D,) n G = S}) 5 n - 1 from the countable sum theorem and ind B ( W )5 n - 1 from the subspace theorem. The induction hypothesis yields Ind B (W) 5 n - 1. As I3 (W) is a partition between F and G, we have Ind X 5 n. Consider a space Y with Ind Y = n that is placed in a hereditarily normal space X. Suppose that F and G are disjoint closed subsets of Y . It follows immediately from the definition of Ind that there is a partition S between F and G in Y with I n d S 5 n - 1. We shall prove now that such partitions may be extended t o partitions between F and G in X.

4. THE LARGE INDUCTIVE DIMENSION Ind

11

4.5. Lemma. Suppose that Y is a subspace of a hereditarily normal space X. Let F and G be disjoint closed subsets of Y . Then for each partition S between F and G in Y there exists a partition T between F and G in X such that S = T n Y .

Proof. For a partition S between F and G in Y there are disjoint open subsets U and V of Y such that Y \ S = U U V , F c U and G C V . The sets U and V have disjoint closures in the open subspace Y' = X \ (clx(S) U (clX(U) n clx(V))) of X. So there are disjoint open subsets U' and V' of Y' (whence of X ) with U C U' and V c V'. The required set T is X \ (U' U V ' ) . 4.6. Proposition. Suppose that X is a hereditarily normal space. Let Y be a subspace of X with IndY 5 n. Then for every pair of disjoint closed subsets F and G of X there is a partition S between F and G in X with Ind (Sn Y ) 5 n - 1.

Proof. Let F' and G' be disjoint closed neighborhoods of F and G respectively. There is a partition T in the subspace Y between F' n Y and G' n Y with In d T 5 n - 1. Note that T is also a partition between F and G in the subspace F U Y U G. By Lemma 4.5 there is a partition S in X between F and G such that T = S n Y . By repeatedly applying the previous proposition we get the following corollary. 4.7. Corollary. Suppose that X is a hereditarily normal space and Y is a subspace of X with IndY 5 n. For each collection of n 4-1 pairs (Fi,G i ) of disjoint closed subsets of X, i = 0 , 1 , . . .,n, there are partitions S; between Fi and G;in X for every i such that Y n So n sl n . . n S, = 0.

We are now in a position to prove the results announced in the Example 3.3.c. 4.8. Theorem. For every positive integer n,

i n d R" = indSn = I n d R" = IndS" = n.

Proof. Denote the interval [-1,1] by K. In view of the inequalities that were established in Example 3.3 and also of the subspace and coincidence theorems, it will be sufficient to establish IndKn 2 n.

12

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

By Corollary 4.7 we need to exhibit a collection of n pairs (8, Gi) of disjoint closed subsets of I n , i = 1,. . .,n, such that the intersection S1 n . . n S, is not empty for all partitions Si between Fi and Gi in I". This property of 1" is the content of the following theorem which is of interest in itself. Henceforth we shall adopt the notation I of the previous proof to denote the interval [-1,1]. 4.9. Theorem. Let { ( F i , Gi) : i = 1,. . . ,n } denote the collection of the pairs of opposite faces of I ,; that is, for i = 1,. . .,n,

Fi = { ( 2 1 , . . . , z n ) E 1 n : 5 i = - 1 } ,

Gi = {(x~,...,x,)E I" : ~ =i 1 ) . If Si is a partition between Fi and Gi for i = 1, . . . ,n, then S1n-nS,

# 0.

Proof, The proof is based on the famous Brouwer fixed-point theorem which reads as follows.

For every continuous map g : I -+ 1 there is an z in I such that g(z) = z, that is, 2 is a fixed-point of g. We shall follow the ingenious proof found in Engelking [1978]. The proof is by way of contradiction. Assume that there are partitions Si between Fi and Gi in In, i = 1 , . ..,n, such that S1 n - -enS, = 0. For each i let 1 \ Si = Ui U V , where Ui and are disjoint open sets with Fi C Ui and Gi C Vi. Consider the continuous map f : 1" ---f I" whose n component functions fi are given by

= Si,ftr'[l]= Fi and f;'[-l] = Gi. It is readily verified that f*r'[O] So f [ F i ]c Gi and f[Gi] c Fi for i = 1 , . . , ,n. As S1 n - - n S, = 8, we have 0 $! f[In]. Now the composition of the map f and the projection of 1" \ { 0 } from 0 onto the combinatorial boundary of In is a fixed-point free map, a contradiction. Let us elaborate on the idea of Proposition 4.6 to obtain a technical lemma which will be useful later on. First we need the following variant of the decomposition theorem.

4.

THE LARGE INDUCTIVE DIMENSION Ind

13

4.10. Theorem. Let { y k : k E N} be a collection o f F,-sets o f a separable metrizable space X . Suppose that indYk = nk 2 0 for k in N. Then there exists an F,-set 2 o f X such that ind 2 = 0 and ind (yj \ 2 ) = n k - 1 for k in N.

Proof. Using the decomposition theorem, we put Yk = Yf U Y: for each k in N, where Yf is an Fu-set of Yk with indYf = 0 and where ind Y : = nk - 1. Because Yk is an Fu-set of X, the set Yf is an Fu-set of X. Let 2 = U{ Y : : k E N}. By the subspace theorem and the sum theorem we have ind 2 = 0. It follows from the addition theorem that ind (Yk \ 2 ) = nk - 1 for k in N. 4.11. Lemma. Let Y be a subspace o f a separable metrizable space X with ind Y 5 n. Suppose that F = { Si : i E N} is a family of closed subsets of X such that

ind (Si, n . . - n Si, n Y ) 5 n - k

+

whenever il < * - < i k and 1 1. k 5 n 1. Then for each pair of disjoint closed subsets F and G o f X there is apartition S between F and G in X such that ind (Sixn

whenever il <

a

a

- n S i k - , n S n Y) 5 n - k

< i k - 1 and 1 5 k 5 n + 1.

The crux of the lemma is that if F is a countable collection of closed sets such that the intersection of each of its k-tuples with some n-dimensional set Y has dimension n - k , then the collection F can be enlarged by adding a partition so as to obtain a new collection with the same dimensional property as the original collection F.

Proof. Write Xo = X and XI, = U{ Si, n - - .n Si, : il < . - - < i k } for k = 1 , . ..,n. Let y k = Y n X k , k = O , l , . . .,n, and note that Yk is an Fu-set of Y. By the sum theorem we have ind y k 5 n - k. The previous theorem provides a subset 2 of Y with ind 2 = 0 and ind ( y k \ 2 ) 5 n - k - 1 for k = 0 , 1 , . . . )n. In particular, we have ind (YO\ 2 ) = ind (Y \ 2 ) 5 n - 1. Now recall Theorem 4.4 to get Ind 2 = 0. Then by Proposition 4.6 there is a partition S between F and G in X such that Ind ( S n 2 ) 5 -1, whence S n 2 = 0. So we have (Si, n - .n Sik--ln S n Y) c X ~ C -n 1 S n Y c Yk-1 \ 2 whenever il < . < i k - 1 and 1 5 k 5 n 1 hold, and the lemma follows.

+

14

I . THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

The point addition theorem for Ind is next. 4.12. Theorem (Point addition theorem). Suppose that X is a regular space consisting of more than one point. For each point p in X , Ind (X \ { p }) 2 Ind X .

Proof. We may assume Ind ( X \ { p } ) < 00. Let F and G be disjoint closed subsets of X . Assume p 4 F . Let V be any closed neighborhood of p with V n F = 8. Define D to be (G U V) \ { p } . Then D and F are disjoint closed subsets of X \ { p } . Observe that X \ { p } must be a normal space because we have assumed I n d ( X \ { p } ) < 00, As any set partitioning D and F in X \ { p } also partitions G and F in X , the theorem follows.

Let us remark here that the example M of Dowker [1955] shows that the inequality of the above theorem cannot be sharpened to an equality. See Isbell [1964] for a detailed discussion. 5 . The compactness degree; de Groot’s problem

Now that our introduction of inductive dimension functions has been completed we shall turn our discussion towards the compactification problem of de Groot. It was mentioned earlier that the compactification problem was posed as a generalization of dimension. The small inductive dimension was the first example of a topological property that was defined by imposing conditions on the boundaries of elements of a suitable base for the open sets (Proposition 3.6). In [1935] Zippin introduced another such property, namely that of a rim-compact space. The terms “semi-compact” or “peripherally compact” have also been used in place of “rim-compact”. 5.1. Definition. A regular space X is called rim-compact if there exists a base B for the open sets of X such that B ( U ) is compact for each U in 13.

Equivalently, a regular space X is rim-compact if for each point p in X and for each closed set G of X with p 4 G there is a compact partition S between p and G. Zippin [ 19351 discovered the following link between compactification and dimension.

5.

THE COMPACTNESS DEGREE; DE GROOT’S PROBLEM

15

Theorem. Let X be a separable completely metrizable space. Then X is rim-compact i f and only i f there exists a metrizable compactification Y of X such that the set Y \ X is countable. 5.2.

It is to be observed that the subspace Y \ X , like any countable metrizable space, is zero-dimensional. This will follow from the countable sum theorem because the small inductive dimension of a singleton point set is zero. The observation also can be proved directly. Indeed, the set { d(x,p) : 5 E (Y \ X ) } of distances is countable for any point p in Y . It follows that, with at most countably many exceptional E ’ S , the &-neighborhood S,(p) has empty boundary. So there are arbitrarily smaIl neighborhoods with empty boundaries. This result of Zippin was modified by de Groot in his thesis [1942].

s,(~)

5.3. Theorem (de Groot [1942]). Suppose that X is a separable metrizable space. Then X is rim-compact i f and only i f there exists a metrizable compactification Y o f X such that ind (Y \ X ) 0.

<

(The proofs of Theorems 5.2 and 5.3 will be presented in Section V1.3.) What makes de Groot’s theorem so interesting is that an external property of a space X , namely the existence of a compactification with special properties, is characterized by the internal property of rim-compactness. There is a strong similarity between the definition of rim-compactness and the characterization of zerodimensionality included in Proposition 3.6. In fact, one obtains the definition of rim-compactness by replacing the empty set by a compact set. In passing, we note that rim-compactness was not the first such generalization of the definition of small inductive dimension. In curve theory, the notions of regular and rational curves already had been introduced. A curve X is called regular (rational)if there exists a base B for the open sets of X such that B ( U ) is finite (countable) for each U in f?. For more details see Whyburn [1942]. In his thesis [1942] de Groot suggested the following possible generalization of Theorem 5.3 which will be called the compactification problem. For its formulation we need two more definitions. The first one is an extension of Definition 5.1. 5.4. Definition. To every regular space X one assigns the small inductive compactness degree cmp X as follows. (i) cmp X = -1 if and only if X is compact.

16

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

(ij) For each natural number n, cmpX 5 n if for each point p in X and for each closed set G of X with p 4 G there is a partition S between p and G such that cmp S 5 n - 1. (iij) c m p X = n i f c m p X < n a n d c m p X $ n - l . (iv) cmpX = 00 if the inequality cmpX 5 n does not hold for any natural number n. 5.5. Definition. For a separable metrizable space X let K ( X ) denote the set of all metrizable compactifications of X . The compactness deficiency of X is defined by

defX = min{ind(Y \ X ) : Y E K ( X ) } . At this moment there is no point in defining def in a more general setting. Consideration of only separable metrizable compactifications has the advantage that all possible dimension functions will coincide. Thus def is unambiguously defined. Because every separable metrizable space can be embedded in the Hilbert cube, there will be at least one compactification of X. The announced possible generalization of Theorem 5.3 is as follows. Compactification problem (de Groot [1942]). For a separable metrizable space X , does the equality cmpX = defX hold? 5.6.

De Groot conjectured the answer to be yes. Indeed, he noted that Theorem 5.3 can be restated as follows. 5.7. T h e o r e m . Let n = 0 or -1. For every separable metrizable space X , cmp X = n if and only if def X = n. Related to the conjecture of de Groot are the following questions that have motivated this book,

1. What are necessary and sufficient internal conditions on separable metrizable spaces X so that defX 5 n? 2. Is it possible to obtain a fruitful generalization of dimension theory by replacing the empty space in the definition with other spaces? 3. What is the special role of the empty space in the theory of dimension?

5 . THE COMPACTNESS DEGREE; DE GROOT’S PROBLEM

17

Let us disclose the end of the story right away. De Groot’s conjecture is not correct. In [1982] Pol gave an example of a separable metrizable space X with cmp X = 1 and def X = 2. (We shall discuss the example in Section 11.) But was this really the end of the story? In the early efforts t o resolve de Groot’s conjecture or to answer Question 1, various functions like cmp were introduced. One of these functions, namely the one introduced by Sklyarenko [1960] (see Section S ) , finally turned out to be successful in answering Question 1. This is the result of Kimura [1988] (see Section 12 and Section VI.5). Now let us return to the conjecture and the related questions. The remainder of the section contains the very earliest results. 5.8. Theorem. For every separable metrizable space X ,

cmpX

defX.

Proof. We may assume, of course, that defX < 00. The proof is by induction on defX. If defX = -1, then X is compact and hence c m p X = -1. Suppose that defX = n. Let Y be any metrizable compactification of X with ind (Y \ X ) = n. Assume that pis a point of X and that G is a closed subset of X with p 4 G. Then cly(G) is a closed subset of Y with p 4 cly(G). The coincidence theorem gives Ind (Y \ X ) = n. From Proposition 4.6 there is a partition S between p and cly(G) in Y with Ind (S fl (Y \ X ) ) 5 n - 1. The set S n X is a partition between p and G in X . As the set S is compact and S = (5’ fl X ) U ( S n (Y \ X ) ) , we have def(S n X ) 5 n - 1 by the subspace theorem. Then cmp (S fl X ) _< n - 1 by the induction hypothesis. The proof is completed. 5.9. Propositions. Before discussing several examples let us list a few propositions. The first one is obvious.

A. (Topological invariance). If X and Y are homeomorphic, then cmp X = cmp Y and def X = def Y whenever the functions are defined.

A simple inductive proof will yield the following result.

B. For every regular space X , cmp X 5 ind X . A proof of the next result is obtained by copying the proof of the subspace theorem.

18

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

C.For each closed su bspace Y o f a regular space X , cmp Y 5 cmp X

.

An analogue of Proposition 3.6 is next.

D. For every regular space X , cmp X 5 n if and only if there exists a base B for the open sets of X such that cmp B ( U ) 5 n - 1 for every U in B. E. For every open subspace Y of a regular space X with cmp X 2 0, cmpY

5 cmp'X.

The easy proof will be omitted. The following proposition has a straightforward proof.

F. For every closed subspace Y of a separable rnetrizable space X , d e f y 5 defX. 5.10. Examples. A common feature of many of the examples in dimension theory is that the examples themselves are easy to describe but the values are hard to compute. The same is true for some of the examples given here. a. Define X to be { (z,y) E R 2 : -1 5 x < 1, -1 y 5 l}. The set X is a closed square with a closed edge removed. As X is locally compact, we find that cmp X = 0 and def X = 0.

b. Form the subspace Y of 112 by adjoining the points p = ( 1 , l ) andq=(l,-l)tothesetXinExamplea,i.e.,Y=XU{p,q}. The space Y is then a closed square with an open edge removed. Let us prove cmpY = d e f y = 1. As Y can be compactified by adding the open edge { ( 1 , y ) : -1 < y < l}, we have d e f y 5 1. By Theorem 5.8, it remains to be shown that cmpY > 0. To this end, consider the sets Vt = { (t,y) : -1 5 y 5 1}, t E [-1,l). Let U be any neighborhood of p that is contained in S i ( p ) . Then B ( U ) n V, is not empty for any t for which ( t ,1) E U because Vt is a connected interval. It follows that B (U)cannot be compact. Thus cmp Y > 0 has been shown. c. cmp Q = def Q = 0. Observe that Q is also not locally compact.

d. Recall that 1 = [-1,1] and let X = Q ' x I n where Q ' = Q nII and n 2 1. As X can be compactified to 11 nfl by adding (I \ Q')x In and as the product theorem yields ind ((I\ Q') x In) 5 n, it follows that def X 5 n. In Example 7.12 we shall prove that cmp X 2 n. It will then follow that cmp X = n = def X .

5 . THE COMPACTNESS DEGREE: DE GROOT'S PROBLEM

19

e. The subspace Z of the n-dimensional cube I", where n 2 2, is defined as follows. We begin with the set

E "-1 = ( ( 2 1 ,...,z,-1,1): - I < zi < 1, i = 1,..., n - 1 ) . Clearly the set E "-l is one of the open faces of 1". Then define 2 to be I" \ E n - 1 , the n-cube with an open face removed. We shall show def 2 = n - 1. Surprisingly the exact value of cmp Z is still unknown when n 2 4. Obviously defZ 5 n - 1; just add En-' t o 2. The inequality defZ 2 n - 1 will be established by way of contradiction. Assume def Z n - 2 and let Z* be a compactification of Z with ind (2' \ 2 ) n - 2. Adopting the notation of Theorem 4.9, we consider the pairs (8'1, GI), . . . , (F,-1, G,-1) of opposite faces of I". These faces are also subsets of 2. By Corollary 4.7, there are partitions St between F; and Gi in 2' for i = 1 , .. . ,n - 1 with (2' \ 2 ) n S; n . n S;-l = 0. It follows that S; n . - . n S;-l is a compact subset of 2. For each St there is a partition S; between Fi and G; in I" with Si n Z = St n 2 (Lemma 4.5). For t in the open interval (-1,l) we let

< <

Vt = { ( 2 1 , . . ., z,-1, t ) : -1 5 z; 5 1, i = 1 , . . . ,n - 1 }. The set Vt is a partition between F, and G, in I" for each t. With the aid of Theorem 4.9 it will follow that ST n n S:-l n V, # 0 for each t. From this we can readily deduce that S; n . . . n S:-l is not compact, a contradiction.

f. Let X be the subset of R defined by X = Sl((0,O)) U { (1, 0) }, an open disc together with one point from its boundary. It can be shown in almost the same way as in Example b that c m p X = 1 = defX. Denote the point ( 1 , O ) by p and let A and B be defined by

& 5 d ( z , p ) 5 & for an odd n in N } U { p } , B = { z E X : & < d ( z , p ) < & for an even n in N } U { p }. A = {z E X :

Both A and B are closed and rim-compact. As X = A U B , this example shows that the (finite) sum theorem does not hold for the function cmp. The example also shows the failure of the (point) addition theorem for cmp. A useful relationship between def and ind can be established by using dimension preserving compactifications. The existence of such compactifications will be established in Section VI.2. We have the following theorem.

20

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

5.11. Theorem. For every separable metrjzable space X ,

def X 5 ind X.

Proof. Let Y be a metrizable dimension preserving compactification for X (Theorem VI.2.7). Then defX 5 ind (Y \ X ) 5 ind Y = ind X . 6. Splitting the compactification problem

Early on, when the original conjecture turned out t o be too difficult t o resolve, splittings of the compactification problem were considered. We shall discuss several of these splittings. The first attempt at splitting used a function which is a large inductive variation of cmp. 6.1. Definition. To every normal space'X one assigns the large

inductive compactness degree K-Ind X as follows. (i) K-Ind X = -1 if and only if X is compact. (ij) For each natural number n, K - I n d X 5 n if for each pair of disjoint closed sets F and G of X there is a partition S between F and G such that K-Ind S 5 n - 1. (iij) K-Ind X = n if K-Ind X 5 n and K-Ind X $ n - 1. (iv) K-IndX = 00 if the inequality K-Ind 5 n does not hold for any natural number n. The following propositions can be proved by easy inductions. 6.2. Proposition. For every normal space X ,

cmp X 5 K-Ind X 5 Ind X. 6.3. Proposition. For every closed subspace F o f a space X ,

K-Ind F 5 K-IndX. That K-Ind is not suitable for the compactification problem will be clear from the next example.

6 . SPLITTING

THE COMPACTIFICATION PROBLEM

21

Example. Let X = (0,1]. Then cmp X = 0 = def X and IC-Ind X = 1 = Ind X . The first equalities are obvious as X is locally compact but not compact. To prove the second equalities, we observe that the disjoint closed sets 6.4.

and F = { 2n+l : n E N, n odd}

G={

:n E

N, n even}

cannot be separated by a compact set. Hence K-IndX ously, IC-Ind X 5 Ind X = ind X = 1.

> 0. Obvi-

In order t o obtain a useful splitting of the compactification problem, we should define a large inductive variation of cmp which agrees with cmp in the values -1 and 0. 6.5. Definition. To every normal space X one assigns the large compactness degree Cmp X as follows. (i) For n = -1 or 0, Cmp X = n if and only if cmp X = n. (ij) For each positive natural number n, C m p X 5 n if for each pair of disjoint closed sets F and G there is a partition S between F and G such that Cmp S 5 n - 1. Cmp X = n and Cmp X = 00 are defined as usual. (The corresponding equalities in the definitions appearing in the remainder of the chapter will no longer be explicitly stated.)

Obviously Cmp is a topological invariant. It also provides a splitting of the compactification problem as the next theorem will show. 6.6. Theorem. For every separable metrizable space X,

cmp X

5

Cmp X 5 def X 5 IC-Ind X.

Proof. The first inequality is easily proved by induction. We shall prove the second by induction on def X. We may assume, of course, that defX < 00. If defX < 1, then c m p X = defX by de Groot’s theorem and therefore c m p X = C m p X by definition. Let n 2 1 and assume that the inequality holds for all values of defX less than n. Consider a space X with defX = n and let Y be any metrizable compactification of X with ind (Y \ X ) = def X = n. For disjoint closed sets F and G of X , let us construct a partition S in X between them with Cmp S 5 n - 1. Denote cly(F) n cly(G) by D. There are two cases to consider, namely D = 8 and D # 8.

22

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

When D = 0,by Proposition 4.6 and the coincidence theorem, there exists a partition S’ between cly(F) and cly(G) in Y such that ind (S’ n (Y \ X)) 5 n - 1. For the resulting partition S = S’ f l X between F and G in X we have def S 5 ind (S’ n (Y \ X ) ) 5 n - 1 because S’is a compactification of S . By the induction hypothesis, Cmp S L n - 1. To contruct S in the contrary case of D # 0, we let 2 = Y \ D. Clearly we have X c 2, and by the subspace theorem we have ind ( 2 \ X ) 5 n. In the same way as in the previous case, there is a partition S‘ between c l ~ ( F and ) clZ(G) in 2 such that ind (S’ n ( 2 \ X)) 5 n - 1. We shall show that the partition S = S’n X between F and G i n X satisfies Cmp S 5 n - 1. The space S’is locally compact. So let a ( S ’ ) denote its one-point compactification obtained by adjoining a new point w . Note that a ( S ’ ) contains a metrizable compactification of S . Consequently we have def S 5 ind ( a ( S ’ )\ X). Using the point addition theorem and the identity a(S’) \ X = (S’n ( 2 \ X))U { w } , we get ind (cr(S’)\X) 5 n - 1 and thereby def S 5 n - 1. We have Cmp S _< d e f S 5 n - 1 by the induction hypothesis. So Cmp X 5 n and the proof of the second inequality is completed. The third inequality is a theorem of de Vries [1962] and will be proved in Section VI.4. In passing, we remark that an easy inductive proof of Cmp X 5 IC-Ind X can be made. 6.7. Example. As in Example 5.10.e we let 2 = I” \ En-’ with n 2 2. We have already computed d e f Z = n - 1. We shall prove Cmp 2 = n - 1. In view of Theorem 6.6, it will suffice to show that Cmp 2 2 n - 1 holds. To this end, we shall introduce in the next paragraph another invariant that is related to compactness. We write CompX = -1 if and only if X is compact. And for a natural number n we write C o m p X 5 n if for any n 1 pairs (Fo,Go), . . . ,(J‘,, G,) of disjoint compact subsets of X there are partitions Si between Fi and Gi in X , i = 0,. . . , n, such that the intersection SOn - n S , is compact. It is easily verified that every rim-compact space X has Comp X 5 0. And Comp X 5 Cmp X is readily established by induction for separable metrizable spaces X , Returning now to the proof of Cmp Z 2 n - 1, we observe that in the discussion of Example 5.10.e it was shown that the assumption Comp 2 < n - 1 leads to a contradiction. So, Comp Z 2 n - 1. Consequently Cmp Z = n - 1 holds as promised.

+

6. SPLITTING THE COMPACTIFICATION PROBLEM

23

It is to be observed that Comp X can be strictly less than Cmp X . Indeed, for the space X defined in Example 5.10.f (the open disc together with a point on its boundary) we find C o m p X = 0 and c m p X = C m p X = d e f X = 1. Another splitting of the compactification problem of a totally different nature was introduced by Sklyarenko in [1960]. He defined a new invariant which is connected to a characterization of dimension by means of special bases for the open sets of a space. 6.8. Definition. Let n = -1 or n E N. A separable metrizable space X is said to have SklX 5 n if X has a base B = { U; : i E N} for the open sets such that for any n -t 1 different indices io, , . , i n the intersection B (Ui,,) n . - n B ( U i , ) is compact.

.

Skl is a topological invariant. There is the following splitting of the compactification problem. 6.9. Theorem. For every separable metrizable space X,

cmpX

5 SklX 5 defX

The proof is divided into two parts. 6.10. Lemma. For every separable metrizable space X ,

cmp X 5 SklX.

Proof, S k l X = -1 means that the intersection of the empty collection of subsets of X , that is X itself, is compact. According t o Definitions 5.1 and 6.8, we have Skl X = 0 if and only if cmp X = 0. The proof of the lemma will be by induction on SklX. The first step of the proof has just been completed. The inductive step is as follows. Suppose that SklX 5 n with n 2 1. Let B = { V , : i E N} be a base for the open sets of X witnessing the fact that S k l X 5 n. For each j in N define

Bj= {&nB(Vj):i#j}. The collection Bj is a base for the open sets of B (Vj). For clarity of notation let d denote the boundary operator in the subspace B (Vj). Then we have d (V n B (V,)) C B ( V )n B (5)for any subset V of X .

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

24

Thus for any n different indices io, . . . ,i,-l from j it follows that

that are all different

: K = o , ..., n - I } C n{B (xk) n B (Vj) : k = 0,. ..,n- 1 ) = B (KO) n n B (Kn-l) n B (Vj).

n{a(K,nB(vj))

*

*

But the last intersection is compact because SklX 5 n. We have established SklB (Vj) 5 n - 1. By the induction hypothesis we have cmp B (Vj)5 n - 1. Since this holds true for any V, in f?, it follows that c m p X 5 n. Thereby the first part of Theorem 6.9 has been proved. Here is the second part.

6.11. Lemma. For every separable metrizable space X , SklX 5 defX.

Proof. We may assume defX < 00. The proof is by induction on defX. As defX = -1 if and only if SklX = -1, we shall focus on the inductive step. Let defX = n and let Y be a metrizable compactification of X with ind (Y \ X )= n. Let ,131 be any countable base for the open sets of Y and consider the countable collection D = { (V,,Wi): i E N} of pairs of elements of ,131 such that cly(K) C Wi. We shall construct inductively a collection S = { Si : i E N } with the properties (1)

Si is a partition between cly(K) and Y \ Wi for i in N

and, when il

< * . - < ik and 1 5 k 5 n t 1,

(2)

ind (Si, n ... n Si, n (Y \ X))5 n

-

k.

This will be done by repeatedly applying Lemma 4.11. Let S: = 8 for all i in N. By induction on m we first construct families S, = {Si :i

< m}u{SI :i 2 m )

satisfying (2) and such that the set Si is a partition between cly(K) and Y \ Wi in Y for i < m. Let So = { S: : i E N}. Suppose that the

6 . SPLITTING T H E COMPACTIFICATION PROBLEM

25

collection S , has already been constructed. By Lemma 4.11 there is a partition S , between cly(V,) and Y \ W , in Y such that Sm+l = { Si : i

5 m}U { S: : i > m }

satisfies (2). The resulting collection S = { Si : i E N} obviously satisfies both conditions (1) and (2). Because of condition ( l ) ,for each Si in S there is an open set Ui of Y such that cly(V,) C Ui C Wi and By(Ui) C Si. Now we observe that the collection B, = { Ui : i E N} is a base for the open sets of Y . This can be seen as follows. For any open set U of Y and any point p in U there is a pair (V,,Wi)in D such that p E V, and Wi C U ; thus, p E Ui C U for this index i. In view of condition (2), this base has the property that the equality ind (By(Uio)n . . . n By(Ui,)

n (Y \ X ) ) = -1

holds for io < -.-< in. Consequently By(Ui,,) n B y ( U i , ) is a compact set contained in X . Let B = { Ui n X : Ui E B2 }, a base for the open sets of X. Note that for each i we have - . e n

Bx(Ui

nX

) = clX(Ui n X )\ Ui = (cly(Ui) n X ) \

ui= By(Ui) n X .

In deducing this formula, we have used the fact that X is dense in Y . It now easily follows that B is a base for X which witnesses the fact that SklX 5 n. We have mentioned already that the invariant Skl is related to a characterization of dimension. The following is this characterization. 6.12. Corollary. Let X be a separable metrizable space. Then ind X 5 n if and only if X has a base B = { Ui : i E N} for the open sets such that for any n+ 1 different indices io, . . .,in the intersection B ( U i , ) f l . . . n B ( U i , ) is empty.

Proof. The proof is quite easy. We shall give the last condition of the corollary the name order dimension and denote it by Odim. Thus Odim X 5 n if X has a base B = { Ui : i E N } for the open sets such that for any n -t 1 different indices io, . . . , in we have B (Ui,) n . . - n B (Ui,) = 8. Copying almost verbatim the proof of

26

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Lemma 6.10, we get ind X 5 Odim X for every separable metrizable space X. We also can redo a large part of the proof of Lemma 6.11 to get OdimX 5 i n d X by making the following modifications. In that proof we replace the superset Y by the space X (the compactness of the space Y is not relevant for the present proof) and the subset Y \ X by the empty set. To obtain the required base B for the open sets of X we take a countable base Bl for the open sets of X and the collection D of pairs of elements (V,,W;)of B1 such that clx(V,) c Wi. Applying Lemma 4.11 (with the set Y as X, resulting in ind Y = ind X = n ) , we then inductively construct a collection S = { 5’; : i E N} with the properties

S; is a partition between clx(V,) and X \ Wi for i in N and, when il

<

< ik and 1 5 k 5 n t 1, ind ( Si, n - n Si, ) 5 n - k. +

Continuing along these lines, we obtain O d i m X completes the proof.

5 i n d X . This

At this point we want to come again to Definition 6.8 of SklX. The definition slightly differs from the original one of Sklyarenko. The condition used in Sklyarenko’s [1960] paper and also in Isbell’s [1964] book is that the intersection of n 1 different boundaries is compact instead of the condition that the intersection is compact for any n 1 different indices that is used in Definition 6.8. We have chosen our definition for two reasons. First, in Kimura’s [1988] paper solving Sklyarenko’s compactification problem-the paper we have alluded to in the comments on Theorem 5.7-Definition 6.8 is used. And second, Definition 6.8 avoids a pitfall that arises from the other definition. The pitfall is best illustrated by analyzing the corresponding one for the function Odim of Corollary 6.12. Consider the following example.

+-

+

6.13. Example. Let X be the subset of R 2 defined by

x = { (5,Y) : z # 0 1u { (070) >. We begin with the following three open sets:

UO = { ( z 7Y) : (5,Y) E X and Y > 0 >, u l = s l ( ( - 1 , 0 ) ) = { ( z , y ) : ( ~ t + ) 2 + Y 2< I } , u 2

= Sl((2’0)) = { (5,y) : (5 - 2)2

+ y2 < 1).

6 . SPLITTING T H E COMPACTIFICATION PROBLEM

27

Note that for any k distinct indicesout of the collection { 0, 1, 2 } the intersection of the corresponding sets in { B ( U O ) ,B ( U I ) , B (U2) } has small inductive dimension at most 2 - k when k 5 3. As in the proof of Corollary 6.12 we can enlarge the collection { U O ,U 1 , U2 } to a base B' = { U; : i E N,i # 3 ) for the open sets of X such that for any 3 distinct indices i l , i2, i3 from N \ { 3 } we have

B ( U ; , ) n B (U;,) n B (Ui,)= 8. (The condition i # 3 is introduced here to facilitate our later calculations.) In the same way as in the proof of Lemma 6.11 it can be shown that for any index j different from 3 the collection B[i = { Ui n B ( V j ) : i 4 { j, 3 } } is a base for the open sets of B ( U j ) such that for any k and 1 that are different from 3 and j and from each other the intersection Bu, ( U k n B ( U j ) ) n Buj (Ul n B ( U j ) ) is also empty, thus witnessing the fact that ind B ( U ; ) = 1 according to Corollary 6.12. Now, to indicate where the pitfall lies in the unindexed version of Odim analogous to the original definition of Sklyarenko, we shall introduce the fourth set U3

= { (z,y) : (x

+ 1)2 + y2 > 1 and z < 0 ) .

From the observation that B ( U 1 ) = B (Us) it follows that for any k distinct sets out of the collection { B (UO), B (Ul), B (Uz), B (U3)) the intersection has small inductive dimension 2 - k when k 5 3. (It is to be noted that one set has been labeled twice.) From the base B' we form the base f3 = { Ui : i E N} by adjoining the open set U3 t o it. Then B has the property that any three different boundaries of members of B have empty intersection. Now let us concentrate on B ( U O ) and its base Bo = { U; n B (Uo) : i # 0 }. One would expect BO to be a base for B ( U O )that witnesses the fact that ind B ( U O )= 1 when one uses Odim. But we shall see that this is not the case. Denoting the boundary operator in B (Uo) by 8, we have V~nB(Uo)={(~,0):-2<~<0}

8 (Ul n B (UO)) = { (-2,O>, (070) >, U2 n B (Uo) = { (z,O) : 1 < 2 < 3 } and

and U3

8 (U2 n B (UO)) = { (1,017 (390) 1,

nB(Uo) = { ( 2 , O ) : z < - 2 ) and

d (Us n B ( U o ) ) = { (-2,0)}.

28

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

It follows that

And we find

Thus it is not true in general that any two different boundaries of Bo have empty intersection. In other words, the base Bo does not witness the fact that ind B (170)= 1. Thereby the pitfall has been exhibited. The approach to dimension via order dimension will be studied in detail in Chapter V. 7. T h e completeness degree

In the discussion of de Groot’s compactification problem 5.6 and Theorem 5.7 we have posed the question whether it is possible to obtain a fruitful generalization of dimension theory by replacing the empty space in the definition of dimension with compact spaces. On examining Example 5.10.f) one will get the impression that the analogy between the functions ind and cmp is rather poor. The example shows the failures of the finite sum theorem and the (point) addition theorem for the function cmp. In this section we shall discuss a dimension function that will show a better analogy with the small inductive dimension. This function, called the small inductive completeness degree, was introduced in [1968] by Aarts. It was the first generalized dimension function for which a substantial theory similar to dimension theory was developed. Before presenting the definition of completeness degree, we shall discuss some results about topological completeness that will play an important role in our development. In this book the term “complete” means “topologically complete”. That is to say, a metrizable topological space is called complete if it is homeomorphic to a space with a complete metric. Such spaces are often called “completely metrizable”. A space Y is called an eztension of a space X if X is a dense subspace of Y . Recall that every metrizable space X has a metrizable extension that is complete. As we shall see, complete extensions are by no means unique. Our investigation will begin by determining which subspaces of a complete

7. THE COMPLETENESS DEGREE

29

space are complete. It is well known that a closed subspace of a complete space is complete. The full determination of the complete subspaces will be settled in Theorem 7.4. Recall that a subset of a space X is called a Gs-set if it can be represented as a countable intersection of open subsets of X . It is to be noted that each closed subset F of a metrizable space X is a Gs-set of X because F = { z : d(z,F ) < } : n E N } for any metric d on X .

n{

7.1. Theorem. Any Gs-set of a complete metrizable space is complete.

Proof. Let X be a complete space. Assume that a complete metric p has been selected. Suppose that A is a Gs-set of X , that is, A = Ui : i = 1,2,. . .} where Ui is open in X for each i. Our tactic will be to find a topological embedding of A which is a closed subset of X x R"0, where R"0 is the countable product of real lines. (It is to be observed that X x R"0 with the usual product metric is complete; thus this copy of A will be complete.) We shall denote pointsofX x R W o b y ( z , y ) w i t h z i n X a n d y = (yI,yz, ...) inR"0. The metric d on X x Rwo is given by the formula

n{

First we define a continuous mapping f from A t o R"0 by defining its component mappings fi: A -+ R by fi(z) = &. As A C Ui for each i, the denominators never vanish. Now we consider the graph G = { (2,f(z)): x E A } o f f . It is well-known that A is homeomorphic t o the graph G and that G is a closed subset of A x R"0. In view of our tactic, it is sufficient to show that G is a closed subset of X x R"0. To this end we let (pn,qn), n = 1 , 2 , . . . , be a sequence in G that converges to a point ( p , q ) in X x R"0. If p 4 A , then p $! Ui for some i. For this i we have fi(pn), n = 1,2,. . . , is a divergent sequence because p ( p , , X \ Ui)-+ 0 as n + 00. In particular, the sequence fi(p,), n = 1 , 2 , . . . , does not converge t o the component qi of q; this denies the continuity of fi. So we must have p E A and, because f is continuous, the sequence qn = f ( p , ) , n = 1,2,. . . , must converge to f(p). It follows that q = f(p) and hence ( p , q ) E G. This shows that G is closed and the proof is completed.

30

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

As a bonus we get the formula

for a complete metric p1 on A. The following lemma is a preparation for Lavrentieff’s theorem which will play a crucial role in our development of the theory. 7.2. Lemma. Suppose that Y is a metric space with a complete metric p and that A is a subset o f a metrizable space X . Iff: A + Y is a continuous map, then there is a subset C of X and there is a continuous map F : C -+ Y such that C is a Ga-set ofX containing A and F is an extension o f f , that is, FIA = f.

Proof. Recall that the diameter S ( B ) of a subset B of Y is defined by

J ( B )= suP{P(Yl,Y2): ( Y l , Y 2 ) E B x B ) . Choose a metric for X and also recall that for each z in clx(A) the oscillation osc (x) of

f at x is defined by

osc(z) = inf{ ~ ( c l y ( f [ S ~ ( ~ ):]E) > ) 0). As f is continuous on A, we have OSC(X) = 0 for each x in A. The set clx(A) being a Gpset of X we have that

C = { x E clx(A) : O S C ( X ) = 0 } is also a Ga-set of X . (It is to be observed that for each n in N the set { 2 E clx(A) : O S C ( X ) < & } is open in the subspace clx(A).) The map F will be defined as follows. Let x E C. Consider the set V ( x ) = cly(f[S,(z)]) : E > 0 ) . Let us show that this set is a singleton. From osc(2) = 0 it follows that S(cly(f[S,(z)])) -+ 0 as E ---f 0. Thus V(X) contains a t most one point. Because Y is complete, the set V ( x )is nonempty, whence a singleton. Now define F by assigning its value F(z) to be the unique member of V(X) for each z in V(X). The continuity of F is readily established. Finally it is obvious that FIA = f .

n{

Now we turn to Lavrentieff’s theorem.

7. THE COMPLETENESS DEGREE

31

7.3. Theorem. Suppose that X and Y are complete metric spaces. Then every homeomorphism between subspaces A and B of X and Y respectively can be extended to a homeomorphism between Gs-sets of X and Y .

Proof. Let f : A + B be a homeomorphism and let g : B + A be its inverse. As clX(A) and cly(B) are complete, we may assume without loss of generality that A and B are dense in X and Y respectively. By Lemma 7.2 there is a continuous extension F : A0 + Y of f such that A0 is a Gs-set of X containing A. Similarly there is a continuous extension G : Bo + X of g such that Bo is a Gs-set of Y containing B. Let Al = A0 n F-'[Bo] and B1 = Bo n G-'[Ao]. Clearly A1 is a Ga-set of Ao. As A0 is a Gs-set of X , so is Al. Similarly B1 is a Gs-set of Y . And obviously F l A l is an extension of f and GJB1is an extension of g . We shall prove that FlAl and GlBl are inverses of each other. The composition G F is defined on A l . As the restriction (GF)IA equals the identity on the set A and A is dense in Al the function G F is the identity on A l . It follows that F[A1] c G-l[Al] c G-'[Ao]. As F[A1] c Bo obviously holds, we have F[AI] C B1. Similarly FG is the identity on B1 and G[BI] c Al. The theorem follows. We now present a few applications of Lavrentieff's theorem. The first one is the classical characterization of con leteness. 7.4. Theorem. A metrizable space X is complete if and only if X is a Gs-set of every metrizable space Y that contains X as a s u bspace.

Because of this theorem, complete spaces are sometimes called absolute Gs-spaces, that is, a space that is a Gs-set of every metrizable space that contains it. Proof. The sufficiency part follows from Theorem 7.1 because any space can be embedded in a complete space Y . To prove the necessity part, we may assume without loss of generality that X has a complete metric. Let be any complete extension of Y . The embedding of X into Y will be denoted by e. By Lavrientieff's theorem, the map e can be extended to a homeomorphism E between Gs-sets. Obviously e = e" and e[X] is a Gs-set of Y , whence of Y . The second application is a lemma that will prove t o be very important. A topological property P will be called hereditary with

32

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

respect to Gb-sets if for each space X having property P every Gb-set of X has property P. According to Theorem 7.1, completeness is an example of a property that is hereditary with respect to Ga-sets. 7.5. Lemma. Let P be a topological property that is hereditary with respect to Gg-sets. If some complete metric extension of a space X has a property P , then every complete extension of X contains a complete extension of X with property P . Similarly, if a space X has a complete metric extension Y with ind (Y \ X ) 5 n, then every complete metric extension of X contains a complete extension Y’ of X with ind (Y’ \ X )5 n.

Proof. Let 2 be a complete extension of X with property P and let Y be any complete metric extension of X . By Lavrentieff’s theorem, the identity map of X can be extended to a homeomorphism between Gg-sets A of 2 and B of Y . Because P is hereditary with respect to Gh-sets, the subspace A has property P. The space B has property P because P is a topological property. And the space B is complete by Theorem 7.1. Thereby the space B is the required extension of X contained in Y . Due to Lemma 7.5, complete extensions will be easier to handle than compactifications. This is also illustrated by the following proof of Tumarkin’s [1926] extension theorem. 7.6. Theorem. Suppose that X is a separable subspace of a metric space Y and let i n d X 5 n.. Then there exists a Gg-set X I of Y with X c X 1 and indX1 5 n. In particular, every separable metrizable space X has a complete extension XI with the same small inductive dimension as X .

Proof. We shall assume the additional condition that Y is complete because the general case can be reduced to the complete case. One makes the reduction by first applying the special case to some complete extension of Y so as to obtain a set 2 1 and then applying the subspace theorem t o the intersection of 2,and Y . As cly(X) is a Gb-set of Y , we may also assume that X is dense in Y . Observe that Y is a separable space under this assumption. The proof will be by induction on n. The statement being obvious for n = -1 we shall assume that the statement has been established for all spaces that are less than n-dimensional. Let X be such that i n d X 5 n. By Proposition 3.6 there exists a base 13 for the

7. THE COMPLETENESS DEGREE

33

open sets of X such that ind Bx(U) 5 n - 1 for every U in f?. For each U in B we select a fixed open set V of Y with V n X = U . For any V obtained in this way the diameter S ( U ) of U = V n X is equal to the diameter S(V). Let Vdenote the collection of all V obtained in this way. For each k in N we let Vk = { V E V :S(V)5 & } and E k = v : v E v k }. Each space E k is separable, whence Lindelof. It follows that there is a countable subcollection { v k j :j E N } Of v k such that E k = U { V k j : j E N } . Let Z = r ) ( E k : k E N } and W k j = Z n v k j for each k and j in N. Obviously X C 2 C Y holds and 2 is a Gs-set of Y . Since each point of Z is a member of elements of B, with arbitrarily small diameters, the collection 81 = { Wkj : k E N, j E N } is a base for the open sets of 2. Consider the natural numbers k and j . The open set u k j = wkj f l X is in B and Bx(Ukj) C BZ(Wkj) holds. By the induction hypothesis there is a Gs-set c k j of Bz(Wkj) with Bx(Ukj) C c k j and i n d c k j 5 n - 1. The set Bz(Wkj) \ C k j is an #,'-set of 2 because Bz(Wkj) is closed. Using these F,-sets, we define the space X1 by

u{

x1 = z\ (U{ B Z ( W k j ) \ c k j : k E M,j

E

N}).

It is readily established that X1 is a Gs-set of 2, whence of Y , and that the collection { wkj nX, : k E N, j E N } is a base for the open sets of X I such that Bx,(Wkj n X I ) c c k j for all k and j in N. By the subspace theorem and Proposition 3.6 it follows that ind X I 5 n. With this recapitulation of the basics of complete extensions we can now give an outline of the theory of the completeness degree. 7.7. Definition. To every metrizable space X one assigns the small inductive completeness degree icd X as follows. (i) icd X = -1 if and only if X is complete. (ij) For each n in N, icdX 5 n if for each point p in X and for each closed set G with p 4 G there exists a partition S between p and G such that icd S 5 n - 1, 7.8. Definition. To every metrizable space X one assigns the large inductive completeness degree Icd X as follows. (i) IcdX = -1 if and only if X is complete. (ij) For each n in N, IcdX 5 7t if for each pair of disjoint closed sets F and G there exists a partition S between F and G such that Icd S 5 n - 1.

34

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Although it would be nice t o see some examples at this point we shall postpone our presentation of spaces with prescribed values of icd and Icd because the required computations will become much easier after the main theorem (Theorem 7.11) has been established. 7.9. Propositions. Just as before, we shall first collect a few propositions whose proofs are quite easy. Obviously the functions icd and Icd are topological invariants. By means of a straightforward induction proof one can get the following comparisons of some of the inductive invariants that have been defined up to now.

A. For every metrizable space X , icd X 5 cmp X

< ind X,

icd X 5 Icd X 5 K-Ind X

5 Ind X .

By copying the proof of the subspace theorem, we obtain the next result.

B. For every Gs-set Y of a metrizable space X icdY

,

5 icdX.

A similar result for Icd (Theorem 7.14) will be proved later with the help of the main theorem. The following is a generalization of Proposition 3.6.

C.Suppose that X is a metrizable space and n is a natural number. Then icd X 5 n if and only if there exists a base B for the open sets of X such that icd B ( U ) 5 n - 1 for every U in B. Our first goal is to resolve the analogue of de Groot’s compactification problem for the completeness degrees. To this end, we define the corresponding deficiency.

7.10. Definition. With C denoting the class of complete metrizable spaces, the completeness deficiency of a metrizable space X is defined by .

C-defX = min{Ind(Y

\ X ) :X

C Y , Y E C}.

In the introduction to the coincidence theorem it was mentioned that the functions ind and Ind do not agree on the class of metrizable spaces. This fact should give rise to two completeness deficiency

7. THE COMPLETENESS DEGREE

35

numbers, one for ind and the other for Ind. At this moment we will not worry about this because we shall restrict our attention to separable metrizable spaces for the remainder of the section; the discussion of dimension theory outside of the realm of separable metrizable spaces is yet to come. 7.11. Theorem (Main theorem for completeness degree). For every separable metrizable space X ,

icd X = Icd X = C-defX. Proof. Proposition 7.9.A gives us icd X shown that Icd X 5 C-def X and C-def X

5 IcdX. It remains to be

< icd X

hold.

Proof of Icd X 5 C-defX. The proof is similar to that of Theorem 6.6 but somewhat easier. It is by induction on C-defX. For the inductive step, let C-defX 5 n and let Y be any complete extension of X with Ind (Y \ X ) = C-def X 5 n. Suppose that F and G are disjoint closed sets of X . With 2 = Y \ D where D = cly(F) n cly(G), the space 2 is complete as it is an open subspace of Y . By the subspace theorem, Ind ( 2 \ X ) n. Note that c l Z( F ) n clz(G) = 8. By Proposition 4.6 there is a partition S between clZ(F) and clZ(G) in 2 with Ind (Sn ( 2 \ X ) ) 5 n - 1. It follows that S n X is a partition between F and G in X with C-defS 5 n - 1. From the induction hypothesis we have Icd (S n X ) 5 n - 1. Hence Icd X 5 n holds.

<

Proof of C-def X 5 icd X . The proof is by induction on icd X. It is very similar to the proof of Theorem 7.6. For the inductive step we assume that icdX n and that Y is a complete extension of X . By Proposition 7.9.C there exists a base B for the open sets of X such that icdBx(U) 5 n - 1 for every U in f?. For each U in f? we select a fixed open set V of Y such that V n X = U . (Clearly the diameter 6(V)of V is the same as that of U = V n X . ) Let V denote the collection of all V obtained in this way. Define the collections Vk = { V E V : S ( V ) & } a n d E k = U{V : V E Vk}for each k in N. Since each space Ek is separable, there is a countable subcollection { V& : j E N } of V k with Ek = Vkj : j € N}. Now define 2 = Ek : Ic E N } and W k j = 2 n Vkj fork a n d j in N. Obviously X c 2 c Y holds and 2 is a Ga-set of Y . Observe that & = { wkj : k E M, j E N} is a base for the open sets of 2. Consider the natural numbers k and j . The open set u k j = W k j r l X

<

<

n{

u{

36

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

is in B and Bx(Ukj) C Bz(Wkj) holds. (It is at the next step that the proof differs from that of Theorem 7.6.) By the induction hypothesis and Lemma 7.5, there is a Gs-set C k j of Bz(Wkj) with ind (Ckj \ Bx(Ukj)) 5 n - 1. Clearly the set Bz(Wkj) \ C k j is an F,-set of 2. Using these F,-sets, we define the space X 1 by

It is readily established that X 1 is a complete extension of X and that the collection { W k j n (XI \ X ) : k E N, j E N} is a base for the open sets of X I \ X . Also one readily verifies

for k and j in N. So ind (Bxl\x(Wkj fl ( X I \ X)))5 n - 1 holds. By Proposition 3.6 we have ind ( X I \ X ) 5 n, whence C-def X 5 n. This completes the proof of the inductive step.

We can now present the promised examples.

7.12. Example. Recall that Q is the space of rational numbers and I is the interval [-1,1]. We shall prove that icd(Q x

I") = icd(Q x R") = n,

n E N.

As an aside, we note that the computations of the compactness degree cmp (Q x I") = n in Example 5.10.d can be concluded with the aid of Proposition 7.9.A. The computation of the above values of the inductive completeness degree follows. The completion R "+l of Q x B" is obtained by adding P x B" t o Q x B",where P is the space of irrational numbers. It follows that C-def ( Q x R ") 5 n. By the main theorem, icd (Q x R") 5 n. And from Proposition 7.9.B we have icd (Q x I ") 5 icd (Q x R"). We shall complete the computations by showing C-def(Q x I n )2 n. The space R x I" is a completion of Q x I" and, in view of Lemma 7.5, there is a completion 2 of Q x I" such that

Q x I"

C2C

R x I" and ind ( 2 \ (Q x I")) = C-def(Q x I").

Let p denote the natural projection of R x I" onto R. Since 1" is compact, the mapping p is closed. Moreover, the set (W x I") \ 2

7. THE COMPLETENESS DEGREE

37

is an &-set of R x I" because 2 is an absolute Gs. So we write R x I" \ 2 as the union of a collection { Fi : i E N} where each Fi is closed in R x I". Then we have

It is easily seen that each p[FJ is a nowhere dense subset of the space P. Hence, by Baire's category theorem applied to P,there is a point y in P\ p[FJ : i E N}. It follows from (*) that p-l[y] c 2. So { y } x 1" C 2 \ (Q x I") holds, whence C-def(Q x I") 2 n.

u{

7.13. Example. Let X be a separable, complete and dense in itself space. It is to be observed that X contains a topological copy of the Cantor set. Suppose that i n d X = n with n 2 2. We shall exhibit a subset Y of X with icdY 2 n - 2. The construction of Y is related to a construction found in the proof of the Bernstein theorem concerning the existence of totally imperfect sets. The construction employs the following two easily verified facts. First, the cardinality of the space X as well as the cardinality of the collection of uncountable F,-sets of X is c (the cardinality of the set a). Second, by the Cantor-Bendixson theorem each uncountable F,-set of X also has cardinality c. Denoting the initial ordinal number of c by R, we arrange the points of X into a transfinite sequence { 2, : a < 0 ) and the family of uncountable F,-sets into a transfinite sequence { F , : a < S Z } . Then we define two transfinite sequences { y, : a < R } and { z, : a < R } as follows. Assuming that { ya : a < /?} and { z, : a < p } have been defined for /3 < R, we let yp and zp denote the first and second element respectively of the sequence { 2, : a < Q } that are in the set Fp \ U{ { y,, z, } : a < 0). The existence of yp and rp follows from the fact Fp has cardinality c while the set U{ { y, z, } : a! < ,h' } has cardinality less than c. Now we define Y to be { y, : a! < S Z } . It follows from the way the set Y has been constructed that neither Y nor X \ Y contains an uncountable F,-set. Let'us show that ind ( X \ Y ) 2 n - 1. From Theorem 7.6 due to Tumarkin, there exists a complete extension 2 of X \ Y with X \ Y c 2 c X and ind 2 = ind ( X \ Y). The set X \ 2 is an &',-set of X contained in Y . It follows that X \ 2 is countable, whence zero-dimensional. From the identity X = ( X \ 2 ) U 2 and the addition theorem, the inequality ind ( X \ Y) = ind 2 2 n - 1follows. Now let us show that

38

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

C-def Y 2 n - 2. By Lemma 7.5 there is a complete extension Y1 of Y with Y C Y1 c X and ind (Y1 \ Y ) = C-defy. The set X \ Y1 is an F,-set of X that is contained in X \ Y and hence is zero-dimensional. Because X \ Y = (X \ Y l )U (Yl \ Y ) ,from the addition theorem we get ind (Y1 \ Y ) 2 n - 2. By the main theorem, icd Y 2 n - 2. The strength of the main theorem will be illustrated with a few applications.

7.14. Theorem. For every Gs-set Y of a separable metrizable space X , IcdY 5 IcdX.

Proof. This is an easy consequence of Proposition 7.9.B and the coincidence of icd and Icd from the main theorem.

7.15. Theorem (Addition theorem). Let X be a separable metrizable space. If X = Y U 2, then i c d X I i c d Y + i c d Z + 1. In particular, the small inductive completeness degree icd of a separable metrizable space cannot be increased by the adjunction of a complete space or a point.

Proof. Let X* be a complete extension of X . By Lemma 7.5 and the main theorem, there are Gs-sets Y * and Z* of X* containing Y and 2 respectively that satisfy icd Y = Ind ( Y * \ Y ) and icd 2 = Ind (2" \ 2 ) . The set Y * U Z* is a Gb-set of X*, whence a complete extension of Y U 2. We have icd X

5 Ind ( ( Y *U Z * ) \ (Y U 2 ) ) 5 Ind ( ( Y *\ Y ) U ( Z * \ 2 ) ) L Ind (Y * \ Y ) + Ind ( Z * \ 2 ) + 1 = icd Y + icd 2 t 1.

In a similar way we obtain the following result.

7.16. Theorem (Intersection theorem). If Y and Z are subX , then

sets of a separable metrizable space

icd(Y

n 2 ) 5 icdY + i c d Z t 1.

7. THE COMPLETENESS DEGREE

39

Proof. With X * , Y * and Z* as defined in the preceding proof we have

icd (Y n 2 ) 5 Ind ((Y*n Z*) \ (Y n 2 ) )

< Ind ((Y*\ Y ) U (2'

\ 2 ) ) 5 icd Y + icd Z + 1.

There is no countable sum theorem for icd. The space Q x IIn of Example 7.12 will serve as a counterexample since it is the countable union of compact components. However, there is a locally finite sum theorem for icd. 7.17. Theorem (Locally finite sum theorem). Let F be a locally finite closed cover of a separable metrizable space X . Then

i c d X = sup{icd F : F E F}.

Proof. In view of Proposition 7.9.B, only sup { icd F : F E F } 1 icd X needs to be shown. Let Y be any complete extension of X . For each point x in X we select an open neighborhood U, of x in Y such that U, meets only finitely many elements of F . Then W = U{ U, : x E X } is an open subset of Y with X C W and the collection F is locally finite in the subspace W . Since W is a complete extension of X , we may assume Y = W . As a locally finite collection is closure preserving, we have Y = U{ c l y ( F ) : F E F } . By the main theorem and Lemma 7.5, there is for each F in F a completion GF of F with F c GF c cly(F) and icd F = Ind (GF \ F ) . Obviously c l y ( F ) \ GF is an Fu-set of Y . Let us show that the subset H = U{ cly(F) \ GF : F E F } of Y is also an F,-set. The collection { cly(F) : F E F } is locally finite in Y . As the space Y is separable, this collection must be countable. Thereby H is an F,-set of Y . It follows that Y \ H is a complete extension of X . Now to complete the proof, observe that

(Y \ H ) \ X We have

C U{GF : F E F } \ X C

U{ G F \ F : F

EF}.

40

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

by the coincidence theorem, the countable sum theorem and the subspace theorem of ind. As icd X = C-def X 5 ind ( ( Y \ H ) \ X ) , the theorem will follow.

It is still an open question whether or not the following decomposition theorem for the completeness degree holds true: icd X 5 n if and only if X = X OU XI - - U X, for n 1 sets Xi with icdXi 5 0, i = 0,1,. . . ,n.. Observe that the sufficiency will follow from the addition theorem for icd. (The decomposition can be made when i c d X = i n d X. ) However, the “dual” of the decomposition statement, called the structure theorem, is readily established.

-+

7.18. Theorem ( S t r u c t u r e theorem ) . Suppose that X is a separable metrizable space. Then icd X 5 n if and only if for some complete extension Y of X the set X can be represented as the intersection of n 1 subsets Yi of Y , i = 0,1,. . .,n, with icd Yi 5 0 for each i.

+

The easy proof will be omitted. We shall conclude this section with a characterization of icd that is suggested by the definition of Skl and the proof of Corollary 6.12. The characterization will require the following definition which is analogous to that of Skl. 7.19. Definition. Let X be a separable metrizable space. The completeness order dimension of the space X , denoted C-Odim X , is defined as follows. C-Odim X 5 n if X has a base B = { Ui : i E N} for the open sets such that for any n -t 1 different indices i o , . . . ,i, the intersection B (Ui,)n - - . n B ( U i , ) is complete. 7.20. Theorem. For every separable metrizable space X,

icd X = C-Odim X.

Proof. The proof will smoothly follow the steps that were taken in Section 6. The proof of the inequality icd X 5 C-Odim X is achieved by a simple replacement of “compact” with “complete” in the proof of Lemma 6.10. The same replacement in the proof of Lemma 6.11 will yield the inequality C-Odim X 5 C-def X . The proof of the theorem is completed by an application of the main theorem.

8.

THE COVERING DIMENSION dim

41

The results of this section have been generalized in two directions. First, they hold for more general spaces X ; second, they hold for classes of spaces P more general than the class C of complete spaces. The investigation of extensions of spaces along the same lines as in this section will be continued in Section V.2. Finally the discussion of order-type characterizations that are analogous to Theorem 7.20 will be continued in Section V.3. 8. The covering dimension dim

In addition to the small inductive dimension and the large inductive dimension there is yet another dimension function called the covering dimension. Unlike the compactness degree, the completeness degrees discussed in the previous section have many similarities with the two inductive dimension functions. In [1968] Aarts indicated that there is a natural generalization of covering dimension that also fits perfectly into the theory of completeness just as the covering dimension fits into the theory of dimension. So this seems to be the proper moment to present the covering dimension. The covering dimension was first introduced by cech in [1933]. It has its roots in a covering property studied by Lebesgue in [1911]. Lebesgue was searching for a topological property that would distinguish the n-dimensional cube In and the m-dimensional cube I". In fact, Lebesgue was working on the same problem as Brouwer (see Section 2). The covering dimension uses the primitive idea of the order of a collection. 8.1. Definition. Let D = { D, : (Y E A } be an indexed collection in a topological space X and let p be a point in X. The order of D at p is the number of indices a in A (possibly co) for which p E D, holds; this number is denoted by ord, D . The order of D is defined by ord D = sup{ ord, D : p E X } . When the collection D has no explicit indexing we shall formally index the collection as { D : D E D } . Then the order of D becomes the largest number n such that the collection D contains n distinct sets with a nonempty intersection. The order of indexed collections has already been implicitly used in earlier sections, notably Section 6. The importance of the indexing set of the collection was shown to be critical when one considered

42

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

the collection of boundaries that coresponded to the given collection of sets, that is, the two collections

{ U, : a E A } and { B ( U , ) :

ct

EA}.

The discussion in Example 6.13 addressed the pitfalls that can arise when the indexing set A of the two collections was permitted to be changed to another indexing set for the collection of boundaries. The next example is a continuation of that example. 8.2. Example. For the Example 6.13 one can readily compute the two values for the indexed collections.

When the indexing is neglected, the collections of boundaries are the same because B ( U l ) = B ( U 3 ) . The following is a rewording of Corollary 6.12. 8.3. Proposition. Let X be a separable metrizable space. Then ind X 5 n if and only if there is a base 13 = { Ui : i E N } for the open sets of X with ord { B ( U i ) : i E N } 5 n.

Recall the definitions: A collection D in a topological space X is a cover if X = U D . A collection D is called open if every member of D is open. A collection D is said to be a refinement of a collection E if for each D in D there exists an E in E with D C E . (It is to be noted that if D is a refinement of E , then the inclusion D c U E holds with the possibility of a proper inclusion being permitted.) We shall now define a special type of refinement.

U

8.4. Definition. Let D = { D, : a E A } be an indexed cover of a space X . A collection E = { E , : a E A } is called a shrinking of D if E is a cover of X and E , c D, holds for every a in A .

We now define the covering dimension. 8.5. Definition. Let X be a space. The covering dimension of X , denoted dim X , is defined as follows. dimX = -1 if and only if X = 0. For n in N, dim X 5 n if for each finite open cover U of X there exists an open cover V of X such that Vis a refinement of U

43

8. T H E COVERING DIMENSION dim

+

and ord V 5 n 1 holds. As usual, d i m X = n means dim X holds but dim X 5 n - 1 fails.

5n

The following proposition exhibits a condition that is equivalent to the definition of dim. 8.6. Proposition. Let X be a space and let n be in

N. Then

dim X 5 n if and only if every finite open cover U of X has an open shrinking V of order not exceeding n 1.

+

Proof. The sufficiency is obvious. To prove the necessity let us consider an open cover U = { Ui : i = 0 , 1 , . , . , k }. As dim X 5 n, there exists an open cover W such that W is a refinement of U and ord W 5 n 1. For each W in W we choose an index i such that W C Ui holds and denote this choice by a ( W )= i. With Vi defined as the open set U{ W E W : a ( W ) = i} one can easily verify that the collection V = { V , : i = 0, 1, . , . ,k } is a shrinking of U such that ord V 5 n 1.

+

+

The three dimension functions ind, Ind and dim reflect different geometrical structures of normal spaces. We have already seen that the functions ind and Ind coincide for separable metrizable spaces (Theorem 4.4). We shall prove that ind and dim also coincide for separable metrizable spaces and thereby show an intimate connection between these geometric structures. We need the following three lemmas. 8.7. Lemma. Suppose that X is a subspace of a metrizable space Y and let U = { U, : a E A } be an open collection in X . Then there exists an open collection V = { V, : cr E A } in Y such that U , C V, n X for every a in A and ord U = ord V.

Proof. To each open set U of the subspace X we associate the open set U* of Y defined by U* = { y E Y : d ( y , U ) < d ( y , X \ U ) }. (Recall that d ( y , 0) = 00 and hence 0* = 8.) Let us prove the identity ( U n V ) * = U* fl V * . This will be done in two steps. First we assume y E (V f l V ) * . It readily follows that d ( y , U ) 5 4%u fVl) < d(Y, ( X \ U ) u ( X \ V ) ) 5 4%x \ U ) . so, Y E u** And y E V * by symmetry. Therefore, ( U fl V ) *c U* n V * . Next assume y E U" n V*. Obviously, d(Y,

x \ ( U n V ) ) = d(Y, ( X \ U ) u ( X \ v>) = min { 4%x \ w, 4%x \ V )1.

44

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Without loss of generality we assume

Because y E U * , we have d ( y , U ) < d ( y , X

\ U ) . Now

Consequently d(y, I/ n V ) < d(y,X \ U ) = d(y,X \ ( U n V ) ) holds. So, y E ( U n V ) * . Thereby ( U n V ) * = U* n V * is proved. Clearly the finite intersection formula (170n . n Uk)*= U,*n - - n U i now holds. We complete the proof of the lemma by defining V, = U z for each cr in A .

-

+

8.8. Lemma. Every finite open cover of a normal space X has a closed shrinking.

Proof. It is somewhat easier to prove the following equivalent statement . Every finite closed collection { Fi : i = 0,1,. . .,n } with empty intersection has an open collection { Vi : i = 0,1,. ..,n } such that F; c V , holds for every i and : i = 0 , 1 , . .. ,n } = 8.

n{

This statement is proved as follows. The sets FO and F ! n .-enF, are disjoint and closed. By the normality of X there exists an open set Vo with FO c VO and cl (VO) n (Fl n . . n F,) = 0. So we can replace the closed set Fo with the bigger set cl(V0) and at the same time preserve the empty intersection of the collection. Repeating the argument a finite number of times, we complete the proof. This shrinking lemma is well known; its proof has been included because similar arguments will be found later. Its generalization to locally finite collections is also known, but a proof will not be given here, The reader is asked to provide a proof or consult standard texts in topology.

8. THE COVERING DIMENSION dim

45

8.9. Lemma. Suppose that X is a normal space with dim X

5 n.

Let ( F i : i = O , l , . . . ) k } and { U i : i = O , l ) ...)k } beaclosed collection and an open collection respectively such that Fi c Ui for every i. Then there are two open collections { K : i = O ) l )..., k }

and

( W i : i = O , l ,...)k }

such that the following hold.

Fi

C V , C .I(&) C Wi c U i , ord { Wi \ cl (V,) : i = 0, 1,

i = O , 1 , ...,k, . . .)k } _< n.

The property which appears in the conclusion of the lemma will actually characterize the inequality dim X I n. Such a characterization will be discussed in a general setting in Section 11.5. We shall give a direct proof of this characterization for separable metrizable spaces immediately after the proof of Theorem 8.10.

Proof. Consider D = A { { U i , X \ Fi } : i = O , l , . . k }. (Recall that the members of D are of the form Ai : i = 0,1,. ..,k } where either Ai = Ui or Ai = X \ Fi for each i.) Being an open cover of X , it has an open shrinking V = { L j : j = 0 , 1 , . . .)I} with ord V 5 n 1 (Proposition 8.6). And from Lemma 8.8, V h a s a closed shrinking { K j : j = 0 , 1 , . . . 1 }. Using the normality of the space, we can find open collections { I<: : i = 0,1,. . . k 1 } for j = 0,1, . . . 1 such that

n{

.)

+

)

)

+

and (2)

cl (I<;) c Ki+1)

i = 0) 1 ) . . .,k.

It should be noted that from (2) we have

Now for i in { 0,1,. . . ,Ic} we define

)

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

46

and

wi = u{K!++': L~ n F~ # 8 }. = 0 , 1 , . . , 1 } is a cover of X and V is a refinement 3

Because { Ir'j : j , of D , one readily proves

Fi

C

6

C cl(V;) C Wi C Ui,

i = 0, l , . , . ,k .

This is the first formula which was to be proved. By way of a contradiction we shall prove the second formula. Assume for some p in X and some distinct indices i, m = 0,1,. . .,n, that p E Wim \ c l ( K m ) : m = (),I,.. . , n } .

n{

Note that we have

Thus, for suitable pairs (i,,jm), we have

It follows from (3) that the indices j , must be distinct. As X is covered by { K j : j = 0 , 1 , . . .,I }, we have p E K, for some s. It follows from (1) that s is distinct from each of the j,'~. Hence

that is, ord, V > n $- 2. This is a contradiction. 8.10. Theorem (Coincidence theorem). For every separable

metrizable space X , dim X = ind X = Ind X .

Proof. The proof will be in two parts. Proof of dim X 5 ind X. To begin, we shall prove the inequality under the additional assumption of ind X = 0. Let U be a finite open

8. THE COVERING DIMENSION dim

47

cover of X . In view of Proposition 3.6 there is a cover Vof X that consists of open-and-closed sets and refines U . As X is separable, we may assume that V is countable; let V = { Vi : i E N}. For each i in N we let Wi = Vi \ (Vo U . . . U K-1). Then W = { Wi : i E N } is a cover of X consisting of open-and-closed sets such that W refines U and ord W = 1. It follows that dim X = 0. For the general situation we argue as follows. Assume ind X = n is positive. By the decomposition theorem (Theorem 3.8) the space X can be partitioned into n i- 1 disjoint subsets X ; , i = 0,1,. . .,n, such that ind Xi = 0 for each i. Let U be a finite open cover of X and let i be such that 0 5 i 5 n. Then the open cover { U n Xi : U E U } of Xi can be refined by an open cover Vi = { V,, : Q E Ai } of order 1. By Lemma 8.7 there is an open collection { W;, : Q E Ai } of order 1 in X such that Wi, n Xi = Via. For each Q in A; we select a U i , in U such that 6, c U i , n X i . It is readily verified that the collection V: = { W;, n U i , : Q E Ai } has order 1 and refines U. Thus the open cover V * = V: U . . U V; has order n 1 and refines U .

+

Proof of ind X 5 dim X. As in the proof of Corollary 6.12 we use the order dimension of the space X , that is, Odim X 5 n if X has a base B = { Ui : i E N } for the open sets such that for any n 1 different indices io, . . .,in we have B (Ui,,) n . . . n B ( U ; , ) = 0. It has already been shown that ind X 5 Odim X.So only Odim X 5 dim X remains to be proved. To this end let B' be any countable base for X. Consider the family D = { (C;,Di) : i E N} of all pairs of elements of B' such that cl(Ci) c Di. For each pair of natural numbers i and j with i 5 j we shall inductively define open sets V j and W,?' such that, when i 5 j ,

+

ci c v;; c cl (q) c w;c Di, cl(v,j) c v,j+', cl(W;+*) c wij and, when j E N, ord { W!

\ cl

(v'): i = 0 , 1 , . . . , j } 5 n.

Let V t and Woo be open sets with COc V l C cl(V$) C Woo C DO. Assuming that and W j have been defined for i 5 j 5 s so that the above conditions are satisfied, we apply Lemma 8.9 to the collections

{ cl ( KS): i = 0 , 1 , . . .,s } u { cl (C,,,) }

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

48

and

{ w;: i = 0 , 1 , . * .,s} u { Ds+* }

so as to obtain the open collections

{l$”+’:i=O,1,

..., s + l }

and

(Wf+’:i=O,l, ...,s + l }

satisfying the above conditions. Finally define V , = U{ y’ : j 2 i } and B = { V , : i E N}. Note that B(V,) c W j \cl(y’) when j 2 i. It readily follows that B is a base such that B (Ui,)n - n B ( V i m )is empty for any n f 1 different indices io,. . .,in. So, OdimX 5 n. Let us return to our observation made after the statement of Lemma 8.9. As promised, we shall prove that the condition of the conclusion of that lemma characterizes the inequality dim X 5 n for separable metrizable spaces X . The characterization follows from the sequence of implications displayed below, where the numbers above the arrows refer to earlier proved statements.

- -

dim X 5 n

(8.10)

(8.9)

conclusion of Lemma 8.9

OdimX

5n

(6.12)

indX

5n

(8.10)

dimX

5 n.

We have already commented earlier that ind and Ind do not coincide for metrizable spaces. But Ind and dim do coincide for general metrizable spaces. Further discussions on coincidence will appear in Sections 11.7 and V.3. 9. The covering completeness degree

The similarity between dimension theory and completeness degree theory for separable metrizable space is very appealing. We have already mentioned in the last section that the covering dimension has its partner in completeness. We shall now introduce the covering completeness degree. The key to our generalization is the proper adaptation of the notion of an open cover of X. Recall that C is the class of complete metrizable spaces.

9. THE COVERING COMPLETENESS DEGREE

49

9.1. Definition. Let X be a metrizable space. A C-kernel of X is a complete space Y with Y c X. A C-border cover of the space X is an open collection U of X such that X \ U U is complete; the set X \ U U , which is a C-kernel, is called the enclosure of U .

The idea of considering objects like C-border covers in the theory of extensions of spaces goes back to the [1965] paper of Smirnov. In that paper Smirnov gave a characterization of the compactness deficiency of a normal space in terms of the existence of a special sequence of seams (which are border covers with compact enclosures). This characterization of Smirnov will be discussed in detail in Section VI.6. In light of the definition of C-border cover the following definition of covering completeness degree is the natural extension of the definition of covering dimension. 9.2. Definition. Let X be a metrizable space. The covering completeness degree of X , denoted ccd X , is defined as follows. ccd X = - 1 if and only if X E C. For n in N,ccd X 5 n if for any finite C-border cover U of X there exists a C-border cover V of X such that V is a refinement of U and ord V 5 n 1. As usual, ccd X = n means ccd X 5 n holds but ccd X 5 n - 1 fails.

+

An obvious modification of the proof of Proposition 8.6 will yield the following lemma. 9.3. Lemma. Suppose that X is a metrizable space and n is in N. Then c c d X 5 n if and only if for every finite C-border cover U = { Ui : i = 1 , . , . , k } of X there exists a C-border cover V = { K : i = l , ..., k } with enclosureG a n d o r d V S n t 1 such that G contains the enclosure of U and V shrinks the collection { Ui \ G : i = 1 , . . . ,k } in the subspace X \ G.

Though ccd is quite different from Icd in its definition we shall show in Section 11.10 that ccd and Icd coincide for general metrizable spaces. Here we shall give a direct proof of this coincidence for the separable case. We do this because the proof of the coincidence theorem for dimension (Theorem 8.10) can be adapted to obtain the analagous one for the three completeness dimension functions. Before proceding to the proof, we shall adopt the following modulo not ation.

50

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Modulo notation. Suppose that F is a C-kernel of X . For subsets A and B of X and a collection Vof subsets of X , (1) A C B mod F means A \ F C B \ F , (2) A n B = 0 mod F means A n B c F , (3) ord V 5 n mod F means ord { V \ F : V E V } 5 n. For the proof we shall need t o establish the following variation of the preparatory Lemma 8.9. 9.4. Lemma. Let n be a natural number and let X be a metrizable space with ccd X 5 n. Suppose that G is a closed C-kernel of X and that { Fi : i = 0 , 1 , . . .,k } and { Ui : i = 0,1,. . . ,k } are respectively closed and open collections of X such that Fa C Ui mod G for each i. Then there is a closed C-kernel H of X and there are two open collections { V , : i = 0,1,. . .,k } and { Wi : i = 0,1,. . ., k } such that the following conditions hold.

Fi

c q c c l ~ ( q c) Wi c

Ui mod H , ord{ Wi \clx(V,) : i = 0,1, ...,k }

G

i = 0 , 1 , . . .,k, 5 n mod H ,

c H.

The conditions

KcX\H

and W i c X \ H ,

i = O , 1 , ..,,k ,

may also be imposed.

Proof. Consider the C-border cover =A

{ { (ui\ G), ((X\ Fi)\ G) } : i = 0,1,. . .,k 1.

There is a finite C-border cover V = { L j : j = 0,1,. ..,1 } with enclosure H such that V refines D and ord V n 1 (Lemma 9.3). Clearly the C-kernel H contains G and is closed. In the normal subspace X \ H , the finite open collection V has a closed (in the subspace X \ H ) shrinking { K j : j = 0, 1, ...,1 }. The calculations of Lemma 8.9 applied to the subspace X \ H will yield open collections { V , : i = 0,1, ...,k } and { Wi : i = 0,1, ..., k } in X \ H such that

< +

9. THE COVERING COMPLETENESS DEGREE

51

and ord{Wi\clx\H(V,):i=O,l,

...,k }


As clX\H(K) = c l ~ ( F $ \) H holds for each i, we have

F;

i = O , l , ..., k, ord{W;\clx(T/,): i = O , l , ...,k } 5 n mod H

c V , c clx(V,) C Wi C U;

mod H ,

and

V ; c X \ H and W i c X \ H ,

i=O,l,

...,k.

Thus the proof is completed. With this preparatory lemma we can prove the coincidence theorem. 9.5. Theorem (Coincidence theorem). For every separable

metrizable space X ,

ccd X = icd X = Icd X. Proof. As in the proof of the coincidence theorem fcr. dimension we shall give the proof in two parts. Proof ofccd X 5 icd X . This part of the proof will differ from that of Theorem 8.10 because it is not known that there is a decomposition theorem for icd. Suppose icd X n. By Theorem 7.11 and the coincidence theorem for dimension, there is a complete extension Y of X with dim(Y \ X) = icdX 5 n. Let U = { Ui : i = O , l , . . .,k } be a finite C-border cover X with enclosure F. From U we form an open collection u* = { UT : i = 0,1,. . .,k } in Y by selecting an open set I!J? of Y such that U: n X = Ui. Denote the subspace X U (U U * ) of Y by Y1. Clearly the equality Yl = F U (U U * )also holds. It follows that Y1 is complete by Theorem 7.15 and that dim (Y1 \ X ) 5 n by the subspace theorem for dimension. By Proposition 8.6 the open cover { U: n (Y1 \ X ) : i = 0 , 1 , , , , , k } of Yl \ X has an open shrinking V = { V , : i = O , l , ..., k } with ord V 5 n 1. By Lemma 8.7 we have an open collection W = { Wi : i = 0,1,. . .,k } in the subspace Yl with ord W n 1 such that Wi n (Yl \ X ) = & for each i. Furthermore, Wi c Ui may be assumed. Since Yl \ U W = X \ U W holds, we have that { Wi n X : i = 0 . 1 , . ..,k ) is a C-border cover

<

+

< +

52

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

+ 1. Thereby we

of X that refines U and has order not exceeding n have shown ccd X 5 n.

Proof of icd X 5 ccd X. In view of Theorem 7.20, it will suffice to prove C-OdimX 5 ccdX. (Although the proof appears to be similar to the second part of the proof of Theorem 8.10, the nature of the construction forces complications that result from the need t o consider C-kernels.) Assume c c d X 5 n and let B1 be any countable base for X . Then let { ( C i , D i ) : i E N} be the family of all pairs of elements of ,131 such that cl(Ci) c Di. For each pair of natural numbers i and j with i 5 j and for each natural number k we shall inductively define open sets V,' and W j and closed C-kernels Gk such that, when i 5 j,

and

Gk

C Gk+l,

ord{ W: \ c l ( V F ) : i = O , l ,

k = 0,1, ... ,

..., k } 5 n

mod

Gk,

k = O , l , ... .

Let V: and Woo be open sets with Co c V t c cl(V:) c Woo c Do and Go = 8. Suppose for some s that the open sets y' and W! have been defined for i 5 j 5 s and the closed C-kernels Gk have been defined for k 5 s so that the above conditions are satisfied. Then we apply Lemma 9.4 to the C-kernel G, and the collections

{ cl ( V t )\ G, : i = 0 , 1 , . . . ,s } U { cl(C,+l)

\ G,

}

and

{ W f \G, : i = O , l , .

..,s} U {

D,+I \ G , }

so as to obtain the closed C-kernel Gs+l and the open collections { Ks+' : i = 0,1,. ,s 1 } and { Wf+l : i = 0,1,. ,s 1 } satisfy-

. . -+

ing the above conditions. From the inclusion

Cic V: u (Gi n Di)

.. +

10.

THE U-COMPACTNESS DEGREE

U (Gi f l

we find that the interior of

Moreover, when i 5 k quently

5 j we have

53

Di),denoted by V,.,satisfies

q j c Wt

mod Gj and conse-

yj \ Gj C W;lC c Dim Finally we define B = { Vi : i E N } by

V , = V;* U (U{ Kj \ Gj : j > i }), It easily follows that

i E N.

B is a base for the open sets. Notice that

Thus it follows that ord{ B(V,) : i = 0'1,

..., k } 5 n

mod Gk+l,

k E N.

+

We have shown for any n 1 different indices io, . . . ,in that the intersection B (V,,) n . * n B (I&) is contained in some G'. It follows that this intersection is a complete space. We have established C-OdimX 5 n. Investigations along the lines of coverings will be continued in Section 11.5 and in the Chapters IV and V. 10. The a-compactness degree

Even with the splitting of de Groot's compactification problem by way of the three functions Cmp, IC-Ind and Skl, very little progress had occurred in the first 25 years of its history. Negative facts such as the failure of a finite sum theorem for cmp had been established. In [1967] Nagata suggested a different approach t o the compactification problem. He proposed that a a-compactness degree be defined analogously to cmp (and icd) and that the intermediate problem of characterizing the a-compactness deficiency by means of the a-compactness degree be studied. The failure of the finite sum theorem should disappear in this context, a very pleasing prospect.

54

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Continuing our historical perspective, we shall present the outcome of the program proposed by Nagata in [1967] as carried out by Aarts and Nishiura in [1973]. It was found that the notion of extension was not as natural as the notion of a kernel for classes like that of a-compact spaces . This discovery led to a negative solution of Nagata’s problem (see Example 10.13). The class of all a-compact spaces will be denoted by S. 10.1. Definition. To every regular space X one assigns the small inductive a-compactness degree S-ind X as follows.

(i) S - i n d X = -1 if and only if X E S. (ij) For each natural number n, S - i n d X 5 n if for each point p in X and for each closed set G of X with p 4 G there is a partition S between p and G such that S-ind S 5 n - 1. 10.2. Definition. To every normal space X one assigns the large inductive a-compactness degree S-Ind X as follows. (i) S - I n d X = -1 if and only if X E S. (ij) For each natural number n, S-Ind X 5 n if for each pair of disjoint closed sets F and G of X there is a partition S between F and G such that S-Ind S n - 1.

<

10.3. Propositions. Again we shall postpone the presentation of examples until after the main result (Theorem 10.5). First we shall collect a few propositions, the proofs of which are easy. Obviously the functions S-ind and S-Ind are topological invariants.

A. (a) For every regular space X , S-ind X 5 cmp X 5 ind X. (b) For every normal space X , S-Ind X 5 K-Ind X 5 Ind X. (c) For every normai space X , S-ind X 5 S-Ind X. There is the following variant of the subspace theorem. The first step of the inductive proof is the observation that an Fu-set of a a-compact space is a-compact.

B. For every Fu-set Y of a regular space X , S-ind Y 5 S-ind X. There is a similar result for S-Ind. A simple version of it will be proved later with the help of the main result of this section.

10. THE m-COMPACTNESS DEGREE

55

The following is a generalization of Proposition 3.6. C . Suppose that X is a regular space. For each natural number n, S - i n d X 5 n if and only if there exists a base B for the open sets such that S-ind B ( U ) 5 n - 1 €or every U in B. In the context of the class S of a-compact spaces it has been stated that the notion of kernel was more natural than that of extension. Let us now make this more explicit. 10.4. Definition. A subset Y of a space X is called an S-kernel of X if Y is a-compact. The S-Surplus of a metrizable space X is defined by

S-Sur X = min { Ind ( X \ Y ) : Y is a S-kernel of X }. 10.5. Theorem (Main theorem). For every separable metrizable space X , S-ind X = S-Ind X = S-Sur X.

Proof. It has been noticed already that S-ind X 5 S-Ind X. We shall establish S-Ind X 5 S-Sur X and S-Sur X 5 S-ind X. Proof of S-Tnd X 5 S-Sur X . The proof is by induction on the value of S-Sur X. The inductive step follows. Suppose that X is a space with S - S u r X 5 n. Let F and G be disjoint closed sets and let Y be an S-kernel of X with Ind ( X \ Y ) 5 n. By Proposition 4.6 thereis apartition S between F and G with Ind ( S f l ( X \ Y ) ) 5 n-1. We have S n Y E S because S is a closed subset of X . Consequently, S-Sur S 5 n - 1 , whence S-Ind S 5 n - 1 by the induction hypothesis. Thus, S - I n d X 5 n. Proof of S-Sur X 5 S-ind X. The initial step of the induction proof is obvious. For the inductive step, let S - i n d X 5 n. In view of Proposition 10.3.C, there exists a base B for the open sets of X such that S-ind B ( U ) 5 n - 1 for every U in B. Indeed, we may assume that B is countable. Let { Ui : i = 0 , 1 , . . .} be an enumeration of B. By the induction hypothesis we have S-Sur B ( U i ) 5 n - 1 for every i in N. So there are disjoint sets Y; and 2; with B ( U i ) = Y , U Zi such that Ind Yi 5 n - 1 and Zi E S. Write X = 2 U B U A where Z = U{ Zi : i E N }, B = U{ B ( U i ) \ 2 : i E N } and A = X \ ( B U 2).

56

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Obviously 2 is in S. Also ind A 5 0 holds as { Ui n A : i E M} is a base for the open sets of A with B A ( U n ~ A ) = 8 for every i. Since B ( U i \ 2 ) c Yi, we have ind (B ( U i ) \ 2 ) 5 n - 1. By the sum theorem for dimension, ind B n - 1. Consequently ind ( A U B ) n holds by the addition theorem. Finally, by the coincidence theorem for dimension, it follows that Ind ( A U B ) 5 n. Thus S-Sur X 5 n has been shown.

<

Now let us show that for each n in a-compactness degree n.

<

N there exists a space with

10.6. Example. Recall that P is the set of irrational numbers and I is the interval [-1,1]. For each natural number n,

S-ind (P x II ") = S-ind (P x

W ") = n.

<

To prove this we first observe S-ind (P x W") ind (P x W") = n. So the equality will be established on showing S-Sur (P x I") 2 n. Denoting the projection of P x I" onto P by p , we have that the image p[C]of any S-kernel C of P x I" is also an S-kernel of P. By the Baire category theorem we find that P \ p[C]is not empty. It follows that p-'[(P \ p [ C ] ) c ] (Px II ") \ C. Hence, S-Sur (P x I") 2 n.. In passing, we observe that the equality cmp (P x In) = n follows with the aid of Proposition 10.3.A. 10.7. Remark. A consequence of Lemma 7.5 and Theorem 10.5 is the following observation. Suppose that X is a compact metrizable space and Y and Z are disjoint subsets o f X with X = Y U Z . ThenC-defy = S-Sur Z. The observation together with Example 7.12 and Theorem 10.5 will result in another proof of the assertion in the previous example. Also in Example 7.13 we have exhibited a subset Y of a compact metrizable space X of dimension n with icd Y 2 n - 2. In view of Theorems 7.11 and 10.5 we have S-ind ( X \ Y ) 2 n - 2.

The close analogies between the a-compactness degree and the dimension function will be exhibited next. The main theorem (Theorem 10.5) is a key element in the exposition. 10.8. Theorem. For every F,,-set Y of a separable metrizable space X , S-Ind Y 5 S-Ind X .

10. THE o-COMPACTNESS DEGREE

57

Proof. See Proposition 10.3.B and Theorem 10.5. 10.9. Theorem (Decomposition theorem). Let X be a separable metrizable space. Then S-ind X 5 n if and only if X is the union o f n 1 sets Xi,i = 0 , 1 , . . .,n, with S-ind Xi 5 0 for every i .

+

Proof. Let S - i n d X 5 n. By the main theorem there is a subset C of X such that C is a-compact and Ind (X \ C) 5 n. From the coincidence and decomposition theorems for dimension we have a partitioning of X \ C into n f 1 at most O-dimensional sets K , i = 0,1,. . . ,n. The main theorem gives S-ind (C U Yi) 5 0 for every i. This proves the necessity. For the sufficiency we suppose that X = X o U X1 U - - - U X,, where S-ind X i 5 0 for i = 0 , 1 , . . . , n. Let X i = u Zi with Y , E S and Ind Zi 5 0. Using the a-compact subset Y = You Yl U . . U Y, of X , we have S-Sur X 5 n because Ind (X \ Y ) 5 Ind (20 U 2 1 U . U Zn)5 n. It follows from the main theorem that S-ind X 5 n. +

+

The addition theorem for S-ind is deduced in the same way as the one for dimension. 10.10. Theorem (Addition theorem). Let X be a separable metrizable space. If X = Y U 2,then S-ind X

5 S-ind Y + S-ind Z + 1.

10.11. Theorem (Countable sum theorem). Suppose that { Fi : i E N} is a countable closed cover o f a separable metrizable space X . Then

S-indX

5 sup { S-ind Fi : i E N}.

with Ci E S Proof. For each i we let Fi be the union Ci U and ind K 5 S-ind Fa = S-Sur Fi and then let C = Ci : i E N}. The collection { Fi \ C : i E N} is a countable closed cover of X \ C with Fi \ C C Y , for each i. By the countable sum theorem of dimension we have ind ( X \ C) 5 sup { ind (Fi \ C) : i E N }. The theorem follows.

U{

It has been clearly shown that both the a-compactness degree and the completeness degree admit analogies to dimension. But

58

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

a dichotomy is also becoming apparent. The function S-ind is a prototype of the functions of the inductive dimensional type which are studied in Chapter I11 and the function ccd (see Section 9) is a prototype of the functions of covering dimensional type which are studied in Chapter IV and V. Also, S-kernels and S-Surplus are keys to the theory of S-ind, and complete extensions and C-def are keys to the theory of ccd. This dichotomy will be underscored by the next two examples. 10.12. Example. Let X be a metrizable space. Recall that Y is a C-kernel of X if Y is complete space that is contained in X. We define the C-surplus of X, denoted C-surX, by

C-sur X = inf { dim (X \ Y ) : Y is a C-kernel of X }. In contrast to the main theorem for S we shall exhibit a space X for which i c d X < C-surX. The space X is the subspace (Rx Q ) U (Qx W) = W 2 \ P 2 of W2. It is clear that C-defX 5 0. Thus i c d X I: 0 holds by Theorem 7.11. As X is of the first Baire category, we have X 4 C. Observe that each dense subspace of a first Baire category space is also a first Baire category space. Consequently no C-kernel of X is dense in X. Because dim U = 1 for every nonempty open subset U of X, it follows that C-surX 2 1. Obviously, C-surX 5 i n d X = 1. So C-surX = 1 holds. 10.13. Example. Let X be a separable metrizable space. We say that a space Y is a an S-hull of X if Y is a metrizable a-compact space that contains the space X. And the S-deficiency of X is defined as S-defX = inf { dim (Y \ X ) : Y is a n S-hull of X}.

In [1965] Nagata conjectured that S-ind X = S-def X. We shall present an example of a separable metrizable space X with S-ind X = 0

and

S-def X = 1.

Observe also that this is in contrast t o the main theorem for C. Let X be the subspace of W 3 given by X = (W x W x 6)) U (Px P x P).

11. POL'S EXAMPLE

59

The space X is of the second Baire category since its dense subset P x P x P is so. It follows that X cannot be u-compact, whence S-ind X 2 0. On the other hand, R x R x Q is a-compact. By the main theorem we find S-ind X = S-Sur X 5 0. So, S-ind X = 0. To help us compute the value of S-def X we note that (0) x R x Q is a closed subspace of X . Thus by Example 7.12 and Proposition 5.9.C we have 1 = cmp (R x Q)5 cmp X . (It is not difficult to prove further that cmp X = 1.) Let us show that ind (Y \ X ) 2 1 for any S-hull Y of X . We may assume that X is dense in Y . As Y is the countable union of compact sets and X is of the second category, there is a compact subset 2 of Y with nonvoid interior. It follows that 2 n X contains a copy of X , say W , that is open in X. As cmp W = 1, by Proposition 5.9.E we have cmp ( 2 n X ) 2 1. It follows from Theorem 5.8 that ind ( c l z ( ( 2 n X ) \ X ) ) 2 1. We have shown that ind (Y \ X ) 2 1 holds. Consequently, S-defX 2 1 by the coincidence theorem for dimension. The equality now follows easily. Results similar to those of this section can be shown to hold for more general spaces X and for classes of spaces P that are more general than S. The investigation along the these lines will be continued in Chapter 111. 11. Pol's example

In [1982] Pol gave an example of a space X with c m p X = 1 and def X = 2 and thereby resolved de Groot's compactification problem in the negative. We shall describe Pol's example and Hart's generalization which was constructed in [1985].

Agreement. Throughout the section, n is a natural number such that n 2 2. Pol's example is a modification of an example constructed by Luxemburg in [1973] of a compact metric space with noncoinciding transfinite dimensions. The descriptions will require the following notation. 11.1. Notation. As usual, I[ will denote the interval [-1'11 and xi will denote the i-th coordinate of a point z in I". Define for 1 5 i 5 n the sets

60

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

The combinatorial boundary of In will be denoted by d I[ ,. That is,

d I " = { z E In : xj = -1 or Finally let C be the subset of

xj

= 1 for some j ) .

R defined by

c = { & : k € N}u{ 0 ) . To construct examples such as Pol's, we must build a feature into the construction that will push up the value of the compactness deficiency as well as another feature that will keep down the value of the compactness degree. Let us first concentrate on the feature that forces the compactness deficiency to exceed n. 11.2. Proposition. For m in N and j in { n i1 , . . . ,2n - 1 } let Ej" be a partition between K j and Lj in 12,-' and let

Em =

n{EJm:j

= n+1,

...,2n-

1).

Define the subspace X of 12n-1x C to be

Then

defX 2 C m p X 2 n..

Proof. We shall make use of the invariant Comp that was introduced in Example 6.7. It will suffice to prove that C o m p X n - 1 fails. To this end, suppose that it does not fail and consider the n pairs of opposite faces (Fi x {0}, Gi x {0}), i = 1,. . .,n, in the space (dII2,-l) x (0). There are partitions Si in X between Fa x {0} and G; x {0}, i = 1,. . . ,n, such that S1 n . n S , is compact. Let Y = 12,-1 x C. Since X is a subspace of Y , by Lemma 4.5 there are partitions Ti between Fi x (0) and Gi x (0) in Y such that Ti n X = Sj, i = 1 , . ..,n. So TI n -.n T, n X will be compact also. As the sets Fi x (0) and Gi x (0) are compact, there is an m' such that each Ti is a partition between Fi x { & } and Gj x { & }

1 1 . POL’S EXAMPLE

61

in Y for m 2 m’ and i = 1 , . . .,n. As Ej” are partitions between Fj and Gj in 127L-1for j = n 1,. . ., 2 n - 1 and for all m, we also } are partitions between Fj x { have that the sets Ej” x { } and Gj x { } in Y . So for m 2 m‘ we have that l12n-1 x { } is contained in Y and that

+

A

and

Ej” x

{A},

j = n + 1, ...,2n- 1,

are, for k = 1,. ..,2n - 1, partitions in I 2n-1 x {

} between

With the aid of Theorem 4.9 we choose points p , such that pm E

X n Tl n . n T, n ( E m x { & >>, m 2 m‘. *

*

It will follow from the definition of the subsets K j and Lj of 12n-1 for j 2 1~ - 1 that the sequence { pm : m E N} cannot have any convergent subsequence in X . Thus we have T1 n * . n T, n X is not compact. This is a contradiction. e

The following is a key lemma in Pol’s construction. The lemma will build into the construction the feature that keeps the value of the compactness degree down. The lemma itself is a special case of Lemma 3.1 of Luxemburg [1981]. 11.3. Lemma. In the cube I3 there is a base f? = { U; : i E N} for the open sets and for each m in N there is a partition Em between K 3 and L3 such that B ( U i ) n Em is a finite disjoint union of compact sets of diameter not exceeding when i 5 m. Moreover, ind Em 5 2 can be imposed for each m.

Proof, Let B = { Ui : i E N} be the base for the open sets of I3 given by Corollary 6.12. Then the intersection B (Ui,) n . . n B (Ui,) is empty for any four distinct indices io, ...,i3. As ]I3 is compact, we may assume that 6( 17;) converges to 0 as i tends to 00 and that S( U;) are bounded by

5.

62

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

Let us construct Em for each m in N. For i 5 m we select finite covers Bi of H 3 consisting of elements of f3 \ { Uo, . .. Urn } such that 1 W )I m+l’

u E Bi

and

BinBj=O,

j
Then define

i 5 m.

Ci = B ( U i ) n (U{ B ( U ) : U E Bi }),

Because ord { B ( U i ) : i E N } 5 3, the sets Ci are pairwise disjoint. Also no set Ci meets both K 3 and L3. Let

(u{

K = K3 U G; : i 5 m and Ci n K 3 # L = L3 U (U{ Ci : i 5 m and Cin K3 =

a}), a}).

Because the sets Ci are pairwise disjoint, the sets Ii‘ and L are disjoint. Let Em be any partition between Ir‘ and L with ind Em 5 2. That Em has the required property easily follows from the observation that the subset B ( U i ) n Em of X \ (U{ B ( U ) : U E Bi }) is 1 the finite disjoint union of open sets of diameter not exceeding m+l whenever i 5 m.

Remark. We have invoked Corollary 6.12 to guarantee the existence of a base f? = { Ui : i E N } such that any four boundaries of elements of B have empty intersection. Since we are dealing with 13, there is an elementary construction. It is a nice exercise t o show that there is a base with this property consisting of “little” cubes. 11.4. Example (Pol [1982]). The example is a space X such that c m p X = 1 and C m p X = defX = 2.

It will be a subspace of I14. By Lemma 11.3 there exists a base B = { Ui : i E N } for the open subsets of I3 and for each m in N there exists a partition Em between K 3 and L3 in II with ind Em 5 2 such that the set B ( U i ) n Em is the disjoint union of compact sets of diameter not exceeding when i 5 m. Define the space to be

&

X

=

((aH 3,

x { 0 }) u (U{ E , x {

*

) : m E N )) .

11. POL’S EXAMPLE

63

From Proposition 11.2 we have that defX 2 C m p X 2 2. As the space X is two-dimensional, it follows from Theorem 5.11 that defX = 2. To show that cmp X = 1 it will suffice to show that cmp X 5 1 because de Groot’s theorem yields c m p X $ 0. That is, we must show that each point z of X has arbitrarily small open neighborhoods whose boundaries are rim-compact . There are compact neighborhoods for each point z not in (aK3)x (0). So let us consider a point z = (y,O) in (all3)x (0). The open ball in I 3 with radius E centered at a point w will be denoted by S,(w). Recall that X is a subset of 14. A neighborhood base of z is formed by the open sets vi = (Ui x [0, n X with y E Ui E B and m E N. In view of Proposition 5.9.D, it suffices to verify H i = (B13(Ui) x [0, nX is rim-compact. Observe that Hi is a subset of

h])

A])

To prove that Hi is rim-compact it is sufficient to prove that a point z = (w, 0) in ((aII ’) x (0)) n Hi has arbitrarily small neighborhoods in Hi with compact boundary. Let 0 < E < +. Because the set (Bn3(Ui) n E m ) x { & } is a finite disjoint union of compact sets of diameter not exceeding can find an open set W of X such that

and

Bx(W) n Hi

We have shown that

Hi

C

for each m with i

5 m, we

(all3)x (0).

is rim-compact.

In Pol’s example the gap between c m p X and defX is one. In [1988a] Kimura published examples of spaces for which the gap between cmp and def is arbitrarily large. Hart indicated in [1985] how such spaces can be constructed by a modification of Pol’s construction which will be given next. The following is a preparatory lemma. 11.5. Lemma. In a normal space X , let F be a finite closed collection with order not exceeding k. Then for i = 1 , . ..,k there

64

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

are collections Hi of disjoint closed sets such that Hi refines F for every i and U F = U{ (UHi) : i = 1 , . . .,k}. Proof. We may assume X = U F. The proof is by induction on k . As the case k = 1 is obvious, we shall discuss the inductive step. For each subset G of F we define the set K G = F :F E G } and we denote the number of elements of G by [GI. The collection { K G : G c F, \GI = k } is disjoint as k 2 2. For each G with G c F a n d /GI = k there exists an open set UG such that K G c UG holds and the collection { cI(UG) : G C F, !GI = k } is disjoint (see the proof of Lemma 8.8). The required collection Hk is defined to be { cl (UG): G C F, JG)= k}. To get the remaining k - 1 collections let U = U{ UG : G C F , IGI = k } and define the closed collection F k - 1 = { F \ U : F E F, F \ U # 0). As ordFk-1 5 k - 1 holds, the induction hypothesis applies and the collections Hi exist for the remaining i = 1 , . . .,k - 1.

n{

The first step towards generalizing Pol’s example is the following adaptation of Lemma 11.3. 11.6. Lemma. In the (272 - 1)-dimensional cube 1 [ 2 n - 1 there is a base 13 = { Ui : i E N } for the open sets and for each m in N there are partitions Ej” between K j and Lj for j in { n 1 , . . . ,2n - 1} such that the set

+

B ( U p )n

(n{Ej” : j

=n

t 1,.. .,2n - 1})

is a finite disjoint union of compact sets of diameter not exceeding mfl when i m.

’

<

Proof. The proof of Lemma 11.3 will be modified. By Corollary 6.12 there is a base f? = { Ui : i E N} for the open sets of I Z n - l such that the intersection B (Up.,) n f l B (Ui2n-l)is empty for any We may assume furset of 2n indices satisfying io < . . < ther that S(Ui) < $ for all i and that S(Ui) + 0 as i + 00. Let us construct the partitions Ej”, j = n 1,. . . ,2n - 1. For i L. m we choose‘a finite cover f?i of l12n-1 with elements of B \ { Uo, . . . ,Urn } such that W )2 m+l’ U E f ? ; ,

---

+

and

ainBj=O,

i<j<m.

11. POL’S EXAMPLE

65

For i 5 m we let

Ci = B (Ui)n (U{ B ( U ) : U

E Bi }).

Then ord { B (Ui): i E N} 5 2n - 1 gives ord{ Ci : i 5 m } 5 n - 1. By Lemma 11.5, { Ci : i E N} has disjoint closed refinements Hi, i = 1,. . . , n - 1, such that

U{ Ci : i 5 m } = U{ UHi : i = 1 , . . .,n- 1 }. Now for each j in { n -t 1,. . . ,2n - 1 } we define

(u{

I(; = K~ u H : H E H ~ - H~ n, rrj # S}), L; = Lj u (U{ H : H E ~ j - H~ n, ~j = S}). As in the proof of Lemma 11.3 the sets K; and L5 are disjoint and the partitions EJm between I<; and L5 will yield the desired partitions. 11.7. Example. The example is a space X for which

c m p X = 1 and

defX 2 C m p X 2 n.

The example will be a subset of 12n. Choose the base B and the sets Ej”, j = n -t 1,. . . ,2n- 1, as provided by Lemma 11.6 and let

Em =

n{E j m :j

= n + 1, ...,2n- 1).

The space is defined to be

By Proposition 11.2, Cmp X 2 n. The equality cmp X = 1 is proved in the exact same way as in Example 11.4. Using this example, we can construct spaces for which the gap between cmp and def has any prescribed value, including 00. The following space will achieve the gap n. Let X @ 2 denote the topological sum of a space X with def X - n = k 2 1 and cmp X = 1 and the space 2 = Q x Ik. Then cmp ( X @ 2 ) = k and def ( X @ 2 ) = n k. Consequently, def (X @ 2 ) - cmp ( X @ 2 ) = n.

-+

66

I. THE SEPARABLE CASE IN HISTORICAL PERSPECTIVE

12. Kimura’s theorem

Although de Groot ’s compactification problem was resolved by Pol’s example, the example did not shed any light on the validity of Cmp X = def X or Skl X = def X . The possibility of these identities is the result of two splittings of the compactification problem. Our historical perspective will now be concluded with a discussion of these splittings. As we have seen, the compactification problem had the splittings

cmp X 5 Cmp X

5 def X

and

cmp X

5 Skl X 5

def X .

In [1985], Aarts, Bruijning and van Mill established the following connection between these two splittings. 12.1. Theorem. For every separable metrizable space X ,

Crnp X 5 SklX

Proof. The proof of the theorem follows a pattern similar to that of the coincidence theorem for dimension. The proof is by induction on Skl X . As a consequence of Theorems 6.9 and 5.7 we have for n 5 0 that Skl X 5 n if and only if def X 5 n. So by Theorem 6.6, Skl X 5 n if and only if Cmp X 5 n when n 5 0. The inductive step for n 2 1 is proved as follows. Let X be a separable metrizable space with SklX 5 n. Suppose that F and G are disjoint closed sets of X . By the definition of SklX 5 n there is a base B = { Ui : i E N } for the open sets of X such that the intersection B ( U i , ) n n B (Ui,) is compact for any n t 1 different indices io, . . . , i n from N. Consider the family { (Cj, Dj): k E N } of all pairs of elements of B with cl (C,) C Dk such that cl (Dk)n G = 0 or DI, n F = 0. Let Vj = Dj \ U{ cl(C,) : m = 0,1,. ..,k - 1 } for each Ic in N. The collection V = { Vj : k E N } is a locally finite cover of X . For each k in N we have B (Vj) C B (CO) U U B (Cj-1) U B (Dk).Let W be the open set U{ Vj : cl (Dj)n G = 0 }. Clearly F c W holds. Because the collection V is locally finite, we have cl ( W )n G = 0 and B ( W )C U{ B ( V k ) : c l ( D j ) n G = S}. It follows that B ( W ) is a partition between F and G. We shall show that Cmp B ( W )5 n - 1. Observe that each B (Vj) is the union of a finite collection of closed subsets of the boundaries of elements of B. It follows that B(W)

-

9

12. KIMURA’S THEOREM

67

is the union of a locally finite collection { Ej : j E No } where No is a subset of N and each Ej is a nonempty closed subset of B ( U j ) . (Here and also in the rest of the proof, the original indexing of f3 is being retained.) We shall select a subset N1 of N \ NO such that { U; n B ( W ): i E N1 } is a base for B ( W ) that witnesses the fact that Skl B ( W ) n - 1. This is done in two steps. The first step follows. For each j in No we choose a point p j from Ej to form the set P = { p j : j E NO}. Observe that P is closed. Define Nz to be { i E N : Ui E f3, B ( U ; ) n P = S}. It is easily seen that f?* = { U; E f? : i E N z } is a base for the open sets of X and that the boundary B ( U ) for each U in f?* is distinct from the boundary B ( U j ) for every j in N O . So in particular NOn Nz = 8. For the second step let G be an open cover of X such that each element of G meets at most finitely many members of { Ej : j E NO}. Let N1 be the set of all i in Nz such that cl(U;) c G for some G in G . We shall show that { U; n B ( W ): i E N l } is a base for B ( W )that witnesses the fact that Skl B ( W ) n - 1. To this end, let io, . ..,i,-l be n distinct indices from N1 and, with d as the boundary operator in B ( W ) ,let

<

<

Then S is a closed subset of R = B ( U i , ) n ..en B ( U ; a - l )n B ( W ) . As each B ( U i j ) is a subset of some G in G and each member of G meets at most finitely many members of { Ej : j E No }, it follows that R = R n Ek : k E N ’ } for some finite subset N’ of No. For each k in N’ the set R n Ek is a closed subset of B ( U k ) . Because the indices k,io, . . . , i,-l are distinct, the set R f l B ( u k ) is compact. It follows that R and, consequently, S are compact. We have proved that Skl B ( W ) n - 1. Then Cmp B ( W )5 n - 1 by the induction hypothesis. It follows that CmpX n holds.

u{

<

<

Remark. The use of the set P in the above proof allows us to avoid the indexing pitfall that was discussed in Example 6.13. This method of avoiding the pitfall will be used again in Theorem V.4.3. A second method will be used in Section VI.5. The following corollary is of interest. It will be generalized in Section V.4.

12.2.

Corollary. For n

2 1 let X

be a separable metrizable

68

I. THE SEPARABLE CASE I N HISTORICAL PERSPECTIVE

space with SklX 5 n. Then between any two disjoint closed sets of X there is a partition S such that Skl S 5 n - 1. Now that we know the inequality Cmp X 5 Skl X holds for every separable metrizable space X it is clear that the invariant Skl is a better candidate than Cmp for internally characterizing the compactness deficiency. In [1988] Kimura made a major breakthrough by proving the following theorem. 12.3. .Theorem (Kimura [1988]). For every separable metrizable space X, S k l X = defX.

We shall present a proof of Kimura’s theorem in Section 5 of Chapter VI, a chapter which is largely devoted to compactifications. In [1990] the picture of the compactness invariants was completed by Kimura with the following example. 12.4. Example (Kimura). There is a separable metric space X with c m p X = C m p X = 1 and 2 5 defX 5 3.

The reader is referred to Kimura’s I19901 paper for a description of the space X. 13. Guide to dimension theory

In this chapter we have presented the most important dimension functions, namely the small inductive dimension ind, the large inductive dimension Ind and the covering dimension dim. The topics discussed in this book were not picked at random but selected with the theory of dimension-like functions in mind. This had the effect that some aspects of dimension theory, notably dimensional properties of topological completions and compactifications, were emphasized while other aspects, notably product theorems and the dimension of subsets of R”,did not get the attention they deserved. So for a balanced overview of dimension theory the reader should consult books on dimension theory proper. For a long time Hurewicz and Wallman [1941] had been the one and only main reference. More recent books on dimension theory that can be consulted are Alexandroff

14. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

69

and Pasynkov [1973], Nagata [1965], Nagami [1970], Pears [1975] and Engelking [1978]. Also, several text books on general topology have introductory chapters on dimension theory, viz., Kuratowski [1958] and [1961], Engelking [1977] and van Mill [1989]. No prior knowledge of dimension theory is required for a successful reading of this book. In a sense the material of Chapter I can be considered to be a first introduction to dimension theory. Except for the sum and decompositions theorems, we have proved many results in great detail. Chapter I1 deals with mappings into spheres and dimension. It is an ab ovo introduction of the generalized theory of large inductive dimension (modulo P ) and covering dimension (modulo P ) and the characterization of these dimension functions by the existence of mappings into spheres. This theory is helpful in computing the values of the dimension-like functions in various situations. In Chapter I11 the fundamental sum and decomposition theorems are discussed for those dimension-like functions that are of inductive type. As the sum and decomposition theorems for dimension are included, this discussion provides a firm base for Chapter I. The covering type functions are discussed in Chapter IV. Excluding compactifications, the theory of extensions and dimension is given in Chapter V. The developed theory uses the basic inductive dimension as a main tool. (Here basic refers to bases.) The final chapter is devoted to the interplay of dimension theory and compactification theory. To place the book in proper perspective, a simultaneous browsing on a book in dimension theory is recommended. 14. Historical comments and unsolved problems

The chapter was motivated by an historical theme. We touched upon the origins of dimension theory in Section 2. An interesting account on this subject is Duda [1979]. In the whole, our historical comments have been concentrated on the history of dimension-like functions rather than on a precise historical account of dimension theory. So we did not give, for example, a detailed history of the sum theorem. We refer the reader to books on dimension theory proper for this, notably Engelking [1978]. Much of the material of Section 5 is in de Groot’s thesis [1942]. It can also be found in de Groot and Nishiura [1966].

70

I. T H E SEPARABLE CASE I N HISTORICAL PERSPECTIVE

The proofs in Section 6 are part of the folklore. The large inductive compactness degree Cmp was introduced in de Groot and Nishiura [1966]. In his [1960] paper Sklyarenko proposed the function Skl as a candidate for the internal characterization of the compactness defect. The splittings of the compactification problem were mentioned in Isbell [1964]. The presentation of properties of topological completeness found in Section 7 closely follows Engelking [1977]. Most of the ideas in this section beginning with Lemma 7.5 are from Aarts [1968]. The proof of Lemma 8.7 is due to Kuratowski [1958]. And the proofs of Lemma 8.9 and the coincidence theorem (Theorem 8.10) go back t o Morita [1950]. As already mentioned in Section 9, the notion of a border cover first appeared in Smirnov [1965]. Although the first part of the proof of Theorem 9.5 is almost verbatim from Aarts [1968], the proof of the second part is new and follows the pattern of the proof of a corresponding result about ordinary dimension in Nagata [ 19651. The results in Section 10 are due to Aarts and Nishiura [1973]. The main result in of Section 11, namely Example 11.4, is due to Pol [1982]. Example 11.7 is due to Hart [1985]; the first (published) space with these properties was constructed by Kimura [1988a]. Theorem 12.1 and Corollary 12.2 are due to Aarts, Bruijning and van Mill [1985]. Theorem 12.3 and Example 12.4 are due to Kimura in [1988] and [1990] respectively.

Unsolved problems 1. Is there a decomposition theorem for icd?

See the comments after Theorem 7.17. 2. In Example 5.10.f we have seen the failure of the point addition

theorem for cmp. That example also showed the failure of the addition theorem. Although the examples of van Douwen [1973] and Przymusiriski [1974] show that the point addition theorem for ind does not hold (see Example III.1.2), the following question remains open. Does the addition theorem for ind hold for normal spaces?

14. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

71

An inductive proof will show that the addition theorem for ind holds for hereditarily normal spaces. See Theorem 111.1.9. 3. The computation of def (I" \ ple 5.10.e., where

= n - 1 was given in Exam-

is an open face of 1". The value of cmp (I" \ E n - ' ) is not known for n 2 4 (see de Groot and Nishiura [1966]).

What are the values of cmp ( I n

\ E n-l

) for n 2 4? This question has been proposed as Problem 417 in van Mill and Reed [1990]. The equality Cmp(Hn \ El'.-*) = n - 1 was established in Example 6.7. 4. The inequalities cmp X

5 Cmp X 5 def X are shown to be sharp

by Pol's and Kimura's examples. But, Pol's example has C m p X = def X and Kimura's example has cmp X = Cmp X. Exhibit a separable metrizable space X such that the inequalities cmp X < Cmp X < def X hold. 5 . Related t o problem

4 is the following gap question. (See Exam-

ple 12.4.) For each n with n 2 1, does there exist a separable metrizable space X such that def X - Cmp X = n? The question for the corresponding gap formula def X - cmp X = n has been answered in the affirmative in Example 11.7. 6. There is an obvious sharpening of the gap statement in Example 11.7.

For E and m in N with k < m , does there exist a separable metrizable space X such that cmp X = k and def X = m? In [1986] Kimura has constructed for all m and k with m 2 k a countably compact, completely regular space Xmk such that cmp Xmk = k and defX,k = m. But the example is not metrizable.

Chart 1. The Absolute Borel Classes

The chart summarizes the inclusion relations among the absolute Borel classes in the universe Mo of separable metrizable spaces. The arrows indicate inclusion. 72

CHAPTER I1

MAPPINGS INTO SPHERES

It is time t o change the perspective. The historical survey of the first chapter was essentially limited t o separable metrizable spaces and to the dimension theory and its generalizations spawned by the conjecture of de Groot, namely the modulo theory for the specific classes K , C and S. The focus of the new perspective will be the discovery of relationships between the various dimensions modulo a general class P and mappings into spheres. It will afford a quick exposure to the general problems and techniques found in the theory. The chapter will end with applications to the absolute Borel classes, the natural generalizations of the classes K , C and S. A discussion of the absolute additive, multiplicative and ambiguous Borel classes of order o will be given in Section 9. The chart on the opposite page outlines the connections among the various absolute Borel classes in the setting of separable metrizable spaces. At this juncture it will be useful t o make an agreement on how general the spaces under consideration will be. The definitions in Chapter I were made for general topological spaces. But as the discussion progressed it became evident that the spaces should be TI-spaces in order to connect ind and Ind. Indeed, the requirements increased from TI-spaces to regular spaces and then t o normal spaces. So the following minimal agreement is made. Agreement. Every topological space will be assumed t o be a

TI-space. As the chapter develops the class of spaces under discussion will gradually be narrowed with further agreements. By the beginning of Section 4 all spaces will be assumed to be hereditarily normal spaces. The discussion up to that point will expose the rationale behind the general restrictions that will be placed on the spaces for the remainder of the book. 73

74

11. MAPPINGS INTO SPHERES

1. Classes and universe

The early development of ind X , Ind X and dim X occurred in the setting of separable metrizable spaces X. Instead of the word “setting” the expression “universe of discourse” would be more appropriate. The first generalizations of dimension theory concerned the extension of the theory t o larger universes. The natural definitions were introduced for arbitrary topological spaces. But the theory did not extend-many counterexamples to the basic theorems were discovered. Thus the spaces X had to be limited in order to derive a coherent theory which reflected the theory for the universe of separable metrizable spaces. The first successful extension was to the universe of metrizable spaces (but with the loss of the equality i n d X = IndX). Other universes such as the class of perfectly normal spaces and the class invented by Dowker called the totally normal spaces (see Chapter I11 where these universes will be discussed further) have also been shown t o yield satisfactory theories. It is now time to define the notions of classes of spaces and universe of discourse. 1.1. Definition. A topologically invariant class (class for short) is a collection of spaces X with the property that if Y is homeomorphic to X then Y is also a member of the class. A universe of discourse (universe for short and always denoted by U )is a class that defines the spaces under discussion.

Once the universe of discourse has been specified, usually at the beginning of each section, all spaces in the subsequent discussion are t o belong to the universe. As classes enter into the discussion, only the members of the class that belong to the universe will be considered.

A topological property will define a class of spaces and conversely a class of spaces will define a topological property. The following are some obvious examples of classes of spaces that will appear in the book. By agreement they are contained in the class of all 7’1-spaces. 1.2. Examples. (1) 7 = { X : X is a topological space} (2) R = { X : X is a regular space } (3) R,= { X : X is a completely regular space} (4)N = { X : X is a normal space}

1. CLASSES AND UNIVERSE

(5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

75

NH = { X : X NT = { X : X

is a hereditarily normal space} is a totally normal space} Np = { X : X is a perfectly normal space } M = { X : X is metrizable} Mo = { X : X is a separable metrizable space}

(0)

K = { X : X is compact }

C = { X : X is a complete metrizable space } S = { X : X is a-compact} L = { X : X is locally compact }

In Chapter I we have seen how the dimension functions ind, Ind and dim are generalized t o the functions cmp, icd, Icd, ccd, S-ind and S-Ind. The inductive generalizations were made by replacing the empty space with spaces from the classes K , C and S respectively to begin the induction. The natural way to generalize these inductive processes is t o use an arbitrary class P t o define the functions P-ind, P-Ind, etc. When the universe U is used in conjunction with a class P , one naturally changes the class P t o the smaller class P n U . For the universe Mo and the class K , the inductive dimensions cmp and K-Ind fortunately agree with the respective inductive dimensions using the smaller class K fl Mo in place of K . The situation will be the same for the universe U and a class P under rather mild conditions on U and P. This will be made evident in the next section. Also in Sections 3 , 5 and 8 we shall be dealing with other topologically invariant functions associated with classes P such as surplus, deficiency and covering dimensions that could be affected by the introduction of the universe U.Again we shall see that the introduction of the universe U (and hence the introduction of the smaller class P n U ) will not affect our results when the same very mild conditions on the universe U and the class P are assumed. At the end of Section 2 the minimal conditions that will be needed for our development are formulated into an agreement that will be in effect for the remainder of the book. It will become clear from the discussion of that section as to why such conditions are needed. The role of the universe will show its greatest influence in Chapter I11 when we embark upon a search for an optimal universe in which a reasonable dimension theory can be developed.

76

11. MAPPINGS INTO SPHERES

2. Inductive dimension modulo a class

P

Agreement. The empty space 0 is a member of every class. This section introduces the inductive approach to the extension of dimension modulo P to an arbitrary class. Obviously the definitions of these extensions are motivated by the usual small and large inductive dimensions ind and Ind. The small and large inductive dimensions were defined in terms of partitions-see the Definitions 1.3.2 and 1.4.1. The extensions of these definitions and their relationships to the universe of discourse will be established here. Since the aim is to extend dimension theory, every class P is assumed to have the empty space as a member. Consequently, the class P will be nonemp ty. The following is the natural extension of the definition of the small inductive dimension ind. 2.1. Definition. Let P be a class of spaces and X be a topological space. One assigns the small inductive dimension modulo P , denoted P-ind X, as follows.

(i) P-ind X = - 1 if and only if X is in P . (ij) For each natural number n, P-ind X 5 n if for each point p and for each closed set G of X with p !$ G there is a partition S between p and G such that P-ind S 5 n - 1. (iij) For each natural number n, P-ind X = n if P-ind X 5 n and P - i n d X $ n - 1. (iv) P-ind X = 00 if the inequality P-ind X 5 n does not hold for any natural number n. Since the class P is topologically invariant, the function P-ind is also topologically invariant. Obviously, { 8 }-ind = ind ; the notation ind will be the preferred one. Our first proposition is an immediate consequence of the definition. Its proof is left t o the reader. 2.2. Proposition. If P - i n d X < 00, then X is in P or X is a regular space.

A consequence of the proposition is that there is no loss in assuming that all spaces are regular.

2. INDUCTIVE DIMENSION MODULO A CLASS

P

77

The easy inductive proof of the next proposition is left to the reader. 2.3. Proposition. If P and Q are classes with P 2 Q, then P-ind 5 Q-ind. In particular, P-ind 5 ind.

Let us now prepare for the generalization of Proposition 1.3.6. That proposition showed a nice connection between i n d X and the boundaries of the members of an appropriate base L? for the open sets of X . The following example will illustrate the difficulties that can arise in generalizing this proposition if no conditions are placed on the class P . 2.4. Example. For this example, let P be the class of spaces X with i n d X = 1 or X = 8. (Note the exact equality!) It should be observed that a closed subspace Y of a space X in P need not be in T’. This class P will contrast the use of general partitions between points and closed sets and the use of partitions between points and closed sets by boundaries of open sets. Consider the space X which is the disjoint topological sum of R and R2.It is obvious that for each point p in X and each closed set G with p 4 G there is a partition S between p and G with ind S = 1. That is, S is in P . Or, loosely speaking, points and closed sets can be separated by members of P. Also X is not a member of P . So, P-ind X = 0. Since R has the property that the boundary of every bounded open set has ind equal t o 0, some point p of X has the property that each neighborhood base B ( p ) has a U in B ( p ) such that B ( U ) is not in T’. So there is no base f? for the open sets of X with B ( U ) E P for each U in B. This example shows that Definition 2.1 will not lead to an analogue of Proposition 1.3.6.

In order to capture the analogue of Proposition 1.3.6 one will need the property that a closed subspace of a space in the class P is also in P. We make the following definitions. 2.5. Definition. The universe is said to be closed-monotone if every closed subspace of each space in the universe is also a member of the universe. 2.6. Definition. For the universe of discourse, a class P is said to be closed-monotone in U if P satisfies the condition: If X is a space in P n U ,then every closed subspace of X is in P .

11. MAPPINGS INTO SPHERES

78

When a class P is closed-monotone in the universe 7 of all topological spaces, the class P will be called absolutely closed-monotone. Observe that the classes Ic, C and S introduced in Chapter I are absolutely closed-monotone. Further observe that if the universe is closed-monotone and the class P is closed-monotone in U ,then P n U is absolutely closed-monotone. The following proposition is easily proved.

Proposition. Let the universe be closed-monotone. The following statements are equivalent for a class P . (a) P is closed-monotone in U. (b) P-ind Y 5 P-ind X holds for every closed subspace Y o f each space X in the universe. 2.7.

Proof. Clearly statement (b) implies statement (a). For the converse we assume that statement (a) holds. We shall prove inductively the statement: If Y is a closed subspace of X and P-ind X _< n, then P - i n d Y 5 n. When n = -1, the statement is obvious. So assume that the statement is true for n - 1 and consider a closed subset Y of a space X with P-ind X 5 n. Let p be a point of Y and G be closed in the subspace Y with p 4 G. Since Y is closed in X , the set G is also closed in X . Consequently there is a partition S between p and G in X with P-ind S n - 1. We have that S is a member of U because U is closed-monotone. By the induction hypothesis, P-ind (S n Y ) 5 n - 1. Since S n Y is a partition between p and G in Y , it follows that P-ind Y 5 n. Thus statement (b) holds.

<

The next example will aid the reader in seeing the connection between the closed monotonicity of the universe U and of the class P in U .

Example. Though the definition of a class of spaces depends only on topological properties, the notion of closed-monotone class in U is dependent on the universe of discourse. To see this we consider the class P defined by separability and consider the universe U = R,of completely regular spaces. The Cech-Stone compactification PN of the space of natural numbers N is completely regular and is also in the class P . Clearly R, is closed-monotone. The closed set PN\N of PN is known to be nonseparable, that is, PN \ N 4 P . So the class P is not closed-monotone in the universe of 2.8.

2. INDUCTIVE DIMENSION MODULO A CLASS

P

79

completely regular spaces. In contrast to this, the class P is closedmonotone in the closed-monotone universe M of metrizable spaces. The following theorem is a useful complement t o Proposition 2.7. It will illustrate why one passes from P to P n U in the context of the universe of discourse. Observe that the theorem requires the class P t o satisfy only the agreement made at the beginning of this section.

& be an absolutely closed-monotone class. For every space X in Q and every class P, 2.9. Theorem. Let

Proof. We already know that P-ind X 5 ( P n Q)-ind X holds for any X. For the reverse inequality we shall prove inductively the statement: If X E Q and P - i n d X 5 n, then (P n Q)-indX 5 n. The statement is obviously true for n = -1. So we assume that it is true for n - 1 and let X be a space in Q with P-ind X 5 n. Suppose that p is a point and G is a closed set with p $ G. Then there is a partition S between p and G such that P-ind S 5 n - 1. Because X is in Q and Q is absolutely closed-monotone, S is in Q. The induction hypothesis yields (P n Q)-ind S 5 n - 1. Thereby we have (P n &)-indX 5 n. With the aid of Proposition 2.7 we can prove the following analogue of Proposition 1.3.6. 2.10. Theorem. Let the universe be closed-monotone and n be a natural number. Suppose that the class P is closed-monotone in U and X is a regular space in the universe. Then the following statements are equivalent. (a) P-ind X 5 n. (b) There exists a base B for the open sets of X such that P-ind B ( U ) 5 n - 1 for every U in B .

Proof. The reader will recognize similarities between the proof of Proposition 1.3.6 and the one given here. Suppose P-ind X 5 n. We must consider two cases. The first is that of P-ind X = -1. In this case Proposition 2.7 will yield that any base f? for the open sets has the required property because the regular space X is a member of the universe. Let P-ind X 2 0 for the

80

11. MAPPINGS INTO SPHERES

second case. For a point p and an open neighborhood U of p there is a partition S between p and X \ U with P-ind S 5 7t - 1. That is, there are disjoint open sets V and W with p E V and X \ U c W such that X \ S = V U W . Observed that S is in U because the universe is closed-monotone. Since B ( V )is a closed subset of S , we have P-ind B ( V )5 n - 1 by Proposition 2.7. For the converse implication we let p be a point and G be a closed set with p $! G. Since X is a regular space, there exist disjoint open sets V and W such that p E V and G c W . In the base B there is an open set U with p E U c V . The set B (U)yields the required partition between p and G to establish P - i n d X 5 n. Let us now give the natural extension of the definition of the large inductive dimension Ind.

2.11. Definition. Let P be a class of spaces and X be a topological space. One assigns the large inductive dimension modulo P , denoted P-Ind X , as follows. (i) P - I n d X = -1 if and only if X is in P . (ij) For each natural number n, P - I n d X 5 n if for each pair of disjoint closed sets F and G there exists a partition S between F and G in X such that P-Ind S 5 n - 1. (iij) For each natural number n, P-Ind X = n if P-Ind X 5 n and P-Ind X $ n - 1. (iv) P-Ind X = 00 if the inequality P-Ind X 5 n does not hold for any natural number n.

Remark. Just as we have found in Chapter I, the topological invariants that appear in dimension theory are often defined by means of inequalities. This leads to the need to define the values of these functions by requirements such as (iij) and (iv) in Definitions 2.1 and 2.11. Continuous repetition of these two conditions becomes somewhat monotonous. So from this point on we shall not give them explicitly and leave it to the reader to supply the missing parts. Clearly P-Ind is a topological invariant and { 0}-1nd = Ind. As with ind, the preferred notation will be Ind.

Example. In [1966] MardeSiC couched an interesting theorem in a form that used the function P-Ind. The theorem concerned a property of ordered cornpacta. The class P in that paper was taken

2. INDUCTIVE DIMENSION MODULO A CLASS

P

81

t o be

P = { X : all components of X are metrizable continua} and the following theorem was proved.

Theorem. Let X be a compact Hausdorff space. If X is the continuous image of an ordered compactum, then P-Ind X 0. Since the proof of this theorem is not directly related to our purposes, it will not be given here. The easy inductive proof of the following theorem is left to the reader. 2.12. Theorem. For every space X

,

P-ind X 5 P-Ind X. The next two propositions also have easy proofs.

If P and Q are classes with P 3 Q , then P-Ind 5 Q-Ind. In particular, P-Ind 5 Ind. 2.14. Proposition. If P - I n d X < co,then X is in P or X is a 2.13. Proposition.

normal space. Thus there will be no loss in assuming that the universe of discourse is contained in the class N of normal spaces. This is similar to the situation for P-ind in which the class of regular spaces was found t o be important. Exactly analogous to Theorem 2.9 is the following. 2.15. Theorem. Let Q be an absolutely closed-monotone class. For every space X in Q and every class P,

P-Ind X = (P n Q)-Ind X. Proof. The proof is a straightforward modification of the one for Theorem 2.9. The subspace theorem for dimension (Theorem 1.3.4) in the universe Mo of separable metrizable spaces has a straightforward proof. It should be no surprise that the subspace theorem does not generalize to P-ind. For example, the function cmp fails to have a subspace theorem. But some form of the theorem can be recovered for P-ind and P-Ind. We shall give three of them.

82

11. MAPPINGS INTO SPHERES

2.16. Theorem. Let the universe be closed-monotone. For classes P the following three conditions are equivalent. (a) P is closed-monotone in U . (b) P-ind Y 5 P-ind X holds for every closed subspace Y of each space X in the universe. (c) P-Ind Y 5 P-Ind X holds for every closed subspace Y of each space X in the universe.

Proof. The equivalence of conditions (a) and (b) has been established in Proposition 2.7. That condition (c) implies condition (a) is easily proved. Finally, that condition (a) implies (c) is proved inductively in a manner similar t o the induction found in the proof of Proposition 2.7. (See also Proposition 1.4.3.) The next theorem concerns the behavior of P-ind for open subspaces. The example K of the class of compact spaces fails to have the property that open subspaces are members of the class. But a rather general open subspace theorem does hold for P-ind. 2.17. Theorem. Let P be any class of spaces. If 0 5 P-ind X, then P-ind Y 5 P-ind X for each open subspace Y of X .

Proof. Let n = P-ind X < 00. Suppose that p is a point of Y and G is a closed set in Y with p @ G. Then p is not in the closed subset G' = G U ( X \ Y ) of X . Since P-ind X 2 0, there is a partition S between p and G' in X with P-ind S 5 n - 1. Clearly S is a partition between p and G in Y. Thus, P - i n d Y 5 n. 2.18. Definition. The universe is said to be monotone if every subspace of each space in the universe is also a member of the universe. A class of spaces P is said to be monotone in U if P satisfies the condition: If X is a space in P n U ,then every subspace of X is in P . When a class P is monotone in the universe 7 of all topological spaces, the class P will be called absolutely monotone.

Clearly the class { 0 } is absolutely monotone. Indeed, the class of all spaces whose cardinality is bounded by a fixed cardinal number is absolutely monotone. In Chapter I we mentioned that rational curves are defined by using countable spaces. For the other extreme, the classes K ,C and S discussed in Chapter I are not monotone in any universe of interest.

2. INDUCTIVE DIMENSION MODULO A CLASS

P

83

2.19. Theorem. Let the universe be monotone. For classes P the following conditions are equivalent. (a) P is monotone in U. (b) P-ind Y 5 P-ind X holds for every subspace Y of spaces X in the universe.

Proof. The proof is similar to that of Theorem 2.7. Suppose that condition (b) holds and let X be a space in P n U . If Y is a subspace of X , then Y is in U because the universe is monotone. From condition (b) we have P-indY 5 P - i n d X = -1. That is, Y is in P . Therefore the class P is monotone in U. Let us prove next that condition (a) implies (b). Let P be monotone in U. We shall prove inductively the statement: If Y is a subspace of X and P-ind X 5 n, then P-ind Y 5 n. When n = -1, the statement is obvious from condition (a) because the universe is monotone. So we assume that the statement is true for n - 1 and consider a subspace Y of a space X with P - i n d X 5 n. That is, assume that the class P’ = { X : P-ind X n - 1 } is monotone in U. We must now prove that the statement is true. But this is exactly the same situation that is found in the proof of the subspace theorem for ind given in Theorem 1.3.4. Indeed, the exact same proof applied in the monotone universe will complete the induction.

<

Concerning the monotonicity of P-Ind, it is known that Ind fails to have a general subspace theorem in the universe NH of hereditarily normal spaces (see Pol and Pol [1979]). In Chapter 111 we shall investigate further the problem of monotonicity. Dual to the subspace concept is that of the ambient space. Our spaces X are often contained in ambient spaces Y . So it will be convenient t o have partition theorems for P-ind and P-Ind in the setting of ambient spaces. 2.20. Proposition. Let the universe be closed-monotone and n be a natural number. Suppose that X is in U and P is a class of spaces. If X is a subspace of a hereditarily normal space Y , then P-ind X 5 n if and only if (a) X E P or (b) for each point p of X and each closed set F of Y not containing p there is a partition S between p and F in Y with P-ind (S n X ) 5 n - 1. Moreover, if P is closed-monotone in U ,the condition (a) can be dropped.

84

11. MAPPINGS INTO SPHERES

Proof. Suppose that P-ind X 5 n and X 4 P hold. Let p be a point in X and F be a closed set of Y with p 4 F . Let F' be a closed neighborhood of F such that p 4 F'. There is a partition So between p and F' n X in X with P-ind SO n - 1. Note that SOis also a partition between p and F in the subspace X U F . By Lemma 1.4.5 there is a partition S between p and F in Y with SO= S fl X . For the converse we assume that X is not in P . Let p be a point of X and let G be a closed subset of X with p 4 G. Then p is not a member of F = cly(G). Let S be a partition between p and F in the space Y with P-ind ( S n X ) n - 1. The set S f l X is a partition between p and G in the space X . Hence, P-ind X 5 n.

<

<

Let us turn to a P-Ind analogue of Proposition 1.4.6. Its statement will contain the notion of separated sets. Recall that two subsets A and B of a space X are separated if clx(A) n B = 8 = A n clx(B).

2.21. Theorem. Let the universe be closed-monotone and n be a natural number. Suppose that Y is in U and P is a class of spaces. If Y is a subspace of a hereditarily normal space X , then P-Ind Y _< n implies (a) Y E P or (b) for each pair of separated subsets F and G of X satisfying

there is a partition S between F and G in X with P-Ind ( S n Y ) 5 n - 1. Moreover, if P is closed-monotone in U ,the condition (a) can be dropped.

Proof. The proof uses the one for Proposition 1.4.6 in the open subspace X \ (clx(F) f l clX(G)). Let us turn now to the promised statement of the agreement that is to hold for the remainder of the book. It should be apparent that the development of any viable theory that generalizes dimension theory will require that the universe be closed-monotone and that the classes P be, at the least, closed-monotone in the universe of discourse. Thus the following agreement will be made for the remainder of the book. With this agreement, reference to the universe will be suppressed unless there are additional assumptions to be made about the universe.

3. KERNELS AND SURPLUS

85

Agreement. For the remainder of the book, the following are assumed t o hold. The universe is closed-monotone. Every class is topologically invariant, has the empty space as a member and is closed-monotone in U . Unless indicated otherwise, X will always be a space in the universe. 3. Kernels and surplus

The concept of a kernel of a set has appeared in many areas of mathematics. As in the definitions of P-ind and P-Ind, our notion of kernel will be phrased in such a way as t o depend only on the class P, namely in the context of the class 7 of all topological spaces. 3.1. Definition. For a class P and a space X in 7 a subset G of X is called a P-kernel of X if G is in P.

The definition of the next function will be in the context of the universe of discourse. 3.2. Definition. Let P be a class of spaces. For a space X in the universe the P-Surplus of X is the extended integer

P-Sur X = min { Ind ( X

\ G) : G is a ( P n U)-kernel

of X }.

Note that the P-Surplus is undefined for spaces not in the universe. Also observe that X \ G is not necessarily in the universe when G is a (P n U)-kernel of X . The concept of surplus has already been encountered in Chapter I when the notion of g-compactness degree was introduced. It should be observed that the definition of P-Sur depends on the universe of discourse. This is in contrast to the definitions of P-ind and P-Ind as has been shown in Theorems 2.9 and 2.15. The definition is designed so as to yield the equality P-Sur X = (P n U)-Sur X . The preferred notation will be P-Sur. Obviously if the universe U O is contained in the universe U , then

(P n U)-Sur X 5 ( P n Uo)-Sur X

for X E U O .

Also the P-Surplus is undefined for spaces not in U O when we are dealing with the smaller universe. Thus one must be careful with the function P-Sur. To underscore these points, consider the following example.

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86

3.3. Example. Consider the class M of metrizable spaces and the class L of all locally compact spaces. Both classes are absolutely closed-monotone. So they are candidates for being the universe according to our agreement at the end of Section 2. For the absolutely monotone class P = { X : X is countable} we have two different values of P-SurR according to whether the universe is M or is L. Indeed, when U = M we have P-SurR = 0. But when U = C we have P-Sur R = 1 because each G in P n .C has isolated points by Baire’s theorem.

The next two propositions are obvious. 3.4. Proposition. For a closed subset F of a space X , the set F is a P-kernel of X if and only if it is a ( P n U)-kernel of X . 3.5.

P-Sur

Proposition. lf P and Q are classes with P 3 Q, then

5 &-Sur. In particular, P-Sur X 5 Ind X .

The next theorem shows the connection between P-Sur and P-Ind.

3.6. Theorem. For every hereditarily normal space X , P-Ind X

5 P-Sur X .

Proof. From Theorem 2.15 we have P-Ind = ( P n U ) - I n d . So the theorem will follow from the stronger inequality for the absolutely closed-monotone class P n U : ( P n U)-Ind 5 ( P fl U)-Sur. The proof of this inequality is by induction. Obviously the case where ( P n 24)-Sur X = -1 follows from the definitions. For the inductive step we assume that X is such that ( P fl U)-Sur X = n < 00. Let F and G be disjoint closed sets and let H be a ( P n U)-kernel of X with Ind ( X \ H ) 5 n. From Theorem 2.21 applied to the class { S } we have a partition S between F and G in X such that Ind (Sn( X \ H ) ) 5 n - 1. Since H is in P n U and S is closed in X , we have H n S E P nU by the absolute closed monotonicity of P fl U. So ( P n U)-Sur S 5 n - 1 holds. By the induction hypothesis we have ( P n U)-Ind S 5 n - 1. Thus ( P n U)-InL X 5 n has been shown. The following closed subspace proposition for P-Sur holds.

4. P-Ind A N D M A P P I N G S I N T O SPHERES

87

3.7. Proposition. For every closed subspace Y o f X ,

P-Sur Y

5 P-Sur X.

Proof. Let P-Sur X = n < 00. Then there is a ( P n 24)-kernel G of X with Ind ( X \ G) = n. Since Y n G is a closed subspace of G and G is in P n U ,we have that Y n G is in P n U.Because { 0 } is absolutely closed-monotone and Y \ G is closed in X \ G, we have Ind (Y \ G) 5 Ind (X \ G) = n. So, P-Sur Y 5 n. We shall close this section with an example that summarizes earlier calculations.

Example. The following have been computed in Chapter I. 0 = c m p R " _< K-IndR" 5 IC-SurR" 5 I n d R " = n. -1 = icdR" = IcdR" = C-SurR" 5 I n d R " = n. -1 5 c m p X 5 K-IndX 5 K-SurX 5 I n d X , X E Mo -1 5 i c d X 5 I c d X = C-IndX 5 I n d X , X EM. The inequalities involving the class of Section 6.

K

will be sharpened at the end

4. P-Ind and mappings into spheres

Agreement. In this section the universe will be contained in the class NH o f hereditarily normal spaces. This section is devoted to developing relationships between P-Ind and mappings into spheres. For this development we need some terminology concerning topological operations in subspaces. (Some of the terms have been defined already in Section 1.9.) The first half of the section will be devoted to a discussion of results not specifically related to dimension theory. 4.1. Terminology. In Section 1.9 we adopted some modulo notation to take care of certain set calculations that occurred in subspaces of a space. For example, the notation A C B mod F was used to mean that A \ F is a subset of B \ F . Loosely speaking, the notation says that A is a subset of B in the subspace X \ F . This modulo notation is a very convenient way of dealing with set

11. MAPPINGS INTO SPHERES

88

properties in the subspace X \ F . Clearly one can devise many others besides those that were introduced in Section 1.9, more than one can reasonably list. We will freely use this terminology. The reader should have no trouble in getting the meaning whenever we use it. 4.2. Propositions. The following two propositions concerning subspaces will be used repeatedly. Of course, these statements do not require that the spaces be hereditarily normal. The proofs will be left to the reader.

A. If F is a closed subset of a space X and A is a subset of X , then ClX\F(A \ F ) = clx(A) \ F.

B. Let F be a subset o f X and let V be an open subset ofX. Then

BX\F(V \ F ) c B x ( V >\ Fa Moreover,

BX\F(V \ F ) = B x ( V ) \ F when F is closed. 4.3. Proposition. Let H be a closed subset of a space X and let V be a finite open cover o f X modulo H . Then there is a closed collection F that shrinks V modulo H and that covers X modulo H . Moreover, it may be assumed that H is contained in each member of F. Proof. Since H is closed, the proposition is an immediate consequence of Lemma 1.8.8 applied to the space X \ H . 4.4. Proposition. Let H be a closed subset of a space X and let F = { Fi : i = 0 , 1 , . . .,n } be a collection o f closed sets. Then there is a collection W = { Wi : i = 0 , 1 , . . . , n } of open sets such that W is a swelling of F modulo H and such that the collection { cl (Wi): i = 0, 1, ...,n } is combinatorially equivalent to F modulo H . That is,

Fi C Wi

mod H ,

and for any finite set of indices

il,

i=O,l,

..., n,

. . . ,i,

Fi, n -4 Fim# 8 mod H if and only if c l ( W i l ) n , . . n c l ( W i , ) # 8 mod H .

4. P-Ind AND MAPPINGS INTO SPHERES

89

Moreover, if U = { Ui : i = 0,1,. . .,n } is an open swelling of F modulo H , then the collection { cl (Wi): i = 0,1,. . .,n } can be assumed to be a shrinking of U modulo H as well, i.e., cl(W;) C Ui mod H for all i. Proof. The proof follows the lines of that of Lemma 1.8.8. Beginning with the closed set Fo, one modifies the proof given there by considering all collections of indices il ,...,,i for which

Fo n Fi, n ... n Fim = 8 mod H Then, working in the subspace X \ H , one finds an open set Wo containing Fo \ H whose closure in the subspace is disjoint with each of the sets (Fi, n - - . n Fi,) \ H . The remaining details are similar. The final statement is obvious. Let us resume our discussion of P-Ind. We shall develop relationships between P-Ind and mappings into spheres that are similar t o those found in dimension theory. The natural ones are those concerned with unstable values of mappings and with extensions of mappings into n-spheres. Let us begin with the following theorem. It is the P-Ind analogue of Corollary 1.4.7. 4.5. Theorem. If P-Ind X 5 n , then for any n + l pairs ( F i , Gj), i = 0 , 1 , . . . ,n, ofdisjoint closed sets Fi and Gi there are partitions Sj between Fi and Gi in X such that Si : i = 0,1,. . . ,n } is a closed P-kernel of X .

n{

Proof. The proof is exactly the same as the one for Corollary 1.4.7 except that Theorem 2.21 replaces Proposition 1.4.6. The above theorem can be used to prove an “unstable value” theorem modulo a class P. To this end we make the following definition. 4.6. Definition. Let P be a class of spaces and let X be in 7 and Y be a metric space. For a continuous map f of X into Y a point y of Y is called an unstable value o f f modulo P if for each positive number E there is a closed P-kernel H of X and there is a continuous map g of X \ H into Y such that

d(f(z),g(z)) < E

and g(4

# Y,

5

E

x \ H.

A point y of Y is called a stable value off modulo P if it is not an unstable value of f modulo P .

11. MAPPINGS INTO SPHERES

90

4.7. Theorem. Let f be a continuous map of X into In+'. If P-Ind X 5 n , then each point of In+' is an unstable value of f

modulo P.

Proof. It is clear that every point of the boundary d II n+' of In+' is an unstable value of f modulo P. So only the interior points of In+' need to be considered. Also it is easily seen that the assertion for interior points will follow from the special case of the Let us prove this special case. The map f has origin 0 of In+'. coordinate maps fk of X into I, k = 0 , 1 , . . .,n. There is no loss in assuming SEJ< 1. For each k define the disjoint closed sets A k = { z : f k ( z ) > ~ } and B k = { z : f k ( z ) < - ~ } . By Theorem 4.5 there are partitions S k between Ak and Bk such that H = s k : k = 0 , 1 , . . . n } is a closed P-kernel of X . By Proposition 4.4 there is a collection W = { Wr, : k = 0 , 1 , . . . , n } of open sets such that s k C TVk mod H and n { d ( W k ) : k = 0 , 1 , ..., n } = 8 mod H . For each k let (Pk be a continuous map of X \ H into [ - E , E ] such that W k 3 { I : p k ( z ) = o}, Ak = { 2 : (Pk(z) = & } and B k = { z : pk(z) = -&}. Defining h k by h k ( 2 ) = fk(z) for z in Ak U B k and h k ( z ) = ( p k ( z ) for z not in Ak U B k , we get continuous maps h k of X \ H into 1. Let h be the map of X \ H into Hn+' whose coordinates are hk. Then llf(x) - h(z)l}5 2 & d mfor z in X \ H . One easily verifies 0 $ h[X \ H I .

n{

In the next lemma it will be convenient to use the supremum norm 11 in IWn+l.

lo

4.8. Lemma. Iff is a continuous map of X into In+' and the origin 0 of In+' is an unstable value o f f modulo P, then there is a closed P-kernel H of X and there is a continuous map g of X \ H into In+' such that

and

Proof. For E = 1/4 let H be the corresponding closed P-kernel of X and let ho : ( X \ H ) -+ be the corresponding continuous map such that

(l)

llf(z)- hO(z)llcc < 1/4

and

# 07

IlhO(~)lloo

5

E

x \ H.

4.

P-Ind AND MAPPINGS INTO SPHERES

91

Define M t o be the open set

Then f-l[8 (I"+')] \ H c M c X \ H . As X \ H is a normal space, there is a continuous function @ on X \ H into [0,1] with

Define h on X \ H by the formula h ( z ) = @(z)f(z)t [ l - @ ( z ) ] h o ( z ) . Then ( l ) ,(2), (3) and (4) will yield

We obtain the required map g : ( X \ H ) -, d (I "+I) by composing h with the the radial projection of In+' \ { 0 } onto O(I"+l). The next theorem concerns a sufficient condition for the existence of extensions of continuous maps into spheres. In later sections (Sections 5 and 6) we shall define the covering dimension P-dim and show that a corresponding extension theorem for P-dim characterizes the function P-dim. In this way we will be able to show that P-dim is smaller than P-Ind. 4.9, Theorem. Let n be a natural number. If P-Ind X 5 n, then for every closed set C of X and every continuous map f of C into 8" there exists a closed P-kernel H of X with H C X \ C such that f can be extended to X \ H.

Proof. Let f be a continuous map of C into S". We can regard 8" as the boundary of I"+'. Since 8" is an absolute neighborhood retract, the map f has a continuous extension to a n open neighborhood U of C. We shall also call this extension f. Let V be a open

set such that C c V c cl ( V )c U . Since X \ V is closed, we have by Theorem 2.16 that P-Ind (X \ V ) 5 n holds. Consider the continuous map flB ( V )on the closed subset B ( V )of X \ V . As In+' is an absolute retract, we may assume that flB ( V )has been extended to a continuous map f of X \ V into 1"+'. We have from Theorem 4.7

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11. MAPPINGS INTO SPHERES

that 0 is an unstable value of this extended map f modulo P . By Lemma 4.8 there is a closed P-kernel H of X \ V and a continuous map g of ( X \ V )\ H into I"+' such that

Clearly, H c X \C. The required extension is now easily constructed since f and g agree on the closed subset B ( V ) \ H of the space

X\H. As an application of the last theorem, we shall prove an extension of a theorem that was first proved by Eilenberg in [1936].

4.10. Theorem. Suppose C is a closed subset of a space X with Ind ( X \ C) 5 n and k is a natural number with Ic 5 n. Then for each continuous map f of C into S k there exists a closed set H contained in X \ C with Ind H < n - k such that f can be extended over X \ H .

Proof. According to our agreement, the universe of discourse will naturally be the class NH of hereditarily normal spaces. Let P be the class of all spaces Y with IndY 5 n - k - 1. Clearly the class P is absolutely closed-monotone. Moreover, we have the equivalence Ind Y 5 n m - k if and only if P-Ind Y 5 m. Now Theorem 4.9 can be applied because P-Ind ( X \ C) 5 k .

+

Theorem 4.9 can be used to compute lower bounds for P-Ind as the next examples show.

4.11. Examples. The examples will use product spaces X x Y and the natural projection T : X x Y -+ X . The map A will be perfect when Y is compact, that is, a closed map with compact point inverses. a. For every noncompact space X , IC-Ind ( X x I") 2 n. To see this we suppose that the inequality fails and consider the closed set C = X x d I and the continuous map f : C + d I" determined by the natural projection 7r of X x I" onto I". Since S"-I is

5 . COVERING DIMENSIONS MODULO A CLASS

P

93

homeomorphic to d I n , by Theorem 4.9 we have a closed K-kernel H with H C (X x 1.) \ C and a continuous extension g of f from the set C to the set ( X x I") \ H . As X is not compact, X \ n [ H ]# 0. So for some point c in X we have ({ c } x I") fl H = 0. The extension g is defined on the set { c } x 1" and can be regarded as the identity map on the set d H ". The composition of g and the reflection of d1" in the origin defines a map of I" into itself without fixed ponts. A contradiction to the Brouwer fixed-point theorem has occurred. b. Recall that S is the class of all a-compact spaces. Since the continuous image of a a-compact space is again a-compact, we have by the same argument as in the above example that S-Ind(X x In) 2 n for every non-a-compact space X . c. For the class C of complete metrizable spaces we claim that C-Ind ( X x I n ) 2 n

for every noncompletely metrizable space X . The proof of this claim follows the same lines as that of the first example. This time the set H is a closed C-kernel of X x I n . As n is a perfect map, we also have that T I H is a perfect map. From Henriksen and Isbell [1958] and Cech [1937] we infer that n [ H ]is in C. Consequently, X \ n [ H ]# 0. The remainder of the proof is the same as in the other two examples. 5. Covering dimensions modulo a class

P

The covering completeness degree was introduced in Section 1.9 for the class C of complete metrizable spaces. Its definition was a natural extension of the definition of the covering dimension dim of dimension theory. We shall now introduce the extension for an arbitrary class P. Our discussion will be focused on generalizing the following characterization of dim due to Morita [195Oa]. 5.1. Theorem. Let X be a normal space and let n be a natural number. Then dim X 5 n if and only if for every locally finite collection { U , : (Y E A } of open sets and every collection { Fa : a E A } of closed sets such that Fa c U , for all a in A there exist collections { W , : a E A } and { V, : (Y E A } of open sets satisfying the

11. MAPPINGS INTO SPHERES

94

inclusions Fa C V, C cl(V,) C W , C cl(W,) ord(cl(W,) \ V, : cy E A } 5 n.

c U,

for cy in A and

Just as the functions P-ind and P-Ind of inductive dimensional types were defined for all topological spaces, the functions of covering dimensional types will also be defined for all topological spaces. We shall begin with the natural extension of the notion of border cover given in Chapter I for the class C. (The reader is reminded that the empty space is a member of every class by the agreement made at the end of Section 2.) 5.2. Definition. Let P be a class of spaces and X be in 7 .An open collection V in X is called a P-border cover of X if X \ U V is in P ; the set X \ U V is called the enclosure of V.

It is clear that the enclosure of a P-border cover of X is a closed P-kernel of X . Using P-border covers, we now define two functions of covering dimensional types. Definition. Let P be a class of spaces and X be a space in 7. Then the small covering dimension of X modulo the class P , denoted P-dim X , is defined as follows. (i) P - d i m X = -1 if and only if X is in P . (ij) For each natural number n, P - d i m X 5 n if every finite P-border cover V of X has a P-border cover refinement of order less than or equal to n -t 1. 5.3.

P be a class of spaces and X be a space in 7. Then the large covering dimension of X modulo the class P , 5.4. Definition. Let

denoted P-Dim X , is defined as follows. (i) P - D i m X = -1 if and only if X is in P. (ij) For each natural number n, P-Dim X 5 n if every P-border cover V of X that is locally finite in the subspace U V has a P-border cover refinement of order less than or equal t o n 1.

+

The discussion of this section will be concentrated on P-dim. So we shall be dealing only with finite P-border covers. The definition of P-Dim uses a local finiteness condition rather than a n arbitrary one. (Obviously these two conditions are equivalent for hereditarily paracompact spaces.) The reason for this bias is Theorem 5.1 which has locally finite collections in its statement. The function P-Dim will be discussed in detail in Chapter IV.

5 . COVERING DIMENSIONS MODULO A CLASS

P

95

Propositions. The following three propositions concern the class 7 of all topological spaces. The proofs of the first two are easy. For the third, see the proof of Proposition 1.8.6. 5.5.

A. P-dim 5 P-Dim.

B. If P and Q are classes with P 3 Q, then P-dim 5 Q-dim and P-Dim 5 Q-Dim. In particular, P-dim 5 dim and P-Dim 5 Dim. C . Let n E N. For classes P and spaces X in 7, P-dim X 5 n if and only if for each P-border cover U = { Ui : i = 0,1, .. .,k } of X there is a P-border cover V = { : i = 0 , 1 , . ..,Ic } with enclosure G such that V shrinks U modulo G and ord V 5 n 1. The corresponding equivalent formulation exists for P-Dim.

-+

We shall prove next a closed monotonicity theorem for P-dim. (Remember the agreement that was made at the end of Section 2). 5.6. Theorem. For every closed subspace Y of a space X ,

P-dim Y

5 P-dim X .

Proof. Let P-dim X = n < 00 and let U be a finite P-border cover of Y with enclosure F. Since Y is closed, the finite open collection W = { ( X \ Y) U U : U E U } is a P-border cover of X with enclosure F . There is a P-border cover W' of X with enclosure G' such that W' refines W and ord W' 5 n 1. Since P is closedmonotone in U ,it is easily shown that V = { Y n W' : W' E W' } is a P-border cover of Y with enclosure Y n G'. Consequently we have P-dimY 5 n.

-+

The following theorem characterizes P-dim. 5.7. Theorem. Let n be a natural number. The following con-

ditions are equivalent for a space X . (a) P - d i m X 5 n. (b) For each P-border cover U = { Ui : i = 0 , 1 , . . . , n 1 } of X with exactly n + 2 indices there exists a P-border cover V = { K : i = 0,1, ...,n + l } such that o r d V 5 n + 1 and V shrinks U modulo G where G is the enclosure of V.

-+

Proof. That (a) implies (b) is obvious from Proposition 5.5.C. So we shall prove (b) implies (a). It will be easier to prove the

11. MAPPINGS INTO SPHERES

96

contrapositive. Suppose that the negation of condition (a) holds. We claim that there is a P-border cover 2 = { Zi : i = 0 , 1 , . , .,k } of X such that ord 2 2 n 2 is true and such that each ?-border cover V = { V, : i = 0 , 1 , . . . , k } that shrinks 2 modulo G is combinatorially equivalent to 2 modulo G, where G is the enclosure of V. By combinatorially equivalent modulo G we mean that for any increasing sequence il, &,.-..,,i of natural numbers not exceeding k the following condition holds.

+

VElnv,,n...n~,#O (1)

modG

if and only if

Ziln Zi,n + -. n Zim# 0 mod G. Indeed, from the negation of (a) and Proposition 5.5.C there is a P-border cover Z = { Zi : i = O , l , ...,k } such that ord V > n + 2 for every P-border cover V = { V , : i = 0, 1,...,k } that shrinks 2 modulo G, where G is the enclosure of V. If 2 does not satisfy condition (l),one just replaces the P-border cover 2 with the P-border cover { V, : i = 0,1,. .. ,k } and continues this process until a P-border cover with the required property is obtained; as the number of subsets of { 0,1,. . . ,k } is finite, the process will come to an end after finitely many such steps. Since n 2 5 ord 2, rearranging if need be the members of 2, we have

+

(2)

n{Zi: i = 0,1, . . . ,n + 1} # 0.

We shall show that the P-border cover U whose elements are Ui = Zi for i = 0,1,. . .,n and Un+l = U{ Zi : i = n 1,.. .,k } has the property that every P-border cover V with n 2 elements that shrinks U modulo G, where G is the enclosure of V, has ord Vexceeding n 2; that is, condition (b) fails. To this end, consider a P-border cover V = { V ; : i = O , l , ...,n + l } such that Vshrinks U m o d u l o G, where G is the enclosure of V. Then the open collection

+

+

{Vo,vl,...,Vn,Vntl nzn+r,Vn+l n z n t 2 , . . . , V n t l

+

n&}

is a P-border cover with enclosure G that shrinks 2 modulo G. By conditions (1) and (2) we have

n V 3 (n{Vj : i = 0 , l,...,n }) n (Vn+ln Zn+l) #0

mod G .

5.

COVERING DIMENSIONS MODULO A CLASS P '

97

Thus we have completed the proof of (a) implies (b). The following corollary is an immediate consequence of the last theorem. 5.8. Corollary. For a normal space X , dim X 5 0 if and only if I n d X 5 0.

We now prove a second characterization theorem. This theorem will yield a generalization of the finite collection version of Theorem 5.1. 5.9. Theorem. Let n be a natural number. The following conditions are equivalent for hereditarily normal spaces X in the universe. (a) P-dim X 5 n. (b) For each closed P-kernel G of X and for each pair of open and closed collections U = { Ui : i = 0 , 1 , . . . ,k } and F = { Fi : i = 0,1, ..., k } of X such that Fi c Ui mod G holds for each i there exists a closed %'-kernel H containing G and there exist open collections V = { Vj : i = 0,1,. . . ,k } and W = { Wi : i = O , l , ...,k } such that

Fi \

Wi c U i , cl(V,) C Wi mod H , cI(Wi) C Ui mod H ,

(b.1)

C Vj C

...,k, i = o , l , ...,Ic, i = o , l , ..., k, i =O,l,

and (b.2)

ord { Wi \ cl(Vi) : i = O , l , .

. ., k } 5 n

mod H .

(c) For each closed %'-kernel G of X and for each collection ofn 1pairs ( A i , Bi), i = 0, 1,. . . ,n, of closed sets Ai and Bi that are disjoint modulo G there is a closed %'-kernel H containing G such that Si : i = 0,1,. . .,n } = H where for each i the set Si is some partition between Ai and Bi modulo H .

+

n{

Proof. The proof of (a) implies (b) is verbatim the same as that for Lemmas 1.8.9 and 1.9.4. Let us now prove that (b) implies (c). Let G be a closed P-kernel and let (Ao,Bo), ...,( A n ,B,) be pairs of closed sets Ai and Bi such

98

11. MAPPINGS INTO SPHERES

that Ai n Bi = 0 mod G. For each i the sets Ui = X \ ( A i U G) and F’ = B; are respectively open and closed and are such that F; c Ui mod G. By (b) there exists a closed P-kernel H containing G and there exist two open collections V = { V, : i = 0,1,. . . ,n } and W = { Wi : i = 0,1,. . . ,n } such that formulas (b.1) and (b.2) hold. By (b.1) the closed sets cl(V;.) and X \ W; are disjoint modulo H . Since X is hereditarily normal, there is a partition 5’: between cl (Vi) and X \ Wi modulo H . Clearly S, = S: u H is a partition between Ai and Bi modulo H . From (b.2) we have the equality H = Si: i = 0,1,. .,n }. Finally let us prove (c) implies (a). This will be done by means of Theorem 5.7. Consider an n + 2 element P-border cover U = { Ui : i = 0,1,. . .,n 1 } of X and denote its enclosure by G. In the subspace X \ G let { Fj : i = 0,1,. ..,n 1} be a closed shrinking of U . The collection F = { Fi U G : i = 0 , 1,. . .,n 1} is a closed shrinking of U modulo G. Let Ai = Fi U G and Bi = X \ Ui for each i. Then A; n Bi = 0 mod G. By (c) there exists a closed collection { Si : i = 0 , 1 , . . .,n } and there exists a closed P-kernel H such that Si is a partition between A; and B, modulo H , G C H a n d n ( S i : i = O , l , ...,n } = H . SoletVi and Wi bedisjoint open subsets of X \ H such that A; \ H c V,, B; \ H c Wi and Si \ H = ( X \ H ) \ ( 6U W;). Obviously V , and Wi are open in X as well. Let V = { K : i = 0,1, ..., n } and W = { Wi : i = 0,1, ...,n } . Observethat ( U V ) u ( U W ) = U { K u W i : i = O , l , ..., n } = X \ H holds because 5’;: i = 0, 1, . .,n } = H . Then the inclusion

n{

.

+

n{

+

+

.

results from Ai \ H C Vi for i = 0,1,. ..,n and An+l \ H C Un+l. So 2 = { Zi : i = 0 , 1 , . . . , n + l}, where Zi = for i = O , l , . . . , n and Zn+l = Un+t n (U W ) , is a P-border cover with enclosure H . Finally

and therefore ord 2 5 n -t 1. Thus (a) holds by Theorem 5.7.

A generalization of the finite collection version of Theorem 5.1 is next.

5. COVERING DIMENSIONS MODULO A CLASS P '

99

5.10. Theorem. Let n be a natural number and let X be a hereditarily normal space in the universe. Then P-dim X 5 n if and only if for each closed P-kernel G of X and for each open collection { U; : i = 0 , 1 , . . .)k} and each closed collection { F; : i = 0 , 1 , . . .,k } such that F; C Ui mod G holds for every i there exists a closed P-kernel H and there exist open collections { W; : i = 0,1, . . . k } and { : i = 0, 1,. . .,k } such that G c H and )

V , C W; C U;, cl(V,) C W; mod H , cl (Wi)C U; mod H , F'\H

(*>

C

i = O,l,.. .,k, i = O , 1 ) . . . )k ,

i=O,l,

...,k,

and

(**)

ord{cI(Wi)\K:i=O,l

)...)k } < n m o d H .

Proof. Only the condition (**) of this theorem differs from condition (b.2) of Theorem 5.9. Suppose P-dim X n holds. Then the condition (b) of Theorem 5.9 and the hereditary normality of X will yield the required closed P-kernel H and the open collections of the theorem. Conversely, the required closed P-kernel H and partitions Si of condition (c) of Theorem 5.9 are easily constructed from the open collections of the theorem because of the condition (**).

<

Remark. The above proofs illustrate the naturalness of the definition of covering dimension modulo a class P . When P is { 0}, the theorems reduce to those of the covering dimension dim. The only unfortunate aspects of the proofs are that the theorems for dim hold in the universe N of normal spaces while some of the proofs here have the additional requirement of hereditary normality. But the full force of hereditary normality is not needed and the following definition may provide a way out of this unfortunate situation. For a class P the universe N [PI,called the normal universe modulo P , is defined as

N [PI = { X : X \ G is normal for each

closed P-kernel G of X }.

Observe that the space X \ G in the definition is not assumed t o be in the universe N [ P ] .The above proofs can be carried out for

100

11. MAPPINGS INTO SPHERES

each X in N [PI.Clearly N [{ 0 }] is the class of normal spaces. In this way, natural generalizations of dim will result. Indeed, observe the following inclusions:

The normal universe modulo P can be used in many other places also. We shall not concern ourselves with this more general approach. 6. P-dim and mappings into spheres

Agreement. The universe is contained in N;i. The function P-dim will be shown to be characterized by means of mappings into spheres just as dim is. The fact that such a characterization theorem exists is no surprise because of Theorem 5.9. Our discussion will begin with unstable maps. Unstable maps were already encountered in Section 4 where connections with P-Ind were made. There the continuous maps f were defined on the whole space X . But for dim we shall see that condition (c) of Theorem 5.9 will introduce a closed P-kernel G of X such that the maps f are continuous on X \ G. The following is a variation of Definition 4.6. 6.1. Definition. Let P be a class of spaces and let f be a map (not necessarily continuous) from a space X into a metric space Y . A point y of Y is called a P-unstable value o f f if for each positive number E and for each closed P-kernel G of X for which f l ( X \ G) is continuous there exists a closed P-kernel H and there exists a continuous map g of X \ H into Y such that H 3 G and

A point y of Y is called a P-stable value of f if it is not a P-unstable value of

f.

Observe that the notion of P-unstable is stronger than that of unstable modulo P of Definition 4.6. Similar to P-Ind we have the following unstable value theorem (compare with Theorem 4.7).

6. P-dim AND MAPPINGS INTO SPHERES

6.2. Theorem. Let f be a map o f X into In+'. IfP-dimX then each point of In+' is a P-unstable value o f f .

101

5 n,

Proof. It is clear that every point of the boundary d I n+l of 1 n+' is a P-unstable value of f. So only the interior points of In+l need to be considered. Also it is easily seen that the assertion for interior points will follow from the special case of the origin 0 of In+'. Let us prove this special case. The proof will follow the lines of that for Theorem 4.7. Let G be a closed P-kernel of X such that f l ( X \ G) is continuous. Denote the coordinate maps o f f by fi. Then the maps f i l ( X \ G) of X \ G into I, i = 0 , 1 , . . .,n, are continuous. There is no loss in assuming that 6~4< 1. For each i we define two closed sets A; = cl({ x : x 4 G, f i ( x ) 2 E } ) and Bi = cl({ x : 2 4 G, fi(x) 5 - E } ) . The closed sets A; and B; are disjoint modulo G because the restriction of f to X \ G is continuous. By condition (c) of Theorem 5.9 there is a closed P-kernel H and there are partitions Si between A; and Bi modulo H , i = 0,1,. . .,n, such that H = Si : i = 0,1,. . .,n } . The remainder of the proof is the same as that of Theorem 4.7 where X is replaced by X \ G and H is replaced by H \ G.

n{

There is the following interpolation lemma for P-dim that is analogous to Lemma 4.8 for P-Ind. Its proof is a simple modification of the one given in that lemma. 6.3. Lemma. let f be a map (not necessarily continuous) of a space X into In+' and let G be a closed P-kernel of X such that fl(X \ G) is continuous. If the origin 0 of In+' is a P-unstable value o f f , then there is a closed P-kernel H of X and there is a continuous map g of X \ H into In+' such that H I)G,

The P-dim analogue of Theorem 4.9 will be labeled as a lemma since its converse will be shown to be true also. 6.4. Lemma. Let n be a natural number and let G be a closed P-kernel of a space X . If P-dim X 5 n and C is a closed set of X , then for every continuous map f of C \ G into 8" there exists a

102

11. MAPPINGS INTO SPHERES

closed P-kernel H with G c H and H a continuous extension over X \ H .

n (C \ G) = 0 such that f

has

Proof. Consider a continuous map f of C \ G into 8". We may extend f to all of C in an arbitrary manner since we are concerned with f being continuous only on C \ G. We shall identify 8" with the boundary d (1n + l ) . Since 8" is an absolute neighborhood retract, we can extend f I(C \ G) to a continuous map of an open neighborhood U of C \ G in the subspace X \ G. Clearly U is open in X . Let V be an open subset of X \ G such that cl ( V )C U mod G and C \ G C V . Then B ( V )U G is a closed subset of X \ V and f restricted to B ( V )\ G is continuous. We have P-dim ( X \ V ) 5 n by Theorem 5.6. Similar t o the proof of Theorem 4.9,by Lemmas 6.2 and 6.3 there is a closed P-kernel H of X \ V and there is a continuous map g of (X \ V )\ H into 8" such that G C H and g and f agree on B ( V )\ H . Obviously H is a closed P-kernel of X . The extension of f I(C \ H ) can now be constructed in a manner similar to that in Theorem 4.9. 6.5. Lemma. Let n be a natural number. Suppose X is a space that satisfies the following property: For each closed set C , for each closed P-kernel G and for each continuous map f of C \ G into 8" there is a closed P-kernel H with G C H and H n (C \ G) = 0 such that f has a continuous extension over X \ H . Then P-dim X 5 n.

Proof. We shall show that condition (c) of Theorem 5.9 holds for X. For i = 0 , 1 , . . . ,n let ( A i , Bi) be a pair of closed sets and G be a closed P-kernel such that Ai and Bi are disjoint modulo G and then select continuous maps fi : X \ G + I with f;' [- I] 3 A; \ G and fcl[l]3 Bi \ G. Identify the boundary of I["+* and 8". Since f = (fo, f 1 , . . . ,f n ) is a continuous map of X \ G into I[ "+', we have that C = f - ' [ S n ] U G is closed and C \ G = f - ' [ $ " ] holds. From the property given in the statement of the lemma there is a closed P-kernel H with H 3 G and H n f [ P I = 0 and there is a continuous extension g = (go,g1,. . ., g n ) of fl(C \ H ) to X \ H . Clearly the set Si = r3:'[0] U H is a partition between Ai and B; modulo H for every i and Si : i = 0, 1, . . .,n } = H , We have shown that condition (c) of Theorem 5.9 holds. Consequently, P - d i m X 5 n.

-'

n{

Combining Lemmas 6.4 and 6.5, we have our characterization theorem.

6. 'P-dim A N D MAPPINGS INTO SPHERES

103

6.6. Theorem. Let n be a natural number. For every class P and every space X , P-dim X 5 n if and only if for each closed set C of X , for each closed P-kernel G of X and for each continuous map f of C \ G into 8" there exists a closed P-kernel H of X with G C H and H n (C \ G) = 0 such that f can be continuously extended over X \ H .

6.7. Example. In Example 4.11 the lower bound n for K-Ind, S-Ind and C-Ind of the space X x I" was computed for spaces X that were noncompact, non-a-compact and, in the case of metrizable spaces, noncomplete respectively. This lower bound was established by means of Theorem 4.9 which relates P-Ind and extensions of mappings into $". With the aid of Theorem 6.6, a straightforward modification of these proofs will yield the lower bounds

K-dim(X x In) 2 n, S - d i m ( X x 1") 2 n,

C-dim(X x

I") 2 n

when X is respectively noncompact, non-a-compact and, in the case of metrizable spaces, noncomplete. Let us now consider the universe U = Mo of separable metrizable spaces and the class K of compact spaces. The next proposition will connect K-Dim and K-Sur. To do this we shall have need of the following lemma connecting Dim and Ind. 6.8. Lemma. For every separable rnetrizable space X ,

dim X = Dim X = ind X = Ind X .

Proof. In view of Theorem 1.8.10 and Proposition 5.5.A we only need to prove Dim X 5 Ind X . The proof is by induction on Ind X . We shall give only the inductive step. Suppose I n d X <_ n and let U be locally finite open cover of X. As X is a separable metrizable space, we have that U is also a countable collection. Indexing U as { U; : i = 0, 1, . . . }, we let F = { F; : i = 0 , 1 , . . .} be a closed cover that shrinks U . Then for each pair (Fi,U;) there is a partition S; between Fi and X \ Ui such that Ind S; 5 n - 1. Let V , and Wi be disjoint open sets with Fi c V,, X \ Ui c Wi and X \ Si = V , U Wi. As Si C Ui holds for each i, { Si : i = 0,1,. . . } is locally finite in X. So the set S = U{ S; : 0 , 1 , . . .} is closed. By the sum theorem for Ind the inequailty Ind S 5 n - 1 holds. From the induction hypothesis

104

11. MAPPINGS INTO SPHERES

we have a collection 2' = { 2: : i = 0,1,. . . } with ord 2' 5 n such that each of its members 2: are open sets of S with 2: c Ui. There is an open collection 2 = { Zi : i = 0, 1, . . .} of X such that 2: = Zi n S for each i and ord 2 = ord 2'. Next let Vd = Vo \ S and for i 2 1 let V,' = Vi \ U{ (Vj U S) : j < i}. Then the open collection V = { V,' : i = 0 , 1 , . . .} satisfies ord V 5 1 and U V = X \ S. So the collection VU 2 satisfies ord( VU 2) 5 n 1 and refines U . Thus we have DimX _< n.

+

6.9. Proposition. For every separable metrizable space X ,

K-dimX 5 K-DimX

5 K-SurX.

Proof. Only the right inequality needs proof. Let K-SurX 5 n. Then there is a K-kernel G of X such that Ind (X \ G) 5 n. Let U be a K-border cover of X with enclosure F such that U is locally finite in X \ F . Then U is locally finite in X \ H , where H = F U G, and Ind (X \ H ) 5 n. By Lemma 6.8 there is an open collection Vwhose union is X \ H such that V refines U and ord V 5 n 1. Since H is in K , we have IC-DimX 5 n.

+

The above proposition will be sharpened to equalities in Theorem 6.11. (See Proposition 7.4 for the inequalities corresponding to more general classes P.) To do this we shall now use the characterization of d i m X n to calculate lower bounds for IC-IndX and K-dim X . We first prove a lemma.

<

6.10. Lemma. Let n 2 1. Suppose that X is a separable metrizable space and { xk : k = 0 , 1 , . . .} is a sequence of closed subsets

o f X such that (a) {xi,: k = 0, 1,. ..} is a discrete collection in X , (b) d i m x k 2 n fork 2 0. Then there exists a closed set C and a continuous map f of C into Sn-I such that for any compact set H the set C \ H is nonempty and the map fl(C \ H) cannot be extended to a continuous map of X \ H into Sn-l.

ck

Proof. For each k there is a closed subset of XI, such that some continuous map fk of c k into 8"-' does not have a continuous extension over xk. The collection { ck : k = 0,1,. . .} is discrete in X because of condition (a). So C = U{ Ck : k = 0, l,...}

6. P-dim AND MAPPINGS INTO SPHERES

105

is closed in X. The sequence of functions fk, k = 0, l , . . . , will define a continuous function f on C in the obvious way. The collection { XI, : k = 0,1,. . .} is an open cover of the closed subspace F = U{ Xk : k = 0 , 1 , . . .} of X . So the compact set H n F is contained in XI, : 0 _< k < k~ } for some natural number k ~ There. fore if fl(C \ H ) h+s a continuous extension to X \ H then fk,, has a continuous extension to X k 0 . This is a contradiction.

u{

The next theorem is intimately connected to the conjecture of de Groot (1.5.6). Many topological invariants were invented in the course of the development of the theory of dimension modulo a class of spaces. The theorem shows that several of these invariants for the class K actually coincide in the universe Mo of separable metrizable spaces, the universe that is most relevant to de Groot's conjecture. Indeed, the theorem rules out many candidates for an internal characterization of the compactness deficiency.

6.11. Theorem. For spaces X in Mo, K-Ind X = K-SurX = K-dimX = K-DimX.

Proof. The equalities are obvious for X in K ; and Theorem 3.6 and Proposition 6.9 yield the case where K-SurX = 0. Because of Theorem 3.6 and Proposition 6.9, we only need t o prove the inequalities K-Sur X 5 K-Ind X and K-Sur X 5 K-dim X . So assume n 2 1 and K-Sur X 2 n. Then dim ( X \ H ) 2 n for every K-kernel H of X . Let us use this property t o establish the existence of a closed set C and a continuous map f from C into Sn-' such that for any compact set H the set C \ H is nonempty and the map f l ( C \ H ) cannot be extended to X \ H . Let A be the set of points p of X such that every neighborhood U of p has dim U 2 n. Then q 4 A implies that some neighborhood U of q has dim U < n. So A is closed in X . Moreover, dim ( X \ A ) < n. To verify this we observe that each point of X \ A has a neighborhood U, such that cl (U,) C X \ A and dim cl (U,) < n. From the Lindelof property there is a countable collection of such sets cl(U,) that covers X \ A . Thus dim ( X \ A ) < n holds by the countable sum theorem of dimension. Let us show next that A \ H # 0 for every H in K . Suppose that H 2 A for some H in K . Then n 5 K - S u r X 5 dim ( X \ H ) 5 dim ( X \ A ) < n, a contradiction. Hence the closed

11. MAPPINGS INTO SPHERES

106

set A is not compact. So there is a sequence p k , k = 0,1,. . . , in A that has no limit point in X . For each k let X I , be the closure of a neighborhood U p , of pk so that {XI, : k = 0 , 1 , . . . } is a discrete collection in X . Clearly d i m x k 2 n for all k. Lemma 6.10 yields the existence of the required closed set C and continuous map f. The proof of the lemma will now be completed by means of contradictions. First assume K-Ind X n - 1. Since n - 1 2 . 0 , by Theorem 4.9 there is a closed K-kernel H of X such that the continuous map f of C into Sn-* has a continuous extension to X \ H and H n C = 0. This is a contradiction since such an extension cannot exist. Finally assume K-dim X n - 1. Let G be a closed K-kernel of X. Then by Lemma 6.4 there exists a closed K-kernel H with H n (C \ G) = 8 and G c H and there exists a continuous extension g of f to X \ H . Then, as in the first case, a contradiction will result.

<

<

In the example at the end of Section 3, a summary was made of various functions modulo the class K. A sharpening of some of the inequalities presented there was promised to appear a t the end of this section. We have the following examples. 6.12. Examples. A straightforward application of the above theorem yields the first example.

a. For each positive integer n,

b. Suppose 1 5 n < m and let X be the disjoint topological sum of R " and I". Then 0 = cmpX

< n = K-IndX = K-SurX < m = I n d X .

7. Comparison of P-Ind and P-dim

Agreement. The universe is contained in NH. The comparison of Ind and dim for normal spaces can be nicely made by means of mappings into spheres. The same comparison will be shown to be true for P-Ind and P-dim under suitable restrictions imposed on the space X. These restrictions occur because

7. COMPARISON OF P-Ind A N D P-dim

107

in general the function P-Ind lacks certain properties. One can assure that P-Ind will have these properties by imposing conditions that will be discussed in Chapter 111; the conditions are related to additive and monotone properties of normal families P for spaces that are in the Dowker universe 27. So for now we shall procede by including these restrictions in the hypotheses of our propositions. 7.1. Proposition. Let X be a space that satisfies the two condi tions: (a) For each closed P-kernel G of X it is true that X \ G is a member of the universe and P-Ind ( X \ G) 5 P-Ind X. (b) For each closed P-kernel G of X and each ( P r l U)-kernel H of X such that G.U H is closed it is true that G U H is a P-kernel of X . Then P - d i m X 5 P-IndX.

In particular, for every X in NH dim X

5 Ind X,

Proof. Suppose n is a natural number and P - I n d X 5 n. Let G be a closed P-kernel of X and C be a closed set. Then by (a) we have P-Ind ( X \ G) 5 P-Ind X 5 n and X \ G is in U. For each continuous map f of C \ G into 8" there is by Theorem 4.9 a closed P-kernel Ho of the subspace X \ G such that Ho n (C \ G) = 0 and f has a continuous extension to ( X \ G) \ Ho. Clearly H = G U Ho is closed in X . So we have by (b) that H is a closed P-kernel of X . Since G C H , we have P-dim X 5 n by Theorem 6.6. Observe that the condition (a) of the proposition fails for the universe Mo of separable metrizable spaces and the class X: of compact spaces. Also when the universe is NH and the class P is { S}, the inequality dim X 5 Ind X will result. (The discussion of the coincidence of dim and Ind will be completed in Section V.3.) With this last observation in mind we make the following definition. 7.2. Definition, Let P be a class of spaces. For a space X in the universe the P-surplus o f X is the extended integer P-sur X = min {dim (X \ G) : G is a ( P n 24)-kernel of X}.

11. MAPPINGS INTO SPHERES

108

Note that the P-surplus is undefined for spaces not in the universe. See also the additional remarks following Definition 3.2. The definition has been made so as to yield

P-sur X = ( P fl U)-sur X for each X in the universe. The next proposition explains the choice of notations in Definition 3.2 and the above definition. 7.3. Proposition. For every space X ,

P-surX 5 P - S u r X . 7.4. Proposition. Let X be a space that satisfies the two conditions: (a) dim 2 5 dim Y whenever 2 c Y c X . (b) For each closed P-kernel G of X and each (Pn U)-kernel H of X such that G U H is closed i t is true that G U H is a P-kernel of X .

Then P-dim X

5 P-sur X .

Proof. Suppose n is a natural number and P-surX 5 n. Then there is a ( P fl U)-kernel Ho such that dim ( X \ H o ) 5 n. Let U = { Ui : i = 0,1,. . , , k } be a finite P-border cover of X with enclosure Go and let Y = X \ (Go U Ho). Then dimY 5 n holds in view of (a). Let 2 be X \ Go = U U and let U' = { U: : i = 0,1,. . . , k } be an open shrinking of U in the space. 2 such that clz(U,!) C Vi for each i. By Propositions 4.3 and 5.5.C)there exists in the subspace Y aclosed c o v e r F = { F i : i = O , l , ...,k } o f Y suchthat F i c U , ! f o r each i and o r d F 5 n f 1. For each i , clz(Fi) c clz(U,'). As F' = { clz(Fi) : i = 0 , 1 , . . .,k} is finite, E = { z E 2 : o r d , F ' 2 n + 2 ) is closed in 2. Also, E n Y = 0. Let Y' = U F ' \ E . Then Y c Y' holds and Y ' is closed in 2 \ E . By Proposition 4.4 there is an open swelling W = { W i : i = O , l ,...,k } in Z \ E of the collection F' with ord W 5 n 1 and Wi c Ui for all i. The set L = X \ U W is contained in Go U Ho. Since the sets G = L n Go and H = L n H o fulfill the requirements of (b), it follows that W is a P-border cover of X . Therefore, P-dim X 5 n.

+

8. HULLS AND DEFICIENCY

109

8. Hulls and deficiency

Agreement. The universe is contained N H . Early discoveries concerning rim-compact spaces lead t o the concept of the deficiency of a space. This concept is associated with the structure of supersets, or hulls, of a space. Hulls and deficiency will be generalized from the class K: to the the class P . By working in the context of the universe 7 of all topological spaces, our notion of a hull will depend only on the class P. 8.1. Definition. Let

P be a class of spaces and X be

in 7. A space Y is called a P-hull of X if X

cY

a space

and Y E P.

The definition of the next functions will be in the context of the universe of discourse. 8.2. Definition. Let P be a class of spaces. For a space X in the universe the small P-deficiency o f X is the extended integer

P-def X = min { dim (G \ X ) : G is a ( P n U)-hull of X } and the large P-deficiency of X is the extended integer P-DefX = min { Ind(G \ X) : G is a ( P n U)-hull of X } . The functions P-def and P-Def are undefined for spaces that are not in the universe in spite of the fact that the defining formulas may be well defined for spaces not in the universe. But the function values are CXI when no ( P n U)-hull of X exists. Obviously P-defX = (P n U)-defX and

P-DefX = ( P n U)-DefX.

Similar t o P-sur and P-Sur is the following. 8.3. Proposition. For every space X,

P-def X

5 P-Def X .

8.4. Proposition. If P and Q are classes with P 3 Q, then P-def 5 Q-def and P-Def 5 &-Def.

The next proposition connects P-dim and P-def. The statement uses the notion of open monotonicity which will be defined first.

11. MAPPINGS INTO SPHERES

110

8 . 5 . Definition. The universe is said to be open-monotone if

every open subspace of each space in the universe is also a member of the universe. A class P of spaces is said t o be open-monotone iii U if P satisfies the condition: If X is a space in P n U ,then every open subspace of X is in P nu. When P is open-monotone in the the universe 7 of all topological spaces, the class P will be called absolutely open-monotone. 8.6. Proposition. Let P be a class that is open-monotone in

U

and satisfies the condition (a) for every Y in P the inequality dim 21 5 dim 22 holds whenever 21 c 2 2 c Y . If a space X has the property (b) for each closed P-kernel G of X and each ( P flU)-kernel H of X such that G U H is closed it is true that G U H is a P-kernel of X , then P-dim X 5 P-def X .

Proof. Assume P-def X 5 n. Let us show that P-dim X 5 n holds by verifying the condition (c) in Theorem 5.9. Consider a closed P-kernel G of X and pairs ( A i , Bi), i = 0,1,. . ., n, of closed subsets Ai and Bi of X that are disjoint modulo G. For each i there are closed sets A: and Bi with Ai c A:, Bi c B: and A: n B: = G such that A: \ G and BI \ G are neighborhoods of Ai \ G and B; \ G respectively in the subspace X \ G. There exists a ( P n U)-hull Y of X with dim (Y \ X ) 5 n. For each i we denote by Ci the subset cly(A:) n cly(B:) of Y . Let 2 = Y \ U{ C; : i = 0,1,. . . , n } . A simple calculation will yield X n 2 = X \ G. Also 2 \ X c Y \ X and 2 n cly(A{) n cly(B:) = 0 for each i. As d i m ( 2 \ X ) 5 n by (a), condition (c) of Theorem 5.9 will produce in the space 2 \ X partitions Si between ( 2 \ X ) n cly(A:) and ( 2 \ X ) n cly(B:) such that 5': : i = 0,1,. . . , n } = 8. Observe that S: is a partition between A; \ G and B; \ G in the subspace (Ai U ( 2 \ X ) U B i ) \ G for each i. By Lemma 1.4.5 there is a partition T,! between Ai \ G and Bi \ G in 2 with Ti' n ( 2 \ X ) = Si. It easily follows that the set Ho = T,! : i = 0,1, , , .,n } is closed in 2 and is contained in X \ G

n{

n{

8. HULLS AND DEFICIENCY

111

because X n 2 = X \ G. Therefore HO is closed in X \ G. We define Si = ( X n T!) U G for every i and H = G U Ho. Then the sets H and S O ,5’1, . . . , S, are closed in X . One easily shows that Si is a partition between Ai and Bi modulo H for each i. It is clear that H = Si : i = 0 , 1 , . . . ,n } holds. Because 2 is an open set of Y and Y is a space in P n U ,we have that 2 is in P n U.So Ho is also in P n U. We have by (b) that H is in P n U.

n{

There is the following connection between P-Ind and P-Def. 8.7. Proposition. Let

P be a class that is open-monotone in U

and satisfies the condition (a) for each Y in P n U the inequality Ind 2 5 Ind Y holds for every open set Z of Y . Then P-Ind X 5 P-DefX

for every space X .

Proof, The proof will be by induction on n = P-DefX. We shall provide only the induction step. Assume P - D e f X 5 n. Let A and B be disjoint closed sets of X and let Y be a ( P n U)-hull of X with Ind (Y \ X ) 5 n. We may assume Y satisfies cly(A) n cly(B) = 8. Indeed, from condition (a) the space Y \ (cly(A) f l cly(B)) can be used in place of Y . Theorem 2.21 applied to Ind will give a partition S between cly(A) and cly(B) with Ind ( S n (Y\ X ) ) 5 n - 1. As ( S n X ) E U and S E P n U and as S \ ( S n X ) = S n ( Y \ X ) , we have P-Def(S n X ) _< n - 1. The induction hypothesis gives P-Ind ( S n X ) 5 n - 1. Therefore, P-Ind X 5 n. Let us end the section with a few remarks. The dimension functions dim and Ind do not satisfy the conditions (a) of Propositions 8.6 and 8.7 in the universe NH of hereditarily normal spaces. But they will for spaces in the Dowker universe which will be defined in Chapter 111. And the property (b) of Proposition 8.6 is related to the notion of strongly closed-additive classes defined in Chapter 111. Observe that the class X: of compact spaces fails to be open-monotone. But the class C of complete metric spaces is open-monotone. So the propositions apply to C but not to K in the universe Mo of separable metrizable spaces. Finally we have K-Def W 2 = K-def W 2 = 0 < 2 = X:-IndR2 = K-dimR2.

11. MAPPINGS INTO SPHERES

112

9. Absolute Borel classes in metric spaces

Agreement. The universe of discourse is the class M of metrizable spaces. The examples up to this point have been the classes K , C and S of compact, complete and a-compact metrizable spaces respectively. This section concerns other classes of this type, namely the absolute Borel classes. These are the next obvious examples and will be used in the next section. The Borel sets in metrizable spaces are wellknown in mathematics today, but the Borel classes of sets are not as well known. Since it is the Borel classes of sets that is needed in the next series of examples, a brief discussion of them will be given here. Let X be a space. The family of Borel sets of X is the smallest family B of subsets of X that satisfies the following conditions. BF1: Every closed set is a member of B. BF2: If H is a member of B, then X \ H is a member of B. BF3: If H,, n = 0,1,2,. . . , is a sequence of members of B, then the set U{ H , : n = 0,1,2,. . .} is a member of 6. Equivalently, the family of Borel sets of X is the smallest family B of subsets of X that satisfies the following conditions. BGl: Every open set is a member of B. BG2: BG2 = BF2. BG3: BG3 = BF3. Because of the De Morgan’s rules, conditions BF2 and BF3 together are equivalent to condition BF2 and the condition BF4 given below. BF4: If H,, n = 0,1,2,. . . , is a sequence of members of B, then the set H , : n = 0,1,2,. . . } is a member of 6. Consequently the family of Borel sets of X is the smallest family B of subsets of X that satisfies the conditions BF1, BF2 and BF4, or equivalently the conditions BG1, BG2 and BG4 = BF4. The family of Borel sets can be generated by an inductive transfinite process. To describe this, the notions of even and odd ordinal numbers must be explained. To this end let 00 denote the first infinite ordinal number. If a is an infinite ordinal number, then a can be uniquely written as Q = ,f n where 5 is a limit ordinal number and n < w o ; so every infinite ordinal number is called even or

n{

+

9 . ABSOLUTE BOREL CLASSES I N METRIC SPACES

113

odd as n is even or odd. (This makes limit ordinal numbers even.) Clearly no discussion of even or odd is needed for ordinal numbers n less than W O . For every ordinal number a the Borel classes F, are defined transfinitely as follows. (i-F) Fo is the family of closed subsets of X . (ij-F) If a is odd, then F, is the smallest family of subsets of X that is closed under countable unions and that contains the family U{ Ft : [ < a } . (iij-F) If a is even, then F, is the smallest family of subsets of X that is closed under countable intersections and that contains the family U{ Ft : 5 < a }. The complementary process gives the Borel classes G , defined transfinitely for every ordinal number a as follows. (i-G) Go is the family of open subsets of X. (ij-G) If a is odd, then G, is the smallest family of subsets of X that is closed under countable intersections and that contains the family U{ Gt : ( < a }. (iij-G) If a is even, then G , is the smallest family of subsets of X that is closed under countable unions and that contains the family U{ GE : ( < a } . Let w 1 denote the first uncountable ordinal number. Then the following is a basic fact about the family of Borel sets and the classes F, and G,. 9.1. Theorem. For any space X ,

B = U{ F, : a < w l }

and

B = U{G, : a < ul}.

Moreover, (a) M E F, if and only if X \ M E G,, (b) both F, and G, are closed under finite unions and under finite intersections, (c) if 2 c Y c X , then Z is a member of the family F, (or G,) for Y if and only if there exists a subset Z’ of X such that 2’n Y = 2 and 2’is a member of the family F, (or G,) for X . Next the classes F, and G, will be identified by their properties of closure under countable intersections and closure under countable unions.

114

11. MAPPINGS INTO SPHERES

9.2. Definition. For a

< w 1 , the multiplicative Bowl class a is

the family F, or G , according as a is even or odd respectively, and the additive Borel class a is the family F, or G, according as a is odd or even respectively. Note that the multiplicative Borel class a is closed under countable intersections and that the additive Borel class a is closed under countable unions. Clearly for a < /3 < w1 we have that both the multiplicative class a and the additive class a are contained in the multiplicative class ,8 and contained in the additive class p. The next theorem has a topological flavor and the proof found in Engelking [1967] will be given. 9.3. Theorem. The union of a locally finite collection of members of an additive or a multiplicative Borel class a is a member o f the same class.

Proof. The proof is by transfinite induction on a. The statement is obviously true for the classes Fo and Go. Let a > 0 and assume that the statement is true when [ < a. Consider a locally finite collection { A , : s E S } of members of the additive Borel class a. Then for each s in S we have A , = U{ A,,, : n = 0 , 1 , 2 , . . .}, where A,,n is a member of FC(,,,) U GC(,,,) for some [(s, n) less than a. For each n and each ,$less than a we consider the collection { A,,, : ((s, n ) < 6 }. For each fixed n the induction hypothesis gives U{ A,,, : ((s, n ) < [ } is a member of F, U G t . As [ < a , we have Ft U Gt C F, n G,. Also one or the other of F, and G, is the additive Borel class a. Because a < w 1 , there is a nondecreasing sequence ,$, m = 0, 1,2, . . . , of ordinal numbers converging to a. For each pair of natural numbers ( m , n ) define B,,, as the set U{ A,,, : ( ( s , n ) < Then U{ Bm,, : m E N,n E N} is a member of the additive Borel class a. Since U{ A , : s E S } = U{ B,,, : m E N,n E N}, the case of the additive Borel class a is proved. Suppose next that { A , : s E S } is a locally finite collection of sets from the multiplicative Borel class a where the indexing by S is injective. As X is metrizable, there exists a locally finite open cover { Vt : t E T } of X such that for each t in T there is a finite subset St of S with Vt n (U{ A , : s $ St }) = 8. So & \ U { A , :s E S} is equal to & \ U { A , : s E S t } and the last set is a member of the additive Borel class a. Consequently the set U{ (Vt \ U{ A , : s E St }) : t E T } is a member of the additive Borel class a. Therefore U{ A , : s E S} is a member of the multi-

tm}.

9. ABSOLUTE BOREL CLASSES IN METRIC SPACES

plicative Borel class a because U{ (Vt \ U{ A , : s E S }) : t E its complement. The transfinite induction is completed.

115

T } is

The following corollary is an easy consequence of the fact that the additive classes are closed under countable unions. 9.4. Corollary. The union o f a a-locally finite collection of sets in the additive Borel class a is a member of the same class.

Although (c) of Theorem 9.1 states that the additive Borel class a and the multiplicative Borel class a both have subspace and superspace relationships, it is not strong enough for the purposes of dimension theory modulo a class 'P. The property defined by membership in P is topologically invariant and hence is independent of any particular ambient space. Therefore a set A in a Borel class a for a space X is suitable for dimension theory modulo a class P if and only if for every space Y and every subset B of Y such that A and B are homeomorphic the set B is a member of the same Borel class a for Y . This motivates the following definition. 9.5. Definition. Let a < w1. A metrizable space X is said to be of absolute multiplicative (additive) Borel class CY provided that X is a member of the multiplicative (additive) Borel class a in Y whenever X is a subspace of a metrizable space Y . The respective absolute Borel classes a will be denoted by M ( a ) and A ( a ) . In a similar way one can define the absolute Borel classes F, and G,, which will be denoted by IF, I and IG, J respectively.

There are natural inclusion relationships among the various absolute Borel classes. These inclusions are diagrammed on page 72. The following is a characterization of M ( a ) and A ( a ) . 9.6. Theorem. (a) X E A(0) i f and only i f X = 8. (b) X E M(0) i f and only i f X is a compact space. (c) X E A ( l ) i f and only i f X is a a-locally compact space. (d) For 1 5 a < w 1 , X E M ( a ) i f and only i f X belongs to the

multiplicative Borel class a in some complete space Y . (e) For 2 5 a < 01, X E A ( a ) i f and only i f X belongs to the additive Borel class a in some complete space Y . Proof. Statements (a) and (b) are easily proved. The proof of the statement (c), which is due to Stone [1962], will not be given here

11. MAPPINGS INTO SPHERES

116

because it is rather involved and lengthy. The proofs of (d) and (e) will be given to illustrate the techniques employed in the universe of metrizable spaces, namely the use of the Lavrentieff theorem. Let us prove the sufficiency parts of (d) and (e). Suppose that X I is a member of the Borel class a of a complete space Y I ,that Xz is X2 is a homeomorphism. a subset of a space Y2 and that f : X1 Since every space has a complete extension, let Y be such an extension for Y2. By virtue of the theorem of Lavrentieff (Theorem 1.7.3) the function f can be extended to a homeomorphism 2 1 -+ 22, where 21 and 2 2 are Gg-sets of Y1 and Y respectively (containing X I and X,).Because X I is a member of the Borel class a of 21 (Theorem 9.1), the set X2 is a member of the Borel class a of 22. By Theorem 9.1 there is a subset 2 2 of Y such that 2 2 n 2 2 = X2 and belongs to the Borel class (Y of Y . Now observe the equality 2 2 f l 22 = ( 2 2 n &) n Y2 = X z . With the assumption on a in statements (d) and (e) of the theorem, 2 2 n 2 2 is a member of the Borel class Q in the space Y because Gg-sets are members of the additive Borel classes a when a 2 2 and are members of the multiplicative Borel classes a when a 2 l. Thus the sufficiency of the conditions have been established. The necessity parts are trivial. So statements (d) and (e) are proved.

s:

z2

9.7. Corollary. In the universe Mo of separable metrizable spaces the following hold.

A(O) = { S},

M(0) = X: n M ,

A ( l ) = S n Mo,

M ( l ) = C.

Proof. Only A( 1) = S n Mo requires proof. The proof will be left to the reader. Another easy consequence of the last theorem is the following. 9.8. Corollary. For a 2 2 the classes A ( a ) and M ( a ) are F,-monotone and Gg-monotone in M.

(The meaning of the two monotonicities should be evident.) The next theorem establishes closure properties under unions and intersections for the absolute additive and multiplicative classes.

9. ABSOLUTE BOREL CLASSES IN METRIC SPACES

117

9.9. Theorem. Let X C Y . Then the following statements are true.

(a) I f X is the intersection or the union of a finite subcollection H of A ( a ) (M((Y)) where each member of H is a subset of Y , then X is a member of A ( a ) ( M ( a ) ) . (b) If X is the union of a a-locally finite subcollection of A((Y), then X is a member of A ( a ) . (c) For 1 5 (Y < w 1 , if X is the union of a locally finite subcollection of M ( a ) , then X is a member of M(o). (d) If X is the union of a a-locally finite subcollection of M(O), then X is a member of A(1). (e) If X is the intersection of a countable subcollection H of M ( a ) where each member of H is a subset of Y , then X is a member of M(a).

Proof. The proofs for the smaller values 0 and 1 of a are easily handled separately. The proof for each of the remaining a's uses Theorem 9.6 and the Lavrentieff theorem. Here one uses a complete extension of Y together with Theorem 9.3 and Corollary 9.4.

A property possessed by nonempty Borel subsets of a complete separable metrizable space is that of being the continuous image of the space P of irrational numbers. This property yields the further property that such Borel subsets which are uncountable must contain a copy of the Cantor set. We shall use the remainder of the section to give a proof of these properties. The space X in the discussion will always be a nonempty Polish space, that is, a complete separable metrizable space. 9.10. Theorem. Suppose X is a Polish space. Then there exists

a continuous map

f of P onto X .

Proof. Assume that complete metrics have been chosen for the spaces X and P. As X is separable, there is a closed cover F(o) = { F(0j) : j E N } of X such that diam F(oj)5 1 for j in N. For each pair (On) there is a closed cover F(O,) = { F(o,j) :j E N} of F(on) such that F(0,j) c F(0,) and diam F(o,j) 5 2-' for j in N. In general, for each finite sequence (Onlnz . . .n k ) there is a closed cover

11. MAPPINGS INTO SPHERES

118

G(0n1n2...lzk)

= { G(0n,n2...nkj)

:j

N}

for the space P with the additional feature that the covers consist of mutually disjoint sets that are simultaneously open and closed in P.Now observe that each point p in P is associated with a unique sequence (Onlnl . . . nk . . .) such that G Q , , , ~ ~ . . :. k~ ~E )N } is the singleton set corresponding to p . As the metric on X is complete, the corresponding set F(on,n,...nk) : k E N } is a singleton subset of X . In this way we have defined a map f : P + X . Since the metric on P is complete, the map f is onto X . The map f is obviously continuous at each point p of P because the sets Gp,,,,,..,nk) are open in P and are mapped into sets of diameter less than or equal to 2 - k .

n{

n{

9.11.

Theorem. Suppose that X is a Polish space and let .. , be a sequence of continuous maps. Then the following s tat em en ts hold. (a) The set S = U{ fn[P]: n = 0, 1,2,. . . } is the image of a continuous map f on P. (b) The set M = fn[P]: n = 0,1,2,. . . } is either empty or the image of a continuous map f on P.

fn

: P -+ X , n = 0, 1,2,.

n{

Proof. To prove statement (a) we observe that P i s homeomorphic to the disjoint topological sum of a countably infinite number of copies of P.Define f to be the union of the continuous function fn. For the proof of statement (b), assume M # 0. It is well-known that the infinite product space P" is a complete separable metrizable space. For the continuous map F : P" -+ X defined by

"

F ( p O , p l , * * - , p* k*,. ) = the set

(fO(p0)) f i ( p I ) ,

*

e . 9

fic(pk),

*

.),

10. DIMENSION MODULO BOREL CLASSES

119

is closed because 2 is the inverse image under F of the diagonal A of X". As A is homeomorphic t o X , we have M is homeomorphic t o F[Z]. And as the subspace 2 is complete, there is by Theorem 9.10 a continuous map g of P onto 2.Define f t o be the composition of F and g . 9.12. Corollary. Let X be a Polish space. If B is a nonempty Bore1 subset of a space X , then there is a continuous map f of P onto B .

Proof. Let A be the collection of subsets of X consisting of the empty set and the continuous images of P. Clearly every Gs-set of X is a member of A (Theorem 1.7.1), in particular the closed sets and the open sets are members of A. Since the collection A is closed under countable unions and under countable intersections, every Borel subset of X is a member of A. Theorem. Every uncountable Bore1 subset of a Polish space X contains a copy of the Cantor set. 9.13.

Proof. Let f : P -+ X be a continuous map such that f[P] is uncountable. Observe that { 2 E X : f-'[x] has nonempty interior} is countable because P is separable. Consequently the set

D = {p

E

P : f-'[f(p)] is nowhere dense }

is an uncountable G6-set of P. It is now a simple matter to construct a copy C of the Cantor set contained in D such that flC is injective. 10. Dimension modulo Borel classes

The agreement of the last section will be continued. Agreement. The universe is the class M of metrizabfe spaces. Let us make some easily verified observations. From the agreement we have A ( a ) = A ( a ) n M and M ( a ) = M ( a ) n M . Both of these classes are closed-monotone, and they are, except for M(O), also open-monotone. Finally if X is a space that is the union of two members of one of these classes, then X is also a member of the same class. In order t o apply the theorems and propositions of the chapter it will be convenient to state the following theorem.

120

11. MAPPINGS INTO SPHERES

10.1. Theorem. Suppose that the class P satisfies the conditions: (a) If X = F U G is such that F is closed and F and G are in P , then X is in P. (b) I f X = U{ X i : i = 0,1,2,.. .} where { X i : i = 0,1,2,.. . } is a countable locally finite closed collection in X with X i in P for all i, then X is in P . Then P-Ind is simultaneously closed-monotone and open-monotone and satisfies the conditions: (c) I f X is the union of two subsets F and G, one o f which is closed, then max { P-Ind F, P-Ind G } 5 P-Ind X. (d) I f { X i : i = 0, 1,2,. ,. } is a countable locally finite closed cover o f X , then

sup { P-Ind Xi : i = 0, 1,2,. ..} = P-Ind X . The proof of the theorem will be given later in subsection 10.3. In addition to the theorems of this chapter we shall need to refer to several theorems from Chapter V which is devoted to basic dimension functions, that is, dimension functions defined by the existence of special bases for a space. The first two consequences of Theorem 10.1 are that A(a)-Ind is open-monotone for 0 5 a < R and that M(a)-Ind is open-monotone for 1 5 a < a. Therefore, on applying Proposition 7.1 t o all absolute Bore1 classes except for M(O), we have

(1)

A(a)-dim 5 A(a)-Ind, M(a)-dim 5 M(a)-Ind,

a 2 0, a 2 1.

Theorems 3.6 and 8.7 give the inequalities

(2)

A( a)-Ind 5 A( a)-Sur, M(a)-Ind 5 M(a)-Def,

Q

Q

2 0, 2 1.

The proof of the next inequality will make use of a consequence of Tumarkin’s theorem (Theorem V.2.12).

5 A(@)-Def, M(a)-Sur 5 M(a)-Def,

A(a)-Sur

(3)

L 1, a 2 2. Q

10. DIMENSION MODULO BOREL CLASSES

121

Proof. We shall only prove the second inequality since the other inequality is proved in the same way. Suppose that n = M(a)-DefX is finite and let Y be a M(a)-hull of X with n = Ind (Y \ X ) . By Tumarkin’s theorem there is a Gs-set 2 of Y such that Y \ X c 2 and n = Ind 2. As X \ Z = Y \ 2 and a 2 2, we have the F,-set X \ 2 is a M(a)-kernel of X by Corollary 9.8. By the subspace theorem for Ind we have Ind ( X \ 2 ) 5 n. Thus we have shown M(a)-Sur X 5 n.

The next proposition takes advantage of the complementary relationships between the additive and multiplicative Bore1 classes. Further discussions on complementary relations and ambiguous classes can be found in Section V.2. 10.2. Proposition. Suppose that 2 is in M ( l ) and that X and Y are disjoint subsets such that 2 = X U Y. Then, for a 2 1, A(a)-SurX 2 M(a)-Defy, M(a)-Sur X 2 A(a)-Defy. Proof. Suppose that n = A(a)-Sur Y is finite and let S be an A(a)-kernel of Y such that n = Ind (Y \ S ) . As a 2 1 and 2 is in M ( l ) , the set T = 2 \ S is a member of M ( a ) that contains X and n = Ind (T \ X ) holds. From this we infer M(cr)-DefX 5 n. We leave the proof of the second inequality to the reader.

We can now prove the following formulas.

(4) A(a)-Sur = A(a)-Def

and

M(a)-Sur = M(a)-Def,

a

2 2.

Proof. From (3) and Proposition 10.2 we have

A(a)-DefX 2 A(&)-SurX 2 M(a)-Defy

2 M(a)-Sur Y 2 A(a)-Def X when 2 is a disjoint union of X and Y and 2 is in M(1). We complete the proof by letting 2 be any completion of X for the first equation of (4) and by letting 2 be any completion of Y for the second equation. The reader is referred to Examples 1.10.12 and 1.10.13 for a = 1 in formula (4)above.

122

11. MAPPINGS INTO SPHERES

Let us now use some more material from Chapter V . The coincidence theorem dim X = Ind X for metrizable spaces X (originally proved by Moritain [1954] and in Kat6tov [1952]) will be established in Theorem V.3.14. This coincidence leads immediately t o the following.

(5)

A(a)-sur = A(a)-Sur

and

A(a)-def = A(a)-Def,

M(a)-sur = M(a)-Sur

and

M(a)-def = M(a)-Def.

The coincidence theorem for dimension is a special case of the sharpening of the inequalities ( l ) , namely for A(0). The special role of the universe M allows us to take advantage of the basic dimension functions discussed in Chapter V. Indeed, Theorems V.2.4, V.2.21 and V.3.12 give the following sharpening of ( 1 ) and (2). A(a)-dim = A(a)-Ind,

(6)

M(a)-dim = M(a)-Ind,

a 2 0, a 2 1.

A(a)-Sur = A(a)-Ind, (7)

M(a)-Def = M(a)-Ind, Results in Section IV.2 will yield the fc lowing two formu .as. A(@)-dim = A(a)-Dim,

(8)

M(a)-dim = M(a)-Dim,

2 0, a 2 1.

Let us turn to the universe Mo of separable metrizable spaces. The connections between the small and large inductive dimensions modulo the absolute Borel classes can be established in the same manner as that in Chapter I for the special cases of C and S. (Also see Corollary V.3.13.) The argument presented there fails only for the absolute Borel class Mo fl M(0). Theorem 10.1 will be needed in the proof.

A(a)-indX = A(a)-IndX,

(9)

M(a)-indX = M(a)-IndX,

X E Mo, a 2 1, X E Mo.

10. DIMENSION MODULO BOREL CLASSES

123

Returning t o arbitrary metrizable spaces, we have that the inclusion relationships between the various absolute Bore1 classes will result in inequalities for the functions under discussion. For example, (10)

A(cr)-sur 5 M(P)-sur I A(y)-sur,

a > ,B > y.

Finally we remark that (11)

A(a)-ind

# A(a)-Ind

and

M(a)-ind

#

M(a)-Ind.

This will be shown in Example 111.1.16. 10.3. Let us now turn to the proof of Theorem 10.1. We have need of a general construction that will be used twice in the course of our proof. Let Y be an open subset of a metrizable space X . Then, since Y is a cozero-set of X, there is a continuous function f : X + [0,1] such that f-'[ (0,113 = Y . For i in N we define the closed sets Hi = { 2 : 2-(i+l) < - f(z) 5 2 4 }

and the open sets

ui= { 2 : 2 - ( i + 2 )

< f(.)

< 24i-1)

1.

Hi C U;c c l ~ ( U i c) Y holds for every i and the collection { Ui : i = 0 , 1 , 2 , . . . } is locally finite in Y .

Then

Proof of Theorem 10.1. The proof is by induction and has the flavor of the normal family argument that is exploited in Chapter 111. For each class P we define a collection PI by

PI = { X E U : P contains a partition between each pair of disjoint closed subsets of X } . With P(-l) = P n U we further define Pen) = (P(n-l))lfor n in N. As U = M , we have P(.)= { X E M : P - I n d X 5 n } for n 2 -1. The theorem will follow easily if it can be shown that PI satisfies conditions (a) and (b) whenever P does. Assume P satisfies the conditions (a) and (b). We shall first prove P and Pr are open-monotone (in U = M of course). Then we shall address the proof that P r satisfies conditions (a) and (b). Let us show that P is open-monotone. Suppose X

11. MAPPINGS INTO SPHERES

124

is in P and Y is an open set of X . Then from the construction immediately preceding the proof we have Y is the union of the locally finite collection { H i : i = 0,1,2,. ..} in Y where each set H i is closed in X. Since P is closed-monotone, each H i is in P. The set Y is in P by condition (b). We infer from Theorem 2.16 that Pr is closedmonotone. Now we can show PI is also open-monotone. To this end we suppose X is in P'and Y is an open set of X . Consider disjoint subsets A and B of Y that are closed in Y . Let Hi and Ui be as in the construction above. Then Ai = A n H i and Bi = B U (X \ U i ) are disjoint closed sets of X . From the inclusions c l ~ ( U i C ) YCX we have clX(Ui) E PI, And from the definition of PI there is a partition Si between A; and Bi in X such that Si fl c l ~ ( U i )is in P. Since Si C U i , we have Si E P. Let V i and Wi be disjoint open sets in X with Ai C V , and Bi C Wa such that X \ S; = V, U Wi. Then the set V = U{ V, : i = 0 , 1 , 2 , . . . } is open in Y and A C V . Also the subset U{ Si : i = 0,1,2,. . . } of Y is in P and

W = n{winY:i = 0,1,2, ...} = y\U{siuv, : i = o,1,2, ...} is an open set of Y containing B because S; U V, c Ui \ B for each i and the collection { Ui : i = 0,1,2,. . . } is locally finite in Y . Finally let S = Si : i = 0,1,2,. . .} \ V . Then

U{

Y\

s = Y \ U{ si \ v : i = 0, 1,2,. . .} = n{(vz u (wi n Y )u v): i = 0,1,2,.. .I = ((7{winy:i=o,i,2,

=wuv.

...}) uv

So S is a partition between A and B in Y and S is in P.Thus we have shown Y E P' and PI is open-monotone. Let us show P' satisfies the condition (a). Suppose F and G are in P' and F is closed in X = F U G. Since P' is open-monotone, we have G \ F is in P'. So we may further assume F n G = 8. Let A and B be disjoint closed sets. Then there is a partition SObetween A n F and B n F in F with So E P. Clearly SOis also a partition between A and B in 2 = F U A U B . Let A0 and Bo be disjoint open sets of 2 such that Z \ S o = A0 UBo, A C Ao and B C Bo. As 2 is closed in X , the sets A0 and Bo are also closed in X \ SO. Let A1 and B1 be disjoint closed neighborhoods of A0 and BO in

10. DIMENSION MODULO BOREL CLASSES

125

X \ SOrespectively. In the space G there is a partition S1 between the disjoint closed sets A1 n G and B1 n G with S1 E P. As Al and B1 are disjoint closed neighborhoods in X \ SO of A0 and Bo, from F n G = 0 we have clx(S1) c SOU S1. Hence S = So U S1 is a partition between A and B in X . Since P satisfies condition (a), we have S is in P . Thus we find P’ also satisfies condition (a). Finally we shall show that PI satisfies the condition (b). Suppose { X;: i = 0,1,2,. . . } is a locally finite closed cover of X such that each Xi is in PI. From (a), Fi = U{ X j : j = 0,1,. . ., i } is in PI for each i. Moreover, from the open monotonicity of P’ we have G;+l = Fi+l \ F; is also in PI. Let A and B be disjoint closed sets. Employing the construction of the preceding paragraph, we can inductively construct subsets Si of Gi that are in P and disjoint sets Vi and Wi that are open in the space Fi satisfying A n Fi c Vi, B n Fi c Wi and Fi \ U{ Sj : 0 5 j 5 i } = Vi U Wi. The local finiteness of { X;: i = 0,1,2,. ..} yields V = U{ : i = 0,1,2,. ..} and W = U { W i : i = O , 1 , 2 ,... } areopen, S = U { c l ( S i ) : i = O , 1 , ... } i s t h e s e t c l ( u { S i : i = 0 , 1 , 2 ,...} ) a n d X \ S = V U W h o l d s . Clearly, A C V and B C W . Because P satisfies condition (a), we have cl ( S i ) is in P ; and, because P satisfies condition (b), we have S is in P . We have shown X E PI. So P r satisfies condition (b). The theorem is now proved. Our final task of the section will be to exhibit for the functions A(a)-Ind and M(cr)-Ind a space for each possible value. The Example 1.7.13 used the existence of totally imperfect sets (= contains no nonempty perfect set) to achieve this for the class C. The discussion of that example contains a proof of the fact that every uncountable complete separable metrizable space X contains a subset Y such that neither Y nor X \ Y contains an uncountable F,-set of X . Moreover, it is proved there that if dimX = n, then dim Y 2 n - 1 and dim ( X \. Y .) 2 n - 1. These facts will be summarized in the following proposition. 10.4. Proposition. Let Z be an uncountable complete separable metrizable space. Then Z can be written as the the union of two disjoint subsets X and Y such that both X and Y are totally imperfect. Moreover, if dim Z = n, then dim X n - 1 and dimY 2 n - 1.

>

11. MAPPINGS INTO SPHERES

126

10.5. Example. Let subsets of In+' such that

A(a)-surX, = n

X, and Y, be disjoint totally imperfect In+' = X, U Y,. We shall prove and

M(a)-surX, = n )

a 2 0.

As both X, and Y, are dense in In+', we have by induction on n that dimX, = n and dimY, = n. Let B be an absolute Borel set contained in X,. We infer from Theorem 9.13 that B is a countable set. The two disjoint sets X , \ B and Y, U B are totally imperfect sets whose union is In+'. Therefore we have dim (X, \ B ) = n for every absolute Borel set B contained in X,. Our assertions are now easily proved. From formulas (5), (6)) (7) and (9) we have

A(a)-ind

X, = A(a)-Ind X, = A(a)-dimX, = A(a)-sur X, = n ,

cy

2 0.

Formulas (3), (4)and (5) yield A(a)-sur X, = A(a)-def X,, n = A(1)-surX,

Q

>_ 2,

5 A(1)-defX, 5 n,

A(0)-defX, = $00.

For the multiplicative absolute Borel classes we find with the help of Proposition 10.2 and Theorem 6.11 M(a)-ind X, = M(a)-Ind X, = M(a)-dim X, = M(a)-sur X, = M(a)-def X , = n,

a 2 0.

11. Historical comments and unsolved problems

The Definitions 2.1 and 2.11 are essentially those of Lelek [1964] who introduced them under the name of inductive invariants. The systematic introduction of a universe of discourse into the general development of dimension modulo a class of spaces appears here for the first time. Earlier papers implicitly used a universe. But also implicit in these papers is the assumption that the class P is contained in the universe. This last assumption does not make the

11. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

127

dimensions P-ind, P-Ind and P-dim independent of the universe. The introduction of the universe into the development also has resulted in cleaner statements of the theorems and propositions. The partition definitions of the inductive dimensions modulo a class P has been chosen over the more commonly used boundaries of open sets. In proofs that use boundaries of open sets, it is the partitioning property of the boundaries that is used. By using the partition definitions, we find that many proofs become much shorter because they arrive more quickly to the essential part of the argument. The early papers on surplus and deficiency (and hence kernels and hulls) used implicitly some universe. Indeed, it was always assumed that the kernels and hulls were to be taken from P n U. This has been formally stated in this chapter. The domain of each of the functions P-sur, P-def, etc., is the universe. The discussion of Section 4 essentially appeared first in Aarts and Nishiura [1972]. The unstable value theorems appear here for the first time. The Theorems 5.7 and 5.9 characterizing P-dim are new. The concept of the normal universe modulo P is new and perhaps can be exploited further. The Theorem 6.6 characterizing P-dim by means of mappings into spheres is also new. In this connection, reference is made to Baladze [1982]. For P-dim in the metrizable setting, the reader is directed to the papers Aarts [1972] and Aarts and Nishiura [1973a]. The equalities IC-Ind = K-dim = K-Dim = K-Sur in Theorem 6.11 are new. The comparison of P-Ind and P-dim by means of mappings into spheres in Proposition 7.1 appears here for the first time. In this chapter the surpluses P-sur and P-Sur and the deficiencies P-def and P-Def are defined in terms of dim and Ind respectively. The need for these distinctions will become apparent in the next two chapters. The question of determining a reasonable universe of discourse in which all of the agreements and special technical hypotheses imposed in the various propositions of this chapter are valid will be addressed in the subsequent chapters. Several classes of spaces have been proposed in the literature in response t o this question for the functions Ind and dim. A historical discussion of these classes will be given in Chapters I11 and IV. The Dowker universe D, which includes all of the previously given classes, is defined In Chapter 111.

128

11. MAPPINGS INTO SPHERES

The examples of the absolute Borel classes first appeared in the papers of Aarts [1972] and Aarts and Nishiura [1973a]. For each a the intersection A ( a ) n M(a) is called the ambiguous absolute Borel class a. The computations found in the proof of Proposition 10.2 are special cases of those related to the concept of complementary dimension functions discussed in these papers. The reader is referred to Sections V.2 for a discussion of this concept. The class C ( = M(1)) of complete metrizable spaces can be generalized to a subclass of completely regular spaces by means of the notion of Cech completeness of a space X (= a G6-subspace of a Hausdorff compactification). This will permit the generalization of the large inductive completeness degree Icd and the completeness deficiency C-def to the universe of completely regular spaces. It was shown by van Mill in [1982] that there is a completely regular space X such that Icd X = 0 and C-defX = 00. Thus it is not possible to generalize the completeness degree results in the universe M of metrizable spaces to the larger universe R, of completely regular spaces.

Unsolved problems 1. What are the values of the gaps that are possible for the inequalities (10) of Section 10 and the corresponding inequalities of the other functions that have been defined in this chapter? 2. Extend in a meaningful way the results of Section 10 t o a universe that properly contains the universe M . In particular, find a universe strictly between the class M and the class R, of completely regular spaces in which the pathology found by van Mill in [1982] fails to exist.

CHAPTER I11

FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Introduced in Chapter I1 were various dimension functions modulo a class P. Just as in dimension theory, there were two distinct types; one used an inductive approach and the other used a covering approach. The discussion of Chapter I indicated that these two approaches gave rise t o a dichotomy for certain classes P in the universe Mo of separable metrizable spaces. It was found that the combination of kernels, surplus and the inductive approach was a natural one for the class S of a-compact spaces and that the combination of hulls, deficiency and the covering approach was a natural one for the class C of complete spaces. This dichotomy reflected the fact that, on the one hand, the class S has closure under countable unions of closed sets and the class C does not and, on the other hand, the class C has closure under countable intersections of open sets and the class S does not. To distinguish the two approaches, the terminology functions of inductive dimensional type and functions of covering dimensional type will be used. In this chapter the functions of inductive dimensional type will be studied. These functions include P-ind, P-Ind, P-Sur and P-Def. The exposition will begin with addition theorems. The -esults, in particular that of point addition, will be used t o study connections between P-ind and P-Ind. The investigation will move next t o the normal families of dimension theory. The functions P-ind and P-Ind lead naturally t o normal families. Normal families were introduced by Hurewicz in [1927] and by Morita in [I9541 in the universes of separable and general metrizable spaces respectively. A general investigation of normal and related families will be carried out in Sections 2 and 3. These investigations are applied in a manner similar to that of Dowker [1953] to create what will be called the Dowker universe. The chapter will end with axiomatics. Menger proposed in [1929] an axiom scheme for the dimension function. His axioms 129

130

111. FUNCTIONS O F INDUCTIVE DIMENSIONAL T Y P E

were strongly influenced by the work of Hurewicz [1927] on normal families. A discussion of Menger-type axioms and other axioms will be given in Section 5 .

Agreement. The universe of discourse is closed-monotone and is contained in N H .Every class has the empty space as a member and is closed-monotone in U. Unless indicated otherwise, X will always be a space in the universe. 1. Additivity Generally speaking, we shall reserve the notion of additivity for discussions concerning finite unions of spaces. Obviously, addition theorems involve relationships between spaces situated in an ambient space. It is important that the ambient spaces as well as the spaces be in the universe of discourse. (This arrangement is similar to that of surplus and deficiency found in Chapter 11.) We shall now define four types of additivity for classes P . 1.1. Definition. A class P is said to be additive in U if each space X in the universe is a member of P whenever (i) X = Y U 2 , (ij) Y and 2 are in P nu. When the additional condition (iij-c) Y and 2 are closed is imposed, the class P is said to be cbosed-additive in U. When the additional condition is (iij-s) Y is closed, the class P is said to be strongly closed-additive in U . And finally when the additional condition is (iij-p) Y is a singleton, the class P is said to be point-additive in U .

The definitions of the various additivities have emphasized the membership of the spaces in the universe. In a sense, this emphasis is redundant. We have included this redundancy because comparisons between results for the universe and results for subcollections of the universe will be made. In the universe Mo we have already seen three main addition theorems in Chapter I. The first of these is the inequality:

1. ADDITIVITY

131

A. For X and Y in Mo, ind ( X U Y ) 5 ind X t ind Y t 1. The second and third theorems concern closed-additive and strongly closed-additive classes. Both are consequences of the countable sum theorem.

B. For each n, the class { X :ind X 5 n } is closed-additive in the universe Mo. C . For each n, the class { X : ind X 5 n } is strongly closed-additive in the universe Mo.

Of course, the strongly closed-additive statement C implies the statement B. With respect to point additivity, recall that there exists a space X in M o with cmp X = 1such that cmp ( X \ { p }) = 0 for some point p in X . Thus we have that the class K' = { X : cmp X 5 0 } fails to be point-additive in the universe Mo.Obviously, from C above we know that the small inductive dimension ind cannot be raised by the addition of a point to spaces X in the universe Mo. In other words, the class { X : ind X 5 0 } is point-additive in the universe Mo. The next example will show that this class is not point-additive in the universe M of all metrizable spaces. 1.2. Example. There exists a space X in M such that ind X = 1 and ind (X \ { p } ) = 0 for some point p in X. To see this, let A be the famous example of Roy [1962] which has the properties that A is a complete metric space, ind A = 0 and Ind A = 1 (see Section 1.4 for other references). There are disjoint closed sets F and G of A such that each partition S between F and G in A is nonempty. The set X is formed from A by identifying the set F to be the point p . With p as a metric on A for which the sets F and G have a distance 1 between them, we shall define a metric d on X by means of a formula. In the formula, p ( x , F ) will denote the usual distance between the point x and the set F .

for x # p , z = p for x = p , z # p for x = p , t = p .

A straightforward calculation will yield that d is a metric for X . Moreover, when x # p and z # p , the inequality d ( x , z ) 5 p ( z , t )

132

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

holds. To see that X \ { p } and A \ F have the same topology we let z and E be such that z # p and 0 < 2~ 5 p ( z , F ) . If p(z,.) < E , then p(.,F) 2 p(.,F) - @,.) 2 2.5 - P(Z,.) 1 E > p ( z , t ) . Thus the definition of d gives d ( z , z ) = p ( z , z ) whenever p ( z , z ) < E . Consequently, the open balls { z € A : p ( z , t ) < E } and { z E X : d ( z , z ) < E } coincide when 2~ 5 p(z, F ) . Clearly the space X has the required properties. Let us return to inequalities like those in A above. We shall begin with P-ind. The following proposition, which is an immediate consequence of the definition, will be needed. 1.3. Proposition. Let q be an isolated point of a space X . Then P-ind X = P-ind ( X \ { q } ) whenever P-ind ( X

\ { q } ) 2 0.

The proposition with P = { 8 } is used in the next theorem.

1.4. Theorem. Let A and B be subspaces of X such that B is closed and X = A U B. Then P-ind X

5 P-ind A + ind B + 1.

It is to be observed that the statement of the theorem is in the context of the universe of discourse. Consequently the spaces A and B are in the universe; arbitrary subsets of X need not be in the universe because the universe is only closed-monotone by agreement.

Proof. The proof is by induction on ind B. Since the inequality is obvious for ind B = -1, we need prove only the inductive step. Suppose that the inequality holds when ind B < n and assume that B is such that ind B = n. Let 2 be a point of X and F be a closed set of X with z @ F . Since B is closed in X , we have B U { z } is closed and ind ( B U { z }) = ind B. Therefore no generality is lost in assuming z is a member of B . Then by Theorem 11.2.20 there is a partition S between z and F in X with ind ( B n S) 5 n - 1. As F-ind ( A n S) 5 P-ind A , we have P-ind S 5 P-ind A in by the induction hypothesis. Thereby the inequality holds when ind B = n. Even though point addition theorems fail for K-ind (= cmp) in the universe Mo and for ind in the universe M , the last theorem

1. ADDITIVITY

133

does provide an upper bound on how badly it may fail. That is, the difference becomes at most 1 by letting B be a singleton subset of X . The large inductive dimension analogue of the above theorem permits the set B to be other than closed in X . 1.5. Theorem. Suppose X = A U B . Then

P-Ind X 5 P-Ind A

+ Ind B + 1.

Proof. The proof is by induction on Ind B = n. We shall indicate only the inductive step. Let F and G be disjoint closed sets of X . Then by Theorem 11.2.21 there is a partition S between F and G in X with Ind ( B n S) 5 n - 1. The induction hypothesis yields P-Ind 5’ 5 P-Ind ( A n S) n. The proof is now easily completed.

+

Concerning the replacement of ind and Ind with P-ind and P-Ind in the above two theorems, we have the following results. The additivity conditions defined in the section will be used. 1.6. Theorem. Suppose that P is closed-additive in U . Let A and B be closed sets of a space X with X = A U B . Then

P-ind X

5 P-ind A + P-ind B + 1.

Proof. Since both A and B are closed, we need to check only the points of A n B. An induction on n = P-ind A P-ind B 1 will be made. The case n = -1 is obvious. For the inductive step, observe that n 2 0 implies P-ind A 2 0 or P-ind B 2 0. Suppose P-ind A 2 P-ind B and let 2 E A n B. Then for any closed set F of X not containing 2 there is by Proposition 11.2.20 a partition S between 2 and F in X such that P-ind ( S n A ) 5 P-ind A - 1. Applying the induction hypothesis to the sets S , S n A and S n B , we have P-ind S 5 n - 1. That is, P-ind X 5 n.

+

1.7. Theorem. Suppose that P is additive in be subspaces of X such that X = A U B . Then

P-Ind X

+

U . Let A and B

5 P-Ind A + P-Ind B + 1.

+

+

Proof. We shall induct on the sum n = P-Ind A P-Ind B 1. As the case n = -1 is obvious, we shall pass to the inductive step.

134

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

Let F and G be disjoint closed sets of X and suppose that n 2 0. Then we have P-Ind A 2 0 or P-Ind B 2 0. The inductive step is completed in a manner similar to that of the proof of the previous theorem. Instead of using Proposition 11.2.20, one uses Theorem 11.2.21. The following analogue for strongly closed-additive classes is established in a manner similar t o the last theorem. 1.8. Theorem. Suppose that P is strongly closed-additive in U. Let A and B be subspaces of X such that A or B is closed and X = A U B . Then

P-Ind X 5 P-Ind A t P-Ind B t 1. With U = NH and P = { 0 }, Theorems 1.5 and 1.7 both yield the second inequality of the following important theorem of dimension theory. The first one is proved by observing that ind X # - 1 # ind Y may be assumed. 1.9. Theorem (Addition theorem). For any hereditarily normal space X U Y ,

ind ( X U Y ) 5 ind X t ind Y t 1, Ind ( X U Y ) 5 Ind X t I n d Y t 1. We shall conclude the section with a discussion on the interrelationships of the finite sum theorem, the addition theorem and the point addition theorem for P-ind in the universe M of metrizable spaces.

Agreement. In the remainder of the section the universe is the class M of metrizable spaces. It is already known that the finite sum theorem, the addition theorem and the point addition theorems do not have analogues for the class K in the universe Mo. For the reader’s convenience we shall list the three inequalities corresponding to these theorems. (A = addition; P = point addition; S = finite sum.)

+

+

Theorem A. P-ind ( X U Y )5 P-ind X P-ind Y 1. Theorem P. P-ind ( X U { p } ) 5 P-ind X when X 4 P, Theorem S. P-ind ( X U Y) 5 max{ P-ind X, P-ind Y } whenever X and Y are closed in X U Y .

135

1. ADDITIVITY

The idea of the proof of the next proposition stems from Example I.5.10.f. 1.10. Proposition. Theorem S implies Theorem P.

Proof. Assume X 4 P and let p be a metric for X U { p } . Then the set X U { p } is the union of the two closed sets

F = { p } U { x 6 X : 2k 5 p ( p , x) 5 2"'

for some odd integer k } ,

G = { p } U { x E X : 2 k 5 p ( p , x ) 5 2'+l for some even integer k}. Since P - i n d X 2 0, from the definition of P-ind one easily sees that P-ind F and P-ind G are both less than or equal t o P-ind X. Theorem S now completes the proof. 1.11. Proposition. If P n M

# {a},

then Theorem A implies

Theorem P. Proof, The proposition follows easily because { p } E P holds as a result of the closed-monotonicity in M of the class P and the existence of a nonempty space in the class P n M . 1.12. Proposition.

If P is an additive class, then Theorem P

implies Theorem A. (Recall that the present discussion has assumed M as the universe. Consequently, additive refers to this universe.)

Proof. The proof is similar to that of Theorem 1.6. Again we shall induct on n = P-ind X P-ind Y 1. The case n = -1 is the statement that P is additive. For the inductive step assume n 2 0 and let 2 = X U Y . Consider a point p and a closed set F in 2 with p 4 F . Without loss of generality we may assume P-ind X 2 0. Then by Theorem P we have P-ind ( X U { p }) 5 P-ind X . There is a partition S between p and F with P-ind (S n X ) < P-ind (XU { p } ) by Proposition 11.2.20. Consequently, as in the proof of Theorem 1.6, we have P-ind S 5 n - 1. Thereby we have P-ind ( X U Y ) 5 n and the induction is completed.

+

+

In the next proposition we shall use the point addition theorem for P-Ind. Its proof is a simple modification of the one for Ind in Theorem 1.4.12 and is left to the reader.

136

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

1.13. Proposition. P-ind = P-Ind if and only if Theorem P holds. Proof. Assume P-ind = P-Ind. Then Theorem P will follow from the point addition theorem for P-Ind (Theorem 1.4.12). Conversely, assume that Theorem P holds. We already know that P-ind 5 P-Ind holds. The reverse inequality will be proved by induction. Assume that the reverse inequality holds for spaces X with P-ind X < n. Let X be such that P-ind X = n and consider disjoint closed sets F and G. We may assume that both F and G are nonempty for the sets can be partitioned by S = 0 in the contrary case. If P-ind (X\ F ) = -1, then there is partition S between F and G in X such that S E P. So assume P-ind ( X \ F ) 0. Let Y = X \ F and identify the closed set F to be p . By Theorem 11.2.17 we have P-indY 5 n. Then, using the metric d defined in Example 1.2, we have that Y U { p } is a metric space such that Y is homeomorphic to X \ F . By Theorem P, P-ind (Y U { p } ) 5 n. There exists a partition S between p and G in Y U { p } such that P-ind S 5 n - 1. The induction hypothesis gives P-Ind S 5 n - 1. Since S c Y , we have that S is also a partition between F and G in X. Thus we have proved P - I n d X 5 n and the induction is now completed.

>

At the beginning of our discussion of Theorems A, P and S we observed that ind and cmp failed t o satisfy Theorem P. That is, the absolute Borel classes { 0) and K: are examples of classes P for which Proposition 1.13 applies. We shall show next that the other absolute Borel classes are also examples. To this end, we shall construct a subspace of Roy’s space A for which the point addition inequality fails for each absolute Borel class a, This will be achieved by showing that P-ind # P-Ind whenever P is an absolute Borel class. A preliminary discussion of the space A will be needed. The space A is complete and dense in itself and has cardinality c = 2No. So it has exactly c countable subsets and hence exactly c separable closed subspaces. Using the classical Bernstein construction referred to earlier in Example 1.7.13, we can decompose A into two disjoint subsets A and B so that every Cantor set contained in A intersects both A and B . 1.14. Proposition. Let A and B be disjoint subsets of A such that A = A U B and such that every Cantor set contained in A inter-

1. ADDITIVITY

137

sects both A and B. Let E be a Bore] subset of A that contains B. Then the sets 2 = E n A and 5 = A \ 2 also have the property that every Cantor set contained in A intersects both 2 and g.

Proof. Let K be a Cantor set contained in A and E be a Borel set that contains B. Then Ir' \ E is an absolute Borel space in M o . Consequently, as in the discussion of Example 11.10.5, the set I( \ E must be countable since it is contained in A. So the uncountable absolute Borel space K n E in Mo contains a Cantor set L . As L n A # 0, we have K n 2 = K n E n A 3 L n A # 0. And as L n B # 0, we also have Ir' n 5 3 K n B # 0. The space A is complete. So if the Borel subset E in the above proposition is a set of Borel class a in A, then it is also a space of absolute Borel class a when a 2 2. And if E is a Gs-set in A, then it is in M(1). 1.15. Proposition. Let A and B be disjoint subsets of A such that A = A U B and such that every Cantor set contained in A intersects both A and B . Then IndA = 1 and I n d B = 1.

Proof. By way of a contradiction we shall show IndA 2 1. Assume Ind A = 0. Let F and G be disjoint closed subsets of A such that every partition between F and G is nonempty. By Theorem 11.2.21 there exists a partition S between F and G such that S n A = 0. Since S is closed and S c B , we have that S contains no Cantor sets. Observe that every dense in itself complete metric space contains a Cantor set. So no closed subset of S is dense in itself. That is, S is a scattered set. Let us show that the closed set S (indeed, any scattered metrizable space) is a-discrete. To this end, let d be a metric on the space A. For each ordinal number a we shall denote by D, the derived set of order a of the set S . As S is scattered, there exists for each point z in S an ordinal number a, such that z E Daz and 2 $! D,,+l. Let X , be the set { 2 E S : d ( z , D a z + l )> & } for n in N. Clearly we have S = X , : n E N}. It remains t o be shown that each set X , is a-discrete. Let Y, be a maximal subset of X , such that no two points of Y, have distance smaller

U{

138

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

&.

&

than If d(y,z) < for a point y in Y, and a point z in X,, then from the definition of the set X , we have cyg = ax. Consequently the sets X,, = { z E X, : a, = a }, where (Y is an ordinal number, form a collection of open subsets of the subspace X , such that the distance between two distinct sets X , , and Xnar exceeds +. 2(n+l Since the set X,, is a subset of D, \ D,+1, the set X,, is a-discrete. It follows that each set X, is a-discrete. By reindexing the sets we have a countable collection { Si : i < wo } of discrete subsets of A such that S = U{ Si : i < wg }. We may also assume that the Si’s are disjoint. We shall construct a neighborhood W of S that is open-and-closed and such that W n ( F U G) = 0. We may assume that the metric for A is complete. Using the facts that A is collectionwise normal and ind A = 0, we can find for each i less than wg a discrete collection { W , : s E Si } of open-and-closed sets such that

(1) s ~ W , a n d W , n ( F U G ) = 0 f o r e a c h s i n S i , (2) diameter W , < 2 - i for each s in Si. Let W = U{ W, : s E Si,i < wo }. Then the set W is open and disjoint from F U G. To show that W is closed, let 5 be in cl ( W ) and let { X k : k < w o } be a sequence in W converging to z. Pick an point sk in S with z k E W,, for each k . On the one hand, if the set { S k : k < wg } fl si is infinite for some i, then z is a member of cl (U{ W , : s E S i } ) and thereby, from cl (U{ W , : s E Si }) = U{ W, : s E Si } C W , is a member of W . And on the other hand, if the sets { s k : k < wo } n are finite for every i, then it follows that lim ,+-too d ( z k , S k ) = 0 and consequently lim k - - + m S k = z because lim k-,W z k = z. Since S is closed, we have z E S C W . Thus W is closed. Let U and V be disjoint open sets in A with F c U and G c V such that S = A \ ( U U V ) . Then U’= U U W and V’ = V \ W are disjoint open sets of A such that A = U ’ U V‘ with F c U‘ and G C V‘. This is contrary to the choice of F and G. So Ind A = 1. The proposition easily follows.

si

It was remarked in Section 11.10 that P-ind # P-Ind held when P was an absolute Bore1 class a in the universe M . The next example shows this to be the case. 1.16. Example. For every countable ordinal number a there

2. NORMAL FAMILIES

139

exists a space X in M such that A(a)-ind X

< A(a)-IndX and M(a)-ind X < M(a)-Ind X.

Such spaces have been already exhibited for a = 0. We shall construct the others by employing decompostions of Roy’s space A that were discussed above. The space A can be decomposed into disjoint subsets A and B with the property that each of these sets intersects every Cantor set contained in A. Let X be the subspace A . Suppose that G is an A(a)-kernel of X . Then, with 2 being the set in Proposition 1.14 that corresponds to E = A \ G, we have X \ G = and hence Ind (X \ G) = 1. Therefore, A(a)-Sur X = 1. From formula (7) of Section 11.10 we have A(a)-IndX = 1. For M ( a ) , a 1, let X = B. We infer from Lemma 1.7.5 that M(a)-DefX = 1. Formula (7) of Section 11.10 yields M(a)-IndX = 1. Finally we have A(a)-ind X = 0 and M(a)-ind X = 0 from Proposition 11.2.3.

>

In view of the above example, it now follows from Proposition 1.13 that when the universe is M the point addition theorem for the small inductive dimension modulo a class (Theorem P) fails for every absolute Borel class a. Consequently, Theorem S fails for every absolute Borel class a by Proposition 1.10, and Theorem A fails for all absolute Borel classes except for the absolute Borel class A(0) = { S} by Proposition 1.11 and Theorem 1.9. 2. Normal families

The inductive theory of dimension naturally derives from { S} two sequences of new classes, namely the classes { X : ind X 2 n } and { X : Ind X 5 n } for n in N. Each of the two partitioning operations of inductive dimension creates a new class from the previous class in the sequence. The study of normal families investigates those properties of the original class that are inherited by the new class, especially the properties of closures under subsets and under sums, that is, unions. In fact, normal families are natural for modelling inductive proofs of these properties. (See the proof given in 11.10.3 of Theorem 11.10.1 for an example of such an argument.) A very successful theory exists when the universe is Mo or M . The aim of this section is to develop a parallel theory for functions of inductive dimensional type in the universe Mo. The proofs of the sum and

140

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

other theorems promised in Chapter I will be provided by means of normal family techniques. The definitions of this section will be given in the general universe U since they will be used in the next section where an optimal universe called the Dowker universe will be constructed. (Of course, we mean optimal in the sense that the subspace theorem and the sum theorems will hold.) The two partitioning operations of the inductive approach t o dimension theory will be formally defined next. But before giving the definitions we note that for each class of spaces P the definitions of P-ind X and P-Ind X apply to any space X in the class 7.The same will be true for the definition of the partitioning operations.

P be a class. (i) The operation H assigns to P the class H[ PI consisting of all

2.1. Definition. Let

spaces X in 7 with the property that for each point p of X and for each closed set F with p @ F there is a partition S between p and F such that S is in P . (ij) The operation M assigns to P the class M [ P ] consisting of all spaces X in 7 with the property that for each pair of disjoint closed sets F and G there is a partition S between F and G such that S is in P. Obviously one can iterate these operations. Indeed, one sees immediately that

H " [ P ] = { X : P - i n d X 5 n - I}, M"[ P ]= { X : P-Ind X 5 n - 1 },

N, n E N,

nE

where, by definition, Ho[ PI = Mo[ PI = P . The next proposition has a straightforward proof. 2.2. Proposition. The following are true.

(4 H P I 2 M P l * (b) IfP c &, then H[P]C H [ Q ] and M [ P ] C M [ (c) H [ P n U ] n U = H [ P ] n U a n d M [ P n U ] n U = M [ P ] n U .

&I.

Observe that only (c) required the universe to be closed-monotone. The proof of the next theorem is related to those found in Proposition 11.2.7 and Theorem 11.2.16.

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141

2.3. Theorem. The classes H[P ] and M[PI are closed-monotone in U .

Proof. We shall give only the proof for the M operation since the proof for the H operation is no harder. Suppose that the space X is in M [ P ] and Y is a closed set of X . Let F and G be disjoint closed subsets of Y . Since Y is closed in X , the sets F and G are also closed in X. From the definition of the M operation there is a partition S in X between F and G such that S is in P . Because the universe is closed-monotone, the space S is also in the universe. Since P is closed-monotone in U ,the space S n Y is in P . Thus we have shown that Y is in M[PI. The corresponding monotonicity theorem requires more than the obvious change of hypothesis in the case of M[PI. But the following is true for the H operation. Theorem. For a monotone or open-monotone universe, let P be respectively a monotone or an open-monotone class in U. Then the class H[ PI is also respectively monotone or open-monotone 2.4.

in U.

Proof. Suppose the universe is open-monotone and the class P is open-monotone U. Let X be in H [ P ] and Y be an open set of X . Consider a point y of Y and a set F that is closed in the subspace Y with y $! F . The point y is not in cl ( F ) . From the definition of the H operation there is a partition S in X between y and c l ( F ) such that S is in P . We have that S n Y is in P n U.Consequently we have Y is in H[ PI. The proof for the monotone case is an easy modification of the above. In the same vein as the proof of Theorem 11.2.21 we have the following theorem concerning ambient spaces. (Recall that the universe is contained in the class NH of hereditarily normal spaces.) Theorem. Suppose that Y is a subspace 0f.X.If Y is in M[P ] then for each pair of separated subsets A and B of X such that clx(A) n clx(B) n Y = 0 there exists a partition S between A and B in X such that S n Y is in P . 2.5.

It has already been shown in Section 1 that closed-additivity of the class K: is not preserved by the H operation in the universe Mo. Also in [1949] Lokucievskii gave an example of a compact space X

142

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

with I n d X = 2 such that X is the union of two closed subspaces each with Ind less than 2 (see also Engelking [1978]). That is, closedadditivity is not preserved by the M operation as well. Consequently one must impose more conditions on the class P to assure that these partitioning operations will preserve the various additivity properties defined in Section 1. We begin by defining other forms of closedadditivity for a class P . 2.6. Definition. A class P is said to be countably (locallyfinitely, a-locally finitely) closed-additive in U when the following statement (respectively) is true: If V is a countable (locally finite, a-locally finite) closed cover of a space X in the universe with V C P , then X is in P .

Using these newly defined forms of closed-additivity, we now define normal and related families in the universe. 2.7. Definition. A class P is said to be a normal family in U if P satisfies

N1: P is monotone in U , N2: P is a-locally finitely closed-additive in U.

A class P is said to be a semi-normal family in U if P satisfies S1: P is closed-monotone in U , S2: P is a-locally finitely closed-additive in 2.4.

A class P is said to be a regular family in U if P satisfies R1: P is closed-monotone in U , R2: P is countably closed-additive in U. A class P is said to be a cosmic family in U if P satisfies C1: P is closed-monotone in U , C2: P is locally finitely closed-additive in U. The conditions S1, R1 and C1 are redundant in view of the agreement made at the beginning of the chapter. We have repeated them for emphasis.

Remark. We have come t o the last of the definitions that are dependent on the universe of discourse. From here on we shall suppress the “in U” whenever the context of the discussion will permit it.

2. NORMAL FAMILIES

143

There are obvious relationships among the various families. In particular, a class that is both regular and cosmic is necessarily semi-normal. And a semi-normal family is both regular and cosmic. In the universe M one can easily verify that the absolute Borel classes A ( a ) are examples of semi-normal families for every (Y and that the absolute Borel classes M ( a ) are examples of cosmic families for (Y 2 1. Historically, normal families were defined to be subclasses of Mo that are monotone and countably closed-additive. Later, normal families were defined precisely as above for the universe M of metrizable spaces. One can easily show for spaces in the universe Mo that locally finite collections are countable ones. Therefore regular families are semi-normal in this universe. Indeed, in the universe Mo a class P is a normal family if and only if P is monotone and countably closed-additive. The operations H and M and the notion of a normal family were introduced to give an elegant proof of the sum and decomposition theorems in the universe M . For now we shall concentrate on the separable metrizable case t o provide the proofs that were promised in Section 1.3. (The decomposition theorem for general metrizable spaces will be discussed in Section V.2.) The universe will be Mo for the remainder of the section. 2.8. Theorem. In the universe M o the class M [ PI is a normal family whenever P is a normal family.

Proof. (Compare the proof to that in Section 11.10.3.) Let us show that M[ P ] is monotone. To this end let us first prove that it is open-monotone. Suppose that X is in M [ P ] and Y is an open set of X . Choose a metric d on X that is bounded by 1. With f(z) = d ( s , X \ Y ) ,for each k in N let H k = f-' [ ( 2 - k - 2 , 2 - k + 2 ) ] and I k = f-' [ [2-"-', 2-k+1]]. Then the collections { H k : k E N} and { I k : k E N } are respectively open and closed in the space X , are locally finite in Y and are covers of Y . Clearly, I k c H k c Y . Now let F and G be disjoint sets that are closed in Y . For each natural number k let F k = F n I k and Gk = \ f f k ) U G. The sets Fk and G k are disjoint closed subsets of X . So there is a partition s k between Fk and G k in such that s k is in P . w e have that A4 = sk : k E N } is in P because P is a regular family. Let uk and v k be disjoint open sets with Fk c u k and Gk c v k

(x

u{

x

144

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

s

such that x \ S k = U!, U v,. Then denote by the set Y \ ( U U V) where U = U{ U k : k E M } and V = v k : k E M}. As the collection { H k : k E N} is locally finite in Y , the set V n Y is open. One easily sees that S is a closed subspace of M and hence S is in P because P is closed-monotone. As 5’ is a partition between F and G in Y , we have shown that M [ P ] is open-monotone. Let us now show that M [ P ] is monotone. Suppose that X is in M [ P ] and Y is a subset of X. Let F and G be disjoint sets that are closed in Y . Clearly, F and G are separated sets in the open subspace 2 = X \ (clx(F) n clX(G)). Since 2 is in M[PI by the first part of the proof, we have by Theorem 2.5 that there is a partition S’ between F and G in 2 such that S‘ is in P . Consequently S = S’ n Y is a partition between F and G in Y . Finally S is in P because P is monotone. It remains to be shown that M [ P ] is countably closed-additive. Suppose that X = XI, : k E N}, where each set x k is closed in X and is a member of M [ PI. There is no loss in generality in assuming X O = 8. We have from the open monotonicity of M [ P ] that the set = Xk \ xj : j < k } is also in M [ P ] , Consider disjoint closed sets F and G of X . Let A0 and BO be open sets with disjoint closures such that F C A0 and G C Bo. By Theorem 2.5 there is a partition 5’1 between A0 and BO such that L1 = S1 n K1 is in 7‘. Let U1 and V1 be open sets in X1 such that X \ S1 = U1 U V1 with A0 n XI C U1 and Bo n X1 c V1. Then the sets F1 = A0 u U1 and GI = BOU V1 are separated and clx(F1) n clx(G1) C L1. Now select disjoint open sets A1 and B1 such that F‘1 c Al, G1 c B1 and clx(A1) n clx(B1) c L1. Obviously, XI \ (A1 u B1) c L1. By Theorem 2.5 there is a partition 5’2 between A1 and B1 in X such that L2 = S2 n K2 is in P. We have that L1 U L2 is in P because L2 is a countable union of closed subsets and P is countably closedadditive. Repeating the above construction, we have disjoint open sets A2 and B2 satisfying the inclusions A1 C A2 and B1 c B2 and such that (XI U X,) \ (A2 U B2) c L1 U L2. Continuing inductively, we have a sequence of disjoint open sets Ak and Bk and F,-sets Lk in P such that

n{

u{

u{

G C Bk C Bk+l, u{xj :j5 k}\(AkU Bk) C Li : i 5 k } , f’ C Ak C Ak+i

and

u{

The set L =

lc E N,

k

E

N.

u{ Lk : k E N} is in P because P is countably closed-

2. NORMAL FAMILIES

145

additive. Let S = X \ U{ (Ak U B k ) : k E N}. Then S is partition between F and G in X and S C L . By the closed-monotonicity of P we have S E P. Thus we have shown that X is in M [ PI. 2.9. Theorem. Suppose that P is a normal family in the universe Mo. The following are equivalent for spaces X in the universe. (a) X E M[P]. (b) X = P U N , where P E P and ind N 5 0. (c) X = P u 2, where P E P and Ind 2 5 0.

Proof. Let us show that (a) implies (b). Let B = { Ui : i E N } be a countable base for the open sets of X . Consider the countable family { (Cj, Dj): j E N} of all pairs of elements of B such that cl ( C j ) c Dj. For each j there is a partition Sj between cl ( C j ) and X \ Dj in X with Sj E P. The set U{ Sj : j E N } is in P because P is a normal family. Obviously, ind (X \ U{ Sj : j € N }) < 0. That (b) implies (c) will follow from the identity H[ { 8 ) J nMo = M[ { 0}] f l Mo. The proof of this identity is the same as the one for the inductive step of Theorem 1.4.4 and will not be repeated here. Indeed, the proof of this identity is much less complicated than that of the inductive step because the open sets will have empty boundaries. Finally, that (c) implies (a) follows easily from Proposition 1.4.6. 2.10. Theorem. In the universe Mo of separable metrizable spaces the dimension Ind satisfies the countable sum theorem and the decomposition theorem.

Proof,As the class { 0 } is a normal family, the classes M"[ P ] are also normal families by Theorem 2.8. We have that Ind X 5 n - 1

if and only if X E M"[{ 0) 1. So the countable sum theorem for Ind follows. Also from the equivalence of (a) and (c) of Theorem 2.9 we have the statement: X E M " [ { 0 } ] i f a n d o n l y i f X = Y U Z whereY E Mn-'[{O}] and Ind 2 5 0. The decomposition theorem for Ind now follows. It has been shown already in Theorem 1.4.4 that ind and Ind coincide for separable metrizable spaces. The proof given there was based on the countable sum theorem (which was assumed to have been proved) together with the construction of a specific locally finite

146

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

cover. We shall give here a somewhat more elegant proof of the same theorem by using the decomposition theorem. The proof is a simple consequence of normal family arguments. 2.11. Theorem. For each separable metrizable space X ,

ind X = IndX.

Proof. We already know that H"[ { S } ] 3 M"[ { S}]. We shall prove the reverse inclusion. The reverse inclusion is true for n = 0 by definition. For the inductive step we let X E Hn[{ S}]. Then we infer the existence of a countable base B = { Ui : i E N} for the open sets of X such that B ( U i ) E Hn-l[ { S}] for every i. From the induction hypothesis we have { B ( U i ) : i E N} C Mn-'[ { S}]. As Mn-'[ { S}] is a normal family, we have P = U{ B (Ui) : i E N} is in M"-'[ { O } ] . Moreover, ind ( X \ P ) 5 0. So by Theorem 2.9 we have X E Mn[{ S}]. Combining Theorems 2.10 and 2.11, we now have the promised proofs of Theorems 1.3.7 and 1.3.8. 3. Optimal universe

The following agreement will be in force for this section. Its content will facilitate the investigations of the properties of the Dowker universe which will be defined later (see Definition 3.22). The focus will be the "optimal" property of the Dowker universe. Agreement. In addition t o the agreement made in the introduction to the chapter, the universe is open-monotone.

The utility of normal families was illustrated in the last section for the universe Mo of separable metrizable spaces. The separability reduced the considerations to monotone regular families. For the universe M of metrizable spaces the full force of normal families is needed to derive the sum theorems of dimension theory. As we shall discover in this section, the development of a sum theorem when the universe is larger than M will require much more than the techniques found in the previous section for separable metrizable spaces. The main task is to find a universe in which the class M[P ] will be open-monotone in 24 (which is contained in the class NH of hereditarily normal spaces by agreement) whenever P is. The operation S

3. OPTIMAL UNIVERSE

147

is introduced to determine a necessary and sufficient condition for the open monotonicity of M[P]. The techniques of this section are essentially those used by Dowker [1953] in his successful study of the class NT of totally normal spaces. (A space X is totally normal if it is a normal space with the property that every open set U is the union of a locally finite in U collection of open &-sets in X . ) The class NT of totally normal spaces includes the class Np of perfectly normal spaces and hence the class M of metrizable spaces. Here we shall use normal families techniques instead of induction which was used by Dowker. The advantage of this approach is that one can isolate those properties of a family which are needed for the inductive step to preserve the sum and subspace theorems on applying the M operation to a normal family. Among these properties are open monotonicity and strong closed additivity. (Both properties are free for the universe M . ) The end result of the investigation is the creation of an optimal universe, called the Dowker universe, in which a satisfactory theory of P-Ind can be developed. The notions of normal and related families and the two operations H and M were defined in the previous section. A class P need not be a normal family in the universe. An obvious procedure would be the expansion of the class to a normal family. This is our first definition. The operation defined here will be used later to create the Dowker universe. 3.1. Definition. For classes P and Q the normal family extension of P in & is the class of spaces X in Q for which X = U F for some subcollection F of P such that F is a a-locally finite closed cover of X. This class of spaces is denoted by N [ P : Q ] .

Obviously one has the inclusions

Pn Q c N [ P : Q ] c &, and, when P is a normal family in the universe,

PnU = N[P:U]. Moreover,

N [ P : U ] c N [ Q : U ]when P c Q . Let us now prove a key relationship between cosmic families and cozero-sets of a space X . The following proof is reminiscent of the

148

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

proof that cozero-sets of a normal space are normal subspaces. Its proof has been foreshadowed in the first part of the proof of Theorem 2.8. 3.2. Theorem. Suppose that P is a cosmic family. Then every cozero-set of a space X in M [ PI is also in M[PI.

Proof. Let Y be a cozero set of X . There is a continuous function f on X into [0,1] such that Y = f - l [ (0,111. For each natural number k, let H k be the open set f - l [ (2-"2, 2-k+2)] and 11,be the closed set f - l [ [2-"-', 2-k+1] 3. The collections { H I , : k E N } and { I I , : k E N} are covers of Y that are locally finite in the subspace Y . Clearly, 1, C HI, C Y . Let F and G be disjoint sets that are closed in Y . For each natural number k, let F k = F f l I,, and G k = ( X \ H I , )U G. The sets F k and G k are disjoint closed sets of X . s o there are partitions s k between Fk and G k in such that s k is in P. Let u k and v k be disjoint open sets such that \ s k = u k u v k , F k c u k and GI,C VI,. Clearly, SI, c H k . So U{ s k : k E N} is in P because P is a cosmic family. It can be shown that U = U k : k E N } and V = Y n n { VI,: k E N} are open sets. Then S = Y \ ( U U V ) is a partition between F and G in Y such that S is in P .

x

x

u{

We shall generate special open sets of a space X , to be called Dowker-open sets, by employing unions of point-finite collections of cozero-sets of X . Before proceeding along this direction we shall give a lemma on point-finite covers of a general topological space. (The collections that are being considered are indexed ones.) 3.3. Lemma. Let { Us : s E S } be a point-finite open cover of a topological space Y . For each positive integer i, denote by K ; the set of all points of the space Y which belong to exactly i members of the cover { Us: s E S} and by T;the family of all subsets of S that have exactly i elements. Then

(a) Y = U{ Ki : i 2 1} , (b) Ki n K j = 8 for i # j , (c) F i = U{ K j : j 5 i} is closed for i 2 1, (d) K;= U{ KT : T E T i } , where thesets K T , defined by letting KT = Ki n U , : s E T } for T E T i , are open in K ; and pairwise disjoint.

n{

3. OPTIMAL UNIVERSE

149

Proof. The equalities (a) and (b) follow directly from the definition of the sets Ki. For statement (c), observe that if z @ Fi, then z E Us, n Us, n - - n Us;,,c X \ Fj, where s1, s2,. .., si+l are distinct elements of S. To establish (d) it will suffice to note that KT C Ki for T in Ti and that whenever T and TI are distinct members of T;the set KT n K p is the empty set since the union T U TI contains at least it 1 elements of S.

-

We shall now resume our development of normal families by defining Dowker-open sets of a space X . 3.4. Definition. A subset U of a topological space X is called Dowlcer-open in X if it is the union of a point-finite collection of cozero-sets of X.

Obviously, if Y is a subspace of X and U is Dowker-open in X, then the trace of U on Y is Dowker-open in Y . 3.5. Lemma. Suppose that P be a strongly closed-additive seminormal family. Let Y be Dowker-open set of a space X in M [ PI. Then Y is in M[ PI.

Proof. Suppose that { Us: s E S } is a point-finite collection of cozero-sets of X whose union is Y . The notation of Lemma 3.3 will be used. Let us first show that KT is in M [ P ] for each T in Ti. The set X i = ( X \ Y )U Fi is closed in X . From the equalities Ii'i = Fi \ Fi-1 = Xi n (Y \ Fi-1) and the property (d) of

Lemma 3.3 we infer that the set KT is open in Xi. Clearly we haven{Us:sET}nFi-l =O. AlsothesetXinn{U,:sET}= Fi n Us : s E T } is an F,-set of X i . So KT is a co-zero set of X i . As X i is a closed set, it is in M [ P ] . By Theorems 2.3 and 3.2, we have that KT is in M [ P ] . As a consequence of this fact, we have from property (d) of Lemma 3.3 that K i is an open set of X i that is in M[ PI because a semi-normal family is a cosmic family . We are ready to use the strong closed additivity of the class P . At this juncture, only the properties of regular families are used. Let F and G be disjoint sets that are closed in Y . By Theorem 2.5 there is a partition SO between F and G in Y such that LO = KOn SO is in P. There are disjoint open subsets UL and Vd of Y such that F C UA, G c Vd and Y \ So = UA U Vd. Since Y is in N H ,there are open sets UO and Vo in Y such that FO = (VA n K O )U F C V O , GO= (Vd n K O )U G c VOand cly(U0) n cly(V0) c LO. Theorem 2.5

n{

150

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

yields a partition S1 between UO and VOin Y with Li = Ir'l n S1 in P . Obviously, K O n S1 c Lo. Since P is strongly closed-additive, we have that L1 = Lo U Li is in P . Inductively, one constructs sequences of disjoint open sets U i , Vi and closed sets Li that are in P such that

ui c Ui+l, K c K+l, F C

u=

Li c Li+l, U{ Ui : i E N}, G C V = U{K : i E N}, Y \ U{ Li : i E N } = u u

v.

Since S = U{ Li : i E N } is closed in Y , we have that S is in Thereby we have shown that Y is in M[PI.

P,

In [1953] Dowker introduced the class of totally normal spa,ces. Subsequently several generalizations of totally normal spaces have appeared. (See Section 6 for further references.) In [1978] Engelking followed the methods of Dowker to define the class of strongly hereditarily normal spaces. We shall give the definition of this class next. The definition finds its origin in the following characterization of hereditarily normal spaces:

A Hausdorff space X is hereditarily normal if and only if for each pair of separated sets F and G in X there is a pair of disjoint open sets U and V such that F c U and G c V . This last condition can be restated in terms of partitions as:

For each pair of separated sets F and G in X there is a partition S between F and G in X . 3.6. Definition. A space X is called strongly hereditarily normal if it is a Hausdorff space such that for each pair of separated sets F and G in X there is a pair of disjoint Dowker-open sets U and V such that F c U , and G c V . The class of strongly hereditarily normal spaces will be denoted by &. The class & is easily shown to be absolutely monotone. Clearly each strongly hereditarily normal space is hereditarily normal. The following equivalence is easily proved.

3.7. Theorem. A space X is strongly hereditarily normal if and only if X is a Hausdorff space such that for each pair of separated

3. OPTIMAL UNIVERSE

151

sets F and G in X there is a partition S between F and G in X such that X \ S is a Dowker-open set of X . The last theorem suggests another operation on classes of spaces based on the concept of partitions that is complementary to that of the operation M. More precisely, we have the following definition. 3.8. Definition. Let P be a class of spaces. The operation S assigns t o P the class S [ P ] consisting of all topological spaces X such that for each pair of separated sets F and G in X there is a partition S between F and G in X with X \ S in P.

Suppose that a space X is in S[PI. Since F = 8 and G = X are separated sets in X , we have that X is in P . Thus we have

S[P ] c P . 3.9. Theorem. For each class P , the class S[PI is closed-monotone in U.

Proof. Let Y be a closed subspace of a space X in S [ P ]flu. Let F and G be separated subsets of Y . As they are also separated in X , there is a partition S between F and G such that X \ S is in P. As the universe is open-monotone, X \ S is in U. Then the set S n Y is partition between F and G in Y such that Y \ S is in P n 1A because the class P is closed-monotone in U.

There is a simple property one can derive for the S operation applied to the universe. Clearly, S [ U ] c U . Even more, as the universe is open-monotone, we have

S [ U ]= U

c NH.

We shall create a new universe from the universe U. Obviously the universe is a class. In this context, consider the normal family extension N[U : N H ]of U in the class N H . 3.10. Theorem. The class N [ U :NH] is absolutely open-mon-

otone and absolutely closed-monotone and

S[N [ U : N H ] = ] N [ U :NH]. Proof. Let Y be an open subset of a space X in N[U :N H ] There . is a a-locally finite closed subcollection F of U such that X = U F .

152

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

Since U is open-monotone, the collection { F n Y : F E F } is also a subcollection of U that is a a-locally finite closed cover of the subspace Y. That is, Y is in N [ U :N H ] . The proof is the same for closed-monotone. The remainder of the proof follows from the properties of the operation S listed above. 3.11. Theorem. For any class Q ,

S[QnU]=S[Q]nU. Proof. Clearly, S[Q n u ] c S[ Q ] n S [ U ]c S[Q ] n U . For the other inclusion suppose that X is in S[Q ]n U and let F and G be separated subsets of X. Then there is a partition S between F and G in X such that X \ S is in Q. Also X \ S is in U because U is open-monotone. Therefore X is in S[Q n U ] . The last two theorems show that N and S operations behave nicely. The next theorem shows the connection between open monotonicity and the operations M and S. 3.12. Theorem. Suppose that the class P is open-monotone. Then S [ M[ P I ] is open-monotone. Moreover,

S [ M[PI

] n U = { X : Y E M [ PI n U

for each open set Y o f X }.

Proof. In view of the last theorem the inclusion of the right-hand collection of the equation in the left-hand one is easily established. We shall prove the opposite inclusion. Let Y be an open subset of a s p a c e X i n S [ M [ P ] n U ] Wemust . s h o w y E M [ P ] n U . Let F and G be disjoint subsets of Y that are closed in Y. Then F and G are separated in X . Since X is in S [ M [ PI there is a partition S between F and G in X such that X \ S is in M[ PI. Let U and V be disjoint open sets with F c U , G c V and X \ S = U U V . Since F c U and F is closed in the space Y , we have that F and Y \ U are separated sets in X and that c l x ( F ) n clx(Y \ U ) n (X\ S ) = 0. By Theorem 2.5 there is a partition L between F and Y \ U in X such that L n ( X \ S) is in P. Clearly L is a partition between F and G in X . Since L n (Y \ U ) = 0, it follows that L n Y = L n Y n U . And so we have Y n L n (X\ S ) = ( L n Y n U ) u ( L n Y n V ) = L n Y n U = Y n L. Because P is open-monotone, we have Y n L is in P . Thus we have shown that Y is in M [ P ]and the opposite inclusion is now established. That S [ M [ PI n U ] is open-monotone follows immediately from the equality of the two collections.

1,

3. OPTIMAL UNIVERSE

153

3.13. Theorem. For each P that is open-monotone, the class M [ P ] is open-monotone i f a n d only if S [ M [ P ] ]n U = M [ P ] nu.

Proof. If M[ PI is open-monotone, then the class M[ PI n U is absolutely open-monotone by definition. Consequently the equality s[M [ P ] n U ]= M[PI n U holds. The converse statement follows from Theorem 3.12. From open monotonicity we go next to monotonicity. 3.14. Theorem. Suppose that the universe is monotone and the class P is monotone. Then the following statements are equivalent for every space X . (a) I f Y i s a s u b s p a c e o f x , t h e n Y i s i n M [ P ] n U . (b) If Y is an open subspace ofX, then Y is in M [ PI nU. Consequently,

S [ M [ P ]n U ]= { X : Y E M[PI n U for each subspace Y of X } and hence the class

S [ M [ PI]is monotone.

Proof. Clearly (a) implies (b). Assume (b) holds and suppose that Y is a subspace of a space X . We must show that Y is in M [ P ]n U. Let F and G be disjoint subsets of Y that are closed in Y. The sets F and G are then separated in X and 2 = X \ (clx(F) n clX(G)) is in M[P ]n U. By Theorem 2.5 there is a partition S between F and G in 2 such that S is in P. From the absolute monotonicity of P n U and the inclusion Y c 2 we have that S n Y is a partition between F and G in Y such that S n Y is in P n U. Thus Y is in M [ P ]i l U and (a) holds. The last statement of the theorem is a consequence of Theorems 3.11 and 3.12. The next theorem concerns the strongly closed-additive property in the universe. 3.15. Theorem. Suppose that P is open-monotone and strongly closed-additive. Then S [ M[P I ] is open-monotone and strongly closed-additive.

Proof. We have from Theorems 3.11 and 3.12 that S [ M [ P ] ] is open-monotone. Let us show that S [ M[PI ] is strongly closed-additive. A s s u m e t h a t X = Y U Z w i t h Y a n d Z i n S [ M [ P ] ] n U a n d Y

154

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

closed in X. Since S [ M[ P I ] is open-monotone, we may assume further that Y and 2 are disjoint, Let F and G be disjoint closed sets in X . By Theorem 2.5 there is a partition So between F and G such that SonY is in P . Let UOand Vo be disjoint open sets in X such that F ~ U o , G c V o a n d X \ S o = U o u V o W . ithF1 = ( Y n U o ) U F and G1 = (Y n Vo) U G we have from Theorem 2.5 a partition S1 between F1 and GI in X such that 5’1 n 2 is in P . Clearly S1 n Y is contained in SOn Y . So S = 5’1 U (So n Y ) is a partition between F and G in X. By the strongly closed-additivity of P we have that S is in P . Therefore X is in M[ PI n U. Moreover, if 2 is an open subspace of X , then ? = 2 n Y and 2 = 2 n 2 are in S[ M [ P ] ] and ? is closed in 2.Since U is open-monotone, we have that 2 is also in M [ P ] . By Theorem 3.12 we have that X is in S [ M [ P ] ] . To prepare for the theorems concerning normal families and its variants, the concept of strong closed additivity will be generalized to a well-ordered collection { X p : ,8 < a } of subsets of a space X in U. The natural requirement on this collection is that U{ X, : y < p } be closed in X whenever ,8 5 a. In general, some further condition must be imposed. For the study of normal families there are two such conditions. The first is that a 5 0 0 , where W O is the first infinite ordinal number. And the second is that { X p : ,8 < a } be a locally finite collection in X . The first condition is related to regular families and the second is related to cosmic families. Clearly the cases for the ordinal numbers a = 0, 1 and 2 are either trivial or the case of strong closed additivity. We shall begin with the regular families in U. 3.16. Theorem. Suppose that P is an open-monotone, strongly closed-additive regular family. Then S [ M [ PI ] is an open-monotone, strongly closed-additive regular family.

Only the condition R2 for regular families remains to be verified. Due t o Theorem 3.12, this is easily achieved from the next lemma. 3.17. Lemma. Let P be a strongly closed-additive regular family. Suppose that X is a space and K = { Ir‘i : i < o0} is a collection of subsets of X such that (a) Ki E M [ P ] n U for i < W O , (b) Fi = U{ ICj : j < i } is closed for i < W O , (c) ICi n K j = 8 for i # j ,

3. OPTIMAL UNIVERSE

155

(d) X = UK. Then X is in M(P].

Proof. The proof of the lemma is found in the last paragraph of the proof of Lemma 3.5. Next we shall prove the analogous statements for cosmic families in U. 3.18. Theorem. Suppose that P is an open-monotone, strongly closed-additive cosmic family. Then S [ M [ PI ] is an open-monotone, strongly closed-additive cosmic family. Only the condition C2 for cosmic families remains t o be verified. Again due to Theorem 3.12, this is easily achieved from the next lemma.

3.19. Lemma. Let P be a strongly closed-additive cosmic family. Suppose that X is a space and K = { ICp : p < a } is a locally finite collection of subsets of X such that (a) K p E M [ P ] nu for ,D < a , (b) Xp = U{ K , : y < p } is closed for p < a , (c) K p n K , = 0 for p # y, (d) X = UK. Then X is in M[PI.

Proof. The proof is by transfinite induction on a. Suppose for each p with /3 < a that the closed subspace X p is in M[P]. Let F and G be disjoint closed sets of X . One can inductively construct sets U,, V, and L, such that (1) L, is a partition between F, = F n IC, and G, = G n I{, in K,, and U, and V, are disjoint open sets of K , with F, C U,, G, c V, and K , \ L , = U, U V,, (2) for p < a the set Sp = U{ L, : y < p } is a partition between F n X p and G n X p in X p and the sets U{ U, : y < p } and U{ V, : y < p } are open in Xp. The construction is straightforward because the collection K is locally finite in Let S be the union U{Sp : p < a } . From (1) and (2) and the local finiteness of the collection K we have that S is partition between F and G in X. To verify that S is in P let us first verify inductively that each Sp is in P. Suppose that Sb is

x.

156

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

in P w h e n 6 is less than p and let y be any ordinal number less than p. Then L, U S, is in P because P is strongly closed-additive. So c l ~ ( L , ) is in P . Since the collection K is locally finite and X p is closed, the equality Sp = U{ clx(L,) : y < /3} holds. So Sp is in P by the fact that P is a cosmic family. It is now a simple matter t o show that S is in P . The induction step for the ordinal number a is now completed and the lemma is proved. Combining Theorems 3.16 and 3.18, we have the following theorem for semi-normal families in U. 3.20. Theorem. Suppose that P is an open-monotone, strongly closed-additive semi-normal family. Then S [ M[P ]] is an open-monotone, strongly closed-additive semi-normal family. From Theorems 3.14 and 3.20 the following normal family theorem will result. 3.21. Theorem. Suppose that the universe is monotone and that P is a strongly closed-additive normal family. Then S [ M[PI ] is a strongly closed-additive normal family.

It is to be noted that a semi-normal family P is necessarily openmonotone and strongly closed-additive whenever the universe is contained in the class M of metrizable spaces. And when M is replaced by the class Mo of separable metrizable spaces, the semi-normal families are necessarily regular families. The operation S was used to single out the part of the class M[PI which is open-monotone (see Theorem 3.12). Indeed, in the development of normal families and its variants it was the class s [ M[PI] that inherited open monotonicity. The class & of strongly hereditarily normal spaces uses the Dowker-open sets in conjunction with the operation S. From Lemma 3.5 we see that the Dowker-open sets are, in a sense, the largest collection of open sets that contains the cozero-sets and are related to a-locally finite unions of closed sets, a property of semi-normal families. Unfortunately the class & is not closed-additive in the universe NH of hereditarily normal spaces. The next definition will correct this deficiency, thereby creating a universe which is, in a sense, optimal with respect to cozero-sets and normal families.

3.22. Definition. The Dowker universe 27 is the normal family extension of the class & in N H .That is, D = N[ &: NH1.

3. OPTIMAL UNIVERSE

157

Clearly the inclusions & C 'D C NH hold, and by the proof of Theorem 3.10 the universe 2, is absolutely monotone. 3.23. Theorem. Suppose that the universe is contained in the Dowker universe. Let P be an open-monotone, strongly closedadditive semi-normal family. Then M [ PI is also an open-monotone, strongly closed-additive semi-normal family.

Proof. From Lemma 3.5 and the proof of Theorem 3.12 we have that M[F ]n & n U is absolutely open-monotone. Let us show that M[P ] n U is absolutely open-monotone. Assume X E M [ P ] n U. Then X = U X,where X is a a-locally finite closed cover of X such that X c &nu. Let X = U{ X, : n = 1 , 2 , . . .} where X, is locally finite in X . Because M[PI n & n U is absolutely open-monotone, we have from Lemma 3.19 that X , = U X, is in M [ P ] n U. Indeed, for each open subspace Y of X the set Y n X , is in M[PI n U. So X is in M[PI by Lemma 3.17. Again each open subspace Y of X is in M[PI nu. Thus we have that M[PI n U is absolutely openmonotone. Theorem 3.13 gives s [ M [ P ] ] n U = M [ F ] n U ;and, Theorem 3.20 completes the proof. Observe at this point that the open-monotone and strongly closedadditive hypotheses on the class P are redundant for the universe Np of perfectly normal spaces since each open set of a perfectly normal space is a cozero-set. The last theorem and Theorem 3.14 give the following one for normal families. 3.24. Theorem. Suppose that the universe is monotone and is contained in the Dowker universe. Let P be a strongly closedadditive normal family. Then M[ P] is also a strongly closed-additive normal family.

With P = { 0) we now have the extension of the theorem of Dowker 119531 from the universe NT of totally normal spaces to the Dowker universe 2). 3.25. Theorem. In the Dowker universe D,the strong inductive dimension Ind has the following properties: (a) Monotone: If Y c X , then Ind Y 5 Ind X . (b) Countably closed-additive: If X = xk : k 2 1 } where each xk is closed, then I n d X is the supremum of the set {IndXk: k 2 1 ) .

u{

158

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

(c) Locally finitely closed-additive: IfX = (JF for a locally finite closed collection F in X , then Ind X is the supremum of the set { Ind F : F E F } . (d) Strongly closed-additive: If X = F U G and F is closed, then Ind X is the maximum of Ind F and Ind G. (e) Additive: Ind (X U Y ) 5 Ind X Ind Y t 1.

+

A consequence of the last theorem is that if a space X in the Dowker universe is such that it is the union of n 1 subspaces with Ind not exceeding 0 then I n d X 5 n. This is one-half of the decomposition theorem in the universe M of metrizable spaces. The decomposition theorem with the additional condition of separability has been shown in Theorem 2.10. The theory of Ind in the universe M will be completed in Section V . l . The condition that P be a normal family can be replaced by cosmic family when the universe is contained in the class Np of perfectly normal spaces.

+

3.26. Theorem. Suppose that the universe is contained in the class Np of perfectly normal spaces. Let P be a strongly closedadditive cosmic family. Then P is open-monotone and M[PI is also an open-monotone, strongly closed-additive cosmic family.

Proof. Each open set of a perfectly normal space is a cozero-set. Consequently P is open-monotone. The remaining part of the theorem follows from Theorems 3.2, 3.12 and 3.18.

3.27. Example. Consider the universe M of metrizable spaces. The class C of complete metrizable spaces is a strongly closed-additive cosmic family that is not a semi-normal family. There are cosmic families that are not strongly closed-additive. In particular, the class L of locally compact spaces is one. We shall conclude the section with a theorem showing the coincidence of P-ind X and P-Ind X for a-totally paracompact spaces X. 3.28. Definition. A space X is said to be a-totally paracompact if for every basis B of X there exists a a-locally finite open cover U of X such that for each U in U there is a V in B such that U c V and B x ( U ) C Bx(V).

3. OPTIMAL UNIVERSE

159

3.29. Theorem. Suppose that the universe is contained in the Dowker universe. Let P be an open-monotone, strongly closedadditive semi-normal family. If X is a a-totally paracompact space, then P-ind X = P-Ind X . Proof. Only the inequality P-ind X 2 P-Ind X requires a proof. The proof is by induction on P-ind X. We shall provide the inductive step of the proof. Assume P - i n d X 5 n and let F and G be disjoint closed sets of X . Let Fo and Go be disjoint closed neighborhoods of F and G respectively. Then the collection B of all open sets V such that P-ind B x ( V ) < n holds and such that V n FO = 0 or V f l Go = 0 holds is a basis for the open sets of X. Since X is a-totally paracompact, there is a a-locally finite open cover U of X such that for each U in U there is a V in f? with U c V and B x ( U ) c Bx(V). The set A4 = U{ B x ( U ) : U E U } need not be a member of the universe. But nonetheless it is covered by the a-locally finite closed collection { B x ( U ) : U E V } whose members satisfy P-ind B x ( U ) = P-Ind B x ( U ) < n. Let U = U{ Uk : k 2 l } where each U k is locally finite. For each Ic we define the open sets

wok= U{ u E uk: U n Fo # 0 } , wlk= U{ u E uk: U n Fo = 0 }, and then form the open sets

Clearly, U{ W o k U W1k: : k 2 1 } = X . A simple calculation will show that @,oi n @1j = 0 for all i and j. So the open sets

EO

and G C @I hold. Deare disjoint and the inclusions F C note the set X \ (@o U El) by S. Let us show that S is contained in M . For z in S let i be the first integer such that z E Woi U Wli.

160

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Since z $ @oi U Eli, we have that z is in U{ B x ( U ) : U E Uj } for some j not greater than i. So x is in M . Since S is a closed set of X , we have that S is in U. From S = U{ S n Bx(U) : U E V}, we have that P - I n d S < n by Theorem 3.23. Thus we have shown that P - I n d X 5 n. The definition of a-totally paracompactness appears t o be rather technical. It can be shown that strongly paracompact spaces are a-totally paracompact. A Hausdorff space is strongly paracompact if each of its open cover has a star-finite open refinement. 4. Embedding theorems

Many axiom schemes for the dimension function have utilized universal n-dimensional spaces in the separable metrizable space setting and universal O-dimensional spaces in the general metrizable space setting. To prepare for the discussion in the next section on axioms for dimension, we shall give a brief account of the relevant embeddings. The first embedding theorem will use the function space (R2n+1)X and the notion of &-mapping.

4.1. Definition. Let E be a positive number and let f : X + Y be a continuous map of a metric space X t o a topological space Y . Then f is said to be an &-mappingif diam ( f - l ( y ) ) < E for every y in Y . 4.2. Theorem. A compact metrizable space X with d i m X 5 n can be embedded into the subspace N;"+l ofW2"+l consisting of all points which have a t most n rational coordinates.

Proof. The complement R 2n+1 \ Nin+' = LZ:' is the union of a countable family of linear n-varieties in lR2"+l defined by the equations zil = T I , ziz = 7 3 , . . . , - ~ , + 1 , where T I , ~ 2 .,. . , r,+l are rational numbers and 1 5 il < i 2 < - - - < in+l 5 2n 1. Arrange this family of n-varieties into a sequence H I , H 2 , . .. , and denote by @i the subset of the function space (R2n+1)X consisting of all (l/i)-mappings whose values miss H i , i = 1 , 2 , . , . . Let us show that @i is a dense open subset of (W 2n+1)X. To prove that @ p i is open, consider an (l/i)-mapping f : X + 1W2"+l such that f[X] n Hi = 0. The closed subspace

+

A = { ( 3 , ~ ' E) X x X : d ( z , x ' ) 2 l / i )

4. EMBEDDING THEOREMS

161

is compact. Since IIf(x) - f( d)II > 0 holds for each pair there exists a positive number S such that

(2,~') in

A,

We may assume

6 < dist (f[X],Hi).

(2)

To see that f is an interior point of @i we observe that every continuous map g: x + R ~ with ~ \If +- gll~ < S/2 is a (I/i)-mapping with g[X] n Hi = 0, where Ilf - gll is the distance between f and g in the function space (R2"+1)x. Indeed, because llf(z) - f(z')ll < S follows from z E g-'[y] and z' E g-l[y], we have from (1) that every pair (z,~') of points of the set g-'[y] is not in A for any y. Thus, diamg-'[y] < l/i for every y in R2n+1,i.e., g is an (l/i)-mapping. The equality g[X]n Hi = 0 is easily proved by way of contradiction by virtue of (2). So, @; is open. It now remains to be shown that Cpi is dense in the function space (R2n+1)x. Consider a continuous map f : X -+ and a positive number E . The uniform continuity of f yields a positive number y smaller than l / i such that

Ilf(z) - f(z')ll

<~

/ 2whenever

d(z,z')

< y.

As dimX 5 n, there is an open cover U = { U j : j = 1 , .. . , r } of X such that (3) (4)

ord U 5 n diam(Uj)

< y,

+ 1, j = 1,..., T .

Consequently, (5)

diam ( f [ U j ] )< ~ / 2 ,

j = 1,. . .,T .

+

Select vertices p l , . .. ,p , in R 2n+1 in general position (i.e., no m 2 of the vertices p j lie in an m-dimensional linear variety of R2n+1, m = 0, 1, ... ,an) such that every n-dimensional simplex in B 2n+1

162

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

formed by these vertices is disjoint from the n-dimensional variety and

Hi

For each point z of X define wj(s) = d(z,X

\ Uj),j

= 1 , . .., T ,

and

~ ( s=)

wj(z)

and form the continuous map g : X -+ W2n+1 by the formula

Using (3)-(6), we will get after a straightforward computation that g is an (l/i)-mapping with Ilf - gll < E and g [ X ]n Hi = 0. We have now proved that @; is a dense, open subset of the function space (R2nt1)X. Since this function space is complete, the Baire category theorem yields that

As every map in CP is an embedding of X into N;"+l, the theorem is proved. It is to be observed that the above argument can also be carried out in the function space (l12n+1)x due to the convexity of 12n+1 in R2n+1. Consequently the set N:n+l of the theorem can be replaced by the set N:"+' n 12n+1.

It will be proved in Chapter VI that every finite dimensional separable metrizable space has a metrizable compactification of the same dimension (see Theorem VI.2.9). Thus we have the following theorem. 4.3. Theorem. A separable metrizable space has finite dimen-

sion if and only if it can be embedded into a finite dimensional Euclidean space. We shall next describe a construction due to Hayashi [ 19901 which will be used t o construct a universal n-dimensional space. The construction is similar t o that of the Menger sponge M," (see Engelking [1978], page 121). For each m-cell I" there are two construction

163

4. EMBEDDING THEOREMS

schemes. We shall call one the exclusion scheme and the other the inclusion scheme. The constructions are made on a base 6 procedure rather than the base 3 one used by Menger [1926]. To help the reader understand how the two schemes come into play, the construction will be illustrated in the 1-cell I1. In both constructions a Cantor set is built. As this special case is somewhat easier than the general one, the constructions can be made on a base 3 procedure. After completing the constructions for 11, we shall show how the base 6 procedure comes into the picture in 1 2 . For the exclusion scheme in the 1-cell I 1 let us consider a subset X of I 1 that is the union of a countable family of compact subsets Xi of the set P of irrational numbers. We want to construct a Cantor set in I' that excludes the set X. We begin the construction by excluding X I . Since { -1, 1 } n X1 = 0, there are rational numbers a1 and a2 such that -1

< a1 < a2 < 1

and

X1

c{

z : q

< x
For each of the 1-cells [-1, all and [a2,1] we perform a similar construction to exclude X 2 . This results in four new 1-cells. Clearly the construction, when continued indefinitely, will eventually exclude each point of the set X from the Cantor set which consists of all points that are in some descending sequence of 1-cells, one from each stage of the construction. To illustrate the inclusion scheme in the 1-cell ltl, let X be a compact subset of P n I1. We want t o construct a Cantor set in I" that includes the set X . Clearly the midpoint 0 of the 1-cell 1' is such that not in X . There are rational numbers 11 and

This is the first step in the construction of the Cantor set. On each of the 1-cells [-1,l1] and [ T I , 11 we perform a similar construction and remove a suitable middle open 1-cell. Obviously the space X is included in the resulting Cantor set. Now let us indicate why the base 6 procedure is needed. The next space in the hierarchy of universal spaces is the 1-dimensional one contained in the 2-cell I z. The exclusion scheme of the construction will require a base 6 procedure. We consider a subset X of I[ that is the union of a countable family of compact subsets X i of the set PxP.

164

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

We want to construct a Sierpinski curve in I that excludes the set X . Since the combinatorial boundary of I2 is disjoint with P x P, there are rational numbers a l , a2, b l and b2 such that -1 < a1

< a2 < 1,

-1

< bl < b2 < 1

and

Xi C { (5,y) : a1 < 2 < ~

2 b ,l

< y < b l }.

For the first step of the the construction of the universal curve we remove the open 2-cell (a1,u2) x ( b l , b2). In this way, the set X1 is excluded from the the union of the remaining eight 2-cells. It is to be observed that there is no guarantee that la1 - a21 and Ibl - b2l are less than 1. In the next stage we want to perform a similar construction for each of the remaining eight 2-cells in order t o exclude X2. The universal l-dimensional space we wish t o construct will consist of all points that are in the the intersection of some descending chain of 2-cells, one from each stage. To guarantee the topological uniqueness of the resulting l-dimensional space, we must force the diameter of the chains of 2-cells to converge to 0. Using the rational midpoints c1, c2, c3 and dl, d2, d3 of the l-cells contained in I" that are formed by the points a l , a2 and b l , b2 respectively, we get the arrangements -1 -1

< c1 < a1 < c2 < a2 < c3 < 1, < dl < bl < d2 < b2 < d3 < 1,

where all of the l-cells have length less than 1 and all of the 2-cells have diameter less that fi. (In this way, a base 6 procedure has emerged.) Now X2 is excluded by removing simultaneously the middle open 2-cells from each of the 32 2-cells. Now we shall describe the complete procedure more systematically. Both the exclusion and the inclusion schemes will use a base 6 construction. Let us describe the base 6 construction. For each positive integer i let Qi = { qi,k : k = 0, 1, .. . ,6 i} be a finite subset of the set Q of rational numbers such that ( h l ) -1 = qi,o < qi,l < * * * < qi,6i = 1, (h2) Q i C Q i + i , 1 (h3) IQi,k - q i , k - i I < p , k = 1,. . . ,ci, (h4) each l-cell [ q j , k , q i , k + l ] formed by Qi is subdivided into 6 parts by Qi+i.

165

4. EMBEDDING THEOREMS

For each i we set Di = { q i , k : 2 divides k}. Then the l-cells determined by Di are subdivided into two parts by Qi. We can now describe Hayashi's modification of the Menger sponge construction. Let m and n be natural numbers with n < m. The m-cell I" is divided into 6im m-cells by the set Qi and into 3im m-cells by the set Di. We shall denote the family of m-cells formed from Q i by Li and the family formed from Di by Ki. In the following way we inductively define a sequence of pairs of collections Fi and Gi, i = 0,1,. . . , and a sequence Fi, i = 0 , 1 , . . . , of closed subsets of I" hold for every i: such that Fi c K i , Gi c Li and Fi 2 Let FO = Go = { I" } and Fo = I". Suppose that Fi, Gi and Fi have been defined. With 5':"' denoting the union of the faces S of the members of GI with dim S 5 n, let

n{

is defined to be the set Fi : i E N}. Clearly a Hayashi sponge depends on the particular sequence Qi, i = 1 , 2 , . . . , possessing the conditions (h1)-(h4). But equally clearly from condition (h3) is the fact that any two Hayashi sponges are homeomorphic for a given pair m and n. Let us construct an exclusion Hayashi sponge in the m-cell 1". Let :I be the set of all points of I" at most n of whose coordinates are irrational. Suppose X is the union of a countable family of compact subset Xi of I" \ 1;. The set Sin'is the union of all n-dimensional faces of I". Since Sin'c I?, there is a positive rational number 60 such that dist ( X I , Sin')> mSo. Define

A Hayashi sponge H:

Then the set F1 of the Hayashi construction and X I are disjoint. Applying the next stage of the Hayashi construction with the compact set X1 U X 2 , one gets a set Fz such that F2 and X1 U X , are disjoint. Proceeding inductively, one can construct a Hayashi sponge H r that excludes X . Let us next construct an inclusion Hayashi sponge. Here we make the assumption that m > n > 0. On inspecting the construction of the closed set F!, one finds that I" \ Fl is contained in the set A1

166

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

which is the union of the family of all m-cells H formed by n factors of 1' and m - n factors of the middle l-cells formed by the set D1. Let X be a compact subset of 1" \ .1 ; Our aim is to find a set Q1 such that A1 n X = 0. Consider the set

PI = { 2 E II " : z k = 0, Ic = 1,2,. . . ,rn - n }. (There are (mTn)such sets formed by the combinations of m - n coordinates.) Since PI c 12, there is a positive rational number 6 less than 1 such that the m-cell

HI = {x E

I[":

lxkl

5 6,k

= 1,2 ,...,m - n }

does not meet X . Indeed, a common S can be used for all (,"_) sets H j which form the set A l . Consequently the set Q1 given q1,2 = -6, 41,3 = 0, 41,4 = 6 and q1 5 = by q l , l = will result in a set Fl such that X c F1. In a similar wai, one can construct the set Q2 so that X C Fz. Proceeding inductively, one can assure that X C Fi, i E N. The resulting Hayashi sponge H," will contain the compact set X . We are now ready to prove that the Hayashi sponge Hin+' is a universal n-dimensional space.

-y)

4.4. Theorem. For n 2 1, every space X in Mo with dim X can be embedded in each Hayashi sponge Hi"+'.

y

5n

Proof. Since each space X with d i m X 5 n can be embedded in a compact space of the same dimension, we may assume that X is compact. By Theorem 4.2, the space Nin+' contains a copy of X . In the inclusion construction we have shown that there is a Hayashi sponge Hin+' which contains this copy, Also we have shown that two Hayashi sponges are homeomorphic. So the theorem follows. The following is an interesting application of the Hayashi sponge to the space I[?+' of all points of I12n+1 at most n of whose coordinates are irrational. (Recall that C is the class of complete metrizable spaces.) 4.5. Theorem. Every C-hull of I[?,"+* contains a copy of H ; " + l .

Proof. Let Y be a C-hull of 1 By Lavrentieff's theorem (Theorem I.7.3), there is no loss in assuming that Y is a subset of

167

4. EMBEDDING THEOREMS

\ Y . In the preceding discussion of the exclusion Let X = construction of the Hayashi sponges, we have shown that there is an Htn+' that excludes X and thus is contained in Y . The next proposition is a technical one whose roots are found in the first paper on axiom schemes by Menger [1929]. 4.6. Proposition. If X is a finite dimensional, metrizable continuum with more than one point, then there is a separable metrizable space Y such that Y is the union of a countable family of copies of X and such that every C-hull 2 of Y contains an arc.

Proof. The construction of Y is a typical example of a construction of a self-similar set or fractal. By Theorem 4.3, the space X can be embedded in a unit cube I". Indeed, there is no difficulty in arranging the embedding to contain a vertex of the cube and the center of the cube. Using this embedding, we can find 2" such embeddings, one for each vertex of I". The union Yo of all of these copies is connected and contains the center and all of the vertices of the cube. On dividing the cube into 2" subcubes, we can use 2" copies of YO,one for each of the subcubes, and take their union to form a connected set Yl. Continuing inductively, we form the connected sets Y, with the property that the 2nm subcubes of I" intersect the set Y, in a geometrically similar way as Yo. Clearly the set Y = Y, : n E N} is connected and each of the 2nm subcubes of I" contains a subset of Y which is geometrically similar to Y and dense in the subcube. Consequently the set Y is uniformly locally connected as well. Let 2 be any C-hull of Y . By Lavrentieff's theorem, we may assume that 2 is a subset of 1". So, 1" 2 2 3 Y . Clearly 2 is connected and, from the construction of Y , uniformly locally connected, As 2 is topologically complete, in a standard way, it follows that 2 is arcwise connected.

U{

Let us turn now to zero dimensional, metrizable spaces. It is not difficult to show that every separable such space can be embedded into the space P of irrational numbers. (Moreover, the space P can be embedded into the Cantor set.) For the nonseparable case, Morita [1954] has shown that they can be embedded into the generalized Baire space N ( D ) of the same weight. For the purposes of the axiomatization of dimension we will need a lemma which is a sharpened version of Morita's theorem. In preparation for this lemma, some preliminary notation will be given.

168

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

Let D denote a discrete space. The Baire space N ( D ) is the count able product

N ( D )=

u{Di

: Di

= D , i = 1,2,. . .},

where the metric p is defined by the formula

Clearly the Baire space is complete. The r-neighborhood S,.((ai)) of ( u i ) is the set {ul} x ..-x { a h } x Di : i > k } whenever -< T 5 Consequently, if S,((ui)) n S , ( ( b i ) ) is not empty, then S,((ai)) = S , ( ( b i ) ) . So the collection

kil

n{

5.

u k

= { U : U = S i ( ( u i ) )for some (ui)}

is an open cover of N ( D ) by pairwise disjoint sets such that u k + l refines v k . Let p be a fixed point in D. Then, for each i, the finite product

Ni(D) =

n{Dj : Dj = D , 1 5 j 5 i}

is embedded into N ( D ) by the map e , given by the formula

Denote the natural projection of N ( D ) onto N i ( D ) by (pi and the natural projection of N ( D ) onto Di by ni. Finally denote by Gi and G the subsets of N ( D ) given by

Gi = e i [ N i ( D ) ] and

G = U{ Gi : i = 1 , 2 , . . .}.

Clearly we have that

G = { z E N ( D ) : n(z) = p except for at most finitely many i ) and G is the union of a a-discrete collection of singletons in N ( D ) .

4. EMBEDDING THEOREMS

169

4.7. Lemma. Let D be an infinite discrete space with cardinality Q and let p be a fixed point of D. With the notation given above, let H be a Gg-set of N ( D ) that contains G. Then N ( D ) can be

embedded into H . Proof. Let E = D \ { p } . Then N ( E ) is a closed set of N ( D ) that is disjoint from G. We may assume without loss of generality that N ( E ) n H = 0. The set N ( D ) \ H is the union of a collection { Fi : i = 1,2, ...} of closed sets of N ( D ) with Fi c Fi+l and F1 = N ( E ) . To avoid confusion, let X denote the space N ( D ) which is t o be embedded into H . From the above discussion, for each i there exists a cover Ui of X such that (al) each member of Ui is open and closed,

(a2) any two distinct members of Ui are disjoint, (a3) mesh Ui 5 I/;, (a4) Ui+l is a refinement of Ui. Because the weight of X does not exceed a , the cardinality of 7Ji also does not exceed a. We shall construct an embedding f of X into H by inductively defining continuous maps f, = r,f. To this end, for each n let 5, be a one-to-one correspondence between U , and a subset of E . Define the continuous map g,: X + E by gn(x) = S,(U), x E U e U,. These maps g, will be used to define the maps f, on each U in U , and to define natural numbers k(n, U ) and l(n,U ) for each U in U,. Let us start by observing that for each k and each (ax,...,a,) there exists an m with (p;l[cp,e,((ul,. ..,a,))] n F k = 8 because e , ( ( a l , . . . ,a , ) ) is not a member of Fk. For n = 1 let f1 = gl. Obviously f1 is constant on each U in U1. For each U in U1 let k(1, U ) = 1 and let l(1, U ) be the minimum natural number 1 2 1 such that the set y;'[plel f ~ ( z )n] Fk(l,~)+1 is empty, where x E U . Let us go to n = 2. For U in U2 let U' be the unique member of U1 with U c U'. If 2 > Z(1, U'), we let f2(z) = g2(z) for z in U and let k(2, U )= k(1, U ' ) t 1. If 2 5 Z(1, U ' ) , we let f 2 ( 2 ) = p for z in U and let k(2, U ) = k(1, U ' ) . Let Z(2, U ) be the minimum natural number 1 2 2 such that

170

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Note that l ( 2 , U ) = l(1, U‘) and k ( 2 , U ) = k(1, U’) when 2 5 Z(1, U’). Continuing inductively, we have for each n, n > 0 , a continuous map f,, and natural numbers k(n,U,) and l(n,U,) that satisfy the following conditions: (b) k ( 1 , U ) = 1 for each U in Ul. (c) f, is constant on each U in U,. (d) For each U in U, the number k(n, U ) is the minimum natural number 12 n such that

where x is any point in U . (e) For each U in U,+l let U’ be the unique member of U, such that U C U ’ . The following hold when x is in U . (el) If n t 1 > l(n, U’), then fn+l(x) = gn+l(x) and k(n t 1, U ) = k ( n , U’) t 1. (e2) If n t 1 I l(n,U’),then f,+l(z) = p and k ( n

+ 1, U ) = k(n,17’).

Observe that when U’ E U,, U E U,+l and U c U‘ hold we have l(n,U‘) = Z(n 1, U ) under condition (e2). In advance of proving f is an embedding of X let us prove that the set { n : n t 1 > l(n,U,),x E U , E U, } is unbounded. Suppose that there is an m larger than any member of this set. Then n 2 m implies that n 1 5 l(n,Un) holds for the unique Un in U, such that x E U,. So we have l(n,U,) = Z(n f 1 , U , + l ) for n 2 m from the earlier observation. That is, l(n, Un) = l(m,Urn) when n 2 m. But this contradicts the inequality Z(n,U,) 2 n of condition (d). To prove that f is an embedding of X ,it is sufficient to show the maps f,, n > 0 , separate points and closed sets. (See Kelley [1955], page 116.) Let x be a point and L be a closed set with 2 $ L . There exists an m such that Urn n L = 0 when z E Um. Let n be such that n > m and n > Z(n - 1 , U ) when x E U E U,-1. Note that U C Urn. Then f,(z) = gn(x) by (el). As gn is one-to-one on U, and gn(x) # p , it follows that f n [ X \ Um] C D \ { f,(x) }. Because D is a discrete space, f, separates x and L; thereby f is an embedding of X into N ( D ) .

+

+

5 . AXIOMS FOR

THE DIMENSION FUNCTION

171

Finally let us show that f [ X ]c H . For 2 in X and k in N \ (0) choose an n such that k(n, U ) = k and 2 E U E U,. (Such an n exists by condition (el).) Then with m = Z(n,U) we have from ( d ) that yil[ymem(flx x fm)(2)]n Fk+l = 0 holds. So f(s)4 Fk follows since Fk c Fk+l. The above proof will apply when X is a metrizable space with weight not exceeding a and I n d X = 0. For, in this situation, the paracompactness of X will permit the construction of the covers Vi satisfying the conditions (a1)-(a4). Thus we have a sharpening of Morita's embedding theorem. 4.8. Theorem. Let D be an infinite discrete space with cardi-

nality (I( and let p be a fixed point of D. With the notation given above, let H be a Gs-set of N ( D ) that contains G. Then every metric space X with weight not exceeding a and I n d X = 0 can be embedded into H . 5. Axioms for the dimension function

An early problem in dimension theory was that of finding suitably simple properties of the dimension functions that can serve as axioms for dimension theory. One property is indisputable. Any such axiom scheme must have as an axiom the topological invariance of the function. That is, if d is a dimension function defined on a universe U ,then d ( X ) must be equal to d(Y) when X is in U and Y is homeomorphic t o X . The other properties that will be emphasized in this section are those related to normal families. This section will be devoted to a discussion of axiom schemes that have been formulated around those of Menger. In Menger [1929] a characterization of the dimension function for the universe of all subsets of the Euclidean plane was given and a conjecture was made for the universe of all subsets of Euclidean n-dimensional space. These axioms by Menger will be called the M axioms. 5.1. The M axioms. Let U, be the universe of spaces X that are embeddable in the n-dimensional Euclidean space W and let d be an extended-integer valued function on U, that satisfies the axioms: ( M l ) d(0) = -1, d({ 0)) = 0 and d(W") = m for m = 1,2,. . . , n . (M2) If Y is a subspace of a space X in U,, then d(Y) 5 d(X).

172

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

(M3) If a space X in U, is a union of a sequence XI, X2, . . . of closed subspaces, then d ( X ) 5 sup { d(X;) : i = 1 , 2 , . . .}. in U, satis(M4) Every space X in U, has a compactification fying d ( 2 ) = d(X). The axioms (M2) and (M3) and the first two conditions of axiom ( M l ) are related to Theorem 3.24 on normal families. The remaining axioms are much deeper properties from dimension theory. In [1929] Menger conjectured that ind is the only function d which satisfies the axioms (Ml)-(M4) for each positive natural number n, and he showed that the conjecture was correct when n 5 2. The conjecture is still unresolved for the cases of n greater than 2. Indeed, it is still not known whether d = ind satisfies axiom (M4) when n > 3. Consider next the universe U, of all spaces X that can be embedded in some n-dimensional Euclidean space and a function d that satisfies the M axioms (Ml)-(M4). Then ind does satisfy all four of the M axioms, However, ind is not the only function t o do so; each cohomological dimension dim G with respect to a finitely generated Abelian group G also satisfy the M axioms in the universe U,. (See Example H5 below.) The next set of axioms first appeared in Nishiura [1966]. The axioms were for functions d defined on the universe Mo of separable metrizable spaces. Clearly U, is a proper subclass of Mo. These axioms will be called the N axioms. 5.2. The N axioms. Let d be an extended-real valued function on the universe Mo that satisfies the axioms:

(N1) d({ S}) = 0. (N2) If Y is a subspace of a space X in M o , then d ( Y ) 5 d ( X ) . (N3) If a space X in Mo is a union of a sequence XI,X2, . . . of closed subspaces, then d ( X ) 5 sup { d(Xi) : i = 1 , 2 , . . .}. (N4) If a space X in Mo is a union of two subspaces X I and X2, then d ( X ) 5 d(X1) t d(X2) t 1. (N5) Every space X in M o has a compactification in M o satisfying d ( 2 ) = d(X). (N6) If z is a point of a space X in Mo, then every neighborhood U of z has a neighborhood V with d (B ( V ) ) d ( X ) - 1 and V C U . (As usual, +co - 1 = +m.)

x

<

5 . AXIOMS FOR THE DIMENSION FUNCTION

173

Clearly all of the N axioms except for (N5) are related to Theorem 3.24 for normal families. Indeed, from Theorems 3.25 and 1.4.4 we see that ind satisfies these axioms. Only the axiom (N5) is a deeper property of the function ind. The next theorem shows that these axioms are independent and characterize the function ind. 5.3. Theorem. For the universe Mo, the N axioms are independent and the function ind is the only function d which satisfies the N axioms.

Proof. Let us first prove that ind is a function d which satisfies the N axioms. From the coincidence theorem (Theorem 1.4.4) and Theorem 3.25 we have that ind satisfies all of the N axioms except (N5). The coincidence theorem and Theorem VI.2.9 show that ind also satisfies (N5). So let us show that ind is the only such function. This will be done in four parts.

Part I. d ( X ) = -1 if and only if X = 0. Proof. From axioms (N6) and ( N l ) we have d(0) I -1. And from axioms (N4) and ( N l ) we have 0 = d({ 0 ) ) = d({ S} U 0) I d({ 0 ) )

+ d(0) + 1.

Hence, d(0) = -1. Now suppose that X # 0. Then by (N2) and ( N l ) , we have d ( X ) 2 d({ 8 ) ) = 0 > -1. Thereby part I is proved.

Part 11. i n d X = 0 implies d(X) = 0.

Proof. By (N3) and ( N l ) we have that the set of rational numbers Q has d(Q) = 0. Axioms (N5) and (N2) then imply that there is a nonempty, compact dense-in-itself space X’ with d(X’) = 0. Let X be a zero-dimensional space. Then X can be embedded in X ’ . From (N2) and part I we have 0 = d(X’) 2 d(X) 2 0. Thus part I1 is proved. Part 111. For each extended integer n with n 2 -1 we have i n d X 5 n implies d(X) I n.

Proof. Suppose that i n d X 5 n < +oo. Then by the decomposition theorem (Theorem 1.3.8) we have X = U{ Xi : i = 0, 1,. . .,n } , where indXi 5 0 for each i. From axiom (N4) and part I1 we have d(X) I d(Xi) 5 n. Part I11 is now proved.

174

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Part IV. For each extended integer n with n 2 -1 we have d ( X ) 5 n implies ind X

5 n.

Proof. The statement is true for n = -1 by part I. Suppose that the statement is true for an integer n and let d ( X ) 5 n 1. By (N6), each point of X has arbitrarily small open neighborhoods U whose boundaries B ( U ) have d(B ( U ) ) _< d ( X ) - 1 5 n. So ind B ( U ) 5 n. Thus we have shown that ind X 5 n 1. The induction is completed and part IV now follows.

+

+

Parts I11 and IV give that ind is the only function d that satisfy the N axioms. The following six examples show the independence of the N axioms.

Example N1. Let d = ind

+ 1.

Example N2. Let P2 be the class of zero-dimensional a-compact spaces and the space 0. Clearly 7 3 is a semi-normal family in the universe Mo. From Theorem 3.29, we have that Pz-ind X = 73-Ind X for X in Mo. So, by Theorem 3.23, P2-ind satisfies the axioms (N3) and (N4). Clearly, (N2) fails. To see that axiom (N5) is satisfied, we observe for X in Mo that the inequality ind X - 1 5 P2-ind X can be shown to hold by a straightforward induction and that the inequality P2-ind X 5 ind X holds by Proposition 11.2.3. And obviously, if X is nonempty and compact, then ind X - 1 = P2-ind X . So, when X is not compact and ind X - 1 = "2-ind X we let be a dimension preserving compactification of X . Let us consider the remaining case where X is not compact and ind X = IPz-ind X holds. Let Xo be a dimension preserving compactification of X and let 20 be a point in the set X O \ X . Let S = { 2, : n E N } be a sequence in X converging to 50 and let D = { zn : n E N } be a dense subset of an absolute retract 2. Observe that if F is a closed subset of Xo that contains at least m points of S then there is a continuous map of F into 2 such that its image contains the first m points of D. Using the metrizability of X o and this observation, one can easily construct an embedding of X O\ { 20} into XOx 2 such that its image Y has a closure 2 with 2 \ Y = { 5 0 } x 2 . Let 2 be the space In+' where n = ind X. Then 2 is a compactification of X such that P2-ind X = P,-ind 2 .

x

5 . AXIOMS FOR THE DIMENSION FUNCTION

175

For the axioms ( N l ) and (N6), define d as follows: d(0) = -1; d ( X ) = PZ-indX 1 if and only if X # 0. This function d satisfies all the axioms except for (N2).

+

Example N3. Let d be defined as follows: d ( X ) = i n d X if and only if X is finite; d ( X ) = ind X 1 if and only if X is infinite.

+

Example N4. Let d be defined as follows: d ( X ) = ind X if and only if ind X 5 0 ; d(X) = ind X 1 if and only if ind X > 0.

+

Example N5. Let P.r, be the class of countable spaces. Then Ps is a normal family in the universe Mo. So by Theorem 3.24, the axioms (N2)-(N4) are satisfied by PS-ind and axiom (N5) fails. For the axioms ( N l ) and (N6), consider the function d defined as follows: d(0) = -1 ; d ( X ) = Ps-indX 1 if and only if X # 0.

+

Example N6. Let d be defined as follows: d(0) = -1 ; d ( X ) = ind X/(indX 1 ) if and only if -1 < ind X < t c o ; d ( X ) = 1 if and only if ind X = +co.

+

The proof of the theorem is now completed. Obviously the N axioms restricted to the universe U, of spaces embeddable in some Euclidean space also characterize ind and the axioms are independent. This fact relies on the equivalence of the finite-dimensionality of a space and its embeddability in some Euclidean space (Theorem 4.3). The axiom (N6) is closely tied to the small inductive dimension, that is, the H operation. The question of replacing this axiom with one more suitable to the covering dimension dim has been answered by Hayashi [1990] for the universe U,. 5.4. T h e H axioms. Let d be a real valued function on the universe U, that satisfies the axioms: ( H l ) d(0) = - 1 , d({ 0}) = 0 and d(lIm) = m for m = 1 , 2 , . . . . (H2) If Y is a subspace of a space X in U,, then d(Y) 5 d(X). (H3) If a space X in U, is a union of a sequence XI, XZ, . . . of closed subspaces, then d ( X ) 5 sup { d(Xi) : i = 1 , 2 , . . .}. (H4) Every space X in U, has a cornpactification 2 in U, satisfying d ( 2 ) = d(X). (H5) If X is a space in U, with d ( X ) < n (where n is a positive integer), then there are n sets X i , i = 1,. ..,n, with d(X;) 0 f o r e v e r y i a n d X = U { X i : i = 1 , ..., n } .

<

176

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

5.5. Theorem. For the universe U,, the H axioms are independent and the function dim is the only function d which satisfies the H axioms.

The proof of this theorem will use the facts that the class U, is precisely the class of finite dimensional spaces in Mo and that Proposition 4.6 holds. These facts will also be used in the discussion of the following axiom scheme which is an extension of the H axioms. A proof of Theorem 5.5 will result from this axiom scheme. 5.6, T h e extended H axioms. Let d be an extended-real valued function on the universe Mo that satisfies the axioms:

(eH1) d(0) = -1, d({ 0 ) ) = 0 and d(1") = m for m = 1,2,... . (eH2) If Y is a subspace of a space X in Mo, then d(Y) 5 d(X). (eH3) If a space X in Mo is a union of a sequence XI, Xa, . . . of closed subspaces, then d(X) 5 sup { d(Xi) : i = 1 , 2 , . . .}. (eH4) Every space X in Mo has a compactification 2 in Mo satisfying d ( 2 ) = d(X). (eH5) If X is a space in Mo with d ( X ) < n (where n is a positive integer), then there are n sets Xi, i = 1,.. . , n , with d(Xi) 5 0 for every i and X = U{Xi : i = 1 , . . . , n } . (eH6) If X is a space in Mo with d(X) 5 0 then X is embeddable in some Euclidean space Rm. 5.7. Proposition. For the universe M o , the extended H axioms (eH1) through (eH5) are independent and dim is the only function d which satisfies the extended H axioms. Consequently, the restriction of the extended H axioms to the smaller universe U, is equivalent to the H axioms.

Proof. We already know that dim satisfies the extended H axioms. So let us show that a function d that satisfies the extended H axioms must be dim. Let us first prove that the equivalence d(X) = 0 if and only if X is totally disconnected holds for every nonempty compact space. Suppose that some nonempty compact space with a nondegenerate component C has d(X) = 0. From (eH2), (eH3), (eH6) and Proposition 4.6 there exists a space Y with d(Y) = 0 such that every compactification of Y contains an arc. From (eH4) we have d ( 7 ) = 0 for some compactification of Y . We infer from (eH2) that d(1) = 0.

5.

AXIOMS FOR THE DIMENSION FUNCTION

177

But this contradicts d(1) = 1 from (eH1). So, d ( X ) = 0 implies X is totally disconnected when X is compact. For the converse implication let us show that d ( X c ) = 0 where Xc is the Cantor set. By (eH1) and (eH3) we have d(Q) = 0, where Q is the set of rational numbers. Axiom (eH4) gives a compactification 2 of Q such that d ( 2 ) = 0. As the space X is a nonempty, dense-in-itself and compact, it contains a copy of the Cantor set Xc. By (eH2) we have d ( X c ) = 0. As every compact totally disconnected space can be embedded in the Cantor set, we have by (eH2) the converse implication. The equivalence d i m X = 0 if and only if d ( X ) = 0 follows immediately from the one in the last paragraph and from (eH2) and (eH4). We have d i m X 5 d ( X ) from the decomposition theorem and axiom (eE35). It remains t o be shown that d(X) d i m X . Suppose is the set of all points that 0 < n = d i m X < 00. Recall that of l12n+1 at most n of whose coordinates are irrational. Axioms (eH1) and (eH3) give d(I?+') = n. Let Y be a compactification of I[?+' given by (eH4). By Theorem 4.5 the space Y contains a copy of the Hayashi sponge which, by Theorem 4.4, contains a copy of X. By axiom (eH2) we have d ( X ) 5 d(Y) = n = d i m X . Thus d = dim has been established. Obviously, axiom (eH6) is redundant in the universe U,. To establish the independence statement in the theorem it will be sufficient to prove the independence statement of Theorem 5.5 since d(X) = 00 if and only if dim X = 00.

-

<

Hi"+'

Example H1. Let d(0) = -1 and d(X) = 0 for every nonempty

space X in U,.

Example H2. Let d be the one given in Example N2. Example H3. If X is embeddable in the interval 1, then define d ( X ) = dim X. If d i m X = 1 and X is not embeddable in 1, then define d ( X ) = 2. For all other spaces X , define d ( X ) = dim X. Example H4. Let d be the one given in Example N5. Example H5. Let Z ( p ) be the cyclic group of order p where p is a prime number. If d ( X ) = c-dimz(,) X , where c-dimz(p)X is the cohomological dimension of X for X in U,, then d satisfies the axioms (Hl)-(H4) by a result of Shvedov (see Kuz'minov [1968]). Now, there is a space X in U, such that c-dimZ(p)X # d i m X . Consequently, d does not satisfy axiom (H5).

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111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Both Theorem 5.5 and Proposition 5.7 are now proved. The N axioms have been generalized to the universe M of metrizable spaces for Ind as follows. 5.8. The A axioms. Let d be an extended-real valued function on the universe M that satisfies the following axioms:

( A l ) d({ 0)) = 0. (A2) If Y is a subspace of a space X in M , then d(Y) 5 d(X). (A3) If a space X in M is a union of a a-locally finite collection F of closed subspaces, then d ( X ) 4 sup { d ( F ) : F E F } . (A4) If a space X in M is a union of two subpaces X I and X z , then d(X) i d(X1) t d(X2) t 1. (A5) Every space X in M has a completion % in M satisfying d ( 2 ) = d(X). (A6) If F is a nonempty closed subset of a space X in M , then every neighborhood U of F has a neighborhood V satisfying V c U and d(B ( V ) ) d(X) - 1.

<

The A axioms are the natural generalizations of the N axioms. The following characterization theorem holds. 5.9. Theorem. For the universe M , the A axioms are independent and the function Ind is the only function d which satisfies the A axioms.

The next lemma will be used in the proof of Theorem 5.9 and in the proof of the characterization by means of the S axioms which will be discussed after the proof of Theorem 5.9. 5.10. Lemma. Let d be an extended-real valued function on the universe M that satisfies the axioms ( A l ) , (A2), (A3) and (A5). Then d(N(D)) = 0 for every discrete space D.

Proof. Because of (A2), we may assume that D is infinite. The axioms ( A l ) and (A3) yield that the set G as defined in the discussion preceding Lemma 4.7 has d(G) = 0. By (A5), there is a completion of G such that d ( 5 ) = 0. By Lavrentieff’s theorem, the identity map of G onto itself can be extended to a homeomorphism of a Gs-set of containing G onto a Gs-set H of N ( D ) containing G. So, d( H ) = 0. By Lemma 4.7 we have that N ( 0 ) can be embedded into H. Axiom (A2) completes the proof.

5 . AXIOMS FOR THE DIMENSION FUNCTION

179

Proof of Theorem 5.9. That Ind satisfies the A axioms follows from Theorem 3.25 and Theorem V.2.12. The needed modifications of the proof of Theorem 5.3 occur in parts 11, I11 and IV. Since the modifications for parts I1 and I11 are obvious, only the statement that d ( X ) = 0 if and only if I n d X = 0 requires a proof. To this end, suppose d ( X ) = 0. Then, by (A6) and the analogue of part I, we have I n d X = 0. For the converse we apply Lemma 5.10 and Theorem 4.8. As for examples to show independence, only the axioms (A2) and (A5) require serious attention. For (A2), consider the absolute Bore1 class A(2) (that is, the Gsg-class). As in Example N2, one can construct the required example by using A(2)-Ind. For (A5), one uses the class of metric spaces that are the union of a a-locally finite collection of singletons in place of the class Ps of countable spaces used in Example N5. (Observe that this class is N[P5 : M 1, the normal family extension of the class 735 in the universe M . ) According to Lemma 5.10, the A axioms imply the following S axioms. 5.11. The S axioms. Let d be an extended-real valued function on the universe M that satisfies the axioms: (Sl) For each discrete space D,d(N(D)) = 0. (S2) If Y is a subspace of a space X in M , then d(Y) 5 d(X). (S3) If a space X in M is a union of two subpaces X I and Xz, then d ( X ) 5 d(X1) t d(X1) t 1. (S4) If F is a nonempty closed subset of a space X in M , then every neighborhood U of F has a neighborhood V satisfying V c U and d(B(V)) d ( X ) - 1.

<

5.12. Theorem. For the universe M , the S axioms are independent and the function Ind is the only function d which satisfies the S axioms.

Proof. Axiom ( S l ) implies axiom (Al), and axiom (S4) is axiom (A6). Lemma 5.10 yields d ( X ) = 0 for every metric space X with I n d X = 0. Axiom (S4) yields d ( X ) = -1 if and only if X = 0, and -1 < d(X) 5 0 if and only if Ind X = 0. The remainder of the proof is the same as that of Theorem 5.9. For the independence of the S axioms, only (Sl) needs attention. In this case let d ( X ) = Ind X t 1.

180

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL T Y P E

The above axioms do not employ anything more than subsets and unions. And, the only maps involved are homeomorphisms. Another very useful tool in dimension theory is that of approximation of spaces by polyhedra. This approximation uses the concept of &-mappings from a metric space to a topological space. The development of functions of dimensional types along the lines of approximations is yet to be made. The discussion on axioms will end with a short discussion of the axioms called the A1 axioms and the Sh axioms. Since proofs of the theorems associated with these axioms will lead us afield, only the theorems will be stated. 5.13. The A1 axioms. Let KO be the universe of compact subspaces of Euclidean spaces and d be an integer valued function defined on K O that satisfies the axioms:

(All) d(0) = -1, d({ 0}) = 0 and d(H") = n for n = 1 , 2 , . . . . (A12) If a space X in KO is the union of two closed subsets X I and Xz,then d(X) = m u { d(XI), d(X2,)). (A13) For every space X in KO there exists a positive number E such that if f : X + Y is an &-mapping of X onto a space Y

in KO,then d ( X ) 5 d(Y). (A14) If X is a space in KO of cardinality larger than one, then there exists a closed subset L of X separating X with d(L) < d(X). 5.14. Theorem. For the universe KO,the A1 axioms are independent and the function dim is the only function d which satisfies the A1 axioms. 5.15. The Sh axioms. Let U, be the universe of subspaces of Euclidean spaces and d be an integer valued function defined on U, that satisfies the axioms:

(Shl) d(0) = -1, d({ 0)) = 0 and d(II") = n for n = 1 , 2 , . . . . (Sh2) If a space X in U, is the union of a sequence X I ,Xa, . . . of closed subsets, then d ( X ) 5 sup { d(X;) : i = 1 , 2 , . . .}. (Sh3) For every space X in U, there exists a positive number E such that if f : X -+ Y is an &-mapping of X onto a space Y in U,, then d ( X ) 5 d ( Y ) . (Sh4) If X is a space in U, of cardinality larger than one, then there exists a closed subset L of X separating X with d(L) < d(X).

6. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

181

5.16. Theorem. For the universe U,, the Sh axioms are independent and the function dim is the only function d which satisfies the Sh axioms.

Finally we mention that the product theorem has not been used in the axioms. Its role appears not to be natural in the development of normal families. The formulation of the product theorems for general functions of inductive type is somewhat uncertain at this time. Perhaps an axiom using the product theorem would be a suitable substitute for the axiom using the addition theorem. In this way, it may be possible to avoid the decomposition theorem which is heavily used in the characterizations of ind and Ind. The decomposition theorem uses strongly the metrizability of the spaces; the product theorem does not use the metrizability and hence may afford a characterization to a universe larger than the metrizable one. In this vein, there is an inductive dimension that uses a subbase for the open sets of a topology (de Groot [1969]). It has been observed that the product theorem is easily proved with a definition that uses subbases. (See also van Douwen [1973] for a brief discussion of this subbasic dimension.) 6. Historical comments and unsolved problems

The terminology of functions of dimensional type was first introduced by Baladze [1982]. Earlier in [1964], Lelek introduced the terminology inductive invariant for the functions of inductive dimensional types P-ind and P-Ind. The Example 1.2 and the material concerning the interconnection between the addition, point addition and the sum theorems (that is, the Theorems A, P and S) in Section 1 are taken from van Douwen [1973]. Similar results, in particular an example showing the failure of Theorem S, can be found in Przymusiliski [1974]. The Proposition 1.15 is due to Hart, van Mill and Vermeer [1982]who showed that the small and large inductive completeness degrees disagreed in the universe of metrizable spaces. Example 1.16 generalizes this last result to other absolute Bore1 classes. These results should be compared with the fact that a normal space X has I n d X = 0 if and only if every normal space Y obtained from X by the adjunction of a single point satisfies the condition indY = 0, Isbell [1964]. The study of normal families began with Hurewicz [1927] in the

182

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

universe Mo of separable metrizable spaces and by Morita [1954] in the universe M of metrizable spaces. The symbols for the operations H and M have been chosen to designate these origins. The various generalizations of normal families discussed in the chapter have appeared in Aarts [1972], Aarts and Nishiura [1973a]. The development found in Section 3 is essentially new. The influence of the work of Dowker [1953] is strongly felt in this development. The introduction of the operation N, the normal family extension, is the result of it. The Dowker universe V is defined for the first time in Section 3. The definition is a culmination of the investigation in Section 3 of the connections between the operation M, S and N. There is a long history associated with this universe. It was known that the subspace theorem for Ind failed in the universe N of normal spaces. Even more, it fails for the universe NH of hereditarily normal spaces. It was shown by Cech [1932] that the universe of perfectly normal spaces worked well for the subspace theorem. Subsequently, Dowker [1953] defined the universe NT of totally normal spaces and proved the subspace theorem there. In Pasynkov [1967] and Lifanov and Pasynkov [1970], a space X is defined to be a Dowlcer space if it is a hereditarily normal space for which every open set is Dowker-open in the sense of Definition 3.4. (The class of Dowker spaces as defined by Lifanov and Pasynkov is not the Dowker universe V defined in this book.) In [1977] Nishiura defined a TI-space X to be super normal if for every pair of separated set A and B there is a pair of disjoint open sets U and V such that A c U and B c V and U and V are the union of locally finite, in U and V respectively, families of cozero-sets of X. This definition was the first recognition of the connection between the operation M and the operation s. In Engelking [1978] the ideas in super normal and Dowker spaces were combined to yield the definition of strongly hereditarily normal spaces (see Definition 3.6). Clearly the Dowker universe contains the collection of strongly hereditarily normal spaces which, in turn, contains the class of super normal spaces. The Dowker universe contains all of the other classes properly. (See Engelking [1978] for a discussion of these inclusions.) Along these lines, Kotkin [1990] defined a new dimension function related to super normal spaces. In Section 5 on axiomatics, the A axioms are due to Aarts [1971] and the S axioms are due to Sakai [1968]. Lemma 4.7 and Theo-

6 . HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

183

rem 4.8 are essentially from Aarts [1971]. The A1 axioms are due to Alexandroff [1932] and the modification of these axioms to the Sh axioms are due to Shchepin [1972]. The reader is referred to Alexandroff and Pasynkov [1973], Engelking [1978] and Arkhangel'skii and Fedorchuk [1990] for further discussions on these axioms. The theorem that cohomological dimension with respect to a finitely generated Abelian group satisfies the M axioms in the universe U, of subspaces of Euclidean spaces was proved by Shvedov; the proof was first published in Kuz'minov [1968]. The H axioms are due to Hayashi [ 19901. The preliminary embedding theorems involving the Hayashi sponge H," found in Section 4 are essentially due t o Hayashi. The extended H axioms are a natural outgrowth of Hayashi's proof of Theorem 5 . 5 . As mentioned in the Section 5 , the additivity axioms (N4), (A4) and (S3) are closely tied to the metrizability of the spaces X . A suitable substitute for these axioms that would yield characterizations in larger universes would be welcomed. In this regard, see Baladze [1982] for a characterization theorem along the lines of the S axioms for nonmetrizable spaces. (The theorem of the last reference suffers from the requirement that the spaces satisfy the Lindelof condition as well as other conditions that imply the additivity theorem.)

Unsolved problems 1. The extended H axioms have not been shown t o be independent. The axiom (eH6) has been used in the proof of Proposition 5.7. This was made necessary by the requirement that the space X in Proposition 4.6 be finite dimensional. Can this requirement be deleted from the hypothesis of Proposition 4.6? It is known that there are infinite dimensional continua that have no positive, finite dimensional, closed subsets. (See Henderson [1967].) 2. Are the extended H axioms independent?

3. Can one prove the equivalence (in the universe U,) of the three axiom schemes N , H and Sh without passing through the dimension function dim? 4. A set of axioms characterizing the covering dimension dim in the

universe of all (not necessarily metrizable) compact spaces whose dimension dim is finite was given by Lokucievskii [1973].

184

111. FUNCTIONS OF INDUCTIVE DIMENSIONAL TYPE

Find axioms characterizing Ind or dim for universes than M .

U larger

5. Are the Menger sponge M," and the Hayashi sponge H r homeomorphic?

6. The class € of strongly hereditarily normal spaces has been char-

acterized as those hereditarily normal spaces X for which every regularly open set U is the union of a point-finite family of open F,-sets in X (see page 168 of Engelking [1978]). Characterize those spaces X which form the Dowker universe D.

CHAPTER IV

FUNCTIONS OF COVERING DIMENSIONAL TYPE

The chapter deals with covering dimension modulo a class P . Parallel to Chapter 111, the functions that are associated with the covering approach will be designated as functions of cowering dimensional type. They include P-dim, P-Dim, P-sur and P-def. The investigation of functions of covering dimensional type that began in Chapter I1 will be continued in this chapter. Theorems on dimension of finite unions will be discussed in the first section. The other theorems of dimension theory such as the countable sum theorems and the subspace theorems are naturally studied, as they were for functions of inductive dimensional type, in the context of cosmic and normal families. These will be discussed in the second section. Also, the coincidence of P-dim and P-Dim will be established in Section 2. This investigation of cosmic and normal families will lead to the conclusion that the Dowker universe ZJ is, in a sense, an optimal universe in which the theory of functions of covering dimensional type can exist without anomalies. Section 3 will be devoted to this discussion of the Dowker universe. The dimensional relationships of the domain and image of closed mappings (or open mappings) are usually stated in terms of the covering dimension. Section 4 has a short discussion on related mapping theorems for dimensions modulo classes P. Although some parts of the discussion do not require the full strength of the agreement for the chapter, the simplicity of the exposition gained from the agreement will compensate for the lack of a fuller discussion.

Agreement. The universe is closed-monotone and open-monotone and is contained in JVH.The classes contain the empty space 8 as a member and are closed-monotone, open-monotone and strongly closed-additive in the universe. Unless indicated otherwise, X will always be a space in the universe. 185

186

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

1. Finite unions

The initial part of this section will be devoted to P-dim. Let us begin with the theorem that has the easiest proof. 1.1. Theorem. If A and B are closed subsets o f a space X such that X = A u B , then

P-dim X = max { P-dim A , P-dim B }.

Proof. In view of Theorem 11.5.6 it is sufficient to prove the maximum is not less than P-dim X . Let max { P-dim A , P-dim B } i n with n 2 0 and suppose that U = { U; : i = 0,1,. . . k } is a finite P-border cover of X with enclosure F . The restricted collection { Ui n A : i = 0,1,. . . , k } is a P-border cover of A and hence it has a P-border cover shrinking V = { V , : i = 0, 1, . . . k } with enclosure F' such that ord V 5 n 1. For each i let U,! = V , U (Ua \ A ) and denote the collection of such U: by U'. Then U' is a P-border cover of X with enclosure F U F' that shrinks U . We now perform a symmetric construction with U' and B to complete the proof.

+

The stronger form of the above finite sum theorem will require a technical lemma on P-border covers. The Proposition 11.4.4 has the following generalization which will hold for normal spaces as well. 1.2. Lemma. In a space X let F = { F, : a E A } be a closed collection and U = { U , : cr E A } be a locally finite open collection such that F, C U , for each a. Then there exists an open collection V = { V, : a E A } such that F, c V, c U , for each a and the collections F and { cl(V,) : a E A } are combinatorially equivalent, that is,

for each finite collection of distinct indices

01,.

. .,a,.

Proof. One merely applies a transfinite induction on the indexing set A to attain the appropriate modification of the proof of Proposition 11.4.4.

1. FINITE UNIONS

187

1.3. Lemma. Let X be a space and Y be a subspace of X. Suppose that U = { U , : a E A } is an open collection of X such that U is locally finite in the space U U and such that the restricted collection U y = { U , n Y : a E A } has ord U y 5 m. Then there exists an open collection W = { W, : a E A } that shrinks U such that ( U W ) n Y = ( U U ) n Y andordW<mhold.

Proof. We may assume X = IJ U without loss of generality. Then Y' = { z : ord, U 5 m } is a closed subset of X that contains Y. In the closed subspace Y' let F = { F, : (Y E A } be a closed cover that shrinks U . Then ord, F 5 m for each x in Y'. The required collection W is obtained by an application of Lemma 1.2. 1.4. Lemma. Let Y be a subspace of a space X and suppose U = { Ui : i = 0,1,. . . , k } is a finite open collection of X such that its restriction U y t o Y is a P-border cover of Y with enclosure F . If P - d i m Y 5 n, then there is a set G and there is an open collection W = { Wi : i = 0,1,. ,k } of X such that the following

..

conditions hold: (a) W i c U i , i = O , l , ...,k. (b)G=Y\UWEP. (c) ord W 5 n 1. Moreover, if Y is also closed in X, then the further condition

+

(u

u

(d) u )\ G = w holds provided the condition (c) is changed t o the following: (c*) There is an open set V of X with ord, W 5 ord. U for each x in X \ V such that the inclusions Y \ G C V C { x : ord, W n l } hold modulo the set H = X \ U U .

< +

Proof. As 8 E P n U ,the case of P - d i m Y = -1 is trivially true. So we shall assume 0 5 P - d i m Y 5 n. Without loss of generality, we may also assume x = (U U ) u Y . The restricted P-border cover U y = { Ui f l Y : i = 0 , 1 , . . .,Ic } has a P-border cover shrinking 2 = { Zi : i = 0, 1, . . .,k } with enclosure G such that ord 2 5 n 1. For each i let U,! be an open set of X such that U,! n Y = Zi and U,! c Ui. For the open collection U' = { U , ! : i = O , l , . . . , k }w e h a v e o r d , U ' < n + l f o r e a c h x i n Y \ G . By Lemma 1.3 there is an open collection V = { V , : i = 0,1, . . .,k } such that V, C Ui for each i, ord V 5 n 1 and

+

+

188

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

Since (U U ’ )n Y = (U 2) n Y = Y \ G, the conditions (a)-(c) are easily verified for the set G and the collection W = V. Suppose that Y is also closed in X . Then we have that the set G is closed in X , the open set X \ G is equal to the set ( X \ Y ) U (U V) and the set Y \ G is closed in the subspace X \ G. So there is an open set V in X such that Y \ G c V c X \ G and clX\G(V) C U V. Define the two collections WOand W1 to be

wo = { v, u (Ui\ (ClX(V) u Y ) ): ui n v # 0 >, w, = { ui\Y : ui n V = S}. Then W = WOU W, is an open collection such that W shrinks U and U W = X \ G = (U U ) \ G. Hence the condition (d) holds. If 2 E V , then ord IW = ord ,Wo = ord ,V 5 n 1. If 5 E X \ V, then ord, W 5 ord, U because W shrinks U . As X = (U U) U Y has been assumed, we have H = F c G. Thereby the condition (c*) holds.

+

1.5. Theorem. If

X = A U B where A is open, then

P-dim X 5 max { P-dim A , P-dim B }.

Proof. From Theorem 11.5.6 there is no loss in generality in assuming the further condition A n B = 0. Let max { P-dim A, P-dim B } 5 n with n 2 0 and suppose that U = { Ui : i = 0, 1, . . .,k } is a finite P-border cover of X with enclosure F . By the closed monotonicity conditions on P and U ,the trace U B of U on B is a P-border cover of B. From Lemma 1.4 we have an open collection W = { Wi : i = O , l , , ..,k } of X that shrinks U, a set G and an open set V such that

\ U W E P nU ,

(U U ) \ G = U W, B \ G C V C { 2 : ord, W 5 n + 1 } mod F, ord ,W 5 ord ,U , x E X \ V,

G=B

Since X \ U W = F U G and since P is strongly closed-additive, we have that W is a P-border cover of X with enclosure F U G. The Collection WA = { Wj n A : i = 0, 1, , ..,k } is a P-border cover of A with enclosure F n A . As P-dimA 5 n, there is a collection

189

1. FINITE UNIONS

W‘ = { W: : i = 0, 1, ...,k } such that W‘ is a P-border cover of A with enclosure G’ that shrinks W, and ord W‘ 5 n 1. The sets A \ (V U F ) and B \ (G U F ) are separated. Let V’ be an open set such that A \ (V U F ) c V’ and ( B \ (G U F ) ) n clX(V’) = 8. For each i, define the open set Zi = (Wi n V )U (W/ n V’). Then the open collection 2 = { Zi : i = 0,1,. . . ,k } is a P-border cover of X with enclosure F U G U (G’ \ V) and ord 2 5 n 1.

+

+

1.6. Corollary. Let A and B be open subsets of a space X such that X = A U B . Then

P-dim X

max { P-dim A , P-dim B }.

Another consequence of Theorem 1.5 is the following. In a sense, the theorem justifies the “modulo a class P” terminology.

1.7. Theorem. For every closed P-kernel F of a space X, P-dim X = P-dim (X\ F ) .

Proof. By Theorem 1.5, P-dim X 5 P-dim ( X \ F ) . To prove the opposite inequality, assume P-dim X 5 n and let U be a finite P-border cover of X \ F with enclosure G. As P is strongly close& additive, U is a finite P-border cover of X with enclosure F U G. Let W be a P-border cover of X with enclosure H such that W refines U and has ord W 5 n 1. Open monotonicity implies that the restriction W’ of W to X \ F is a finite P-border cover of X \ F of order less than or equal to n 1. Therefore P-dim ( X \ F ) 5 n.

+ +

In passing, we state the analogous theorem that provides the same justification for P-Ind. Its proof is an immediate consequence of Theorems 111.1.8 and 111.3.23. 1.8. Theorem. Suppose that the universe is contained in the Dowker universe 2, and that the class P is a semi-normal family. For every closed P-kernel F of a space X ,

P-Ind X = P-Ind ( X

\ F).

190

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

1.9. Theorem (Point addition t h e o r e m ) . If X $!

P,then

P-dim ( X U { p } ) 5 P-dimX.

Proof. As the inequality is trivial when p E X , we assume p Theorem 1.5 will complete the proof.

4 X.

Finally we have an addition theorem.

1.10. Theorem (Addition theorem). Suppose that the universe is monotone and that P is monotone and additive. Let A and B be subsets of a space X such that X = A U B . Then P-dim X 5 P-dim A

+ P-dim B + 1 .

Proof. We may assume that m = P-dim A and n = P-dim B are finite because the inequality is trivial in the contrary case. Let U be a finite P-border cover of X with enclosure F . Since U is monotone and the class P is monotone, the restriction U A of U to A is a P-border cover of A . By Lemma 1.4 there is a set G A and an open collection WA such that WA refines U and satisfies G A = A \ U WA E P n U and ord WA 5 m 1. Similarly, there is a set Gg and an open collection WB such that W Brefines U and satisfies Gg = B \ (J WB E P nU and ord W g 5 n 1. Let W = WA U WB. Clearly, W refines U and ord W 5 m -t n t 2. Obviously, X \ U W = F U ( G A\ U W g ) U (GB \ U WA). As P is monotone and additive, we have that W is a finite P-border cover of X . Therefore, P-dim X 5 m n 1.

+

+

+ +

Let us now go to P-Dim. The closed monotonicity of the function P-Dim is not immediately obvious without further conditions. So some of the corresponding statements to the above theorems cannot be proved by analogy. The following lemma and theorems do not rely on the closed monotonicity of P-Dim and so their proofs are straightforward modifications of the corresponding ones for P-dim.

1.11. L e m m a . Let Y be a subspace of a space X and suppose that U = { U, : (Y E A } is an open collection of X such that U is locally finite in the subspace U U and the restriction U y of U to Y is a P-border cover of Y with enclosure F . If P-Dim Y 5 n, then

1. FINITE UNIONS

191

there is a set G and there is an open collection W = { W , : a E A } of X such that the following conditions hold: (a) W , C U , for each a in A . (b) G = Y \ U W E P . (c) ord W 5 n 1.

+

Moreover, if Y is also closed in X , then the further condition

q

u

(4 (U \ G = w holds provided the condition (c) is changed to the following: (c*) There is an open set V of X with ord, W 5 ord, U for each x in X \V such that the inclusions Y \ G c V c

{ x : o r d , W ~ n + l } h o l d m o d u l o t h e s eH t =X\UU. 1.12. Theorem. If A and B are closed subsets of a space X such that X = A U B , then

P-Dim X

5 max { P-Dim A , P-Dim B }.

Corresponding to Theorem 1.5 is the following. Notice that there is the stronger hypothesis of A n B = 0. 1.13. Theorem. Let A and B be disjoint subsets of a space X such that X = A U B and A is open. Then

P-Dim X 5 max { P-Dim A, P-Dim B }.

Proof. The proof of Theorem 1.5 assumes A n B = 0 because this condition is a consequence of Theorem 11.5.6. Since the analogous theorem for P-Dim is not known, we must assume this added condition. Otherwise, the proof is the same when the obvious changes have been made. 1.14. Theorem (Addition theorem). Suppose that the universe is monotone and that P is monotone and additive. Let A and B be subsets of a space X such that X = A U B . Then

P-DimX 5 P-DimA

+ P-Dim B + 1.

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IV. FUNCTIONS O F COVERING DIMENSIONAL TYPE

2. Normal families

The functions of covering dimensional type will be studied in the context of normal families. The main concerns are the monotonicity theorems and the various closed additivity theorems for P-dim and P-Dim. It will be shown by cosmic family arguments that the equality P-dim = P-Dim holds and thereby the closed monotonicity will be establish for P-Dim in the Dowker universe ID. The following theorem concerns the monotonicity of functions of covering dimensional type.

2.1. Theorem. Let the universe be monotone and P be monotone. For each space X , P-dim Y P-dim X holds for every subspace Y of X if and only if P-dim U P-dim X holds for every open subspace U o f X .

< <

Proof. As the proof of one of the implications is obvious, we shall give only the other one. Suppose that P-dimU 5 n for each open subset U of X and let Y be a subset of X . We may assume 0 n = P-dim X because the contrary case follows from definitions. Let W be a finite P-border cover of Y with enclosure F . For each W in W let W‘ be an open set of X such that W = W’ f l Y . Denote the collection of these open sets W’ by W’. Then we have Y \ F C U = U W’. Recalling that 0 E P n U ,we have that W’ is a finite P-border cover of U . From P-dimU 5 n, there is a P-border cover V‘ of the open subspace U with enclosure G’ such that ord V ’5 n 1 and V‘ refines W’. Denote { V‘ n Y : V‘ E V ’} by V and let G = G‘ n Y , Since the universe U is monotone and P is monotone and strongly closed-additive, we have that G U F is in P. It follows that V is a P-border cover of Y with ord V 5 n t 1.

<

+

Let us precede our investigation of normal families by studying regular and cosmic families separately. We extend Lemma 1.4. 2.2. Lemma. For a space X let U = { Uj : j = 0 , 1 , . . . ,k } be a finite P-border cover with enclosure F and let A be an open set contained in { 2 : 0 < ord, U 5 n 1}. Suppose that Y is a closed set o f X with P-dimY 5 n. Then there is a P-border cover W = { Wj : j = 0,1,. . . ,k } with enclosure G and there is an open set B that satisfy the conditions

+

(a) Wj

c U j for j = 0 , 1 , . - .,k,

2. NORMAL FAMILIES

193

(b) A c B c U W = ( U U ) \ G , (c) G = F u ( Y \ B ) a n d Y \ G = Y n B , (d) Wj n A = U j n A for all j , (e) ord { Wj n B : j = 0, 1, . ,k } 5 n 1.

..

+

Proof. As A is open, we have P-dim (Y \ A ) 5 n. By Lemma 1.4 there is an open collection W’ = { Wj’ : j = 0,1,. . .,k } of X with G’ = Y \ ( A U U W’) E P and there is an open set V such that W‘ shrinks Umodulo G’, V C U W’ and (1)

(U U ) \ G’= U W’,

(2)

ord, W’ 5 ord. U,

(3)

c Y \ A, x E X \ V,

G’

Y \ (AUG’) C V C { x : ord, W‘ 5 n + 1 ) mod F.

For each j let Wj = ( U j n A) U Wj’ and denote the collection of open sets so constructed by W. Also let B = A U (V \ G’).As F n A = 0, we have from (1) that A fl G = 0 and X \ U W = F U G’. Since G‘ is in P fl U ,the collection W is a P-border cover of X with enclosure G = F U G’ that shrinks U . So conditions (a) and (b) hold. We infer from (3) that Y \ ( A U G) C V holds. So it follows that Y = ( Y fl ( A U G ) ) U ( Y \ ( A U G)) c ( A U G ) U V = B U G. Consequently Y \ G = (Y n B ) \ G = Y \ G, where the last equality follows from (b). As B fl G is empty, we have Y \ G = Y fl B and thus Y \ B c G. Moreover, as G \ F c Y , we have G \ F C Y \ B . So we have arrived at G c F U ( G \ F ) C F U ( Y \ B ) C G and thereby (c) is verified. Condition (d) follows from the definition of Wj ; and condition (e) is a consequence of (d) and (3). 2.3. Theorem. Suppose that P is a regular family and let { Xi : i = 1,2,. ..} be a countable closed cover of a space X . Then

P - d i m X = s u p { P - d i m X i : i = 1 , 2 , ...}.

Proof. We must show that the supremum is no less than P-dim X. By Theorem 1.1 we may assume Xi c Xi+l for each i. Let VO = { U0,j : j = 0 , 1 , . . ., k } be a finite P-border cover of X with enclosure Fo. Define A0 = 0. With the aid of Lemma 2.2 we can construct for each i a P-border cover Ui = { U i , j : j = 0,1,. . ., k } with enclosure Fi and an open set Ai that satisfy the following conditions: For i 2 0 (a-i) Ui+l shrinks Ui mod

194

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

and for i 2 1 (b-i) A;-1 C Ai C U 77;= (U Ui-I) \ F;, (c-i) Xi \ Fi = X i n A;, (d-i) Ai-l n Ui,j = n Ui-l,j, (e-i) ord{ U;,j n A; :j = 0,1,. ..,k} 5 n 1. For each j let Uj = U{ U;,j n Ai : i = 1 , 2 , . . . } and let U denote the collection of open sets so constructed. By (a-i), (c-i) and (d-i) we have U j = U;,j: i = 1,2,. . .} and hence it follows that X \ U U = U{ F; : i = 0,1,2,. . . } by condition (b-i). Since P is a regular family and U; C Uo,+,the collection U is a P-border cover of X that shrinks UOmod U{ F; : i = 0,1,2,. ..}. Let 2 be a point of X that is not in the enclosure of U. Then z is in Xi n A{ for some i by condition (c-i). Consequently ord, { U;,j n A; : j = 0,1,. . . ,k } n 1 for that i by condition (e-i). So ord U 5 n 1 by condition (d-i). Thereby we have shown P-dim X n.

+

n{

<

< +

+

Cosmic families will be considered next. The following lemma will prove useful in the next section as well as in the present one.

Lemma. Suppose that P is a cosmic family and that n is in N. Let X be a space and F be a locally finite closed cover of X such that P-dim F 5 n for each F in F . If U = { Us : s E S } is a P-border cover of X with enclosure G such that each member of F meets only finitely many sets Us,then there is a P-border cover with enclosure H that shrinks U modulo H and has order not exceeding n 1. 2.4.

+

Proof. For the purposes of a transfinite inductive construction, adjoin the set FO = 0 to the collection F and arrange the members of this collection into a transfinite sequence F,, a [ , of type [ 1. Let us inductively define for each a a closed P-kernel G, of F, and a P-border cover U, = { U,,s : s E S} of X such that G,, V, and the closed set H , = U{ G, : y 5 (Y } satisfy the condition

<

(4)

X\UU,=GUH,,

and satisfy the next conditions modulo the set H ,

+

2. NORMAL FAMILIES

195

Every condition will be satisfied for Q = 0 if we define UO,,= Us for all s in S and define Go = 0. Assume that the P-border cover U , and closed P-kernels G, of Fa satisfying (4)-(7) are defined for all Q less than Q O , where QO 2 1. Let us first consider the collection Ub, = { U;,,, : s E S} and the set HA, defined by

for each s in 5’. We shall show that this collection is open in X . This is clear when (YO = q + 1, because we will then have that HAo = H, is a closed set and U;,,, = U,,, \ H,. So we shall assume that a0 is a limit ordinal number. Consider a point 2 that is not in the closed set G U Hk,. As P is a cosmic family and F is a locally finite collection in X , we have that G U is in P n U. Also there exists a neighborhood U in X of x and an ordinal number ,B less than QO such that U n F . = 0 whenever ,B 5 y < Q O . As Up is a P-border cover of X with enclosure G U H p , there exists an s in S such that x is in Up,s. It follows from (7) that z E Ua,, whenever /3 < a < QO, so x E U;,,,. Hence Uh, will be a P-border cover of X with enclosure G U HA, once it is shown that each U;,,, containing the point x is a neighborhood of z. To this end, consider the open set U and the ordinal number p determined above for the point x. From ( 5 ) we have x E ( U n Up+) \ Hp. Since 5 is not in the closed set HAo, we have that z is in the open set ( U n Up,,) \ HAo and this open set is contained in UAo,s by (7). Thereby, it has been shown that Ub, is a P-border cover of X . The restricted collection { F a , n U:,,, : s E S} of Ub, to Fa, is a finite P-border cover of F,, because each member of the family F meets at most finitely many members of the collection U and condition ( 5 ) holds. Consequently, from P-dim Fa, n, this P-border cover of Fa, has a P-border cover shrinking { V, : s E S } with enclosure G,, of order not exceeding n + 1. Define Ua,,, to be the open set (ULOIs\ Fa,) U V, for each s in S . Then U,, = { UAo,s : s E S } is readily seen to be a P-border cover of X with enclosure H;, U G,, that satisfies the conditions (4)-(7) for Q = (YO. Hence the construction of the P-border covers U , of X satisfying the conditions (4)-(7) for Q 5 ( is completed.

<

196

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

Now it follows from ( 6 ) that ord Ut 5 n. By virtue of ( 5 ) , Ut is a shrinking of U modulo H and the lemma is established. The next locally finite sum theorem for P-dim is an immediate consequence of Lemma 2.4. 2.5. Theorem. Suppose that P is a cosmic family and let X be a space and F be a locally finite closed cover of X. Then

P-dim X = sup { P-dim F : F E F}. Combining Theorems 2.3 and 2.5, we have the following theorem on semi-normal families. 2.6. Theorem. Suppose that P is a semi-normal family. Then the class { X : P-dim X 5 n } is a semi-normal family for each n.

Finally we come to the analogue of Dowker’s theorem that characterizes dim by means of locally finite open covers. 2.7. Theorem. Suppose that P is a cosmic family. Then

P-dim X = P-Dim X .

for each space X .

Proof. Obviously P - d i m X 5 P-D im X . So let n be in N and assume P-dim X 5 n. Let U be a P-border cover of X with enclosure G such that U is locally finite in X \ G. By Theorem 1.7 we have P-dim ( X \ G) 5 n. The collection U = { Us: s E S } is a locally finite open cover of X \ G. Denote by T the family of nonempty finite subsets of S and for every T in T define

We have P-dim FT 5 n because FT is closed in X \ G. The collection F is a locally finite closed cover of X \ G. By Lemma 2.4 there is a P-border cover V of X \ G with enclosure H that shrinks U modulo H and has ord V < n+ 1. As X \ U V = G U H and G n H = 0 hold and P is strongly closed-additive, we have that V i s a P-border cover of X. Therefore, P-Dim X 5 n.

Remark. As we have found in the remark at the end of Section 11.5, the spaces under consideration are required to be hereditarily normal due to the possibility that P # { 8) is true. It was

3. THE DOWKER UNIVERSE

D

197

found there that many of the arguments found in that section held true for all normal spaces when P = { 0). This observation is also true for the present section. Indeed, this is especially true for the last three theorems. We shall state these facts next. 2.8. Theorem. Let space X . Then

F

be a countable, closed cover of a normal

dimX=sup{dimF:FEF}. 2.9. Theorem. Let F be a locally finite, closed cover ofa normal space X . Then

dimX = s u p ( d i m F : F E F}. 2.10. Theorem. For every normal space X

,

dimX = DimX. 3. The Dowker universe Z' Y

It was shown in the last section that the function of covering dimensional type P-dim behaves very nicely for semi-normal families P . Indeed, the family { X : P- d i mX 5 n } is guaranteed to be semi-normal with fewer conditions than the corresponding family for the function of inductive dimensional type P-Ind. But, as it is for P-Ind, the situation for normal families will require a stronger condition on the universe. The difficulty is due to the lack of open monotonicity of these function in the case of the general universe. Thus the Dowker universe 2) will come to the fore in that the universe will be required to be contained in 2). Agreement. In addition to the agreement made in the introduction of the chapter, the universe will be contained in the Dowker universe. 3.1. Lemma. Suppose that P is a cosmic family. If Y is a cozero-set in a space X , then

P-dim Y 5 7'-dim X .

198

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

Proof. Select a continuous function f : X + [ 0,1] such that Y is f-' [ (0,111. As the sets f-' [ [2-i+1, 2-a]], i = 0,1,. . . , are closed in X and form a locally finite cover of the subspace Y , we have P-dim Y 5 P-dim X by Theorems 11.5.6 and 2.5. The next lemma deals with Dowker-open sets. 3.2. Lemma. Suppose that P is a semi-normal family. If Y is a Dowker-open subset of a space X , then

P-dim Y

5 P-dim X .

Proof. Assume 0 5 P - d i m X = n, By definition, the set Y is the union of a point-finite collection { Us: s E S } of cozero-sets U s of X . For each positive integer i let Ti be the family of all subsets of the index set S that have exactly i elements. By Lemma 111.3.3, the collection of subsets Ki of Y consisting of the points of X that are in Us for exactly i members s in S, where i is a positive integer, has the following properties: (1) Y = U{ Ki : i 2 1 }. (2) K in li'j = 0 for i # j . (3) Fi = U{ K j :j5 i } is closed in Y for i 2 1. (4) Ki = U{ KT : T E Ti }, where the sets KT,defined by letting KT = li'i n Us : s E T } for T E Ti,are open in Ki and pairwise disjoint.

n{

As each U , is a cozero-set of X , the set KT is a cozero-set of by (4). From (2) and (4)and from the identity

Ir'i

we have that KT is a countable union of closed sets of X for T in Ti.By Theorem 2.6 we have P-dim KT 5 n. So P-dim Ii'i 5 n by the same theorem because of (4). Theorem 1.5 gives P-dim Fi 5 n. Finally P-dimY 5 n is established by Theorem 2.6. In passing, we comment that the first two lemmas do not use the condition U C 2). The next lemma deals with the class & of strongly hereditarily normal spaces.

3. THE DOWKER UNIVERSE 2,

199

3.3. Lemma. Suppose that the universe is contained in the class & and that P is a semi-normal family. If Y is an open subspace of a space X , then P-dim Y 5 P-dim X.

Proof. Assume 0 5 P - d i m X = n. Let U = { Ui : i = 0 , 1 , . . . ) k } be a finite P-border cover of Y with enclosure F . The sets Ui and X \ clx(Ui) are separated for each i. Since X is in & 7 there is a Dowker-open set V , of X such that Ui = V , n Y . Let ? be the openset u { V ; : : i = O , l ,..., k } . Then V = { V , : i = O , l , ..., k } i s

a finite P-border cover of y . By Lemma 3.2 and Corollary 1.6, we h a v e P - d i m F S n . Let V ‘ = { V , ‘ : i = O , l ,..., k } be a P - b o r d e r cover of ? with enclosure F’ such that V’ shrinks V modulo F’ a n d o r d V ’ S n + l . Thecollection U ’ = { K ’ n Y : i = O , l 7 . . . , k } clearly satisfies ord U’ 5 n 1 and shrinks U modulo F‘. Since P is open-monotone, we have that G = F‘ n Y is in P and hence U’ is a P-border cover of Y with enclosure F U G. So P-dim Y 5 n.

+

3.4. Theorem. Suppose that P is a semi-normal family. Then { X : P-dim X 5 n } is an open-monotone, strongly closed-additive

semi-normal family for each n in

N.

Proof. By definition, the Dowker universe is the normal family extension of the class & in the universe /v;r. (See Definition 111.3.1.) So X is the union of a a-locally finite collection of dosed sets of X that are in the class &. The theorem follows easily from Lemma 3.3 and Theorems 1.5 and 2.6. 3.5. Theorem. Suppose that the universe is monotone and P is a normal family. Then { X : P-dim X 5 n } is a strongly closedadditive normal family for each n in N.

Proof. The theorem follows easily from Theorems 2.1 and 3.4. The aim for the rest of the section is to prove a characterization of P-dim by means of partitions. We shall use the following definition. 3.8. Definition. Let G be a P-border cover of a space X with enclosure G and let Y be a subspace of X . Then the relative dimension of Y with respect to G , denoted P-dimG Y , is at most n if the

200

IV. FUNCTIONS O F COVERING DIMENSIONAL T Y P E

restricted collection GIY admits a shrinking by a P-border cover H of Y \ G with order at most n 1.

+

The following general covering lemma will prove useful. 3.7. Lemma. Let X be a normal space and Fl and Fz be a pair of disjoint closed sets of X . If G is a locally finite open cover of X refining the binary cover { X \ F1, X \ Fz }, then there exists a closed set S of X and a locally finite open cover H of X satisfying the conditions: (a) S is a partition between F1 and Fz. (b) Hshrinks G. (c) ord,H < ord, G for any point x in S.

Proof. Since G = { G, : a E A } is a locally finite open covering of the normal space X , there is an open shrinking U = { U , : a E A } of G such that cl (U,) C G, for every a , Let U1 = { U , : a E A1 } be the family of all members of U that meet F1. Define

Then S is a partition between F1 and Fz. Let x be any point of S. Obviously 2 is not in U , for each a in A l . But there is an a in A1 with 2 E cl(U,) since G, whence U , is locally finite. Since U covers X there exists a ,O that is not in A1 with x E Up. From this observation it follows easily that

satisfies conditions (b) and (c). 3.8. Lemma. In aspace X let G = { G, : a E A } be a?-border cover with enclosure G and let B be a closed set such that G is locally finite in the space X \ G and P-dimG B 5 n. Then there exists a P-border cover H = { Ha : cr E A } of X with enclosure H and there exists an open set V such that H shrinks G mod H , B c V mod H and ordHIV 5 n 1.

+

Proof. As P-dimG B 5 n, there is a P-border cover Go of B with enclosure Go such that ord Go 5 n 1 holds and Go is a shrinking of G ( B . We write Go = { G, : a E A ’ } with A’ c A . Because B is closed in X and P is strongly closed-additive, we have that G U Go

+

3. THE DOWKER UNIVERSE 2,

201

is a closed P-kernel of X . We shall perform the remainder of the construction in the subspace X \ (G U Go). There will be no loss of generality if we let G U Go be the empty set. Select a closed cover K = { I<, : a E A ’ ] of B that shrinks Go with o r d K 5 n t 1. By Lemma 1.2 there is for each a in A‘ an open set HL of X with K , C HL c G, such that ord { HL : a E A‘ } 5 n 1. Let V be an open set of X with B c V c cl(V) c U{ HL : a E A ’ } and let H , = HL U (G, \ cl ( V ) )for (Y E A’. Then V and the collection H = { H , : a E A } satisfy the desired conditions.

+

The next theorem provides a characterization of P-dim by means of partitions. 3.9. Theorem. For every space X , P - d i m X 5 n if and only if for each pair o f closed sets F1 and F2 of X and for each finite P-border cover G with enclosure G such that F1 n F2 = 0 mod G there exists a closed set S such that S is a partition between Fl and F2 modulo G and P - d i m G S 5 n - 1.

Proof. Suppose P - d i m X 5 n. Since Fl n F2 = 0 mod G, there are closed sets Fi and Fi with Fl C Fi and F2 C Fi such that F: \ G and Fi \ G are neighborhoods of Fl \ G and F2 \ G respectively in the subspace X \ G and

Let GI = G A { X \ ( F { U G ) , X \ ( F i U G ) ) . Since G1 is a finite P-border cover of X with enclosure G, there is a finite P-border cover G2 with enclosure G2 that refines GI such that ord G2 5 n 1. In the subspace X \ G2 there exist by Lemma 3.7 a closed subset B that is a partition between Fi \ G2 and Fi \ G2 and a finite open cover H of X \ G2 that shrinks G2 with o r d H ( B 5 n. From (1) we infer the existence of a closed set S that separates F1 and F2 modulo G and such that S n ( X \ G2) C B . Clearly H is a P-border cover of X that refines G. Since S is closed, we have that HIS is a P-border cover of S with enclosure G2 n S . Therefore HIS is a P-border cover of S that refines GIS with ord HIS 5 n. That is, P-dimG S 5 n - 1.

+

202

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

Conversely, suppose that the condition is satisfied and consider a finite P-border cover G = { Gi : i = 1,. . .,k } of X with enclosure G. Since X is hereditarily normal, there is a closed collection { Fi : i = 1,. . .,k } that covers X modulo G with Fi C Gi modulo G for each i. For the purposes of the construction let HO = G and let the enclosure of HObe denoted by Ho. Select an open set U1 of X that satisfies F1 C U1 C

Ho,

cl ( U I ) c GI mod

P-dimHo B ( U 1 ) 5 n - 1. By Lemma 3.8 there is a P-border cover HI = { H1,i : i = 1,. . . , k } of X with enclosure H1 and an open set V1 of X that satisfies

H I shrinks HO mod H I , B(U1) C V1 mod H I , ordHlIV1

5 n.

Analogously, by repeated application of the condition of the theorem together with the repeated application of Lemma 3.8, we can select for i = 2,. . . , k an open set Ui of X and we can get a P-border cover Hi = { H i , j :j = 1,. ..,k } of X with enclosure H i and an open set Vi of X that satisfies

Fi

c Ui c cl(Ui) c Gi

mod

Hi,

P-dimHi-l B ( U i ) 5 n - 1,

Hi shrinks Hi-1 mod H i , B(Ui) c V , mod H i , ordHilV, 5 n. Clearly we have

With

Li

={

ua, x \ Cl(Ui)},

i = 1,. ..,k,

3. THE DOWKER UNIVERSE 2,

203

set

L = A { L i : i = I , ...,k}.

<

Then L covers X \ U{ B ( U i ) : i = 1,. . .,k } and ord L 1 holds. Moreover, L refines G modulo H I , since X \ cl ( U i ) : i = 1, . . .,k } is contained in H k . Define W to be the collection

n{

{ L \ H k :L

EL

} U { H k , j n (U{ V , : i = 1, ...,k })

. }.

: j = 1, . . ,k

Then W is a P-border cover of X with enclosure H I , that refines G for which ord W 5 n 1 holds. Therefore, P-dim X 5 n.

+-

We are now able to prove the following useful lemma. 3.10. Lemma. For a space X let F be a closed set and suppose that G is a P-border cover of X with enclosure G such that (a) the collection G is locally finite in X \ G, (b) P-dimG F 5 n, (c) sup { P-dim 2 : 2 is closed and 2 n F E P} 5 n. Then P - d i m G X n.

<

Proof. Denote by { G, : a E A } the given P-border cover G of X. Since P-dimG F n, by Lemma 3.8 there is a P-border cover H I = { H I , : a E A } with enclosure HI and there is an open set V1 such that El1 shrinks G modulo H I , ord HIIVl 5 n 1 and F c Vl modulo H I . The sets F n Vl and X \ V1 are separated modulo H I . There is an open set U of X such that F n VI c U c cl ( U ) c VI modulo H I . As F \ U € P, from (c) we have P-dim ( X \ U ) n. Again by Lemma 3.8 there is a P-border cover H2 = { Hzm : a E A } with enclosure H2 and there is an open set V2 such that H Z shrinks HI modulo H z , ord H2 IV2 5 n 1 and X \ U c V, modulo H2. Let W be the collection of open sets

<

+

<

+

wa =

n v1)u ( H ~v2)) ~\ H ~ ,

Q

EA

Clearly W shrinks G and W is a P-border cover of X with enclosure H z . Let us show that ord W 5 n 1. To this end let z be in V1 \ H2 and let il, . . . ,,i be distinct indices such that z is in Wij for j = 1,. . .,m. Since H2 shrinks HI, we have

+-

Wij c (Hlij n Vl) u ( H 2 i j n V2)= Hlij,

j = 1,. .. , m .

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

204

So we have m 5 n t 1, that is, ord, W 5 n t 1. Finally suppose that 2 is in ( X \ V1) \ H2 and let il . . . ,i, be distinct indices such that 2 is in Wi,f o r j = 1 , .. .,m. From 2 E (Wij \ VI) c Wij n V2 we h a v e t h a t z i s i n H 2 i j n V , f o r j = 1 , ..., m. S o w e h a v e m < n + l , that is, ord, W 5 n. This completes the proof of ord W 5 n -t 1. Thereby we have shown P - d i m G X 5 n. Note that the discussion of P - d i m G X 5 n did not use the condition U C D. Also, Lemma 3.8 can be proved in the universe "PI, the normal spaces modulo P defined in Section 11.5. The next theorem, due to Morita [1953], concerning ordinary covering dimension will be used in the following section. As its proof is quite long, we shall not present it here; in addition t o the original proof in Morita [1953] there is another one on pages 127-129 of Nagami [1970]. We shall use d i m G for P - d i m G when P = { @ } . 3.11. Theorem. Let X be a space, G be a locally finite open covering of X , F = { F, : a E A } be a closed covering of X and U = { U , : a E A } be a locally finite open collection satisfying the following conditions: (a) F, C U , for each a in A. (b) dim F, 5 n for each a in A. (c) dim ( F , n Fp) 5 n - 1 whenever a

,-J

# p.

Then d i m G X 5 n.

4. Dimension and mappings

The agreement of the last section will be continued.

Agreement. In addition t o the agreement made in the introduction of the chapter, the universe will be contained in the Dowh-er universe. Mappings were already encountered in Chapter 11. Theorem 11.6.6 characterized P-dim by means of mappings into spheres for classes P that satisfied rather mild conditions. Also theorems concerning mappings into spheres and P-Ind were proved in Chapter 11. These theorems can be used to compare the functions P-dim and P-Ind.

4.

DIMENSION AND MAPPINGS

205

4.1. Theorem. Suppose that P is a semi-normal family. For every space X , P-dim X 5 P-Ind X . 4.2. Theorem. Suppose that the universe is contained in the class Np of perfectly normal spaces and that P is a cosmic family. For every space X,

P-dim X 5 P-Ind X .

Proofs. Theorem 4.1 is an immediate consequence of Proposition 11.7.1 and Theorem 111.3.23. For Theorem 4.2 we observe that a semi-normal family is cosmic but not conversely. Consequently, more restrictions have been placed on the universe. When U is contained in the class Np of perfectly normal spaces, a cosmic family P is automatically open-monotone and strongly closed-additive. Thus P-Ind Y 5 P-Ind X follows by Theorem 111.3.26 whenever Y is an open subspace of X. Proposition 11.7.1 will complete the proof. The equivalence of the inequalities I n d X 5 0 and dimX 5 0 for normal spaces X is given in Corollary 11.5.8. We also have the following theorem from Theorem 11.5.7. 4.3. Theorem. Under the conditions of Theorems 4.1 or 4.2, for each space X ,

P-IndX

5 0 if and only if P - d i m X 5 0.

Proof. Only 0 2 P - d i m X 2 P - I n d X requires a proof. Suppose P-dim X = 0 and let FO and Fl be disjoint closed subsets of X . Let Uo and U1 be open sets with F1 n cl (Uo) = 0 and FOn cl ( U l ) = 0 such that X = UOU U l . The P-border cover U = { Uo, Ul } has by Theorem 11.5.7 a P-border cover shrinking V = { VO,V1 } with enclosure G such that ord V 5 1. Define S to be X \ (Wo U W1) where WO= VOU ( X \ cl(U1)) and W1 = fi U ( X \ cl(U0)). As S is a closed subset of G, we have that S is in P. Also FO C Wo and J’l C W1. So S will be a partition between Fo and F1 in X if WOn W1 = 8. This last condition obviously holds. Thereby we have shown P-Ind X 5 0.

206

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

In passing, let us remark that the equality I n d X = d i m X for every metrizable space X will be established in Chapter V by means of basic dimension theory, that is, a dimension theory that relies on the existence of certain bases for the open sets of a space X . Let us turn to other mapping theorems. In dimension theory, there are theorems on dimension raising maps and dimension lowering maps. These theorems usually deal with mappings that satisfy additional conditions such as being closed maps. Some analogues of these theorems also hold for other functions of dimensional type. In another line of development, there are theorems that require no additional conditions on the mapping. These theorems establish relationships between dimensions and other functions of dimensional type. The remainder of the section will be a short exposition on these mapping theorems. 4.4. Definition. Let f : X -+ Y be a continuous function. The order o f f , denoted d j , is the least cardinal number Q such that the cardinality of f-'[y] is less than or equal to Q for every y in Y . The dimension of f, denoted dim f , is the extended integer sup { dim f-'[y] : y E Y }.

The following dimension-lowering closed mapping theorem is due to Morita [1956]. 4.5. Theorem. Let f be a closed, continuous mapping of a normal space X onto a nonempty paracompact space Y . Then

dim X

5 dim f -t Ind Y.

Proof. Obviously we may assume Y = f [ X ] . Recall that the equivalence I n d X 5 0 if and only if d i m X 5 0 holds for the universe N of normal spaces (Corollary 11.5.8). So when I n d Y 5 0 the inequality that is to be proved becomes dim X 5 dim f dim Y . We shall give here a straightforward proof of this inequality. Clearly we may assume that dim f is finite. Let G be a finite open cover of X . For each point y of Y we have dim f-* [y] 5 dim f . We infer from Lemma 3.8 the existence of an open subset H , of X such that f-'[y] C H , and dimG H , 5 dim f. With V, = Y \ f[X \ H , ] for each y in Y , we have that V = { V, : TJ E Y } is an open cover of Y . As Y is a paracompact space, it has is a locally finite open

+

4. DIMENSION A N D MAPPINGS

207

cover { U , : a E A } that refines V.We have Dim Y = dim Y by Theorem 2.10. Consequently there is an open cover W = { :a E A } of Y such that ord W 5 1 and W, c U, for each a in A . Now for each a with W , # 0 we select a member y, of Y such that W , c VYa. Then f-' [W,] C H Y a . So { f - l [Wa]: cr E A 1is an open cover of X by the mutually disjoint sets f-' [We] with dimG f-l[W,] 5 dim f. We infer from this that the finite open cover G of X can be refined by an open cover whose order does not exceed dim f 1. Thereby we have shown that dim X _< dim f Ind Y when Ind Y 5 0. We shall now complete the proof by establishing the inductive step of the induction on IndY. Assume that dimf is finite and let Ind Y = m 2 1 and G be a finite open cover of X . The proof will begin in the same way as in the preceding paragraph. For each y in Y let H , and V, be as above and let { U , : a E A } be a locally finite open cover of Y that refines V = { V, : y E Y }. Assume that the indexing set A is well-ordered. Let F = { Fa : (Y E A } be a closed cover of Y such that F, C U , for each a. Then choose for each a an open set W , of Y such that

w,

+

+

Fa

c W , c C1Y(Wa) c u,,

5 m - 1.

Ind By(W,)

Set

Ii-0 = w,,

I<, = w,\ U{ cly(Wp) : p < a } ,

a

> 0.

It follows that Ind (cly(K,) n cly(Kp)) 5 m - 1 whenever a # ,8 and that Y = U{ cly(Ka) : a E A } . By the induction hypothesis we have dim (f-'[cly(K,)]

n f-'[cly(Kp)]) 5 dimf

Since dimG f-'[cly(IC;,)]

cr

# p.

5 dim f, we have by Theorem 3.11 that

dimG X _< dim f Hence dim X 5 dim f

+ m - 1,

+ m.

+ m and the induction is completed.

In general the classes P n u will contain nonempty spaces. Consequently, extensions of the last theorem to general classes P become more complicated. To get an easy extension, we make the following definition.

208

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

4.6. Definition. For a class P the dimension o$set in the universe is the least nonnegative extended-integer @ ( P )such that

dimX

5 dim f t @ ( P )

holds for every closed map f : X

-+

Y with X E U and Y E P n U.

Obviously the only interesting values of the offset are the finite ones. In this case, we have that dim X 5 @ ( P for ) each X in P n U. 4.7. Theorem. Suppose that P is a semi-normal family. Let f : X -+ Y be a closed continuous map, where Y is a paracompact space. Then dim X

5 dim f

+ P-Ind f [XI + 1 t @ ( P ) .

Proof. The proof will be by induction on P-Ind f [ X ] . The case where P-Ind f[X] = -1 follows from the definition of the P-offset. Suppose P-Ind f[X] = m 2 0 and let n = dim f t rn t 1 t @(P). The inductive step of the proof will follow those of Theorem 4.5. Let G be a finite open cover of X . In exactly the same manner as in Theorem 4.5 we can find a locally finite collection { K , : a! E A } of open sets of Y such that

By the induction hypothesis we have dirn(f-*[cl~(K,)]n f-'[cly(Kp)])

5 n-1

a!

# 0.

Now we have by Theorem 3.11 that dimG X 5 n holds. As G is an arbitrary finite open cover of X , we have d i m X 5 n. 4.8. Examples.

a. Let the universe be the class Mo of separable metrizable spaces and P be the class of countable spaces. Then @(P)= 0. So we have dim X 5 dim f f P-Ind Y 1 for every closed continuous map f:X+Y.

+-

4. DIMENSION AND MAPPINGS

209

b. Let the universe be Mo and Q be a semi-normal family. Define P to be the class { X E Q : Ind X 5 m }. Then P is a semi-normal family and @ ( P )= m. Therefore, for a closed continuous map f : X -+ Y we have dim X 5 dim f P-Ind f [ X ]+ m + 1.

+

In particular, suppose that Q = A(1), the class of absolute additive Bore1 class 1, and suppose that Y is a space with the property that each of its compact subsets K has d i m K 5 m. Then we have dim X 5 dim f A( 1)-Ind Y m 1 for closed continuous maps f : X Y.

+ +

+

---f

4.9. Definition. For a class

P the order offset in the universe is the least nonnegative extended-integer R ( P ) such that dimY

5 d j t R(P)-

2

holds whenever f is a closed map of a nonempty space X in P n U onto a space Y in U. The order offset R({ S}) is defined to be 0. Using a singleton space for Y , we observe that R ( P ) 2 1 if there exists a nonempty space X in P n U . Also, when R(P) is finite we have dim Y 5 R ( P ) - 1 for each Y in P n U.

4.10. Theorem. Suppose that P is a semi-normal family. Let f : X ---f Y be a closed, continuous, onto map of a nonempty space X . Then dim Y 5 P-Ind X d f a ( P )- 1.

+ +

Proof. The proof will be by a double induction on n = P - I n d X and k = d j - 1. If P nu contains a nonempty space, then the case n = -1 and k 2 0 follows from the definition of R(P). Suppose k = 0 and n 0. We shall prove dimY n t R ( P ) by applying Theorem 3.9. Let G be an open cover of Y and let FO and FI be disjoint closed sets of Y. Note that f is a homeomorphism. There is a partition S between f [Fo]and f [ F I ]in X such that P-Ind S 5 n - 1. So f[S]is a partition between Fo and F1 in Y . As n R ( P ) 2 0, it is sufficient to show dim f[S]5 n t R ( P ) - 1 when S # 0. This follows by induction on n for k = 0. We have dimG f[S]5 n R(P) - 1. Theorem 3.9 yields dimY 5 n t R ( P ) when k = 0. Finally suppose n 2 0 and k 2 1. Let G be an open cover of Y and let Fo and Fl be disjoint closed sets of Y. There is a partition S between f - l [Fo]and f-' [ F ! ]in X such that P-Ind S 5 n - 1.

>

<

-'

+

+-

-'

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

210

Then, as n t k iR ( P ) 2 1, we have dim f[S]5 n t k t R ( P ) - 1 by the induction hypothesis. Let Uo and U1 be disjoint open sets of X with f-l[Fo] c UOand f-'[Fl] c U1 such that X \ S = UOU U1. Because f is a closed map, we have that Vo = X \ f[S U U l ] and V1 = X \ f[SU Uo] are disjoint and open in Y with FOC VOand F1 c V1. Denote X \ (& u V1) by I<. Then I( = ( ~ [ UnOf ] [ U l ] )U f[S]. We shall use Lemma 3.10 to show dim IC; 5 n k R ( P ) - 1. Let M be a nonempty closed subset of K (and hence of Y ) that is contained in I( \ f[S]. The set UOn f-I[y] consists of at most k points for each point y in M because

+ +

c \ f IS1 c fPo1 f-lf[hI and UOn U1 = 0. The restriction of f to the induction hypothesis give

= clx ( U O )n f

[MI and

5 n+k+Q(P)- I because P-Ind 2 5 P-Ind X 5 n. From Lemma 3.10 we have d i m G K 5 n + k + R ( P ) - 1. Theorem 3.9 yields dim Y 5 n + k t R ( P ) and the induction is comdimM

pleted. The appearance of the offset functions in the last two theorems permits inductive proofs. The finiteness of these offset functions forces an upper bound on the dimensions of the spaces in P n U.The next theorem replaces the dimension offset function with functions of dimensional type and permits the use of continuous maps that are not necessarily closed. We shall use the class C, of locally compact spaces to define the function of dimensional type that will replace the dimension offset. 4.11. Notation. For each space X let

loccom X = L-ind X .

Also, for a continuous map f : X + Y let loccom f = sup { loccom f - l [y] : y E Y } and loccom

(af)= sup { loccomBx(f-l[y])

:y E

Y }.

Agreement. The universe is the class Mo for the remainder of the section.

4. DIMENSION AND MAPPINGS

4.12. Theorem. Let f: X dimX

+Y

211

be a continuous map. Then

5 dim Y + max{ dim f , def X } + loccom f 4- 1.

The proof of this theorem will rely on the existence of a special closed extension of the mapping (Theorem 4.15 below). Consequently its proof will be delayed. Clearly the class L is closed-monotone and open-monotone. But it fails to be strongly closed-additive. The next lemma is easily proved. Its proof will be left to the reader. 4.13. Lemma. Let f : X + Y be a closed continuous map. Then (a) Bx(f-l[y]) is compact for each y in Y , (b) loccom(df) = -1, (c) if U{Bx(f-'[y]) : y E Y } c 2 c X , then is a closed map o f 2 onto f[Z].

flz

The proof of our closed extension theorem will follow from the next lemma.

4.14. Lemma. Suppose X and 2 = C n X , then

c Y . If C is a closed subset of Y

dimC<max{dimZ, dim(Y\X)}+loccomZtl.

Proof. Select a metric for Y . The lemma is inductively proved by claims involving the following two statements: Statement A,. Let X , Y , C and 2 be as in the hypothesis of the lemma. If loccom 2 5 n, then for each t in 2 and for E > 0 there is a subset U of C such that t E U , U is open in C, diam U < E and dimBc(U) 5 max{dimZ,dim(Y \ X ) } + n .

Statement rn. Let X, Y, C and 2 be as in the hypothesis of the lemma. If loccom 2 5 n, then dim C 5 max { dim 2, dim (Y \ X ) }

+ n + 1.

212

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

Claim. r-1 is a true statement. Proof, If loccom 2 = -1, then Z is locally compact and hence is open in c l ~ ( 2 ) .So each of the sets C \ c l c ( Z ) , c l c ( Z ) \ Z and 2 are F,-sets in C. Consequently, from the sum and subspace theorems for dim we have d i m C 5 max{ dimZ, dim(Y \ X ) } . Claim. The validity of rn-l implies the validity of A, for n 2 0. Proof. Suppose loccom Z 5 n, z E Z and E > 0. There exists by Proposition 11.2.20 an open neighborhood U of z in the space C such that diam U < E and loccom (Bc(U) n 2 ) 5 n - 1. We have from statement rn-l that for C‘ = B c ( U ) and 2’ = B c ( U ) n 2 the inequality dim C’ 5 max { dim Z’, dim (Y \ X ) } n holds. The inequality dim B c ( U ) max { dim 2,dim (Y \ X ) } n also holds because 2’ c 2. Thereby statement A, is valid.

+ +

<

Claim. The validity of A, implies the validity of I?,

for n 2 0.

Proof. Suppose that A, is valid. Let loccomZ 5 n. Then for each positive natural number m there is countable base B, for the open sets of C with diamBc(U) 2 such that dimBc(U) 5 max { dim 2,dim (Y \ X ) } n for each U in B,. Define the sets G and H by

+

n{UB,

6

= 1,2, . . .}, H = U{ U{ Bc(U) : U E B,} : WL = 1,2, ...}. Then G is a Gs-set of C containing Z and H is an F,-set of C. We have dim (C \ G) 5 dim (C \ 2)5 dim (Y \ X ) by the subspace theorem for dim. And dim H 5 m u { dim 2, dim (Y \ X ) } n by the sum theorem. As C \ G is an F,-set of C , we have by the addition G=

:m

+

theorem that dimC5dim((C\G)UH) +dim(G\H)+l

+ + +

2 m u { dim 2, dim(Y \ X ) } n 1 d i m ( G \ H ) . That dim (G \ H ) 5 0 remains to be shown. This will follow from Proposition 1.3.6 and Theorem 1.8.10 with the aid of the fact that the restriction of B, to the set G \ H for each m will yield a base for G \ H. The proof the lemma is completed by the observation that valid for n 2 - 1.

r,

is

4. DIMENSION AND MAPPINGS

213

We are now ready to prove the closed extension theorem. Of course, the existence of a closed extension of a continuous map is not surprising. It is the inequality in the next theorem that is of interest.

Theorem. Let f : X + Y be - -a continuous, onto map. Then there is a closed continuous map f : X -+ Y , where X is dense in 2,such that f = TlX and 4.15.

dim f

5 dim 7 5 max { dim f, def X } + loccom (af)+ 1.

Proof. Let Ir' be a metrizable compactification of X such that defX = dim ( K \ X ) . The natural projection p : K x Y + Y is a closed map. Let G be the graph o f f and H be the closure of G in K x Y . Define g to be p l H . Then g is a closed mapping of H onto Y . With HO = U{ B~(g-'[y]) : y E Y } define 2 to be the set Ho U G and f t o be p 1 2 . By Lemma 4.13 one readily sees that is a closed map of 2 onto Y that extends f and that X is dense in 2. We must prove that dim f satisfies the required inequality. Let y E Y . Then

f

f'-"Y1

= BH(g-'[YI)

u (f-'[YI

x {Y

1).

It is clear from the sum theorem that dimf='[y] 5 max{ dimBH(g-'[y]), dimf}. We shall compute an upper bound for dim B~(g-'[y]). Observe that

in G Also we have from the denseness of G in H that a point (z,~) is an interior point of g-* [y] n G in the space G if and only if ( 5 , y) is an interior point of g-'[y] in the space H . So BH(S-l[Yl)

n ( X x { Y 1) = BG(g-%l n G) = Bx(f-"Yl)

x

Y 1.

214

IV. FUNCTIONS OF COVERING DIMENSIONAL TYPE

We can now apply Lemma 4.12 to the sets X x { y}, K x { y}, B ~ ( g - l [ y ] )and Bx(f-l[y]) x { y } to conclude that dim B H ( P [Y])

5 max{dimBx(f-'[y]), d i m ( K \ X ) } t loccom Bx(f-'[y]) t 1 < max { dim f, def X } t loccom (af)t 1. Let us now prove Theorem 4.12. M

I

Proof of Theorem 4.12. Let f : X Y be the closed extension of f provided by Theorem 4.15. Theorem 4.5 applied t o will complete the proof because dim X 5 d i m 2 and loccom 5 loccom f. The last inequality follows from the fact that loccomBx(f-'[y]) 5 loccom f-' [y] for every y in Y .

7

---f

(af)

Let us now eliminate the invariant loccom from Theorems 4.12 and 4.15. We will need the following proposition. Proposition. Let Z be a space with d e f Z 2 0. Then loccom Z 5 cmp 2. 4.16.

Proof. The proof is by induction on cmpZ. Since c m p Z = 0 if and only if defZ = 0 by de Groot's theorem, the induction must start with cmp Z = 0. Suppose that cmp 2 = 0. Then each point z of 2 has arbitrarily small neighborhoods with compact boundaries. So loccom 2 5 0. The inductive step is proved in the same manner.

4.17. Lemma. Let f : X loccom

--+

Y be a continuous map. Then

(af) 5 min { dim f, def X

}.

Prooj. Let y be a point in Y . That IoccomB (f-l[y]) 5 defX is obvious when defB (f-'[y]) = -1. If def B (f-'[y]) 2 0, then the above proposition yields loccomB (f-'[y])

5 cmp B (f-'[y]) 5 defB (f-'[y]) 5 defX,

where the last inequality holds because B (f-l[y]) is closed in X. So loccom 5 def X follows. Obviously, loccom (Of)5 dim f. The lemma is proved.

(af)

The next theorem is now easily established.

5. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

215

4.18. Theorem. Let f : X Y be - -a continuous, onto map. Then there is a closed continuous map f : X +. Y , where X is dense in f,such that f = and ---f

dim f 5 d i m f s dim

f + defX + 1.

Consequently, d i m X 2 dimY

+ dim f + d e f X + 1.

The following application of Theorem 4.18 will use the fact that for every space X there is a continuous map f of X into the Cantor set such that f-'[y] is a quasi-component of X for each y in f [ X ] .(See Section VI.3 where quasi-components are discussed in more detail.) 4.19. Theorem. Let X be a space such that dim Q quasi-component Q of X. Then

5 n €or each

dimXsdefX+n+l. Proof. Let f : X + f [ X ] be a continuous map into the Cantor set such that f-'[y] is a quasi-component of X for each y in f [ X ] . Then dim f 5 n. Theorem 4.18 will complete the proof. An immediate consequence of Theorem 4.12 follows. 4.20. Theorem. Let X be a space such that

(a) each quasi-component of X is locally compact, (b) dimQ 5 n for each quasi-component Q of X, (c) d e f X 2 n. Then d i m X 5 n. P~oof.As in the proof of the previous theorem, we have from Theorem 4.12 that d i m X n loccom f 1 because of conditions (b) and (c). Condition (a) gives loccom f = -1.

< +

+

5 . Historical comments and unsolved problems

The development of P-dim and P-Dim found in the chapter is essentially new. The reader is referred to the earlier work of Baladze [1982] concerning P-dim for general topological spaces and Aarts

216

IV. FUNCTIONS OF COVERING DIMENSIONAL T Y P E

and Nishiura [1973a]. One can see that the theory of the covering dimension dim in the universe N of normal spaces has strongly influenced the development of the chapter. Of particular importance are the works of Dowker and of Morita. Theorems 4.7 and 4.10 concerning closed, continuous mappings are new in the sense that the offset functions are new. (Although one can find closed, continuous mapping theorems in Baladze [1982] which do not use these offset functions, the proofs found there seem incomplete.) There are many more mapping theorems in dimension theory, in particular, mappings into polyhedra. These have not been investigated in the context of P-dim. The Hurewicz closed mapping theorems for continuous maps on compact metrizable spaces have been generalized to Theorems 4.12 and 4.15. They originally appeared in Nishiura [1972]. Theorems 4.19 and 4.20 are the main theorems of Nishiura [1964]. Results related to the last two references can be found in Lelek [1964] and Reichaw [1972].

Unsolved problems 1. The Morita-Hurewicz closed mapping theorems have been generalized as Theorems 4.7 and 4.10. The statements of these theorems have a mixture of dim and P-dim. Find suitable offset functions so that the function dim in these theorems can be replaced by P-dim. 2. If possible, develop a theory of mappings into polyhedra for P-dim analogous to that of dim.

3. Inverse sequences have been successfully exploited in the study of dim X . Is P-dim X amenable to similar exploitations?

CHAPTER V

FUNCTIONS O F BASIC DIMENSIONAL TYPE

One of the subjects of this chapter is the relation between the dimension-like functions and the existence of extensions that satisfy some dimensional properties. Theorem 1.7.11 is a typical example of the results that will be derived. Because of their different nature, the compactness dimension functions and compactifications will be discussed in the next chapter instead of here. Another subject is the theory of excision. The notion of excision is complementary to that of extension. Theorem 1.10.5 may serve as an example of the results that are discussed in this context. The idea of complementarity has already been seen in Proposition 11.10.2. What unifies excision and extension is the basic inductive dimension function modulo a class P, This and other basic dimension functions, the definition of which involves the existence of special bases, will be exposed in this chapter. With the help of the basic dimension functions we are able t o line up the dimension functions modulo P under mild conditions on P. Metrizability will play a dominant role in the development of this chapter; it is in fact an implicit part of the definition of the basic dimension functions.

Agreement. Every space is metrizable, that is, U = M . 1.

T h e basic inductive dimension

Let us begin with the basic inductive dimension as a motivation for the basic inductive dimension modulo a class P. These basic dimensions are the key t o the excision and extension theorems of the next section. At the end of this section we shall prove some fundamental theorems for Ind. We have seen earlier in Proposition 1.3.6 a characterization of the small inductive dimension ind that uses a base for the open sets. 217

218

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

A well-known characterization of the large inductive dimension Ind reads as follows: Let n be a natural number. A space X has I n d X 5 n if and only if there exists a a-locally finite base L3 for the open sets such that Ind B ( V ) 5 n - 1 for every U in B. This characterization will be a natural by-product of our development (Theorem 1.13 below). The definition of the basic inductive dimension is a mixture of this characterization of Ind and the characterization of ind by means of bases. 1.1. Definition. Let P be a class of spaces and let X be a (metrizable) space. One assigns the basic inductive dimension modulo P, denoted P-Bind X , as follows.

(i) P - B i n d X = -1 if and only if X E P. (ij) For each natural number n , P - B i n d X 5 n if there exists a g-locally finite base B for the open sets of X such that P-Bind B ( U ) 5 n - 1 for every U in B.

Since the class P is topologically invariant, the function P-Bind is also topologically invariant. The function {@}-Bind will be denoted by Bind. In view of the Nagata-Smirnov-Bing Metrization Theorem, the metrizability of the space X is part of the definition of P-Bind X . The first proposition easily follows from the characterization of ind in Theorem 11.2.10. The proof is by induction on P-Bind and is left to the reader. 1.2. Proposition. For every space X ,

P-ind X

5 P-Bind X .

For separable spaces we have the equality of P-ind and P-Bind. 1.3. Proposition, For every separable space X,

P-ind X = P-Bind X.

Proof. In view of the preceding proposition it will suffice to prove that P-Bind X 5 P-ind X for every separable space X . For the inductive step of the proof, suppose that the theorem has been proved for all spaces X with P-ind X 5 n - 1. Assume P-ind X 5 n. By

1. THE BASIC INDUCTIVE DIMENSION

219

Theorem 11.2.10 there is a base f? for the open sets of X such that P-indB ( U ) 5 n - 1 for each U in B. As X is second countable, we may assume that f? is countable. By the induction hypothesis, P-Bind B ( U ) 5 n - 1 for all U in I?. It follows that P-Bind X 5 n. The easy inductive proof of the following proposition will be left t o the reader. 1.4.

P-Bind

Proposition. If P and Q are classes with P 2 Q, then

5 &-Bind. In particular, P-Bind 5 Bind.

The agreements made on the classes P allow easy inductive proofs for the following propositions. 1.5. Proposition. For every closed subspace Y of a space X,

P-BindY 5 P-BindX. 1.6. Proposition. The class P is monotone if and only if the dimension function P-Bind is monotone, i.e., for every subspace Y of a space X , P-Bind Y 5 P-Bind X .

The corresponding statements hold for F,-, Gs- and open-monotone classes. The next open subspace theorem is similar to Theorem 11.2.17. 1.7. Proposition. Let X be such that 0 5 P-Bind X . For every open subspace Y of X , P-Bind Y 5 P-Bind X .

Now we shall compare P-Bind and P-Ind. To this end, we introduce the notion of a framework of a base. 1.8. Definition. A pair ({ U, : y E I?},{ F, : y E I?}) is called

a framework: of a base of X if

(i) { U, : y E I'} ({ F, : y E I'}) is an open (closed) collection, (ij) 0 # F, c U, for every y in r, (iij) if { V, : y E r } is any collection of open sets of X such that F, C V, C U, for each y in r, then it is a a-locally finite base for the open sets of X .

We have the following existence proposition.

220

V. FUNCTIONS OF BASIC DIMENSIONAL T Y P E

1.9. Proposition. For each space X there exists a framework of a base.

Proof. For each i in N let Wi be the collection of all open balls with diameter less than Let { U: : cr E A * } be a locally finite cover of X that refines Wi. Because this covering is locally finite, it has a closed shrinking {FA : a E A; }. We may assume that each F: is nonempty. Then ({ U i : cr E Ai, i E N}, { J': : Q E Ai, i E N}) is a framework of a base of X.

A.

The relation between P-Bind and P-Ind is as follows. 1.10. Theorem. For every space X,

P-Bind X

< P-Ind X .

Proof. The proof is by induction on P-Ind. Suppose that the theorem holds for spaces X with P-Ind X n - 1. Let P-Ind X n and let ({ U, : y E r }, { F, : y E r }) be a framework of a base. For each y in r select an open set V, such that F, c V, c U, and Ind B (V,) 5 n - 1. We have P-Bind B (V,) 5 n - 1 by the induction hypothesis for each y in I' and hence P - B i n d X 5 n follows.

<

<

We shall discuss the special case P = { S} to conclude this section. In the discussion we shall use only the sum theorem for Ind (see Theorem 111.3.25 (b) and (c)). This is permissible because the semi-normal family arguments of Section 111.3 do not require the subspace theorem nor the addition theorem. So in this way we can find a second proof for the subspace and addition theorems in the universe M . We shall state here for convenience the sum theorem for Ind. 1.11. T h e o r e m (Sum theorem). If X = U F for a a-locally finite closed collection F i n X , then

IndX

< s u p { I n d F : F E F}.

Using the sum theorem, we now can prove the following important theorem.

1. THE BASIC INDUCTIVE DIMENSION

221

1.12. Theorem. For every space X , Bind X = Ind X .

Proof. Only Ind X 5 Bind X requires proof. Assume Bind X 5 0. By definition, there is a a-locally finite base B such that B ( U ) = 8 for every U in B. We may write B = { U: : a E Ai, i E N} where the collection { Uk : Q E A i } is locally finite for each i in N. Suppose that F and G are disjoint closed sets of X . For each j in N define Uj

=X

\U{ UA : Q E Ai, i S j ,

U&n F = S}.

Observe that { U i : Q E Ai, i 5 j , UA n F = 8 } is locally finite. It follows that U j is an open-and-closed set. In this way we get a sequence { Ui : i E N} of open-and-closed sets such that

In a similar way, we can construct a sequence { and-closed sets such that

V o 3 V i > V 2 > . - - > G and

:i E

N} of open-

n{K:iEN}=G.

A straightforward argument will show that W = U{ Ui \ V , : i E N} is an open-and-closed set such that F c W C X \ G. Hence 8 is a partition between F and G. Now suppose for n > 0 that the inequality has been proved for all spaces X with Bind X 5 n - 1. Assume Bind X 5 n. Choose a o-locally finite base B for the open sets with Bind B ( U ) 5 n - 1 for each U in B. By the induction hypothesis, Ind B ( U ) 5 n - 1 for each U in B. We consider the set D = U{ B ( U ) : U E f?}. As the collection { B ( U ) : U E f?} is a-locally finite, by the sum theorem we have Ind D 5 n - 1. Obviously, Bind ( X \ D) 5 0. By the induction hypothesis, Ind ( X \ 0 ) 5 0. Now suppose that F and G are disjoint closed subsets of X . By Proposition 1.4.6 there is a partition S in X between F and G such that Ind (S n ( X \ 0)) = -1, that is, S c D. It follows that Ind S 5 n - 1, whence Ind X 5 n. There are several consequences of this theorem. An obvious one is the sum theorem for Bind. The second one is the characterization theorem.

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

222

1.13. Theorem. For every space X and for every natural number n, I n d X 5 n if and only if there exists a a-locally finite base B for the open sets such that Ind B ( U ) 5 n - 1 for every U in B.

The subspace theorem for Ind follows from Proposition 1.6. A more general result has been obtained in Theorem 111.3.25 (a). 1.14. Theorem (Subspace theorem). For every subspace Y of a space X , IndY 5 IndX. The decomposition theorem can be established quite easily. The ideas of the proof will be generalized in the next section.

1.15. Theorem (Decomposition theorem). For each natural number n, Ind X 5 n if and only if X can be partitioned into n f 1 disjoint subsets X;, i = 0,1,. . . ,n, with Ind Xi 5 0 for every i.

Proof. To prove necessity, using Theorem 1.12 and Definition 1.1, we select a a-locally finite base B for the open sets of X such that Bind B ( U ) 5 n - 1 for each U in B. Write XI = U( B (U): U E f? }. The sum theorem for Bind gives Bind XI 5 n - 1. Clearly we also have Bind (X \ XI) 5 0. And thus Ind (X \ XI) 5 0 holds by Theorem 1.12. Repeating this process, we find that X can be partitioned into n 1 zero-dimensional subsets. The proof of the sufficiency is by induction and makes use of Propositions 1.4.3 and 1.4.6. The reader is asked to provide the details.

+

Finally the addition theorem and the product theorem for general metrizable spaces can now be easily established. A proof of the product theorem can be based on Theorem 1.13. We leave the details to the reader. 1.16. Theorem (Addition theorem). If X = Y U 2, then

IndX

5 IndY + I n d Z

f 1.

1.17. Theorem (Product theorem). Let X x Y be the topological product of two spaces X and Y, a t least one of which is not empty. Then I n d ( X x Y ) 5 I n d X IndY. Due to Theorem 1.12 we shall henceforth freely interchange Bind and Ind in the sum, subspace, decomposition, addition and product theorems.

+

2. EXCISION AND EXTENSION

223

2. Excision and extension

The relations between P-Sur and P-Bind and between P-Def and P-Bind will be discussed, The similarities as well as the differences of the notions of surplus and deficiency will be clearly exposed. Let us discuss the surplus function first. The fundamental result is the excision theorem. To prove this theorem we need the next two lemmas. 2.1. Lemma. Let { B, : y E r } be a a-locally finite closed collection of subsets of a space X . For each y in r let { B; : S E A, } be a a-locally finite (closed) collection of subsets of the subspace B,. Then { B{ : y E I', 6 E A, } is a a-locally finite (closed) collection of subsets of X .

Proof. It is sufficient to prove the lemma for locally finite covers { B, : y E I? }. We may assume that the indexing of the collection by is one-to-one. As the collection { B t : 6 E A, } is a a-locally finite collection of subsets of the subspace B, for each y, we may write A, = U{ A; : i E M} where { B; : 6 E A; } is locally finite for each i. We only need to show for any i that { B t : y E r, 6 E A; } is locally finite. Let x be in X . First select a neighborhood U of z such that U n B, = 0 for each y in r \ where rZ is some finite subset of r. (Here we have used the fact that the indexing is one-toone.) Because B, is a closed set, for every y in rZ we can select a neighborhood U, of z such that the set U, n B, meets only finitely many B$ with S in A;. It follows then that U n U, : y E I'I }) is a neighborhood of z meeting at most finitely many members of { B; : y E I?, 6 E A;}.

rz,

(n{

2.2. Lemma. In each space X with P-BindX 5 n there exists F o f subsets of X such that

a a-locally finite closed collection

(a) every member of F belongs to P , (b) Bind (X \ UF ) 5 n.

Proof. The proof is by induction on P-Bind. For the first step, suppose P-BindX 5 0. By definition there exists a a-locally finite base B for the open sets of X such that B (V) E P for every V in f?. Clearly F = { B (V) : V E B } satisfies all of the required conditions. Now suppose that the lemma holds for all spaces X

V. FUNCTIONS OF BASIC DIMENSIONAL T Y P E

224

<

with P-Bind X n - 1. Assume P-Bind X 5 n. Then there exists a a-locally finite open base f? of X with P-Bind B (V) 5 n - 1 for every V in f?. By the induction hypothesis, for every V in f? there exists a a-locally finite closed collection Fv of subsets of the subspace B (V) such that

FV C P,

Bind (B (V) \ U Fv) 5 n - 1.

The collection F = U{ FV : V E 13) is a-locally finite and closed by the previous lemma. Observe that B (V) \ F is a closed set in X \ U F for every V in 13. From the sum theorem for Bind we have Bind ((U{ B ( V ): V E f?}) \ U F ) 5 n - 1. It is clear that Bind(X \ U F ) 0. So Bind ( ( X \ U{ B(V) : V E a})\U F ) 5 0 holds by the subspace theorem for Bind. Finally Bind (X \ U F ) 5 n follows from the addition theorem for Bind.

U

<

The excision theorem is fundamentally related to P-Sur. Its statement will use classes that are semi-normal families. 2.3. Theorem (Excision theorem). Suppose that P is a seminormal family. Then for every space X with P-Bind X n there exists an F, P-kernel Y with Ind (X \ Y ) 5 n.. In particular,

<

P-Sur X 5 P-Bind X . Proof. By Lemma 2.2 there exists a a-locally finite closed collection F of P-kernels such that Bind ( X \ U F ) 5 n. As P is a a-locally finitely closed-additive class of spaces, we have that F is in P. Theorem 1.12 will complete the proof.

U

The additive absolute Bore1 classes may serve as examples of seminormal families (Corollary 11.9.4). The excision theorem has many interesting consequences. 2.4. Theorem (Coincidence theorem). Suppose that P is a semi-normal family. For every space X,

P-Bind X = P-Ind X = P-Sur X.

Proof. This follows from Theorems 1.10, 11.3.6 and 2.3. Having established the coincidence theorem, we easily obtain the sum and decomposition theorems. We state these theorems for P-Ind only. Of course they are also valid for P-Bind as well as for P-Sur.

2. EXCISION AND EXTENSION

225

Theorem (Sum theorem). Suppose that P is a seminormal family. If X = U F where F is a a-locally finite closed collection in X , then 2.5.

P - I n d X = sup{P-Ind F : F E F } .

Proof. By the excision theorem there exists for each F in F a n F, P-kernel YF such that Bind Z F 5 n where ZF = F \ Y F . The set Y = U{ YF : F E F } is a member of P by Lemma 2.1. As F \ Y coincides with ZF \ Y and is closed in X \ Y for each F i n F, we have by the sum and subspace theorems for Bind that Bind ( X \ Y ) 5 n. It will follow from Theorem 1.12 that P - S u r X 5 n. The theorem follows by the coincidence theorem. 2.6. Theorem (Decomposition theorem). Suppose that P is a semi-normal family and that n is in N. Then P-Ind X 5 n if and only if X can be partitioned into n 1 subsets X i , i = 0,1,. ..,n, such that P-Ind Xo 5 0 and Ind X i 5 0 for i = 1,. . .,n.

+

Proof. To prove the necessity, by the excision theorem we select an F, P-kernel Y such that Ind (X \ Y )5 n. Then by Theorem 1.15 the set X \ Y can be partitioned into n 1 sets of dimension at most 0. One of these sets is then added to Y to obtain the desired decomposition. The sufficiency will follow from the 0-dimensional case of Proposition 1.4.6 and induction.

+

In connection with the last theorem there is an example which shows the failure of the addition theorem for the large inductive dimension modulo a class of spaces, even in the case where the class is a semi-normal family. 2.7. Example. Let 2 = { X : Ind X 5 0 }. From the sum theorem it follows that the family 2 is semi-normal. An easy computation will show Z-IndR3 = 2. Because R 3 is the union of two one-dimensional sets, it is the union of two sets with 2-Ind equal to zero. This shows the failure of the addition theorem.

Another consequence of the coincidence theorem is the open subspace theorem for P-Bind, P-Ind and P-Sur.

226

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

2.8. Theorem, Suppose that P is a semi-normal family. For every open subspace Y of a space X,

P-IndY 5 P-IndX.

Proof. From the conditions on P it easily follows that P is openmonotone. The theorem now follows from Proposition 1.6 and the coincidence theorem. We shall turn now to extension theorems. First we introduce the operation of expanding open sets of a space to open sets of an extension of the space. This operation will apply to any topological space. 2.9. Definition. Suppose that Y is an extension of a space X . For each open subset of the space X the expunsion of U in Y is defined to be the open set ex ( U ) = Y \ cly(X \ U ) .

The set ex ( U ) depends on Y . If confusion is likely to arise, a subscript will be used to indicate the space in which the expansion operator is applied. Some properties of the expansion operator will be collected in the next lemma.

2.10. Lemma. Suppose that X is a dense subspace of a space Y . Let U and V be open subsets of the space X . The following properties hold. (a) If W is an open set in Y with W n X c U , then W c ex(U). Thus ex ( U ) c cly( U ) holds and ex ( U ) is the largest open set in Y whose trace on X is U . (b) ex(@)= 0 and e x ( X ) = Y . (c) e x ( U n V ) = e x ( U ) n e x ( V ) . (d) e x ( U U V ) 3 ex(U)Uex(V). (e) By(ex(U)) n X = Bx(U). ( f ) If 2 is an extension of Y , then exZ( U ) n Y = exy( U ) . (g) When a metric for Y is restricted t o X , the diameters of U and ex ( U ) are equal. Proof. We have W n ( X \ U ) = 8 for any an open set W of Y with W f l X C U . So W n cly(X \ U) = 0 follows. Now we have W C ex ( U ) and (a) is proved. The formulas in (b), (c) and (d) are

2. EXCISION AND EXTENSION

227

verified by straightforward computations. Formula (e) is verified by the following computation. B y ( e x ( U ) ) n X = (cly(ex(U))\ex(U)) n X

= (cly(U) \ ex(U)) n X = Clx(U) \ U Bx(U). For the proof of formula (f), observe that exz(U) n Y is an open set in Y whose trace on X is U . By (a) we find exz(U) n Y c exy(U). If W is any open set in Z with W n Y = exy(U), then W n X = U , whence W c exz(U). So we have exz(U) n Y 2 W n Y = exy(U). The last statement follows from the formula in (a). We shall discuss various extension theorems. The following is the key lemma.

2.11. Lemma. Suppose that the space Y is an extension of the space X . Let B be a a-locally finite base for the open sets of X. Then there exists a Ga-set Z of Y such that X C Z C Y and { exz(U) : U E t?} is a a-locally finite base for the open sets of

z.

Proof. W r i t e f 3 = { U v : y E l ? ~ , i E N } w i t h {U,:yEI'i}being locally finite and the indexing by I'i being bijective for each i. First, we find for each i in N an open set Vi of Y such that X c Vi C Y and { exy(U,) : y E I'i} is locally finite in V,. This is done as follows. For each 2 in X there is an open neighborhood B, in X such that B, meets U, for only finitely many y in I'i. By (c) of the previous lemma we have exy(B,) n exy(U,) # 8 for only finitely many y in ri. Then V , = U{ exy(B,) : z E X } is the required open set. Next, for each i in N we let Wi be the set of all points y in Y with the property that y is contained in exy(U) for some U in B with diameter less than As f3 is a base it easily follows that X C Wi. Finally, define 2 by

&.

z= (n{v, : i E N}) n (n{wi: i E N}) and the result easily follows from Lemma 2.10.(f). The first extension theorem that will be discussed is Tumarkin's extension theorem which has been stated for separable metrizable

228

V. FUNCTIONS OF BASIC DIMENSIONAL T Y P E

spaces as Theorem 1.7.6. To place emphasis on the role of the basic inductive dimension we shall use the basic inductive dimension in the statement of the theorem even though we know from Theorem 1.12 that the basic inductive dimension and the large inductive dimension coincide in the universe of metrizable spaces.

2.12. Theorem. Suppose that the space X is a subspace of the space Y and that n is in N. If Bind X 5 n, then there exists a Gs-set Zin Y such that X c 2 and Bind Z 5 n. In particular, every space X has a complete extension 2 with the same basic inductive dimension as X . Proof. By replacing Y with cly(X) if necessary, we may assume that X is dense in Y . As Bind X 5 n, there exists a a-locally finite base for the open sets of X such that Bind B x ( U ) 5 n - 1 for each U in B. By the previous lemma there exists a Gs-set 20 in Y with X C 20 such that { exz,(U) : U E B} is a a-locally finite base for the open sets of 20. By the induction hypothesis, for each U in f? there exists a Ga-set ZU such that

B x ( U ) C Zu C BZ,(exZ,(U))

and

Bind Zu 5 n - 1.

Let E = U{ BZ,(exz,(U)) \ 2, : U E B}. As E is a subset of 20\ X and as BZ,(exz,(U)) \ ZU is an F,-set in BZ,(exz,(U)) for each U in B, it will follow from Corollary 11.9.4 that E is an F,-set in 2 0 . Finally we define 2 = 20\ E . Obviously 2 is a Ga-set in Y such that X c 2 . Also { exz,(U) n Z : U E B } is a a-locally finite base for the open sets of 2. It is easily verified that for each U in B we have Bz(exz,(U) n 2) c Bz,(exzo(U)) n c Z u . As Bind Zu 5 n - 1 for each U in B, we have Bind Z 5 n.

z

The second extension theorem will generalize a key result of Section 1.7 concerning the completeness degree Icd to general metrizable spaces. 2.13. Theorem. Suppose that X is a subset of a complete space Y and let n be in N. If C-Bind X 5 n, then there exists a complete subspace 2 of Y such that Bind ( 2 \ X ) _< n and 2 is an extension of X .

Proof. Because C-BindX n, there exists a a-locally finite base B for the open sets of X such that C-Bind B x ( U ) 5 n - 1 for each U

2. EXCISION AND EXTENSION

229

in B. As in the proof of the previous theorem we may assume that X is dense in Y and that { exy(U) : U E B } is a a-locally finite base for Y. For each U in B,By(exy(U)) is a complete space. By the induction hypothesis there exists a complete space Zu such that Bx(U) c Zu c By(exy(U)) and Bind (Zu \ Bx(U)) 5 n - 1. For each U in B, the set By(exy(U)) \ ZU is an &-set in Y . We define E = U{ By(exy(U)) \ Zu : U E B}. From Corollary 11.9.4 it follows that E is an F,-set in Y . Finally we define 2 = Y \ E . As Z is a Gs-set in Y , it is a complete extension of X . The collection { exy(U) n ( 2 \ X ) : U E B } is a a-locally finite base of 2 \ X . An easy computation will show that, for each U in B, BZ\x(exy(U) n

(z\ X ) ) c B y ( e x ~ ( U )n) ( 2 \ X ) c zu.

It follows that Bind ( 2 \ X ) 5 n. Following the pattern of Section 1.7, we first prove a coincidence theorem for the completeness dimension functions. 2.14. Theorem (Main theorem for completeness degree). For every space X ,

C-Bind X = Icd X = C-DefX.

Proof. Recall that Icd = C-Ind. The theorem will follow from Theorem 1.10, Proposition 11.8.7 and Theorem 2.13. Now that the main theorem has been established one can prove various adaptations of the results for icd from Section 1.7. As the proofs of the Theorems 2.15 through 2.19 are simple modifications of the proofs in Section 1.7, they will be left for the reader to complete. 2.15. Theorem. For every Gs-set Y in a space X ,

IcdY

5 IcdX.

2.16. Theorem (Addition theorem). If X = Y U 2, then

IcdX

5 IcdY + I c d Z + l .

In particular, the large inductive completeness degree Icd of a space cannot be increased by the adjunction of a complete space or a point.

In contrast with this result for Icd, note that icd can be raised by the addition of one point as we have shown in Section 111.1.

230

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

2.17. Theorem (Intersection theorem). If Y and 2 are subsets of a space X, then

Icd (Y n 2 ) 5 Icd Y

+ Icd 2 i1.

2.18. Theorem (Locally finite sum theorem). Let locally finite closed cover of a space X. Then

F be a

Icd X = sup { Icd F : F E F } . 2.19. Theorem (Structure theorem). For every space X and every natural number n, Icd X 5 n if and only if there exists a complete extension Y o f X such that the set X can be represented as the intersection of n 1 subsets Yi of Y , i = 0,1,. . .,n, with Icd Yi 5 0 for every i.

+

The results of Theorems 2.13 and 2.14 will be generalized by replacing C with the multiplicative absolute Borel classes M ( a ) , Q 2.

>

2.20. Theorem. Let o be an ordinal number with (Y 2 2. Suppose that X is a subset of a complete space Y and that n is in N. If M(a)-Bind X 5 n , then there exist a subspace 2 of Y such that X c 2 c Y , 2 E M(Q) and Bind ( 2 \ X ) 5 n.

Proof. Because M(cr)-BindX 5 n, there exists a a-locally finite base B for the open sets of X such that M(cr)-BindBx(U) 5 n - 1 for each U in B. In view of Lemma 2.11 we may assume that X is dense in Y and that { exy( U ) : U E B } is a a-locally finite base for Y . For each U in B the space By(exy(U)) is complete. By the induction hypothesis there exists a space ZU with B x ( U ) C ZU C By(exy(U)), Zu E M(Q) and Bind (2, \ Bx(U)) 5 n - 1. For each U in B the set By(exy(U)) \ 2, belongs to the additive Borel class Q by Theorem 11.9.1. Let E = U{ By(exy(U)) \ ZU : U € a}. From Corollary 11.9.4 it follows that E is a member of the additive class Q in Y . Finally we define 2 = Y \ E . As Z is a member of the multiplicative class Q in the complete space Y , we have 2 E M((Y)by Theorem 11.9.6. The collection { exy( U )n ( 2 \ X ) : U E B } is a a-locally finite base for 2 \ X. An easy computation will show that for each U in B

2. EXCISION AND EXTENSION

231

So Bind (exy(U) n ( 2 \ X ) ) 5 n - 1 by Theorems 1.12 and 1.14, whence Bind ( 2 \ X ) 5 n. The natural generalization of Theorem 2.14 reads as follows. 2.21. Theorem. For every space X and every ordinal a 2 1,

M(a)-BindX = M(a)-Ind X = M(cr)-DefX. The proof of the following result is an application of Theorem 2.12. 2.22. Lemma. Suppose that the class P is F,-monotone.

P-Sur

Then

< P-Def.

It is to be noted that the classes A ( a ) are F,-monotone for each a and that classes M(a) are F,-monotone when Q 2 2. These properties follow from Theorems 11.9.1 and 11.9.6. The concept of complementarity of dimension functions that is illustrated by Proposition 11.10.2 can be developed a little further. 2.23. Definition. A space Z is said t o be ambiguous relative to the classes of spaces P and Q provided that X E P if and only if Y E Q whenever X and Y are complementaryin 2 , that is, whenever 2 = X U Y and X n Y = @.For classes of spaces P,Q and R, the classes P and Q are called complementary with respect to R if every 2 in R is ambiguous relative to P and Q.

In Section 11.9 we have seen that the additive Borel class A ( a ) and the multiplicative Borel class M(a) are complementary with respect t o the class C of all complete spaces when a! 2 2. We shall make a small digression from our discussion of absolute Borel classes t o collect some results concerning complementary classes. The first result follows directly from the definitions. 2.24. Lemma. Suppose that a space 2 is ambiguous relative to the classes P and Q. Then

Q-DefX

5 P-SurY,

whenever X and Y are complementary in 2. By combining the Lemmas 2.22 and 2.24, we get the following result.

232

V. F U N C T I O N S OF BASIC DIMENSIONAL T Y P E

2.25. Theorem. Let P and Q be F,-monotone classes. Suppose that the space Z is ambiguous relative to P and &. Then

P-Def X = P-Sur X = Q-Defy = Q-Sur Y whenever X and Y are complementary in 2. With the help of this theorem we find for each ordinal a with a 2 2 the equalities A(a)-Sur = A(cr)-Def,

M(a)-Sur = M(a)-Def

and ( M ( a ) n A(a))-Sur = ( M ( a ) n A(a))-Def.

For the covering dimension functions modulo a class there is the following surprising theorem. 2.26. Theorem. Suppose that the space 2 is ambiguous relative t o P and &. Then

P-Dim X = &-Dim Y

and

P-dim X = &-dim Y

whenever X and Y are complementary in 2.

Proof, The proof will easily follow from the following observation. If X and Y are complementary sets in 2 and if Vis an open collection in 2, then the trace of Von X is a P-border cover of X if and only the trace of V o n Y is a &-border cover of Y . Let us return t o the discussion of the basic inductive dimensions and complementarity. 2.27. Theorem. Suppose that the classes P and Q are complementary with respect to a class R. If X and Y are complementary subspaces in a space Z in R, then

&-Bind Y 5 P-Ind X.

Proof. The proof is by induction. Let n be a natural number and assume P - I n d X 5 n. By Proposition 1.9, the space Z has a framework ({ U, : y E I'}, { F, : y E 'I }) of a a-locally finite base

2. EXCISION AND EXTENSION

233

for the open sets. As P-Ind X 5 n, by Proposition 1.4.6 there exists for each y in I? an open set V, such that P-Ind (Bz(V,) n X ) 5 n - 1 and I?, C V, c U,. Observe that Bz(V,) is in R. By the induction hypothesis, we have &-Bind(Bz(V,) n Y ) 5 n - 1. It now follows that Q-Bind Y 5 n because By(V, n Y ) c Bz(V,) n Y . Returning to Theorems 2.13 and 2.14, we can generalize these theorems along other lines. The following is a typical example. 2.28. Theorem. Let X be a space and let Y be a complete space. Suppose that F and G are a-locally finite collections of closed subsets ofX and let { fi : i E N} be a countable family of continuous maps of X to Y . Then there exists a complete extension 2 of X with the properties: (a) cla(Fi) : i = 1 , .. . , m } = clz(n{ Fi: i = 1,. . , m } ) for all finite subcollections Fl, . . . , F, of F. (b) Bind clz(G) = BindG for each G in G . (c) Bind (clz(G) \ X ) = C-Bind G for each G in G . (d) Each fi has a continuous extension : 2 + Y .

n{

.

f;

Proof. Let 2 be a complete extension of X . The space 2 will be obtained by trimming F,-sets off 2. As F and G are a-locally finite closed collections of X , we can find a Gb-set in 2 containing X such that F and G are a-locally finite collections in this G6-set. Thus we may assume that Z is an extension of X such that the collections F and G are also a-locally finite in 2. For each finite subset { F l , . . .,F, } of F the set

D=

n{ &(F~): i = I , . ..,m}\clz(n{

Fi: i = 1,. . . , m } )

is an F,-set in 2 that is contained in 2 \ X . The family D of all sets D that can be obtained in this way is a-locally finite. By Corollary 11.9.4 the set U D is an F,-set in 2. For each G in G , by Turnarkin’s theorem (Theorem 2.12) there exists a Gs-set EG

in clZ(G) such that G C EG and Bind EG = BindG. Denote the collection { clZ(G) \ EG : G E G } by E. As c l ~ ( G\) EG is an F,-set in clz(G), by Corollary 11.9.4 the set U E is an F,-set in 2. Furthermore, by Theorem 2.13, for each G in G there exists a Gg extension ZG of G such that ZG C clZ(G) and Bind (ZG\ X ) = C-Bind G. (The equality follows because the set G is closed in X . ) Let 2 be

234

V. FUNCTIONS OF BASIC DIMENSIONAL T Y P E

the collection { clZ(G) \ ZG : G E G } . By Lemma 1.7.2, for each i in N there is a set Ci and there is a continuous map : Ci + Y such that Ci is a G6-set in 2 containing X and is an extension of fi. Now the space 2 is defined by

fi

fi

The verification of properties (a) through (d) is straightforward. 3.

T h e order dimension

The order dimension function modulo the class P, P-Odim, will be studied in this section. The function Skl, which was introduced in Section 1.6, is of this type and will be used in Section VI.5 to characterize K-def. Also the order dimension P-Odim will play a key role in the proof of the inequality P-Bind 5 P-Dim. The definition of P-Odim for general metrizable spaces is a natural generalization of the definitions of C-Odim (1.7.19) and Skl (1.6.8). As in the definition of P-Bind given in Section 1.1, the metrizability of the space will be part of the definition of P-Odim. 3.1. Definition. Let P be a class of spaces and let X be a (metrizable) space. For n = -1 or n E N the space X is said to have P-Odim 5 12 if there exists a a-locally finite base { U, : y E I?} for the open sets of X such that B (U,,) n . - .n B ( UYn)belongs to P for any n i- 1 different indices yo,. . . ,yn from r. (Of course, this gives X E P for n = -1.)

We shall start by collecting some fairly simple properties. 3.2. Proposition. If P and Q are classes with P 3 Q, then P-Odim 5 Q-Odim. In particular, P-Odim 5 Odim.

Recall that for any subspace Y of a space X and for every subset U of X we have by Proposition II.4.2.B that By(U n Y ) c Bx(U). So, easy inductive proofs will give the following propositions. 3.3. Proposition. For every closed subspace Y of a space X , P-OdimY 5 P-OdimX.

3. T H E ORDER DIMENSION

235

3.4. Proposition. The class P is monotone if and only if the function P-Odim is monotone, i.e., for every subspace Y o f a space X ,

P-Odim Y 5 P-Odim X The corresponding statements hold for Fg-,Gs- and open-monotone classes. The next open subspace theorem is similar to Theorem 11.2.17. 3.5. Proposition. Let X be such that 0

5 P-Odim X . For every

open subspace Y of X , P-Odim Y 5 P-Odim X . Our next goal is to show the relation between P-Dim and P-Odim. The relation between P-dim and P-Odim for separable spaces will follow as a corollary. But first we need to extend Lemma 1.9.4 to infinite collections. Notice that in the extended theorem the hypothesis that X be metrizable can be weakened t o hereditarily paracompact. The corresponding result for P-dim has been stated for hereditarily normal spaces in Theorem 11.5.9. 3.6. Theorem. For a natural number n let X be a metrizable space with P - D i m X 5 n. Suppose that G is a closed P-kernel of X and suppose that { F, : y E I?} and { U, : y E r } are locally finite collections of closed and open sets of X respectively such that F, c U, mod G, y E I?. Then there exists a closed P-kernel H of X and two open collections { V, : y E I' } and { W, : y E r } with

F, C V,

c

G c H, clx(V,) c W, c U, mod H ,

ord{W,\clx(V,):yd?}

y E

r,

S n modH.

Moreover, the conditions

v,Cx\H

and W , c X \ H ,

y ~ r ,

may also be imposed.

Proof. We may assume that the indexing by sider the P-border cover

r is one-to-one. Con-

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

236

This border cover is locally finite because { U, : y E I'} is locally finite. Let E be an open cover of the subspace X \ G such that each member of E meets U, for only finitely many indices y from r. We may further require that E be locally finite. By Proposition II..5.5.C, there exists a locally finite P-border cover L = { L , : a E A } with enclosure H such that L shrinks E modulo H and ord L 5 n -t 1. Clearly the P-kernel H contains G and is closed. The locally finite open collection L in the subspace X \ H has a closed (in X \ H ) shrinking { K , : a E A } . As L is a refinement of E, for each a in A there are only finitely many y in r such that L , fl F, # 0. We list the indices y for which L , n F7 # 0 in a one-to-one way as y(0, a ) , . . . ,y(n(a),a). Using the normality of the subspace X \ H , we can find for each a in A a collection { ICL : i = 0 , 1 , . . . , .(a) i- 1 } of open sets such that

(1)

K,

i = 0,1,. ..,n(a)

c K ; c clx\H(K;) c L,,

+ 1,

and

(2)

c

clx\H(Ic;)

i = 0,1,. .+(a).

It should be noted that from (1) and (2) we have

(3)

(Ic:+I

\ clx\H(K:)) n (KZ+' \ clx\H(zt-;,)

Now for y in

= 0,

i < i'.

r we define

v,

= U{ K& : y = y ( i , a ) } , = U{ 1i-:+,+': y = y(i,Cl)}.

w,

Because L = { L , : Q E A } is a cover of X of E , one readily proves the inclusions

F,

c V, c cl(V,) c W, c U,

\H

and L is a refinement

mod H ,

y E

r.

Thus the first two formulas have been established. Now, by way of a contradiction, we shall prove the last of the formulas. Assume for some p in X \ H and some distinct indices ym, m = 0,1,. . .,n, that

3. THE ORDER DIMENSION

237

Note that we have

Thus for suitable pairs ( i , , ~ , ) we have

It follows from (3) and the definition of y(k, a ) , k = 0 , 1 , . . . , n ( a ) , that the indices a , must be distinct. As { K , : Q E A } covers X , we have p E IC, for some 7 in A . From (1) it follows that 7 is distinct from each of the am's. Hence

that is, ord,L> n + 2 . This is a contradiction. Now let us disclose the relation between P-Odim and P-Dim.

3.7. Theorem. For every space X , P-Odim X 2 P-Dim X.

Proof. Let n be a natural number and suppose P - D i m X 5 n. Let ({ U, : y E }, { H , : y E I' }) be the framework of a base of X . We may assume that the indexing of the collection { U, : y E I'} by I' is one-to-one. For each y in 'I we select an open set C, such that

H,

c c, c F7 c u,,

U{

where F, = cl (C,). We write r = i'I : i E M} in such a way that { U, : y E I'i} is locally finite for each i. Observe that for U rj } is locally finite. Ineach j the collection { U, : y E roU ductively on j , we shall define a closed P-kernel Gj and, for each y in I'o U . - .U rj, open sets V i and W; possessing the properties:

238

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

For each j and each y in

(4)

U

+

. U rj,

V; c X \ Gj, W; C X \ Gj, F, C V; C cl(V;) C W; C U , mod G j

and

and, for each j ,

(6)

ord{ W;\cl(V,”)

: y ~ r ~ u . . . U I ’ ~ }m < ond G j .

As the first step of the definition is similar to the inductive step, we shall describe only the inductive one. To this end, suppose that s is such that there have been defined for each j a closed P-kernel G j and open sets V; and W$ for each y in ro U - . U rj so that (4)and (6) hold for j 5 s and (5) holds for j < s. To define Gs+l and V;+’ and WySs1 for each y in ro U * - U rS+l,we apply the preceding theorem to the P-kernel G, and the collections

-

and

so as to obtain the closed P-kernel G,+1 and the open collections

{ V;+’ : y E rou

u rsSl}

satisfying (4)and (6) for j = s pleting the inductive step. From the inclusion

H, c

and { W;+’ :

E

ro u ...u rSs1}

+ 1 and (5) for j = s, thereby

c, c vyi u (u, n Gi),

yE

com-

ri,

we find that the interior of V; U ( U , n Gi), denoted by Vi*,satisfies

(7)

H,

c c, c v,i*c u,,

E

ri,

3. THE ORDER DIMENSION

and

Vi = V,"*\ Gi,

E

239

ri.

for each i in N. Moreover, when i _< Ic 5 j we have from (4)and ( 5 ) that Vy C W," modulo G j for each y in ro U U r j , and consequently VYj \ G j c W," c U,. e

-

.

Finally we define B = { Vy : y E r } by

V, = Vi*U( U{ V:

\ G j : J' > i}),

yE

ri.

As the pairs (Hy, U,) are from the framework of the base, we have from (7) that f? is a base for the open sets. Notice

B (V,)

c (w," \ cl (V,")) u G/c+l,

Ic 2 i.

Thus it follows that

We have shown for any n -+ 1 different indices yo, . . .,yn that the intersection B (V,,) n n B (Vyn) is contained in some Gk. It follows that the intersection is a member of P. Thereby, P-Odim X 5 n. The following corollary of the above proof is easily derived. For, in that proof, Theorem 3.6 can be replaced with Theorem 11.5.9 (b) and the framework of the base for X used there can be chosen in such a way that ri is the collection { i}. See also the second part of the proof of Theorem 1.9.5. 3.8. Corollary. For every separable space X ,

P-Odim X 5 P-dim X . Applying this corollary to the class /c of all compact spaces, we get the following important result. 3.9. Corollary. For every separable space X ,

Skl X 5 IC-dim X. The next theorem compares P-Odim and P-Bind.

240

V. FUNCTIONS OF BASIC DIMENSIONAL T Y P E

3.10. Theorem. For every space X ,

P-Bind X 5 P-Odim X.

Proof. The proof is by induction on P-Odim X . As the inequality is obvious for P-O dim X = 0, let us go to the inductive step. Let n > 0 and assume P-Odim X 5 n. There exists a a-locally finite base B = { V, : y E r } such that ord{ B ( V . ) : y E r } 5 n. For each rl in r the collection B, = { V, n B (V,) : y E r \ { q } } is a 0-locally finite base for the open sets of B (V,). From the fact that B B ( ~ ~ ) (nVB. (V,)) is a closed subset of B (V,)n B (V,) we easily see that P-Odim B (V,) 5 n - 1 for each 77 in r. By the induction hypothesis we have P-Bind B (V,) 5 n - 1, whence P-Bind X 5 n. It is to be noted that the Theorems 3.7 and 3.10 have been proved under the minimal requirement that the class P be closed-monotone. When the class P is a cosmic family, a nice lining up of the dimension functions modulo P will result.

3.11. Theorem. Suppose that P is a strongly closed-additive cosmic family. Then for every space X P-Bind X _< P-Odim X

5 P-dim X

= P-Dim X

5 P-Ind X.

Proof. The equality of the covering dimensions has been established in Theorem IV.2.7. The last inequality in the theorem follows from Theorem IV.4.2. The other inequalities follow from the previous theorems. 3.12. Theorem. Suppose that P is a normal family or that P is a multiplicative Bore1 class M(cy), cy > 0. Then for every space X

P-Bind X = P-Odim X = P-dim X = P-Dim X = P-Ind X .

Proof, Note that P is a cosmic family. The theorem follows from the previous result and Theorems 2.4, 2.14 and 2.21. 3.13. Corollary. With the hypothesis of the theorem, for every separable space X , P-ind X = P-Ind X .

The previous theorem includes the coincidence theorem for the ordinary dimension functions.

4. THE MIXED INDUCTIVE DIMENSION

241

3.14. Theorem (Coincidence theorem). For every space X ,

Bind X = Odim X = dim X = Dim X = Ind X . 3.15. Remark. We have found in the preceding theorem that d i m X = I n d X for every X in M. We can use Theorems 111.3.25 and IV.3.5 to get the same equality for every X in N[ M : N H ] the , normal extension of M in N H . As M is contained in the class & of strongly hereditarily normal spaces, we have that N[M :N H ]is contained in the Dowker universe D. It is obvious that N [ M :NH] properly contains M . 3.16. Corollary. For every class P,

P-sur = P-Sur 4.

and

P-def = P-Def.

The mixed inductive dimension

The purpose of this short section is to place the compactness dimension function Cmp into a proper perspective by presenting the natural generalization of the function Cmp, namely the function P-Mind called the mixed inductive dimension modulo a class P , Most of the theory about this dimension function is yet to be developed. The function Cmp is defined as a large inductive variation of cnip which agrees with cmp in the values -1 and 0. The function ?-Mind will be defined in a completely analogous manner. 4.1. Definition. Let P be a class of spaces. To every space X one assigns the mized inductive dimension, P-Mind X , as follows. (i) For n = -1 or 0, P-Mind X = n if and only if P-ind X = n. (ij) For each positive natural number n, P-Mind X 5 n if for each pair of disjoint closed sets F and G there is a partition S between F and G such that P-Mind S 5 n - 1 .

Obviously the mixed inductive dimension modulo a class P is a topological invariant which coincides with Cmp when P = K . Although the mixed inductive dimension has been defined for all topological spaces, a theory for non-metrizable spaces does not seem very promising at this time. But the following proposition can be established by an easy inductive proof which will be left to the reader.

242

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

4.2. Proposition. For each closed subspace Y of a space X

P-Mind Y

,

5 P-Mind X .

As we shall see in the next theorem, the proof of Theorem 1.12.1 can be carried out in a more general setting. The theorem concerns the existence of certain partitions between closed sets. In Theorem IV.3.9 we have already encountered a partitioning theorem in terms of the relative dimension with respect to P-border covers. 4.3. Theorem. Suppose that P is closed-additive in the universe of separable metrizable spaces. Let X be a space with P-Odim X 5 n where n 2 1. Then between any two disjoint closed sets of X there is a partition S such that P-Odim S _< n - 1.

Proof. The proof is by induction on P - O d i m X . To prove the inductive step let us suppose that X is a space with P - O d i m X 5 n where n 2 1. Let F and G be disjoint closed sets of X . Only the case where both F and G are nonempty need be considered. We shall construct first a partition S. From the definition of P - O d i m X 5 n there exists a countable base B = { Ui : i E N } for the open sets of X such that the intersection B (Ui,) n - . . n B ( U i , ) belongs to P for any n 1 different indices io, . ..,in in N. Consider the collection { (Ck,Dk): k E N } of all pairs of elements of B with cl (Ck) C DI, such that cl (Dk)n F = 8 or cl (Dk)n G = 0. Observe that the indexing for this collection of pairs is unrelated to that of B. Now for each Ic in N let

+

The collection V = { vk : k E N} is a locally finite cover of X , and

for each Ic in N. Let W = U{ V k : cl(Dk) n G = S}. Obviously the set W is open and F c W . From local finiteness of V we have cl(W) n G = 0 and B ( W ) c U{ B(Vk) : cl(Dk) n G = S}. It follows that B ( W )is a partition between F and G. Let us show P-Odim B ( W )5 n - 1. As the inequality is obvious when B ( W )= 0 holds, we assume B ( W )# 8. Observe that each B(Vk) is the union of a finite collection of closed subsets of

4.

THE MIXED INDUCTIVE DIMENSION

243

boundaries of elements of B. It follows that B ( W )is the union of a locally finite collection { Ej : j E NO} where NO is a subset of N and each Ej is a nonempty closed subset of B ( U j ) . From here on we shall be returning t o the original indexing of f?. Next we shall select a subcollection B' of B such that B' is still a base for X and has the added property that the collection { U n B ( W ): U E B'} is a base for the open sets of B ( W )witnessing the fact P-Odim B ( W )5 n - 1. This will be done in two steps. The first step follows. For each j in NO a point p j from Ej is selected. Let P = { p j : j E No } and observe that P is closed. With N1 = { i E N : U; E B, B (Ui)n P = S} we form the subcollection B* = { Ui : i E Nl } of B. It is easily seen that B' is a base for the open sets of X and that the boundary B ( U ) of each U in B* is distinct from the boundary B ( U j ) for any j in NO. So, in particular, NOn N1 = 0. For the second step let G be an open cover of X such that each element of G meets at most finitely many members of the locally finite collection { Ej : j E NO}. Let B' = { U; : i E N2 } be the set of all U; in B* such that cl ( U ) C G for some G in G . Obviously Nz c Nl holds and B' is also a base for X . Let us show that the base { U n B ( W ): U E f?' } for the open sets of B ( W )witnesses the fact P-Odim B ( W )5 n - 1. To this end, let io, . . , , i,-l be n distinct indices from N2 and, with d denoting the boundary operator in B ( W ) ,let

~ = a ( ~ i , n ~ ( W ) ) n . . . n d ( U ~ ~ - ., n ~ ( W ) )

--

Clearly S is a closed subset of R = B (U;,) n - n B fl B ( W ) . As each B ( U i j ) is a subset of some G in G and each member of G meets Ej for at most finitely many j in N O , it follows that the equality R = U{ R n EI, : k E N ' } will hold for some finite subset N' of N O . For each j in N ' , the set R n Ej is a closed subset of B ( U j ) . Now R n B ( U , ) belongs to P because the indices j , io, . . .,i,-l are distinct. Because P is closed-additive, it follows that R and, consequently, S belong to P. We have shown P-Odim B ( W ) 5 n - 1.

Remark. The set P in the above construction was introduced to avoid the indexing pitfall mentioned in Example 1.6.13. In Section VI.5 we shall introduce another method to serve the same purpose. With this result the next statement follows by an easy inductive proof.

244

V. FUNCTIONS OF BASIC DIMENSIONAL TYPE

4.4. Corollary. Suppose that P is cfosed-additive in the universe of separable metrizable spaces. For every space X ,

P-ind X = P-Bind X 5 P-Mind X 5 P-Odim X. Observe that the closed-additivity of P is only required t o establish the last inequality. Obviously the class K is closed-additive. In addition to the class Ic, the class L: of locally compact spaces is closed-additive. Indeed, L: is a cosmic family.

For later use we state a variation of Theorem 4.3 in which the condition P - O d i m X 5 n has been replaced by a weaker condition. The proof is essentially the same. 4.5. Theorem. Let P be closed-additive in the universe of separable metrizable spaces. Suppose that X is’a space for which there exists a collection V of open sets satisfying the following conditions. (a) { V : V E V or ( X \ cl ( V ) )E V} is a base for the open sets. (b) B (KO) n . n B (Vi, ) is in P for any n i1 different indices io, . . . , in in N.

--

Then P-Mind X

5 n.

5 . Historical comments and unsolved problems

Several of the results in this chapter are forerunners of other results in this book. For example, the proof of Theorem 2.13 is almost the same as the proof of the main result in Aarts [1968] which appeared early in history of the subject. Other examples come from the notion of basic inductive dimension which had been only marginally discussed in the [1973] paper by Aarts and Nishiura and many earlier versions of these results had been found by using this dimension function. The excision theorem (Theorem 2.3) was established in Aarts and Nishiura [1973] in essentially the same way as in Section 2. The first extension theorem can be found in Aarts [1968]. Results like Theorem 2.24 can be found in Aarts [1971a], Nagami [1965] and Wenner [1969], [1970]. The proofs in Section 3 are new and follow the pattern of the proofs of corresponding results in Nagata [1965].

5. HISTORICAL COMMENTS A N D UNSOLVED PROBLEMS

245

Unsolved problems 1. This problem concerns an ind analogue of Tumarkin’s theorem for Ind. Suppose that X is a subset of a metrizable space Y and that i n d X 5 n holds. Does there exist a Gs-set 2 of Y with X c 2 and ind 2 5 n ? 2. Does there exist an example of a strongly closed-additive cosmic family P for which P-Bind X < P-Ind X for some space X ?

3. Observe that Theorem 3.12 is not applicable to the absolute Bore1 class M(O), that is, the class = K n M . Also Kimura’s theorem yields def X = Skl X only for separable metrizable space X . Is there a meaningful generalization of the theory of Skl for general metrizable spaces?

Chart 2. Compactness dimension functions

Ind = ind = dim

K-Sur

I

K-Ind

Cmp = K-Mind

'I

1.11.4 I

I

1.6.6

cmp = K-ind The chart summarizes the relations among the compactness dimension functions in the universe Mo of separable metrizable spaces. The solid arrows indicate inequalities and the dashed arrows indicate counterexamples. For example, the solid arrow from K-Ind t o IC-def indicates that K-def X 5 K-Ind X holds for every separable metrizable space X , and the number next t o the arrow shows where the proof is found. The dashed arrow from cmp t o Cmp indicates that there exists a space X such that c m p X < C m p X . 246

CHAPTER VI

COMPACTIFICATIONS

The sole topic that remains is the compactness deficiency of a space. Thus the central theme of this chapter will be compactifications. The chart on the opposite page gives an overview of the relationship among the various compactness dimension functions in the universe Mo of all separable metrizable spaces. By now, most of the arrows have been validated and all counterexamples have been presented. The following arrows are the ones that still require validation.

(A) Skl

-+ K-def, that is, K-def X

-

5 Skl X for every separa-

ble metrizable space X . (B) K-dim K-def. (C) K-Ind + K-Def. Each of these arrows requires a proof of the existence of a compactification, It is to be noted that the equalities def = K-def = K-Def hold in the universe Mo. So statement (A), which will be proved in Section 5, is the essential remaining part of Kimura’s theorem. Observe that (B) and (C) will follow from (A) because Corollary V.3.9 and the two equalities Skl = K-def and K-dim = K-Ind are valid for separable metrizable spaces. Nonetheless, the statements (B) and (C), or proper modifications of them, can be proved in a universe that is much larger than the universe Mo of separable metrizable spaces. This will be done for (B) in Section 6 and for (C) in Section 4. The proofs will require rather delicate constructions. In advance of the presentation of these proofs, a discussion of some of the simpler results in compactification theory will be given. The first section will deal with the Wallman compactification. This compactification method is especially well suited for relating various covering properties, including that of dimension and homology, of a space 247

VI. COMPACTIFICATIONS

248

and its compactification. In the second section a modification of the Wallman compactification and several theorems about dimension and weight preserving compactifications will be discussed. This modification will be used in the next section for a presentation of the Freudenthal compactification. Included in this presentation is de Groot's Theorem 1.5.3 which is central t o this book.

Agreement. T h e universe is the class of TI-spaces. 1. Wallman compactifications

A self-contained development of lattice representations of topological spaces will be given. The theory of the representation of topological spaces by lattices was introduced by Wallman in [1938]. His approach was similar to that of the representation of zero-dimensional spaces by Boolean rings as developed by Stone in [1937]. The main result of our development is the construction of the Wallman compactification W X for each space X . It will be shown that for every normal space X the dimensions, both Ind and dim, of X and W X are the same. We shall begin with a discussion of some simple notions in lattice theory. The notation and terminology of Birkhoff [1973]will be used. 1.1. Definition. A lattice is a nonempty set L with a reflexive partial ordering such that for each pair ( x , y ) of elements of L there is a unique smallest element x V y, called the join of x and y, satisfying x 5 x V y and y 5 x V y and there is a unique largest element x A y, called the meet of x and y, satisfying x A y x and x A y y. A lattice L is distributive provided 5 V (y A z ) = (z V y) A ( x V z ) and x A (y V z ) = (x A y) V ( x A z ) hold for all x , y and z.

<

<

<

The collection of all subsets of a given set X ordered by inclusion is an example of a lattice, the join and meet are the union and intersection respectively. The same holds true for any collection F of subsets of X that is closed under the formation of finite unions and finite intersections. It might be useful t o keep these examples in mind in what follows. Various properties can be deduced easily from the definition. We shall mention just a few. The easy proofs will be left to the reader.

1. WALLMAN

COMPACTIFICATIONS

249

1.2. Proposition. For all x , y and t of L , (a) x A x = x , zVx=x, (b) ~ A y = y A s , zVy=yVz, (z V y) V z = x V ( y V z ) , (c) (Z A y) A Z = z A ( y A z ) , (d) 2 A (Z V y) = z V (Z A y) = z, (e) z 5 y if and only if z A y = 5 , (f) z 5 y if and only if z V y = y. 1.3. Definition. The lattices L and L' are said to be isomorphic if there exists a bijection 79: L + L' such that for all z and y in L

(i) 79(Z A Y) = d(.> A d(Y), (ij) fl(z v y) = 79(z)v 79(y). Such a bijection is called an isomorphism. A function h from L t o L' that satisfies conditions (i) and (ij) is called a homomorphism. In view of property (e) above, for an isomorphism 19 we have z if and only if O(z) 5 79(y).

5y

Our first goal is to construct a topological space out of a lattice. The points of the space will be generated by the maximal dual ideals which we shall now define. 1.4. Definition. A nonempty subset D of a lattice L is called a dual ideal if the following hold for all z and y: (i) If z E D and y E D, then z A y E D. (ij) If 5 E D and z 5 y, then y E D. A dual ideal D in a lattice L is said t o be maximal if D # L and either D = E or E = L for each dual ideal E with D C E.

Concerning the conditions (i) and (ij), one can readily show that each set Do satisfying the condition (i) is contained in the dual ideal D = { y : z 5 y for some z in Do }. As a special case, for each a in L the set { y : y E L , a 5 y ) is a dual ideal. We shall see that dual ideals in the lattice L correspond t o closed filters in the space generated by L . Let us collect some properties of dual ideals. If a lattice'has a smallest element (that is, an element a for which a 5 z for every z in L ) , this element must be unique. When it exists, the smallest element will be denoted by 0. Dually, when it exists, the largest element of a lattice is unique and is denoted by 1. For a dual ideal E in a lattice L with 0, it is easily seen that E = L if and only if 0 E E.

250

VI. COMPACTIFICATIONS

1.5. Lemma. Let L be a lattice with 0. Every dual ideal D in L with D # L is contained in a maximal dual ideal.

Proof. Let E be the collection of all dual ideals E with the property that D c E and E # L . The collection E is partially ordered by inclusion. Each chain in E has an upper bound, namely the union of the chain. By Zorn’s lemma, E has a maximal element which is the maximal dual ideal containing D. There is the following criterion for maximality of dual ideals. 1.6. Lemma. Let L be a lattice with 0. A dual ideal D of L is a maximal dual ideal if and only if the condition

x E D if and only if z A x # 0 for every z in D holds for each x in L. Proof. To prove sufficiency suppose that D is a dual ideal for which (*) holds. Clearly, from (*) we have D # L. Let E be a dual ideal such that D C EandO 4 Eandlet x E E. A s 0 # E, wehavea: A t # 0 for each z in D. By (*) we have x E D thereby showing D = E. That is, D is maximal. For the proof of the necessity, let us assume for some z in L that 2 A z # 0 for every z in D. Define the subset E of L by E = { y : z A z 5 y for some z in D}. Then E is a dual ideal such that D C E, x E E and 0 4 E. As D is maximal, we have D = E and thus x E D. The rest of the proof is

(*I

obvious. The condition in the next lemma is the lattice counterpart of a well-known property of ultrafilters.

1.7. Lemma. Suppose that D is a maximal dual ideal in a distributive lattice L with 0. Then the condition (x V y) E D if and only if x E D or y E D holds for every x and y in L .

Proof. The sufficiency part of the condition is obvious. For the necessity suppose that 2 V y E D and x D. Then by Lemma 1.6 there is a z in D such that z A x = 0. Now from y 2 z A y = z A (z V y) and z A (x V y) E D we have y E D.

e

There is yet another property of lattices that will play a role in the representation of lattices by topological spaces.

1. WALLMAN COMPACTIFICATIONS

251

1.8. Definition. Suppose that L is a lattice with 0. Then L is said to have the disjunction property if for all x and y in L with x # y there exists a z in L with the property that one of the meets z A z and y A z is 0 and the other is not.

Having collected the relevant parts of lattice theory, we now can discuss the representation of lattices by spaces. In the preceding chapters we have always used bases f? for the open sets of a space X . For the purposes of the theory of Wallman representation and compactification it is much more convenient to work with bases for the closed sets of a space. 1.9. Definition. A collection 3 of closed sets of a topological space X is called a base for the closed sets of X if each closed subset of X is the intersection of some subcollection of F.

Note that a collection 3 of closed sets of a topological space X will be a base for the closed sets of X if X E 3 and if for each closed subset G of X and for each point p of X with p $! G there is an F in 3 such that p $! F and G c F . Also, observe that a collection F of subsets of a set X will be a base for the closed sets of a uniquely determined topology for the set X if X E 3 and F has the property that for all F and G in 3 the set F U G is the intersection of some subcollection of F.In particular, 3 will be a base when 3 is a lattice of subsets of X and X E F. We can now formulate the representation theorem. 1.10. Theorem. Suppose that L is a distributive lattice with 0 and 1, and suppose that L has the disjunction property. Then there exists a compact TI-space W L and there exists a base F for the closed sets of u L such that 3 is a lattice that is isomorphic with L .

The space W Lis called the Wallman representation of L. Of course, the proof will consist of two parts, where the first consists of the constructions of the pair W Land 3 and the isomorphism 6 : L -, 3 and the second consists of verifying that F generates a compact 7'1-space topology on w L . Since the need to refer t o the construction will occur often, this part of the proof will be set off from the other. 1.11. Construction of the Wallman representation. Consider the collection of all maximal dual ideals of L. Let this collection be indexed in a bijective fashion and denote the indexing set

252

V1. COM PACT1FI CATIONS

by WL. Then the collection of maximal dual ideals can be written as { D, : z E w L } . For each t in L define the subset Bt of wL. to be

Bt = { x : t E D,}, and define the collection

F t o be

It will be shown that the collection 3 possesses the properties (1) (2)

if u and v are distinct members of L , then B,

B, u B, = Bzlvv and

B, n B, = Bull,,

# B,,

u,v E L.

That is, F is a lattice. Observe that the natural map 19: L -+ 3 defined by d ( t ) = Bt is an isomorphism. Let us verify property (1). Suppose that u and v are distinct members of L. Because L has the disjunction property, there exists a z in L such that one o f t A u and z A w is 0 and the other is not. We may assume that z A u = 0 and z A v # 0. By Lemma 1.5 there is a maximal dual ideal D, such that ( z A v) E D,. It follows that v E D, and u 4 D, whence y E B, \ B,. Thereby property (1) is verified. Turn now t o property (2). We shall prove only the first formula since the second formula is proved in a similar way. Clearly B, U B, c BuVv holds. So, let x E Buvv. By definition (uV v) E D, holds, whence u E D, or v E D, must hold by Lemma 1.7. Thereby property (2) is verified.

Proof of Theorem 1.10. Because F is a lattice, it will generate the closed sets of a topology on the set wL. We shall prove that w L with this topology is compact. Let G be a collection of closed subsets of w L with the finite intersection property. We need G # 0. We may assume that G is closed under finite to show intersections. Define the dual ideal H of L by H = { t E L : there exists a G in G with Bt 3 G}. Because { Bt : t E L } is a base for the closed sets, we have

253

1. WALLMAN COMPACTIFICATIONS

Now choose a maximal dual ideal D, such that H C D,. Then for every t in H we have t E D,. So y E Bt for all t in H and hence y E Bt : t E H } = G . Next suppose that z and y are distinct points of wL. Then D, and D, are distinct and, moreover, neither D, nor D, contains the other. By Lemma 1.6 there are s and t from L such that s E D, t E D, and t A s = 0. It follows that z E B,, y E Bt and B , n Bt = 0, that is, W Lis a TI-space.

n{

n

The following theorem is very important because it provides a lattice criterion for the Wallman representation t o be Hausdorff. 1.12. Theorem. The Wallman representation W L of a distributive lattice L with 0 and 1 is a Hausdorff space if and only if s A t = 0 implies the existence of u and v in L such that the condition

(*>

sAv=0,

uAt=O

and u V w = 1

holds. It should be noted that s 5 u follows from the condition (*) because s = s A 1 = s A (uV v) = (s A u)V (s A v) = s A u.By symmetry, t 5 v also holds.

Proof. To prove the sufficiency of the condition let z and y be distinct points of wL. In view of Lemma 1.6 there are s and t in L such that z E B,, y E Bt and s A t = 0. With u and v as in (*), from the isomorphism between L and { Bt : t E L } we get

Bt n B, = 8,

B , n B, = 0 and B,

u B,

= wL.

It follows that W L\ B, and W L\ B, are disjoint neighborhoods of z and y respectively. To prove the necessity we assume that W L is a Hausdorff space. From the compactness of W L it follows that W Lis normal. Suppose that s and t satisfy s A t = 0. Then B , and Bt are disjoint closed subsets of w L . Let V and U be disjoint open neighborhoods of B, and Bt respectively. Consider the collection H = { B, :W L\ U c B, }. By way of contradiction we shall show that there is a B, in H such that B, n Bt = 8. If B, n Bt # 8 for every B, in H, then by the compactness of W L the set Bt n H is not empty. Let 5 be a point

n

VI. COMPACTIFICATIONS

254

in this set. As { B, : T E L } is a base for the closed sets, there is a B, in H such that z $ B,. This is a contraction. Similarly, there is a in L such that w L \ V c B, and B, n B, = 0. Obviously, B, U B, = w L . As the lattice L and { Bt : t E L } are isomorphic, the condition (*) holds. Let us now turn our attention to a topological space X. There is always a natural distributive lattice with 0 and 1 associated with its topology, namely the lattice of all closed subsets of X . As X is a TI-space, this lattice has the disjunction property because singleton sets are closed. Often we shall be dealing with sublattices L of the lattice of all closed subsets of X. Under such a setting the collection D, = { F : F E L , z E F } is a dual ideal but it may not be a maximal dual ideal of L . For the moment let us assume that D, is a maximal dual ideal and that F E L : z E F } = { z } for each z in X. Then there is a natural injective map

n{

(3)

cp: X

+WL

such that

Dv(,l = { F E L : z E F}.

Obviously the existence of cp will be established by the equality F E L : z E F } = { z } . This assumption is valid when L is the lattice of all closed subsets of a space X . We shall define next the Wallman compactification of X .

n{

1.13. Definition. Let X be a space and let L be the lattice of all closed subsets of X . The Wallman representation W Lof L is called the Wallman compactification of X . The Wallman compactification of X will also be denoted by w X .

That W X is a compactification of X will follow from the next theorem whose proof relies only on the theory that has been developed thus far. 1.14. Theorem. Let X be a space. Then the Wallman representation W X o f the lattice L = { F : F is a closed subset o f X } is a cornpactification of X . Also, there is a canonical embedding cp o f X into W X with the following properties. (a) Dv(,) is the maximal dual ideal { F : F E L , z E F }. (b) The collection FL = { clwx(cp[F]): F E L } is a base for the closed sets o f w X . ( c ) The function d defined by g ( F ) = c l w x ( q [ F ] )F , E L , is a bijection from L to FL.

1. WALL MAN COM PACTIFICATION S

255

(d) When F and G are in L ,

Consequently 29 is a lattice isomorphism. Moreover, W X is the Cech-Stone compactification of X whenever X is normal. The bijective function 29 will be called the canonical isomorphism. Proof. We shall employ the construction of the Wallman representation found in Section 1.11 above. Because { BG : G E L } is a base for the closed sets of w X , we have for all F in L

The properties (a)-(d) are now obvious. For the proof of the final statement of the theorem, we begin by noting that the lattice L satisfies the condition (*) of Theorem 1.12 when X is normal. Hence W X is Hausdorff. Recall that a zeroset of X is a set of the form { z : f(z) = 0 ) for some continuous real-valued function f. As disjoint zero-sets, like any disjoint closed subsets of X , have disjoint closures in w X , from the characterization given by Cech in [1937] it follows that W X coincides with the maximal Hausdorff compactification of X (see also Gillman and Jerison [1960], in particular Section 6.5).

A nice feature of the Wallman compactification is that it carries over to closed subspaces.

1.15. Theorem. Suppose that Z is a closed subset of X and let c p : X + W X be the canonical embedding o f X onto the Wallman compactification W X of X . Then there is an embedding ?I, of W Z into w X that is an extension of the inclusion map i: 2 + X . And the image ?I,[wZ] coincides with cl,~(cp[Z]). Proof. We shall use the notation of the preceding proof. There L denoted the lattice of all closed subsets of X. The canonical embedding of 2 into its Wallman compactification W Zwill be denoted by (PI. Let D be a maximal dual ideal of L z , the lattice of all closed

VI. COMPACTIFICATIONS

256

subsets of 2. Then define E = { G E L : G n Z E D } . It is easily seen that E is maximal dual ideal of L. Thus for a unique C in w Z and a unique [ in W X we have D = D( and E = E,. The required map $ is defined by 5 = $((). We shall show that the following diagram is commutative, where i: Z + X is the inclusion map.

(4)

1.

x cp w x

Let t be a point in 2. Then we have Dpl(z) = { F E L z : z E F } and E+(9pl(z)) = { G E L : i ( z ) E G } = E9(qz)), which shows commutativity. Let us show that $I is a bijection between cl,x(cpl[Z]) and cl wx(cp[ i[Z]]). This will follow from the fact that Z is a member of amaximaldualideal Eof Lif and onlyif E is { G E L : G n Z E D } for some maximal dual ideal D of Lz. In particular, we find that the image $I[wZ]coincides with cl,x(cp[ i[Z]]).

Remark. With very few exceptions, we shall be identifying the space X via the canonical embedding cp with its image VEX] thus making X a subset of w X . This will occur for the first time in the next lemma.

A virtue of the Wallman compactification is the existence of a simple relation between dimension (and homology) of a space and the dimension (and homology) of its Wallman compactification. The closure, expansion and boundary operators behave rather nicely in the Wallman compactification. (See Definition V.2.9 for the definition of the expansion operator.) 1.16. Lemma. If A is a subset of X and if U and V are open subsets of X , then the following are true.

(a) cl,x(A) = CLX(C1X(A)). (b) ex,x(U U V ) = ex,x(U) U ex,x(V). ( c ) Bwx(exwx(U)) = C L x ( B x ( U ) ) .

Proof. The equality (a) is obvious. To prove (b) let F = X and G = X \ V . Then ex,x(U u V ) = W X \ cl,x(X

\ (U u V ) )

\U

1. WALLMAN COMPACTIFICATIONS

257

=wX\cl,x(FnG)

= w x \ ( c i U x ( ~n) CLX(G)) = ex,x(U) u ex,x(V).

For (c) consider the closed sets X \ U and cIx(U) of X . As X is dense in w X , we have U c ex ,x ( U ) C cl ,x( clx(U ) ) . From (a) we infer clwX(exwX(U))= cl,x(clx(U)). Now we have

The maximality of the Wallman compactification, which follows from the next theorem, is the most important property of these compactifications. 1.17. Theorem. Suppose that f : X + K is a continuous map from a space X into a compact Hausdorff space K . Then f can be extended to a continuous map w X + K.

F:

The proof of this theorem, which is left to the reader, is essentially the same as that of Theorem 2.5 below since I( = wh' holds for every compact space K . To conclude this section we shall show that the Wallman compactification of a normal space X has the same large inductive dimension and the same covering dimension as X . 1.18. Theorem. For every normal space X ,

Ind X = I n d o X

and dim X = dimwX.

Proof. To establish the equalities we shall prove four inequalities by induction. As w 0 is the empty space 0, the initial step of the induction is trivial. Only the inductive steps remains to be shown. Ind X 5 Ind w X : Assume Ind w X 5 n and let F and G be disjoint closed subsets of X . By Theorem 1.14, cl,x(F) and cl,x(G) are disjoint. There is a partition S between cl,x(F) and cl,x(G)

VI. COMPACTIFICATIONS

258

in wX with Ind S 5 n - 1. We have w ( S n X) = cl,x(S n X ) c S by Theorem 1.15. In view of Proposition 1.4.3, Ind w ( S n X ) n - 1, whence Ind (S n X ) 5 n - 1 by the induction hypothesis. Thus we have I n d X 5 n. Ind X 2 Ind wX: Assume Ind X 5 n and let F and G be disjoint closed sets of wX. Let U1 and V1 be respective open neighborhoods of F and G with disjoint closures. We infer from the definition of Ind the existence of a partition S between X n U1 and X n Vl in X with Ind S 5 n - 1. Select disjoint open sets U andVofXsuchthatX\S=UUV,XnUl c UandXnV1 cV. By Lemma V.2.10 we have F c U1 C exwX(U), G C VI C ex,x(V) and ex,x(U) n ex ,x(V) = 0. And from Lemma 1.16 and the definition of ex,x we have WX\ cl,x(S) = ex,x(U) U ex,x(V). Consequently, cl,x(S) is a partition between F and G in w X . By the induction hypothesis, IndwS 5 n - 1. Hence Indcl,x(S) 5 n - 1 by Theorem 1.15. Thereby, Ind WX5 n. dim X 5 dimwX: Assume dim wX 5 n and let U be a finite open cover of X. By Lemma 1.16, { ex,x(U) : U E V } is a finite open cover of wX. This cover has an open refinement V such that ord V 5 n 1. The trace of V on X is an open refinement of U of order less than or equal to n 1. It follows that d i m X 5 n. dim X 2 dim wX: Assume dim X 5 n and consider a finite open cover U = { U 1 , . . ., u k } of wX. As WX is compact, we may assume that each Ui is regularly open in wX. Then we infer from Lemma V.2.10 that Ui = ex ,x( U i ) holds for each i. There is an open shrinking W’ = { W1,. . .,w k } of the cover { U1 n X, . . . , UI, n X } of X with ord W‘ 5 n + 1. By Lemmas 1.15 and V.2.10 we have that W = {ex,x(Wl), ..., ex,x(Wk)}isanopen coverofwX such that ex,x(Wi) C Ui and o r d W 5 n t 1. So, dimwX 5 n.

+

-+

2. Dimension preserving compactifications

It was indicated by Frink in [1964] that Wallman’s theory has a wide range of applications in the theory of compactifications. The short discussion of the Wallman compactification given in the first section illustrated the lattice approach of Wallman. The Wallmantype (or more simply, Wallman) compactifications will be the topic of this section. This method of compactification will prove to be very useful for constructing special compactifications. Indeed, many of the compactifications presented in this chapter will be constructed

2. DIMENSION PRESERVING COMPACTIFICATIONS

259

by means of Wallman compactifications. The section will include theorems on dimension and weight preserving compactifications and on extensions of mappings to such compactifications. Basic to the theory of Wallman compactifications is the replacement of the lattice of all closed sets by lattices formed from suitably chosen bases for the closed sets of a space. Following Frink, we shall introduce the notion of a normal base. 2.1. Definition. A base F for the closed sets of a space X is called a normal base if the following conditions are satisfied. (i) F is a ring: F is closed under the formation of finite unions and finite intersections. (ij) F is disjunctive: If G is a closed set and z is a point not in G, then there is an F in F such that z E F and F n G = 8. (iij) F is base-norma2: If J and K are disjoint members of F , then there exist G and H in F such that

J n N = Q ) , G f l K = B and G u H = X . The pair (G, H ) is called a screening of ( J , K ) . A space X is called base-normal if it has a normal base for the closed sets. The name “base normality” reflects the requirement that disjoint members of F are contained in disjoint open sets whose complements are in F. As we shall see shortly, the base-normal spaces are precisely the completely regular spaces. Consequently base normality can be regarded as a separation property that naturally fits in between regularity and normality. In passing, we remark that Frink used in [1964] the name “semi-normal” for our “base-normal” . 2.2. Theorem. Suppose that X is a base-normal space with a normal base F. Then the collection F with the partial order given by inclusion is a lattice with 0 and 1 that is distributive and has the disjunction property. The Wallman representation u ( F ,X ) of the lattice F is a Hausdorff compactification of X . There is a canonical embedding cp of X into u ( F ,X ) with the following properties. (a) Dv(z) is the maximal dual ideal { F : F E F ,z E F } . (b) The collection { cl,(~,x)(cp[F]): F E F } is a base for the closed sets of w ( F , X ) .

260

VI. COMPACTIFICATIONS

(c) The function 19 defined by 6 ( F ) = c l , ( ~ , ~ ) ( c p [ F ] )F, E 3,is a bijection from F to { cl,(~,x,(cp[F]): F E F}. (d) When F and G are in F ,

The compactification w ( F , X ) in the above theorem is called the Wallman compactification of X with respect to the base 3.

Remark. Just as we have remarked after Theorem 1.15, the space X will usually be considered to be a subset of w ( 3 , X ) via its identification with cp[X]that is provided by the canonical embedding cp. Observe that the weight of w ( F ,X ) has the cardinality of F as an upper bound when X is an infinite set. The following corollary will provide a characterization of complete regularity by means of separation properties of closed sets. Its proof is obvious. 2.3. Corollary. A space X is completely regular if and only if it is base-normal.

Proof of Theorem 2.2. The construction of the Wallman representation discussed in the last section will be used. Let L = F. Obviously 0 and X are in F by the ring property of the normal base F. Hence L is a distributive lattice with 0 and 1. To show that L has the disjunction property we let F and G be distinct members of F such that F c G and then choose an z in G \ F . From the disjunctive property of the normal base F there is an H in F such that z E H and H n F = 0. Obviously, H n G # 0. It follows that the lattice L has the disjunction property. Consequently the representation space w ( 3 , X ) exists. The remainder of the proof is the same as that of Theorem 1.14 as only the lattice properties of L were used there. Because F is base-normal, the lattice 3 has property (*) of Theorem 1.12. Consequently the space w ( F , X ) is Hausdorff. 2.4. Examples. It is not difficult to see that the collection 2 of all zero-sets of a completely regular space X is a normal base. The compactification 4 2 , X ) coincides with the tech-Stone compactification P X . This will follow from the argument found in the proof of Theorem 1.14.

2. DIMENSION PRESERVING COMPACTIFICATIONS

261

It is also quite easy to show that the one-point compactification a X of a locally compact Hausdorff space X is a Wallman compactification for an appropriate normal base. Another exploitation of the lattice structure of 3 is the following theorem on the extension of continuous mappings. This theorem will be used many times. 2.5. Theorem. Suppose that X and Y are base-normal spaces with respective normal bases 3 and 6 . Suppose that f : X Y is a continuous mapping such that f-l[G] E 3 holds for each G in 6 . Then f can be extended to a continuous map u ( 3 ,X ) + u(6,Y ) . --$

7:

Proof. We shall use the notation of Theorem 2.2. For notational convenience we shall also use 2 = w ( 3 , X ) and ? = w ( 6 , Y ) and use cpx and (py as the respective canonical embeddings of X and Y . The maximal dual ideals in 3 will be denoted by D's with subscripts and the maximal dual ideals in 6 will be denoted by E's. Let us define the map that will make the following diagram commutative.

7

x (5)

'px

fl Y 1py

ii

=u(3,X)

F = u(6,Y)

Let [ be a member of 2. Then [ is the index of a maximal dual ideal Dc of the lattice F. Let H = { G E 6 : f-l[G] E D, }. It is easily seen that H is a dual ideal in the lattice 6 . By Lemma 1.5, H is contained in a maximal dual ideal E. We shall show by way of a contradiction that there is only one such maximal dual ideal. Assume that E' is another maximal ideal of 6 containing H. By Lemma 1.6, there is a G in E and there is a G' in E' such that G n G' = 0. Let ( H , H ' ) be a screening of (G,G'). Observe that H 4 E' holds because H fl G' = 0. Similarly, H' $ E. By Lemma 1.7 we have that f-l[H] or f-'[H'] is in Dt, whence H or H' is in H. As the collection H is contained in E n E', a contradiction will result. Let 7 be the unique member of ? which corresponds t o the maximal dual ideal that contains H and define to be 7. It is easily verified that f i s an extension o f f and thereby the diagram is commutative. To show that y i s continuous, we shall prove for every G in G that f='[Y\ c l ~ ( p ~ [ G is ] ) open ] in 2.Suppose that [ is a member

Y([)

VI. COMPACTIFICATIONS

262

\ clg(yy[G])] and write 7 = ?(<). Then 7 4 c l ~ ( y y [ G ] ) . It follows that G is not in E, the maximal dual ideal of G with index 7. By Lemma 1.6, we have G n H = 0 for some H in E,. The base normality of G yields a screening ( J , I<) of (G, H ) . Let us show that the open set U = 2 \ c l ~ ( f - ~ [ J in ] ) 2 is a neighborhood of 5 with ?[U] c \ cly((~y[G]).As H fl J = 8, the sets clg(H) and clg(J) are disjoint. So 7 4 clg(f-l[J]) follows, whence [ E U . For [' in U let 7' = ? ( E l ) . As (' 4 c l ~ ( f - ' [ J ] ) holds, we have that E' E c l ~ ( f - * [ K ] )also holds by Lemma 1.7, and consequently f - ' [ K ] is a member of DE,. From the definition of we get K E E,,. Thus we have G 4 D,, or equivalently 7' 4 clg(cpy[G]). The continuity of has been established.

of ?-I[?

r

7

7

We shall now discuss compactifications that preserve both the weight and the dimension. In particular, dimension preserving metrizable compactifications of separable metrizable spaces, which were used in the proofs of Theorems 1.5.11 and 111.4.3, will be obtained. The following is a preparatory lemma. Let us recall that the weight w ( X ) of the space X is the minimal cardinality of a base for the open sets of X . 2.6. Lemma. Suppose that X is a regular space with infinite weight. Then there is a disjunctive base F for the closed sets such that 1 7 1= w ( X ) .

Proof. Let { 17,: cy E A } be a base for the open sets with \A1 equal to w ( X ) . Then F = { cl(U,) : a E A } U { X \ U , : cy E A } is a disjunctive base for the closed sets with (31 = w(X). Observe that the above disjunctive base F is not a ring. We shall often be faced with such a situation. To rectify this we shall employ certain operations to form rings from F. The following notations will be adopted for these operations.

Notations. Let X be a space and 7 be a nonempty collection of subsets of X . Then the operations A , V, T , c and I on the collection F are defined as follows. (i) F A is the collection of all sets that are intersections of finite subsets of F. (ij) F" is the collection of all sets that are unions of finite subsets of 3.

2. DIMENSION PRESERVING COMPACTIFICATIONS

263

(iij) F T= F"A = FA". (iv) F c = { X \ F : F E F}. (v) ' F = { c l ( F ) : F E F'"}. Observe that if

IF1 2

No, then

IF1 = IF'l

= IFc/.

The first construction will be for the large inductive dimension in the universe of hereditarily normal spaces. A bonus will be the simultaneous dimension preserving compactification for a large prescribed collection of subsets of a space. 2.7. Theorem. Let X be a hereditarily normal space such that I n d X < 00. Suppose that E is a collection of closed subsets of X with [El _< w ( X ) . Then there exists a Hausdorff compactification Y of X with w ( X ) = w ( Y ) and such that Ind Y 5 Ind X and Ind cly(E) Ind E for each E in E .

<

Note that the usual Wallman compactification requirement w ( X ) = w ( Y ) is dropped.

W X will

do f the

Proof. We may assume that w ( X ) is infinite and X E E . Our task is to construct an appropriate normal base F. Observc that any base for the closed sets that contains a disjunctive one is also disjunctive. In light of Lemma 2.6 there is a disjuntive base FO for the closed sets such that E c Fo and IF01 = w ( X ) . Let us inductively define a sequence of collections Fi,i = 0 , 1 , . . . . Assume for 0 5 i that Fi has been defined. With the aid of Proposition 1.4.6, for each nonempty set F in Fi and for each pair ( A , B ) of disjoint members of F;we choose a partition S between A and B such that Ind (S n F ) 5 Ind F - 1. For this partition S let U and V be disjoint open sets with A C U and B C V such that X \ S = U U V . Then, with C = S U U and D = S U V , the pair (C, 0 )is a screening of ( A , B ) for which Ind (C n D n F ) Ind F - 1. Let Si be the collection of all sets C and D that are obtained in this way and define Fi+l = (Fi)' U Si. As E c FO C Fi c F;+lfor all i, the collection F = U{ Fi : i = 0,1,2,. . .} is a normal base with IF1 = w(X). Let Y be the Wallman compactification o ( F , X )with repect t o F. To complete the proof we shall show that Ind cly(F) 5 Ind F for each F in F.The proof is by induction on Ind F. Only the inductive step will be given. Let Ind F _< k. Suppose that G and H are disjoint closed subsets of Y . Because Y is compact and { cly(F) : F E F } is a base for the closed sets of Y , there are disjoint members A

<

VI. COMPACTIFICATIONS

264

and B of 3 such that G c cly(A) and H C cly(B). Let (C, 0)be a screening of (A, B ) with C and D in 3 and Ind ( F fl C fl 0)5 Ic - 1. Then cly(C) n cly(D) is a partition between G and H and cly(C) n clY(D) n cly(F) = cly(C n D n F ) . The inequality Ind (cly(C) n cly(D) n cly(F)) 5 k - 1 holds by the induction hypothesis. Thus we have shown Ind cly(F) 5 k.

For the covering dimension we shall use the following lemma which converts the condition dim X 5 n into a condition expressed in terms of closed sets. The straightforward proof which is based on Proposition 1.8.6 and the De Morgan formulas will be left to the reader. 2.8. Lemma. Let X be a normal space and let n be in N. Then d i m X 5 n if and only if for each finite collection F of closed sets such that F = 0 there is a finite collection G of closed sets such that (a) for each G in G there is an F in F such that F C G, G = 0, (b) (c) each subcollection of G with more than n t 1 distinct members is a cover of X.

n

Theorem. Let X be a normal space with d i m X < 00. Then there exists a Hausdorff compactification Y of X such that w ( Y ) = w ( X ) and dimY 5 d i m X . 2.9.

Proof. Let n be a natural number such that d i m X 5 n. We may assume that w ( X ) is infinite. As in the proof of the previous theorem we can find a disjunctive base FO for the closed sets such that 1301 = w(X) and X E Fo. We shall inductively define a sequence of closed collections Fi)i = 0,1,2,. . . . Assume that 3 i has been defined. For each finite collection F of elements of 3 i with F = 8 we select exactly one collection G of closed sets satisfying (a), (b) and (c) of Lemma 2.8. Let Xi be the union of all such collections that are obtained in this way. For each pair ( A ,B ) of disjoint members of 3 i we choose a screening (C, 0 )of ( A , B ) and use the resulting collection Si of all sets C and D chosen in this way for the screening collection. We let 3 i + l = ( 3 i ) ' U Si U Xi and the induction is completed. The collection F = U{ Fi : i = 0,1,2,. . . } is a normal base with 1 3 1= w(X). Let Y = w ( F ,X). Obviously w(Y)is

n

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265

equal to w(X). That dim Y 5 n is verified with the aid of Lemma 2.8 in the following way. Suppose that F' is a finite family of closed subsets of Y with (-) F' = 0. In view of the compactness of Y we may assume that F' is a subcollection of the base { cl,x(F) : F E F}. That is, F' = { cl,X(Fj) : j = 0 , 1 , . . . ,k } . It follows that there is a n i such that F = { Fj : j = 0 , 1 , . . .,Ic } c Fi. Then there is a subcollection G of 'FIi that corresponds t o the collection F. Finally we have that the collection G' = { cl,x(G) : G E G } satisfies (a), (b) and (c) of Lemma 2.8. It follows that dimY n.

<

There are many possible variations and combinations of the above compactification theorems. We shall present two, one for inductive dimension and the other for covering dimension. 2.10. Theorem. Suppose that X is a hereditarily normal space. Let CP be a collection of continuous maps o f X into a compact space Y with 1CP1 < w ( X ) and w ( Y ) 5 w ( X ) . Then there is a Hausdorff compactification 2 of X with w ( 2 ) = w ( X ) such that Ind 2 < Ind X

holds and each cp in @ can be extended to a continuous map (p of 2 into Y .

Proof. We may assume w ( X ) 2 No and I n d X < 00. Let G' be a normal base for the closed sets of Y with 161 = w(Y).As Y is a compact space, we have Y = w ( ( 7 , Y ) . We shall use the normal base F constructed in the proof of Theorem 2.7 with E being the collection { p-'[G] : G E G', cp E CP } whose cardinal number is at most w ( X ) . Then we may apply Theorem 2.5 to each cp in CP. The theorem will follow easily. 2.11. Theorem. Suppose that X is a normal space. Let CP be a collection of homeomorphisms of X into X with 5 w ( X ) . Then there is a Hausdorff compactification 2 of X with w ( 2 ) = w(X) such that dim 2 5 dim-X holds - and each cp in CP can be extended to a homeomorphism Cp: X + X .

Proof. We may assume that w(X)2 No and d i m X < 00. We may further assume that 9-l E @ whenever cp E @. The proof of Theorem 2.9 is modified in the following way. Let FO be chosen as in the proof of Theorem 2.9. For i 2 0 let

J, = { cp-'[F] : F E F;,cp E @ } , Fi+i+1= (qT u s; u 'FI; u J-;,

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where the collections Si and Hi are defined in the same manner as in the proof of Theorem 2.9. Then 2 = w ( F , X ) satisfies the weight and dimension conditions of the theorem. The extensions of the homeomorphisms follow from Theorem 2.5. One begins to recognize common features in the above constructions of normal bases. The collection (Fi)T is used to induce the ring structure and the collection Si is used t o induce the base-normal structure. We shall call Si the screening collection in all subsequent constructions of this type. 3. The Freudenthal compactification

Is there an ideal compactification for a topological space? In the forties many people, most notably Freudenthal in [1942], [1951] and [1952], tried t o provide a definitive answer t o this question. For locally compact spaces the one-point compactification has many natural features, but is it natural? For example, which is the natural compactification for the complex plane, the Riemann sphere or the Poincark disc for the hyperbolic geometry? Such questions are difficult t o answer, but through the years the ideas have converged t o the following definition given by de Groot in [1942].

A compactification 5 of a space X is called ideal if (i) the dimension of \ X is less than or equal t o zero, where (ij)

the dimension function is still t o be specified, to the property in (i).

2 is maximal with respect

With the usual dimension functions in mind, in view of (ij), the twopoint compactification [ - 1 , 1 ] of the open interval ( - 1 , l ) is ideal and the one-point compactification is not. But, in view of (i), the Riemann sphere is the ideal compactification of the complex plane and the Poincark disc is not. In [1942] and [1951] Freudenthal constructed ideal compactifications for rim-compact spaces. In this section we shall present the theory of Freudenthal compactifications and its ramifications. The fundamental result of de Groot that originated the theory in this book will be included in the discussion. At the end of the section we shall prove the theorem of Zippin that inspired some of de Groot's work, notably his [1942] thesis.

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Recall that a set F in a space X is said to be regularly closed if it is the closure of its interior. Similarly, a set U is said to be regularly open if it is the interior of its closure. The complement of a regularly closed set is regularly open and conversely. Also the closure of an open set is regularly closed. The union of two regularly closed sets is again regularly closed but the intersection need not be. Throughout this section the symbol 6 will be reserved for the collection of all regularly closed sets with compact boundaries. We shall use the notations adopted in the previous section. The collection of regularly open sets with compact boundaries is clearly the collection 6" = { X \ G : G E G}. Observe that 6 = G1. The collection G" is the collection of all sets that are finite intersections of members of 6 . As 6 = G", we have 6' = 6". If X is a rim-compact space, then the ring G" will be a base for the closed sets of X. 3.1. Lemma. Let X be a rim-compact space. Then 6" is a normal base. Moreover, i f D is a subset o f X with compact boundary and i f V is an open neighborhood o f D ,then there is a member G o f 6 such that D n G = 8 and X \ V c G.

Proof. It has already been observed that G" is a ring. To see that G" is disjunctive, we observe that each point of X has arbitrarily small open neighborhoods V with compact boundaries. Since X is a regular space and since cl ( V )is a regularly closed set it follows that each point has arbitraily small regularly closed neighborhoods with compact boundaries. To complete the proof of the first statement of the theorem we observe that the base-normal property will follow from the second statement because cl (X \ G) E 6 whenever G E 6 . To prove the second statement consider the open cover of B (D) consisting of all open sets U with cl ( U ) C V such that B ( U ) is compact. As B (D) is compact, there is a finite subcover U of this cover. It is easily seen that W = D U U U is an open neighborhood of D such that d ( W ) E 6 and cl(W) C V . The regularly closed set G = cl ( X \ cl ( W ) )satisfies the required property. 3.2. Definition. The Freudenthal compactification F X of a rimcompact space X is the Wallman compactification of X with respect to the base G", that is, F X = w ( G " , X ) . The base 6" will be called

the Freudenthal base. The following theorem contains the key properties of the Freudenthal compactification.

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3.3. Theorem. Suppose that X is a rim-compact space. Let U and V be regularly open sets with compact boundaries. Then (a) e x f x ( u u V) = e x r x ( U ) u e x f x ( V ) , (b) B f x ( e x f x ( U ) ) = B x ( U ) .

Proof. It is to be observed that the complements of regularly open subsets U and V of X with compact boundaries are members of the Freudenthal base G". Using the lattice G", we can copy the proof of Lemma 1.16 t o get a proof of (a) and (b), where the second formula will require the additional fact that B x ( U ) is compact. Our first task will be t o show that the Freudenthal compactification of a rim-compact space is an ideal compactification. To this end, we shall define yet another dimensional property. In contrast with the usual dimension functions this property will not be an absolute property but will be related t o ambient spaces. 3.4. Definition. A subset S of a space Y is zero-dimensionally embedded in Y if Y has a base B for the open sets such that B y ( U ) is disjoint from S for each U in B.

Observe that ind S 5 0 holds whenever S is zero-dimensionally embedded in Y . But the converse is not true as the following example will show. 3.5. Example. In Example 111.1.2 we have exhibited a metrizable space X with ind X = 1 and ind (X \ { p } ) = 0 for some point p in X . Clearly all sufficiently small neighborhoods U of p have nonempty boundaries Bx(U). It follows that X \ { p } is not zerodimensionally embedded in X . There is also a compact space with a zero-dimensional subset that is not zero-dimensionally embedded in the compact space. Consider the compact space PA where A is Roy's example. Each point 2 of the subspace A has arbitrarily small open-and-closed neighborhoods U . For each such U we have that clpa(U) is an open-andclosed neighborhood of 2 in PA. Thus each point 2 in A has a base for its neighborhoods in PA such that each member of the base has empty boundary, whence disjoint from A. This property is somewhat stronger than the property of ind A = 0. We shall show the existence of a point p in PA \ A such that ind (A U { p } ) = 1. It will then follow that A is not zero-dimensionally embedded in PA. Let A and B

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be disjoint closed subsets of A for which the empty set is not a partition between A and B in A. Observe that c l p ~ ( An) d p A ( B ) = 8. There must exist a point p in c l p ~ ( A such ) that Bpa(U) n A # 8 for each of its neighborhoods U with cl p A ( U ) fl clpA(B) = 8. If this were not the case, then from the compactness of c l p ~ ( A we ) would have the contradiction that the empty set is a partition between A and B in A. There is an intimate relation between zero-dimensionally embedded subsets of a compact space and rim-compact spaces. 3.6. Theorem. A space X is rim-compact if and only if for some Hausdorff compactification Y of X the set Y \ X is zero-dimension-

ally embedded in Y . Proof. Let Y be a compactification of X such that Y \ X is zerodimensionally embedded in Y . Choose a base B for the open sets of Y such that By(U) is disjoint from Y \ X for each U in B. The trace of B on X witnesses the fact that X is rim-compact. For the converse let X be rim-compact and consider its Freudenthal compactification f X . We infer from Lemma 3.1 that the collection { e x f x ( U ) : U E 6 " )is a base for the open sets of F X . By Theorem 3.3 we have B f x ( e x f x ( U ) ) = B x ( U ) c X for each U in 6". This proves that f X \ X is zero-dimensionally embedded in f X . Now we can show that the Freudenthal compactification is an ideal compactification. The following theorem will also provide a characterization of the Freudent ha1 compactification. 3.7. Theorem. The Freudenthal compactification F X of a rimcompact space X has the following properties: (a) f X \ X is zero-dimensionally embedded in F X . (b) f X is maximal with respect to (a). That is, if yX is a compactification of X such that yX \ X is zero-dimensionally embedded in y X , then the identity map of X 'has a continuous extension from f X to yX.

Proof. As (a) has already been proved in the previous theorem, only (b) requires a proof. Let 'H be the collection of all regularly closed subsets H of yX with B y ~ ( Hc ) X . Since y X \ X is zero-dimensionally embedded

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in yX, the collection 7-1 is a base for the closed sets of y X . Moreover, 3.1' is a ring because 'H is closed under finite unions. We assert that this ring is a normal base for yX. This assertion will follow from the analogue of Lemma 3.1 which is the subject of the next paragraph . Let us prove that if A is a subset of y X such that B,x(A) is compact and if V is any open neighborhood of A , then there is a member H of 'H such that A n H = 8 and y X \ V c H . To this end, let U be the collection consisting of all regularly open sets U of yX such that c l , ~ ( U ) c V and B y x ( U ) c X. Since yX \ X is zero-dimensionally embedded in y X we have that U is a cover of B,x(A). The remainder of the proof is the same as the one for Lemma 3.1. Consider the collection R* = { H n X : H E 'HA}. Because the boundaries of the members of 3-1* are contained in X, we have for HI and H2 in 'H that H I n H 2 n X = 8 if and only if HI n H 2 = 8. It will follow that 'H* is a normal base for X. To complete the proof we shall show that y X is canonically homeomorphic with w(7-1*,X ) . Once this is established, the existence of a continuous extension of the identity map is a direct consequence of Theorem 2.5. We define a map $ : yX i u('H*,X ) in the following way. The notation of Theorems 1.10 and 1.14 will be used. Let [ be in yX. Define the collection D by D = { H n X : H E 'H, [ E H } . Because 'H* is disjunctive, by Lemma 1.6 the collection D is a maximal dual ideal of 3-1'. Thus D = D, for some unique q in u(7-1*,X). Define 77 = $([). It is easily seen that $ is injective. Let H E R*. The basic closed set in w ( ' H * , X ) corresponding t o H is B H n X . It is not difficult t o verify

It follows that $ is a homeomorphism. Theorem 3.7 characterizes the Freudenthal compactification by its maximality. We shall briefly discuss another characterization that uses perfect compactifications. 3.8. Definition. A compactification Y of a space X is called perfect if for each point y of Y \ X and for each neighborhood U of y

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in Y the set U n X is not the disjoint union of two open subsets V and W of U n X such that y E cly(V) n cly(W). It is well known that the Cech-Stone compactification ,OX is a perfect compactification of X . There is the following characterization of F X . 3.9. Theorem. Suppose that y X is a compactification of a space X such that yX \ X is zero-dimensionally embedded in yX. Then yX = F X if and only if yX is a perfect compactification of X .

Proof. Let us show that F X is perfect. Assume for some point y of F X \ X that there is a n open neighborhood U in F X and there are disjoint open subsets V and W of X such that U n X = V U W and y is a member of cl F x ( V ) n c l f x ( W ) . One can easily show that each open neighborhood of y contained in U also has the same property. So we may assume that U is the open set F X \ c l f x ( G ) for some G in G. As V and W are disjoint open sets of X whose union is X \ G , we have that clX(V) and clx(W) are in (7. We infer from G E (7 that clX(V) and clx(W) are disjoint. Theorem 2.2 implies cl f x ( V ) fl c I F x ( W ) = 0.This leads to the contradiction y E 0. Conversely, suppose that yX is a perfect compactification of X . As F X is maximal, there is a continuous map c p : F X -+ y X which extends the identity map. Let us show that cp is injective. Assume for some point z in yX that the set cp-'(z) consists of more than one point. As F X \ X is zero-dimensionally embedded in F X , we have Indcp-'(z) = 0. So the set cp-'(z) is the disjoint union of two nonempty closed subsets, say A and B . Let V and W be disjoint open sets such that A c V and B c W . The set cp[V U W ] is a neighborhood of t, and for every open neighborhood U of z with U C cp[V U W ] the set U n X admits a splitting into the disjoint open sets U n X n V and U n X n W . But this is a contradiction to z E cl,x(U n X n V )n cl,x(U n X n W ) ,a contradiction. In the introductory discussion of ideal compactifications we did not specify the dimension function. Let us consider ind. Related t o Example 3.5 is the following quite natural question. Suppose that a space X has a compactification 2 such that ind ( 2\ X ) 5 0. Under what condition does this imply that X is rim-compact? Theorem 3.12 below will provide a partial answer t o this question.

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3.10. Definition. A completely regular space X is said to be Lindelof at infinityif each compact subset F of X is contained in a compact subset I< with a countable base for its neighborhood, that is, there is a countable family of open sets containing K such that each open set containing K necessarily contains an open set from the countable family.

The next lemma will explain the name Lindelof at infinity. 3.11. Lemma. Let X be a completely regular space. Then X is Lindelof a t infinity if and only if the subspace Y \ X of Y is Lindelof for every compactification Y of X .

Proof. Assume that X is Lindelof at infinity. For a compactification Y of X let V b e an open cover of Y \ X . The set F = Y \ U V i s a compact subset of X . Let K be a compact subset of X containing F for which a countable base U = { U; : i E N} for the neighborhoods in X of I< exists. For each i the set cly(X \ U ; ) is a compact subset of (J V and therefore is covered by finitely many members of V. Because U is a neighborhood base for K , the set Y \ X is covered by { cly(X \ U i ) : i E N}. It will follow that Y \ X is a Lindelof space. For the converse suppose that X has a compactification Y such that Y \ X is a Lindelof space. Let F be a compact subset of X. Consider the open cover V o f Y \ X consisting of cozero-sets V of Y with V n F = 8. As Y \ X is Lindelof, there is a countable subcover V‘ = { Vj : j E N} of V. Let K be the compact set Y \ U V’. Clearly, F C Ii’ = X \ U V’. Let us exhibit a countable base for the open neighborhoods of Ii’. As Vj is a cozero-set, there exists a sequence v j k , k = 0,1,2,. . . , of open subsets of Y with cly(Vjk) C Vj for every k and Vj = U{ vjk : k E N}. For each i define Ui to be the open set Y \ U{ Cly(vjk) : j 5 i, k 5 i}. Clearly U‘ = { U; : i E N} is a collection of neighborhoods of A’ whose intersection is K . From the fact that { Vj, : j E N,k E N} is an open cover of Y \ K we infer that U = { U; n X : i E N} is the required base for the neighborhoods in X of the compact set A’. Thereby the converse is proved. Observe that Y \ K is a-compact. In addition to justifying the name Lindelof at infinity, the above characterization will be useful in the proof of the next theorem which characterizes rim-compactness for these spaces.

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3.12. Theorem. Suppose that the space X is Lindelof a t infinity. Then X is rim-compact if and only if there is a compactification Y such that ind (Y \ X ) 5 0.

Proof. We have already shown that ind ( f X \ X ) 5 0 when X is rim-compact . Let Y be a compactification of X with ind (Y \ X ) = 0. We have that Y \ X is a Lindelof space from Lemma 3.11. In view of Theorem 3.6, it is sufficient to prove that for each point p of Y and for each neighborhood U of p there is an open set V such that p E V c U and By(V) n (Y \ X ) = 0. To this end, it would be useful to have Ind (Y \ X ) = 0. This will be so by the assertion in the next paragraph. We assert that the equivalence ind Z = 0 if and only if Ind 2 = 0 holds for Lindelof spaces 2. We only need t o prove that ind Z = 0 implies Ind Z = 0 when Z is a Lindelof space. Let F and G b e disjoint closed sets of 2. Consider the open cover of Z consisting of all open-and-closed sets U such that U n F = 0 or U n G = 0. Such a cover exists because i n d Z = 0. This cover has a countable subcover { Ui : i E N}. Let Vo = Uo and let V, = U , \ U{ Ui : i < n } when n 2 1. The collection V = { V, : n E N } is a cover of 2 consisting of mutually disjoint open-and-closed sets. It is easy t o see that W = U{ V, : V, E V, V, n F # 0 } is an open-and-closed set with F c W and W n G = 0. Returning to the proof of the theorem, we have Ind (Y \ X ) = 0. Let p and U be given. The space 2 = (Y \ X )U { p } is Lindelof. By Theorem 1.4.12 we have Ind 2 = 0. It follows that Z is a disjoint union of open-and-closed subsets A and B of Z such that p E A C U . Let F = cly(A) n cly(B). Since 2 is a Lindelof space we have by Lemma 3.11 that Y \ 2 is Lindelof at infinity. As F C Y \ 2 , there is a compact subset K of Y \ 2 such that F c Ir' holds and Ir' has a countable base for its neighborhoods in Y \ 2. Moreover, Y \ Itmay be assumed t o be g-compact, whence Lindelof and therefore normal. As the set cly(A) \ Ir' and cly(B) \ Ir' are disjoint in Y \ Ir', they have disjoint open neighborhoods V' and W' in Y respectively. Let V be the set V' n U . It is easy to see that A C V and B c W'. Consequently, By(V) n (Y \ X ) = 0 and the proof is now completed. In Section 1.5 we announced the result that a separable metrizable space X is rim-compact if and only if there is a metrizable compactifi-

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cation Y of X with ind (Y \ X ) 5 0. The if part of the statement follows from Theorem 1.5.8 or Theorem 3.12. But the converse will not follow from the Freudenthal compactification because the Freudenthal compactification need not be weight preserving. In particular, the space f X need not be metrizable for a separable metrizable space X. For example, the Freudenthal compactification f N for the space N coincides with ,ON and N o = w(N) < w(,ON) = c. A necessary condition for the Freudenthal compactification t o be weight preserving will be presented. But at this point we shall make a brief digression to prove the following weight preserving modification of the Freudenthal compactification. The theorem includes de Groot's result as a special case (see Theorem 1.5.7). This is the first of two proofs of de Groot's theorem; the second one is found in Subsection 3.20. 3.13. Theorem. Suppose that X is a rim-compact space. Then there is a compactification Y of X such that Y and X have the same weight and Y \ X is zero-dimensionally embedded in Y .

Proof. The proof is a modification of some of the previous ones. We may assume that w ( X ) is infinite. Let B = { U, : cx E A } be a base for the open sets of X with IAl = w ( X ) and consider all pairs (U,, U p ) of members of f3 with clx(U,) c Up. When (U,, Up) is such that there exists a regularly open set G with compact boundary and clx(U,) C G C clX(G) C Up, choose one such G and denote it by G,,p. Otherwise, choose Ga,p = 8. The collection FO = { Ga,p : Q E A , ,O E A } is a disjunctive base with cardinality w(X) that is contained in the Freudenthal base G", We may assume that X is in F,.We shall define a sequence Fi,i = 0 , 1 , 2 , . . . , of subcollections of the Freudenthal base G" with cardinality w ( X ) . Assume that Fi has already been defined. With the aid of Lemma 3.1 we form a screening collection by selecting for each pair ( A , B ) of disjoint members of F;a screening (C, 0 )of ( A , B ) by regularly closed sets C and D with compact boundaries. As usual, Si is the collection of all such sets C and D that were selected in this way. Define the collection Fi+1 to be the subcollection (Fi)T U (.Fi)*U Si of G". Clearly, IFi+ll = w ( X ) . The collection F = U{ Fi : i = 0 , 1 , 2 , . . . } is a normal base with IF1 = w(X)and F C G". Let Y be the Wallman compactification 43,X ) of X with respect to F. The verification of the fact that Y \ X is zero-dimensionally em-

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bedded will be done in the exact same way as in Theorem 3.6. We infer from the way the collections S, were constructed that the collection { exy(X \ G) : G E (G n F)} is a base for the open sets of Y. Moreover, if U denotes any e x y ( X \ G ) in this base, then we also have By(U) = B x ( X \ G). Consequently, Y \ X is zerodimensionally embedded in Y. Let us return to the discussion of weight preserving Freudenthal compactifications. Whether or not the Freudenthal compactification of a space is weight preserving depends on the collection of open-andclosed sets of the space. To make this more precise we shall briefly discuss the notion of the quasi-component space. Suppose that X is a space. The quasi-component of a point z of X is the intersection of all open-and-closed sets that contain 5. The collection of quasicomponents of a space is pairwise disjoint, and each quasi-component is closed. The quasi-component space Q ( X ) is the set of all quasicomponents of X endowed with the topology generated by the openand-closed subsets of X. This topology need not be the quotient topology. But it will coincide with the quotient topology when X is compact. The projection map AX will induce a bijection between the respective collections of open-and-closed sets of X and Q(X). It is not difficult to show that &(X)is a totally disconnected Hausdorff space. The quasi-component space can be a rather powerful tool, especially when it is compact. Some properties are collected in the following theorem. 3.14. Theorem. Let X be a space with infinite weight and compact quasi-component space Q ( X ) . Then w(Q(X)) 5 w ( X ) and the cardinality of the family of open-and-closed subsets of X is less than or equal to w ( X ) . Moreover, if Y is a closed subset of X such that Bx(Y) is compact, then Q(Y) is also compact.

Proof. Let B = { U , : a E A } be a base for the open sets such that / A [= w(X). If for a pair ( U a , U p ) of disjoint elements of B there exists an open-and-closed set H such that U , C H C X \ clx(Up), we choose one such set and denote it by H,,p. If no such set exists, we choose H,,p t o be 8. The collection Q of finite intersections and finite unions of all finite subfamilies of { 7rx[Hol,p]: a,P E A } , where AX is the projection map, is a base for the topology of Q ( X ) because of the compactness of Q ( X ) . Consequently we have w(Q(X)) 5 w(X). Let W be an open-and-closed subset of Q(X). Then W is compact,

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and therefore W can be written as a finite union of basic open sets from &. Thus the cardinality of the family of all open-and-closed subsets of Q ( X ) is finite or equals w(&(X)) and the same holds for the cardinality of the family of all open-and-closed subsets of X. For the proof of the last statement of the theorem, suppose that Y is a closed subset of X such that B x ( Y ) is compact. Let us show that if R = { R, : cr E A } is a collection of closed subsets of &(Y) with the finite intersection property, then R is not empty. Denote the projection map of Y onto & ( Y )by K Y . From the compactness of B x ( Y ) it will follow that nR # 8 when ny'[R,] f l B x ( Y ) # 8 for every a. So suppose that 7rY1[Rp]n B x ( Y ) = 8 for some P in A . It is easily verified that if z is in Rp, then 7r;'[z] is not only a quasicomponent of Y but also a quasi-component of X . Consequently the set n;'[R, n Rp] is a closed subset of X that is the union of quasi-components of X for each a. By the compactness of & ( X )the intersection of { T X [ ~ ; l [ R n p R,]] : cr E A } is nonempty. Consequently R has a nonempty intersection. Thereby &(Y) is compact. It will turn out that the Freudenthal compactification is weight preserving if the quasi-component space is compact. 3.15. Theorem. Let X be a rim-compact space such that Q ( X ) is compact. Then w ( f X ) = w ( X ) .

Proof. We may assume that w(X) is infinite. Choose a base f? = { U , : cr E A } for the open sets such that IAl = w ( X ) . We may assume that f? is closed under finite unions. If for the pair (U,,Up) of disjoint elements of B there exists a regularly closed set H with compact boundary such that U, c H c X \ U p , we select one such set and denote it by H,,p. If no such set exists, we select H,,p t o be 0. Define

Let H E F0. The quasi-component space & ( H ) is compact by Theorem 3.14. By the same theorem the cardinality of the family of all open-and-closed subsets of H does not exceed w ( H ) . We define F1 t o be the collection of all open-and-closed subsets of elements of Fo. Let F2 = Fl",the collection of all finite unions of elements of Fi. Obviously, IF.15 w(X). Recall that 6 denotes the collection of all regularly closed subsets of X and note that F2 C G.

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Let us first prove the assertion: For any two disjoint elements F and G of G there exists an H in 3 2 such that F C H C X \ G. For a proof we first note that by Lemma 3.1 one can select from the collection G a pair of disjoint closed neighborhoods F' and G' of F and G respectively. As B x ( F ) is compact, there is an U in B such that B x ( F ) c U c F'. Similarly there is a V in f? such that B x ( G ) c V C G'. So there is a J in 30with U C J C X \ V . Letting K = clx(X \ J ) , we have V C IC C X \ U . As B x ( G ) C V and V n J = 0, both J n G and J \ G are open-and-closed subsets of J . Similarly K n F and K \ F are open-and-closed subsets of I(. It follows that J n G, J \ G, IC n F and K \ F are members of 3 1 . Define H = (J \ G) U (K n F ) . Then H E 3 2 and F c H c X \ G. Thereby the assertion is proved. Since G" is a normal base we can easily derive from the assertion that the collection 3 2 " is a normal base. By Theorem 2.5 there is a continuous map 'p: F X i w(.F2",X) that is the extension of the identity mapping of X. To complete the proof of the theorem we only have t o show that 'p is injective. Let [ and 77 be distinct points of FX and denote their respective maximal dual ideals of 6" by DE and D,. There are disjoint sets F and G in G such that F E DE and G E D,. By the above assertion there are disjoint sets J and K in 3 2 such that F C J and G C K. It will follow that ~ ( 5E)c ~ ~ ( F ~ A and , xy)((7J)E) clW(FzA,X)(IC).As the intersection d w ( F Z A , X ) ( J ) n Clu(F2A,X)(IC)is empty, 9 is injective. 3.16. Corollary. Let X be a rim-compact, separable metrizable space. Then F X is metrizable if and only if Q ( X ) is compact.

Proof. When Q ( X ) is compact, we have w ( f X ) = w ( X ) 5 No. Consequently f X is metrizable. Before proceding t o the converse we shall have need of the connection between the quasi-components of X and the components of F X . As F X is compact, we have that the components and quasicomponents agree in F X . So each quasi-component of X will yield a component of FX. Conversely, each component of f X which meets X yields a quasi-component of X . Now suppose that f X is metrizable. Denote the quotient map of F X onto Q ( F X ) by 7r and the natural embedding of X into FX by cp. The composition 7rv can be factored through the space Q ( X ) . Let g be the factor map from Q ( X ) into Q ( F X ) . Then g is injective

VI. COMPACTIFICATIONS

278

and the topology of Q ( X ) makes g into an embedding. As F X has been assumed to be metrizable, we have that its image under the surjective map ?r is also metrizable. Suppose that the factor map g is not surjective. Then there exists a point q in Q ( F X )\ Q ( X ) . As Q ( X ) is dense in Q ( F X ) and Q ( F X ) is a compact zero-dimensional space, there is a sequence U i , i E N, of open-and-closed sets of Q ( F X ) with Ui+l C Ui such that V , = Ui \ Ui+l is not empty for each i and V , : i E N} = { q } . Since Q ( F X ) is compact, we have that U{ V , : i E N } is open-and-closed in the subspace Q ( F X )\ { q } for each subset N of N. So x i 1 [g-'[U{ V , : i E N}]], N E N,is an uncountable collection of open-and-closed subsets of X , where KX is the projection of X onto F ( X ) . Thus a contradiction to Theorem 3.14 has appeared. Thereby we have that g is surjective.

n{

The extension of the identity map to the Freudenthal compactification played a role in Theorem 3.7. More general situations are considered in the following two theorems. It will turn out that the class of rim-compact spaces with closed continuous maps whose point inverses have compact boundaries forms a category. The Freudenthal compactification then becomes a reflection of this category to the category of compact Hausdorff spaces with continuous maps.

3.17. Theorem. Let X and Y be rim-compact spaces. Suppose that f : X -+ Y is a closed, continuous map such that point inverses have compact boundaries. Then there is a continuous extension f f : F X + FY for the map f .

Proof. Let us show that if G is a closed subset of Y with compact boundary, then f-l[G] has a compact boundary. For notational convenience let H be the compact set Bx(f-'[G]) and g be the restriction off to H . Note that g is closed and continuous. It is easily verified that g [ H ] C By(G) and g-'[y] C Bx(f-l[y]) n H for each y in By(G). Suppose that U is an open cover of H . Let y E By(G). The set g-'[y] is covered by a finite subfamily of U. Denote by W, the union of this finite subfamily and let V, = Y \ f [ X \ W,]. As f is closed, V, is a neighborhood of y. And as g[H] is compact, it is covered by finitely many V,. Consequently U has a finite subcover. The theorem will now follow from Theorem 2.5. Theorem. Suppose that X and Y are metrizable rimcompact spaces. I f f : X -+ Y is a closed continuous map, then f has a continuous extension F f : F X FY. 3.18.

---$

3. THE FREUDENTHAL COMPACTIFICATION

279

Proof. It is sufficient t o show that point inverses have compact boundaries. The fact that closed maps of metrizable spaces have this property has been observed in Lemma IV.4.13. In our discussion of complete metric extensions, we have shown in Theorem V.2.28 that for a metrizable space X there are metrizable extensions 2 that preserve dimension and at the same time satisfy the requirement that Ind (2\ X) be minimal. More specifically, for every metrizable space X there is a completion 2 such that Ind 2 = Ind X and Ind (2\ X) = Icd X . The situation for metrizable compactifications is somewhat different as the following example will show. 3.19. Example. We shall present a rim-compact subspace X of 8" with n 2 2 such that (a) i n d X = n - 1, (b) i n d 2 = n for every metrizable compactification 2 of X for which ind (2\ X ) 5 0. Let us make a preliminary observation. If a subset S of 8" is a partition between two points 5 and y, then z has arbitrarily small neighborhoods whose boundaries are homeomorphic with S . Such neighborhoods can be obtained by removing the point y and shrinking 8" \ { y }. As ind 8" = n, it will follow that no (n-2)-dimensional set can be a partition between any two distinct points. Let X = 8" \ Q where Q is a countable dense subset of 8", and note that X is complete. We have i n d X > n - 1 by the addition theorem. Though we have not proved it, i n d X 5 n - 1 also holds (see, for example, Hurewicz and Wallman [1941], pege 44). Let 2 be any metrizable compactification of X with ind (X \ X ) 0. We shall show ind 2 = n. To the contrary, let us assume ind 2 5 n - 1. Let 5 and y be distinct points of X. As X is complete, 2 \ X is an F,-set of 2. By Lemma 1.4.6 there is a partition S between z and y such that ind S n - 2 and ind ( S fl(2\ X)) -1, whence S c X . Let X \ S = U U V where U and V are open sets of X with z E U and y E V such that U n V = 8. By the sum theorem, ind (S U Q) n - 2. The interior of S U Q in 8" is empty. So 2 = U U V is a dense subspace of 8". Applying Lemma V.2.10 t o this subspace 2 , we have e x g ( U ) n exsn(V) = 8. Consequently S' = 8" \ (exSn( U ) U e x p (V)) is a subset of S U Q. That is, S' is a

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<

<

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VI. COMPACTIFICATIONS

280

<

partition between x and y in 8" with ind 5'' n - 2, a contradiction. There is another proof of ind 2 = n. The preliminary observation will lead to the fact that 8" is a perfect compactification of X . So by Theorem 3 . 9 , s " = F X . From Theorem 3.7 we have a continuous with dimf = 0. Theorem IV.4.5 and the coinmap f : 8" + cidence theorem yield n = dim $" Ind 2 = ind 2 . The addition theorem gives ind 2 ind X 1 = n.

<

<

+

We shall conclude the section by presenting a construction of a metrizable compactification 2 of a separable metrizable rimcompact space X such that ind (2\ X ) 0. The construction has the additional feature that 2 \ X is countable whenever X is complete.

<

3.20. T h e construction. Let X be a separable metrizable rimcompact space and let d be a metric on X . When X is complete, the metric d will be chosen to be complete as well. Let us construct a collection F that is the union of countably many collections Fi,i E N,where Fi satisfies the conditions (1) through (6) below for every positive i. (1) Each member F of Fi is a regularly closed set with compact boundary. (2) Each pair of sets F and G in Fi satisfies the requirement

F n G = Bx(F) n Bx(G). (3) Each collection .F; is a locally finite, countable, closed cover of (4) The collection { F : F n C # 8, F E Fj } is finite for every compact set C. (5) mesh 3;5 2 - i . (6) Fi is a refinement of Fi-1. Define Fo = { X } . The collection F1 is obtained in the following way. In view of the separability of X , there exists a countable open cover { u k : k E N} of X consisting of regularly open sets u k with compact boundary and diameter less than 2-'. Define for each j in N the set

x.

Flj = clX(uj \

u{clX(Uk) : k < j } )

and let Fl = { Flj : j E N}. It is easily seen that conditions (1) through (6) are satisfied for i = 1. Suppose that Fo,. . . , Fn have been constructed. Our construction will now be directed toward

3. THE FREUDENTHAL COMPACTIFICATION

281

the subspace Fnj. In the subspace F,j there is a countable, closed collection '?-tj satisfying the conditions (1) through (6), where X and 3; have been replaced by Fnj and I i j respectively, and the condition meshXj 5 2-(;+'). Returning to the space X , we let 3n+1 = U{ 'l-tj : j E N}. Then enumerate 3 n + l as { Fn+l,j : j E N}. It is not difficult to see that conditions (1) through (6) are satisfied for i = 1,. . . ,n f 1. Thereby the collection 3 has been constructed. Inductively define a sequence of bases Ji as follows. Let 30= 3. Assume that J;has been defined. Then let Ji+l = J[ U 3:. Finally we let 3 = U{ 3 i : i E N}. By inducting on i, we can show that each member of Ji has a compact boundary. If A is a member of 3 i and V is a neighborhood of A , then there is an m such that m 2 1 and 2-(m-1) is less than the distance between B ( A ) and X \ V . Using (4),one can easily find a screening of ( A ,X \ V )with elements 3 1= No. Finally of Jm. It follows that J is a normal base with 1 we let 2 = u ( J , X ) . This completes the construction of the special Wallman compactification. This construction was used by de Groot in [1942] for his proofs of Theorem 1.5.7 and Theorem 3.13. The details of this part of these proofs will be left to the reader, but we shall indicate how the construction can be used for a proof of Theorem 1.5.2 which states that a separable metric space that is complete and rim-compact can be compactified by the addition of a t most countably many points.

Recall that we have assumed in the construction that the metric d is complete. We shall show that there are at most countably many maximal dual ideals of 3 that are not fixed. Let H be a maximal dual ideal in J that is not fixed. Define k = sup { i : F;n H # S}. We shall show by way of a contradiction that k is finite. Suppose that k = 00. Then 3;n H # 8 for every i. It follows that H contains arbitrarily small members. As X is complete, the set nH consists of exactly one point of X and H is fixed. As H is not fixed by assumption, we have a contradiction. So, k < 00. In view of (2) there is exactly one member of 3 k in H. Denote this element by G. For each F in Fk+l with F C G we have by Lemma 1.7 that clx(X \ F ) E H because H is maximal and F is not a member of H. It can be seen that there is exactly one point in the set

VI. COMPACTIFICATIONS

282

n{

clz(G) n c l z ( X \ F ) : F E .Fk+l, F c G}. Thus we have established a correspondence between the added points and a collection of finite sequences of natural numbers (namely, H corresponds to the indices of the unique members from .Fi n H for i = 0,1,. . . ,k). Observe that the above construction yields the one-point compactification for the space X formed by removing the end points of the Cantor fan and not f X , the Cantor fan. 4. The inequality K-Ind 2 K-Def

The condition that a subset be zero-dimensionally embedded in a space turned out to be pivotal for rim-compact spaces. This condition can be inductively raised to higher dimensions to yield the notion of a subset being ( 2 n)-Inductionally embedded in a space. There is a strong connection between this notion and K-Ind, where K is the class of compact spaces. Conditions of this type were introduced by de Vries in [1962]. The first result will tie together the condition K-Ind X n and the condition that W X \ X be ( 5 n)-Inductionally embedded in the Wallman compactification W X of X . Connections between the relative property that a set S is (< n)-Inductionally embedded in a space Y and the absolute property I n d S n will be discussed; the picture here is far from complete. We shall end the section by continuing our discussion of the interrelations among the compactness dimension functions by proving the inequality in the title of the section for separable metrizable spaces.

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<

4.1. Definition. Let Z be a subset of a normal space X and let n 2 -1. Then the expression Z is (< n)-Inductionally embedded in X , denoted Ind [ Z ;XI 5 n, is inductively defined as follows.

(i) Ind [ Z ;XI = -1 if and only if 2 = 0. (ij) For n 2 0, Ind [ Z ;XI n if for any two disjoint closed sets F and G of X there is a partition S between F and G in X such that Ind [Sn 2 ; S ] 5 n - 1.

<

It is easily verified that I n d [ Z ; X ] 5 I n d X for every normal space X . It is also not difficult to prove that for a closed subset Z of a normal space X the inequality Ind Z 5 Ind [Z; XI holds. Let us prove the following lemma by induction.

4. T H E INEQUALITY K-Ind

2 K-Def

283

Y is a closed subspace o f a normal space X. Let Z be a subset o f X . Then 4.2. Lemma. Suppose that

Ind [ Z n Y ;Y ] 5 Ind [ Z ;XI. Proof. Let n be Ind [ Z ;XI. The lemma is obvious when n = -1. For the inductive step of the proof we let F and G be two disjoint closed subsets of Y . There is a partition S between F and G in X such that Ind [S n 2 ; S ] 5 n - 1. By the induction hypothesis we get the inequality Ind [ S n Z n Y ;S n Y ] 5 n - 1. As S n Y is a partition between F and G in Y , Ind [ Z n Y ;Y] 5 n will hold. Our first result characterizes the condition IC-IndX 5 n by the condition that w X \ X is ( 5 n)-Inductionally embedded in wX. The proof should be compared with that of Theorem 1.18. 4.3. Theorem. Suppose that X is a normal space and let n be in N. Then IC-Ind X

5 n if and only if Ind [wX \ X ; wX] 5 n.

Proof. The proof will be by induction. As the statement is obviously true for n = -1, we shall only discuss the inductive steps. Suppose Ind [wX\ X; wX] 5 n and let F and G be disjoint closed subsets of X . By Theorem 1.14, cl,x(F) n cl,x(G) = 8. In wX there is a partition S between cl,x(F) and cl,x(G) such that Ind [ S n ( w X \ X); S ] 5 n - 1. And w ( S n X ) = cl,x(S fl X ) c 5’ follows from Theorem 1.15. Thus, from the previous lemma, we have Ind [w(S n X ) n (wX \ X ) ; w ( Sn X)] 5 n - 1. By the induction hypothesis IC-Ind ( S n X ) 5 n - 1 holds. It follows that IC-Ind X 5 n. To prove the inductive step of the other implication we assume IC-Ind X 5 n and let F and G be disjoint closed sets of wX. Let UI and V, be open neighborhoods of F and G with disjoint closures. There is a partition S in X between Ul n X and Vl n X such that IC-IndS 5 n - 1. Lemmas V.2.10 and 1.16 imply that cl,x(S) is a partition between F and G in w X . By Theorem 1.15, cl,x(S) = US and wX \ X ) n wS = w S \ S hold. The induction hypothesis gives Ind [wS \ S;wS] 5 n - 1. Hence Ind [wX\ X;wX] 5 n. Recall that IndwX = I n d X (Theorem 1.18). The result of the previous theorem is then in contrast with Example 3.19, but similar

284

VI. COMPACTIFICATIONS

to that of Theorem V.2.28. In general the Wallman compactification is not weight preserving. But for hereditarily normal spaces there is a modification of Theorem 4.3 in which the compactification is weight preserving. 4.4. Theorem. Suppose that X is a hereditarily normal space

with K-Ind X 5 n. Then there exists a weight preserving compactification Y of X with

I n d Y 5 Ind X,

Ind [Y \ X ; Y ] 5 n.

Proof. We may assume that w ( X ) is infinite. Let .To be a disjunctive base for the closed sets such that I 3 0 l= w ( X ) . We shall inductively define a sequence of collections 3;. Assume that 3;has already been defined. Let us first define a screening collection Si for the invariant K-Ind. If 0 5 k 5 n and if an E in 3 i has K-Ind E 5 k, then for any pair ( F , G ) of elements of 3 i with F n G = 0 we select a partition S between F and G with K-Ind ( E n S ) 5 k - 1. Such a partition exists by Theorem 11.2.21. Write X \ S = U U V with disjoint open sets U and V such that F c U and G C V . Let J = U U S and K = V U S . The pair ( J , K ) is a screening of ( F ,G) with K-Ind ( J n I< n E ) 5 k - 1. Let the screening collection Si consist of all sets J and Ir' that are obtained in this way. Next we define a second screening collection S: for Ind in the corresponding way. That is, by using Proposition 1.4.6, for each k in N, each E in Fi with Ind E 5 k and each pair ( F , G ) of disjoint members of 3 i we choose a screening ( J , Ir') of ( F ,G) such that Ind ( J n Ir' n E ) 5 k - 1. Let the screening collection S: consist of all sets J and Ir' obtained in this way. Finally we define 3;+1 = 3; U Si US,! t o complete the inductive construction. Let 3 = U{ 3;: i E N}. Then the collection F is a normal base with 1 3 1= w(X). Let Y be the compactification u ( 3 ,X). Obviously, w(Y ) = w ( X ) . As in Theorem 2.7, it can be shown that Ind Y 5 I n d X . We shall show that Ind [cly(E) \ E ; cly(E)] 5 k holds whenever IC-Ind E 5 k and E E 3 hold, where -1 5 k 5 n. The proof will be by induction on k. Only the inductive step is discussed. Assume IC-Ind E 5 k and I3 E 3. Then let F and G be disjoint closed subsets of Y.

4. THE INEQUALITY K-Ind

2 #-Def

285

By the compactness of Y there are disjoint sets A and B in 3 such th[at F c cly(A ) and G c cly(B). Let ( J , K ) be a screening of ( A , B ) with J and K in 3 and K-Ind ( J n K n E ) k - 1. Then c l y ( J ) n c l y ( K ) is a partition between F and G. We have (cly(J) n cly(K)) n cly(E) = c l y ( J n K n E ) . The induction hypothesis gives

<

Ind [cly(J n I( n E ) \ ( J n K n E ) ;cly(J n K

n E)] 5 k - 1.

It follows that Ind [cly(E) \ E ; cly(E)] 5 k. The strict inequality Ind Y as the next example shows.

< Ind X is possible in the last theorem

4.5. Example. We shall again use Roy's example A. Let F be the collection of all open-and-closed subsets of A. As ind A = 0, the collection F is a base for the closed sets of A. One readily sees that F is a normal base. Denote w(F,A) by yA. As F = F", we find that cl,a(F) is an open-and-closed subset of y A for each F in F.It follows that ind yA = 0. As yA is compact, we also have Ind y A = 0. So Ind yA < Ind A. A slight adaptation of this construction will result in a weight preserving compactification with the same property.

Now we shall address the problem of finding relations between the relative notion of ( 5 n)-Inductionally embedded and the absolute notion of Ind 5 n. The next theorem follows from Proposition 1.4.6 by an easy inductive argument. 4.6. Theorem. Suppose that Z is a subset of a hereditarily normal space X . Then

I n d [ Z ; X ] 5 IndZ. The reverse inequality need not be true as the following example will show. 4.7. Example. There exists a hereditarily normal space X with the properties that Ind X = 0 holds and for every n in N there exists a subspace 2, with IndZ, = n. Such a space was constructed by Pol and Pol in [1979]. From Ind X = 0 we have Ind [Z,; X]5 0.

It is not known whether Theorem 4.6 holds for normal spaces.

VI. COMPACTIFICATIONS

286

To find the relation between K-Ind and K-Def we shall first prove the following proposition. 4.8. Proposition. Suppose that Z is a subset of a normal space X. Then ind Z Ind [Z; XI.

<

<

Proof. We prove the inductive step only. Suppose Ind [Z;X I n. Let G be a closed subset of 2 and p be in Z\ G. Then { p } and clx(G) are disjoint closed sets of X. There is a partition S between { p } and clX(G) such that Ind [S n 2 ; S ] n - 1. By the induction hypothesis ind (S n 2 ) 5 n - 1. It follows that ind Z 5 n.

<

We shall prove two results concerning the relation between K-Ind and K-Def. 4.9. Theorem. Suppose that X is a separable metrizable space. Then K-Ind X 2 K-DefX.

<

Proof. Suppose K-IndX n. Theorem 4.4 gives a metrizable compactification Y with Ind [Y \ X ;Y ] 5 n. By the previous result the inequality ind (Y \ X) n holds. The coincidence theorem gives Ind (Y \ X ) 5 n, whence K-Def 5 n.

<

From the proof of the theorem it is clear that the equality of ind and Ind played an important role. It is not clear whether the detour via ind cannot be avoided. 4.10. Theorem. Let the space X be Lindelof a t infinity. Suppose K-Ind X 1 . Then

<

K-Ind X

2 K-DefX.

Proof. Suppose that K-IndX = n with n = 0 or 1. Theorem 4.3 gives Ind [ w X \ X; w X ] n. By Proposition 4.8, ind (wX \ X ) 5 n. The space W X \ X is Lindelof. We have shown in the proof of Theorem 3.12 that for the case n = 0 the inequality Ind (wX \ X ) 0 holds, whence K-DefX 0. Suppose that n = 1. It will follow from a theorem of Vedenisoff [1939] (see Engelking [1978],Section 2.4) that Ind ( w X \ X ) 1 and consequently K-DefX 1.

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5. KIMURA'S CHARACTERIZATION OF K-def

287

5 . Kimura's characterization of K-def

This section is almost entirely devoted to the proof of Kimura's theorem which states that Skl = IC-def in the universe Mo of separable metrizable spaces. In Section 1.6 it has already been proved that Skl 5 def = IC-def. So it still remains to prove that if Skl X 5 n , then there exists a weight preserving compactification Y of X such that ind (Y \ X) 5 n. Suppose that B = { Ui : i E N} is a base for the open sets of X that witnesses the fact that SklX 5 n , i.e., the intersection B ( U i , ) n - n B (Ui,) is compact for every n 1 indices io < . < in. Kimura's proof uses the beautiful idea of constructing a completely new base out of B instead of the instinctive idea of enlarging B to another base by some means. The intricate construction of this new base yields a countable normal base F that has a "dense" subcollection (7 with the property that every n + 1 of its boundaries has a compact intersection. It will then follow that ind (w(.F, X )\ X) 5 n.

+

Agreement. The universe of discourse in this section is Mo. The nature of the problem under consideration requires the introduction of a substantial number of special notations. 5.1. Notation. Suppose that X is a space and that H is some collection of its subsets. Let us recall the notation given in Section 2 that was used to form the following new collections from H : 7-l" is the collection of all sets that are finite intersections of elements of H ; 7-l' is the collection of all sets that are finite unions of elements of H ; 7-l" is the collection { S : X \ S E H } ; H' is the collection of all sets that are closures of elements of H " . It is to be observed that 'FI'" = H"'. The notation for the topological operators of boundary and closure will be extended to collections H by

B ( H )= { B (S):S E H }

and

cl ( H ) = { cl ( S ) : 5' E H }.

We shall assume in the next definitions that H is countable and has been indexed by the set N as H = { Si : i E N}. For each n define the collection

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Then write K (Z) 5 n if for each member { Si,, . . . ,Sim} of [Z]"+' the intersection B (S,,) f l - . fl B (Si,) is compact. With this notation, the definition of SklX 5 n can be rephrased as: Skl X 5 n if there exists a base f? for the open sets such that ~ ( f ? 5) n. As before we shall reserve B, possibly with index, to denote a base for the open sets of X and the letter F to denote a base for the closed sets of X . In addition, we use A to denote a locally finite open collection and Z to denote the collection of all one-point sets { z } formed from the isolated points z of X . It is to be noted that both A and Z are at most countable and thus may be indexed by a subset of N. Let us remark that the use of script letters in this section has deviated from their earlier use as classes of spaces. As this section will be dealing only with the class K of compact spaces, there should be no confusion arising from the new convention in this section. We shall reserve the use of script letters to denote collections that are related to bases.

-

We begin by stating two lemmas whose easy proofs will be omitted. 5.2. Lemma. In a space X let V = { V, : Q E A } be an open collection that admits a locally finite refinement { W , : Q E A } by a collection of nonempty sets such that W , c V,, Q E A . Let 13 be a base for the open sets. Then (f? \ V) U (Vfl Z) is also a base.

5.3. Lemma. If L is a locally finite collection of sets in a space X and B is any base for the open sets of X , then there is a subcollection f31 of B such that B1 is a base and the collection { L : L E L, L fl U # S} is finite for each U in f?,.

The following lemma has been implicitly established in the proof of Theorem 1.12.1. 5.4. Lemma. Let B be a base for the open sets of a space X . Suppose that F is a closed set and V is an open set such that F c V . Then there is a locally finite open collection A such that (a) F c U d c Ucl(A) c V , (b) for each U in A there is a finite subcollection E ( U ) off? such t h a t B ( U ) c U B ( E ( U ) ) and U n ( U B ( E ( U ) ) ) = 8.

5. KIMURA'S CHARACTERIZATION OF K-def

289

In the proof of Theorem 1.12.1 we used a trick to avoid the indexing pitfall discussed in Example 1.6.13. The same pitfall must be avoided in the proof of Kimura's characterization. The next lemma provides a second trick.

Lemma. For each base f? there are subcollections B(l) and fY2)o f f? such that and B(') are bases and n fY2)= Z. 5.5.

Proof. The first two steps of the inductive proof will be indicated. Let f ? ~be the collection of all sets in f? with diameter less than 1. The collection I?, is a base. Since ,131 is a cover, it has a locally finite open refinement W1. There is a subcollection V1 of B1 such that Wl is a shrinking of V1. By Lemma 5.2 the collection (f?, \ V1) U (V1 n Z) is a base for the open sets. Let f?2 be the collection of all elements of this base with diameter less than As above, there is a subcollection V2 of f?, that has a locally finite shrinking. The induction is completed in the obvious way. When the inductive construction has been completed, we define B(l) = Z U (U{ Vi : i is odd }) and B(') = Z U (U{ Vi : i is even }) .

fr.

The following is the key lemma. It is, in fact, the inductive step in the construction of the base that was mentioned in the introduction.

Lemma. Let A be a locally finite collection of regularly open sets of a space X. Suppose that t3 is a base for the open sets of X consisting of regularly open sets such that K ( A U f?)5 n. Further suppose that E is a finite subcollection of A, F is a closed set and V is an open set such that 5.6.

Then there exist a regularly open set U and a subcollection such that

B1

of f?

(a) F c U c cl(U) c V , (b) f?, is a base for the open sets, (c) K ( d u { U } UB1) 5 n. Proof. In view of Lemma 5.2 we may assume A n f? C Z. Let and f?(') be bases as in Lemma 5.5. Define H t o be the collection { B ( A ' ) : A' E [A \ El" }. As A is a locally finite collection, it will follow that H is also locally finite and therefore countable as

290

VI. COMPACTIFICATIONS

well. .Since E is a finite subcollection of d such that B (V) c U B ( E ) and K (A)5 n, we have that H n B (V) is compact for each H in H . Let { H ; : i E N } be an indexing of the collection H . If H happens to be finite, then we complete the listing with empty sets. We make two observations. First, each point of an H i is a limit point of X . Second, there exists a locally finite open cover of X such that for each element D of this cover we have cl ( 0 )n H ; # 0 for only finitely many i. Using these observations and an inductive construction, we have a sequence Vi, i E N, of disjoint finite subcollections of \Z such that for each i in N (i) H i n B (V) c U V;, (ij) U{ Vj : j E N} is a locally finite collection, (iij) ( d ( U Vi)) n F = 0, (iv) (cl (U Vj)) n ( H j \ (U Vj)) = 8 for each j that is less than i. Let F' = F U U{ ( H i n cl(V)) \ (U Vi) : i E N}. By the first observation made above we may assume that U{ Vj : j E N } and Z are disjoint. Then by (i) and (ij) we have F c F' c V . Since H is a locally finite collection, it will follow that F' is closed. By (ij) and Lemma 5.2 we have that = a(')\ U{ Vj : j E N} is a base. There exists by Lemma 5.4 a locally finite subcollection W of with the property that if W = U W, then - F'CWCcl(W)CV, - {B ( W ) n B (D) : D E W} is a locally finite collection whose union is B ( W ) . Since B ( W ) n B (D) # 0 implies that D has more than one point, we may assume W c (a(')\ 1)because the set F' n (U( W n 1))is an open-and-closed set contained in F' which can be added back in at the end of the argument. Let

U = W\U{d(UVi):iEN}* From (ij) we have that U is open and { B (D) : D E U{ V; : i E N} } is a locally finite collection. Let V be the union of the two collections U{ V; : i E N } and W. Then V C \ Zand also the collection L = { B ( U ) n B (D) : D E V} is locally finite and U L = B ( U ) . By Lemma 5.3 there is a subcollection of E 2 )such that a, is a base and the collection { E E L : E n D # S } is finite for each D in B1. It follows that condition (b) is satisfied. We can verify condition (a) by using (i) and (iij) and the inclusion F C F'. Condition (c)

5. KIMURA'S CHARACTERIZATION OF K-def

291

remains to be verified. Let

G

E

{ B ( A ' ) : A'

E [A U { U } U

}.

Only those D with B ( U ) E G need to be considered. For such a G we have either G n B ( B l ) = 0 or G n B (Bl) # 8.

The case 6 n B (al)= 0. In this case, 6 \ { B ( U ) } is a member of { B ( A ' ) : A' E [A]" }. As the inclusions V c X \ U B ( E ) and G c B ( U ) c V hold, we have that G \ { B ( U ) } is a member of H . So there is an i in N such that Hi= \ { B ( U ) 1) holds. Let US show 6 = H in B ( U ) is compact. Since for each i in N

n

n

n(G

we have

Hin B ( W )= Hin (cl(W) \ W )

By (iv) we have Hi n B (U V j ) Consequently,

c

U Vi for each j

that is larger than i .

H i n B ( U ) c (Hin(B(W)uU{B(UVj):jE N}))\UVi

n U{ B (U V j ) : j 5 i } Hi n U{ B ( D ) : D E U{ Vj : j 5 i}}.

C Hi C

\ Z and ri ( A U Since U{ Vj : j E N } c pactness of 6 = H i n B ( U ) will now follow.

n

_< n hold, the com-

The case G n B (a,) # 0. Let B(D0)E 6 n B (&). We may assume

8 # B (Do)n B ( U ) = U{ B(Do) n B ( U ) n B (D) : D

E

V}.

VI. COMPACTIFICATIONS

292

Note that B ( U ) n B (D) E L for each

D in V. The collection

V’ = { D : D E V, B (Do)n B (U) n B (D) # 0 } is finite by the construction of B1. Now consider the collection G’ = G \ { B (Do), B ( U ) }. We have that 6’ is a member of the col}. If D E V’, then B (D) # 0 ; lection { B ( A “ ): d“ E [ A U so D E \ f?(’) holds. Also B (DO) # 0 yields DO E f?(’) \ P ) .Finally

nG

and the compactness of follows from K ( A U f?) ness of V’. The lemma is completely proved.

5 n and the finite-

Before embarking on the proof of Kimura’s theorem we shall state another lemma whose easy proof will be omitted.

A and B be as in Lemma 5.6. Suppose that E is a finite subcollection of A and let U = U E . Then 5.7. Lemma. Let

K

( ( A\ E ) u { u } u a) 5 n.

Kimura’s theorem will be proved by the existence of an appropriate Wallman-type compactification. The next lemma provides the required normal base. 5.8. Lemma. Let n be a natural number and X be a space such that 0 5 Skl X 5 n. Then there are collections F and Q of regularly closed subsets of X such that

(a) (b)

F is a countable normal base for the closed sets, G C .F holds and G can be represented as G = {G; : i

E N}U{cl(X \ G i ) : i E N}, each pair of disjoint elements of 3 there is a screening by a pair (Gi,cl ( X \ G;)) of elements of G, (d) K ( ( X \ G ; : E~ N}) 5 n. (c) for

5. KIMURA’S CHARACTERIZATION OF K-def

293

TABLE

...............................................

The proof of the lemma is not very difficult, but it will require a lot of careful bookkeeping. To make the bookkeeping easier, the above table will be used.

Proof. Let B be a countable base for the open sets of X such that K ( B )5 n. We may assume that each element of B is regularly open. Define ‘H = {cl(U) : U E B } U { ( X \ U ) : U E B}. Then list the family of all ordered pairs of disjoint elements of ‘FI as { (Si0,TiO) : i E N} and use this listing to form the 0-th column of the table. The remaining columns of the table will be inductively defined. To aid the reader, the construction will be briefly outlined. The idea of the proof is to inductively define screenings of disjoint pairs. The elements of these screenings will be the members of the base F that is to be constructed. The listed pairs in the 0-th column will guarantee that F will be a base for the closed sets. The other columns will be used t o assure that F is a normal base. That is, immediately after a screening has been defined, a new column is made by listing all of the newly formed disjoint pairs for which a screening must be defined at a later stage of the construction. To accomplish this, the canonical ordering of pairs of natural numbers is used. That is, the pair (Ski, T k l ) is indexed by j = cp (k,Z), where

Cp(k,Z) = 0 t 1 t 2 t - * *(kt-I-1 ) t 1. With this outline in mind, we continue the proof.

VI. COMPACTIFICATIONS

294

Inductively on j = 'p(k,Z)we shall define (i) a collection { (Sijtl,Tijtl) : i E N} (which will become the ( j 1)-th column), (ij) a pair of regularly closed sets Fj and G j = cl (X \ F j ) , (iij) a locally finite collection dj of regularly open sets, (iv) a base Bj for the open sets consisting of regularly open sets so that they collectively satisfy the following conditions for each j in N: (1) Sij+l and Tij+l are disjoint closed sets for each i in N. (2) If j = 'p (k,1) and 1 > 1, then the pair ( F j ,Gj) is a screening of (Ski, Tkl), so Skl n G j = 0, Fj n T k l = 0 and Fj U Gj = X. (3) X \ G j E Aj. (4) d j c dj+i and Bj+l c Bj. (5) K (djU B j ) 5 71. (6) For each i in N the pair (Sij+l,T;j+l) is of one of the following two types: Type 1: There is a finite subcollection E of d j such that Sij+l c \ Gj+ r c \ B (E)and B (Tij+l) c B (E). Type 2: There is a finite collection of distinct pairs ( i v , j v ) , 0 5 v 5 u,each of whose indices cp ( i v , j v ) are at most j and whose corresponding pairs (Sivj, ,Tiyj , ) are of type 1 such that Sij+l = U{ Si"j , : 1 L I u}, Tijtl = n { T i e j v: 1 5 5 (That is, the pair (Si jtl ,Ti j+l) is constructed out of finitely many pairs of the first type that have been previously defined.)

+

x

x u

u

Suppose j = 0. Then j = p (0,O). By Lemma 5.4 there is a locally finite open collection db such that

c Udb C UCl(Ab) C x\Too, for each U in dI, there is a finite subcollection E ( U ) of B such that B ( U ) c U B ( E ( U ) )and U n (U B ( E ( U ) ) )= 0. We may assume further that each U in At, is a regularly open set. Because AI, is locally finite, it is countable and can be denoted as dI, = { Ui : i E N}. There is a closed shrinking LO = { Li : i E N} of db that covers 5'00. The collection of all pairs ( L ; n SOO, X \ Ui) is then listed in the 1-th column as (S;1,T;l). It can also be assumed that each S;1 is regularly closed. Note that all of these - so0 -

5 . KIMURA'S CHARACTERIZATION OF IC-def

295

pairs are of type 1. We let F' = 0, Go = X , do = (dh n B)U { 0 } and BO= (f? \ do) U (do n 1).By Lemma 5.2, ,130 is a base. It is easily seen that the conditions (1) through (6) are satisfied for j = 0. Suppose that j 2 1 and that the constructions and definitions have been made for all natural numbers smaller than j. There are two cases to consider. Case 1 : j = c p ( k , l ) and 1 = 0. This case is similar to the case j = 0. By Lemma 5.4 there is a locally finite open collection dg such that -

s,oCUdgCUcl(dg)~X\Tk~,

for each U in di there is a finite subcollection E ( U ) of t? such that B ( U ) c U B ( E ( U ) ) and U n ( U B ( E ( U ) ) ) = 8.

Write .A$ = { Ui : i E N}. Let Lo = { Li : i E N } be a closed shrinking of dg that covers S ~ O List . the pairs ( L i n So0,X \ U i ) in the ( j 1)-th column as (Sij+l,Tij+l). We may also assume that these pairs consist of regularly closed sets. Note that all of these pairs are also of type 1. Define the collections dj = dj-1 U (A: n B j - l ) , Bj = (Bj-1 \ dj) U (dj n Z), and the sets Fj = 8, Gj = X. Again it is easily verified that the conditions (1) through (6) are satisfied for j . Case 2: j = c p ( k , l ) and 1 > 0. Here the pair ( S k l , T k ~ )has been determined at an earlier stage. If Skl = 0,then we let ( F j , G j ) = (Sij+I,Tij+l) = ( 8 , X ) for each i , dj = dj-1 and f?j = Bj-1. This is the trivial case. There are two more cases t o be considered. Case 2A: The pair ( S k l , T k f ) is of type 1. By the construction there is a finite subcollection Ek of dl-1 such that S ~ cI X \ T Mc X \ U B ( E k ) and B ( T ~ c I )U B ( E k ) . Note that dl-1 C dj-1. By Lemma 5.6 there exists a regularly open set U and a subcollection Bj of f?j-l such that

-+

-

Skl

-

f3j

x

c u c Cl(U) c \ Tkl, is a base for the open sets,

~(dj-IU{U}Uf?j)
is easily seen that the conditions (2), (3), (4) and (5) are satisfied.

Define

Dj = { Fi : i 5 j } U { Gi : i 5 j } .

VI. COMPACTIFICATIONS

296

Let us consider the finite collection of all disjoint nonempty pairs of the collection DjvA. Let ( S , T ) be such a pair. Then T can be represented as the intersection T = 21' n n T, for some finite subcollection T,, 1 I w 5 u, of D j V . Let ( L 1 , . . . , L k } be a closed shrinking of { T I , .. .,T k } that covers S . Then ( L , n S,T,) is a pair of type 1 for each w. So we have that ( S , T ) is a pair of type 2 for the index j 1 when u > 1. We list the collection of all such pairs ( L , n S,T,) formed from all disjoint nonempty pairs of Vj"" in the ( j + 1)-th column and complete the listing with the pair (0,X ) . Case 2B: The pair ( s k l , T k l ) is of type 2. There is a finite collection of distinct pairs (i,,,j,,), 0 5 w I u , with type 1 pairs (Si"j, ,Tivj , ) having indices cp (i,, j v ) at most j such that

+

By a simple inductive process similar to that of case 2A we can construct regularly open sets UO,. . . , U, and a subcollection Bj of Bj-1 such that S i v j v C Uv C cl(Uu) C X \ T i v j v , 0 5 21 I U , Bj is a base for the open sets, 0 6(Aj-1 u { U O } U * * . U { U,}uBj) 5 7 ~ . Let U = UOU . U u k . Then in view of Lemma 5.7, - Skl c u c d ( U ) c x \ Tkl, - K(A~-~u{U}UB~)<~L. 0

o

The rest of the construction, in particular the definitions of the pair ( F j , G j ) , the collection Aj, the collection D j and the listing of the disjoint pairs arising from Vj"" in the ( j 1)-th column, is the same as in case 2A. This completes the inductive construction. Finally, we put

+

F = u{ DjV A

: j = cp ( k , Z), IG E

N,1 E N \ { 0 } } G = {Gi i E N } U { c l ( X \ Gi):i E N}.

The only condition that might not be clear at the first glance is that F is a disjunctive base. Suppose that K is a closed set and x is not a member of Ii'. Let ( S k O , T h o ) be the pair with minimal k

5. KIMURA’S CHARACTERIZATION OF X-def

297

such that 2 E S ~ and O K c X \ T k o hold. Write j = cp (k,0). Then let (Sij+l,Tij+l) be the pair with minimal i such that x E Sij+l and T k o C Tij+l hold. With 1 = cp(i,j t 1) we have that (Fl,Gl) is a screening of (Sij + l , Ti j + l ) and therefore of ({ 2 }, K ) as well. This completes the proof of Lemma 5.8. In the proof of the last lemma we have done all the hard work for a proof of the final solution of de Groot’s problem. 5.9. Theorem (Kimura 119881). Let n 2 0. Then

SklX 5 n if and only if

defX 5 n.

Proof. As we have remarked in the introduction, it remains t o be proved that defX 5 n when SklX 5 n. Suppose that F and E are the collections that are provided by Lemma 5.8. Let Y = w(F,X). In view of (a) and (b), we have that B (G) is in F for every G in G. As Bx(G) = G n clx(X \ G), by Theorem 2.2 we have cly(Bx(G)) = d y ( G ) n cly(X \ G) = By(Cly(G)). The sets cly(G), cly(X \ G), exy(intxG) and e x y ( X \ G) all have the same boundary, namely cly(Bx(G)). From (c) of Lemma 5.8 it follows that E is a base for the closed sets of X and also that cly(E) is a base for the closed sets of Y . In view of Theorem 2.2, we have

+

for any n 1 indices io < . < i n . The last equality holds because the last set is compact. So { exy(X \ Gi) n (Y \ X): i E N} is a collection of open sets of Y \ X that satisfies the requirements of Theorem V.4.5 and Corollary V.4.4 with P = { S}. It follows that ind (Y \ X ) 5 n. The last result can also be deduced directly along the same lines as the proof of Lemma 1.6.10.

VI. COMPACTIFICATIONS

298

There are two corollaries. The first one is an immediate consequence of Kimura’s theorem and Proposition V.3.5. 5.10. Corollary. If X is a space with 0 5 def X , then for every open subspace Y of X d e f y 5 defX.

In many examples, notably Pol’s example, a space is constructed by attaching a locally compact space t o a compact space. The next corollary gives an upper bound for the deficiency of the resulting space. 5.11. Corollary. Let not compact. Then

X

= Y U 2,where Y is closed and 2 is

defX < i n d Y + d e f Z + l .

Proof. The theorem is obvious for i n d Y = -1. So we may assume ind Y = n 0. Let def Z = m. As 2 is not compact, we have m 2 0. By the previous corollary, def (2\ Y) 2 def 2 5 m. Consequently we may assume that Y and 2 are disjoint. Suppose that B is a countable base for the open sets. Consider the countable collection { (Fi, U i ) : i E N } of all pairs Fi from clx (B) and Ui from B such that Fi c Ui and Ui n Y = 0 or Fi n Y # 0. If Fi n Y # 0, then we replace Fi with Fi n Y . It is easily seen that if { Vi : i E N } is an open collection such that Fi c V , C cl(V,) C U i , then { V , : i E N} is a base for the open sets. The indexing set N is the disjoint union of N1 = { i : Ui n Y = 0 } and N 2 = { i : Y n Fi # 0 }. Let B’ be a base that witnesses the fact that SklZ 5 m. For each i in N1 for which there exists a V in B’ such that Fi C V c Ui we select one such V and denote it as Vi. The collection formed in this way will be denoted by Bz = { Vi : i E N3 } where N3 C N1 holds. We shall construct inductively a collection { Si : i € Nz } satisfying the following conditions.

>

(1)

Si is a partition between Fi and X \ Ui,

i E N2.

5. KIMURA’S CHARACTERIZATION OF K-def

When i l

299

< . . < in+2,

The construction is similar to the one in the proof of Lemma 1.6.11. Note that from (2) we have

when il < - - < in other words, the sets Si, n - - - n Sin+, and Y are disjoint. Suppose for m that the sets Si, i 5 m, have been constructed. Define

and note that Fm+l C Uh+l. Applying Lemma 1.4.11, we get a partition Sm+l between Fm+land X \ U&+, such that (2) holds whenever il < - < i k 5 m 1 holds. From the construction it follows that (3) holds whenever il < - < in+2 5 m 1 holds. In view of ( l ) ,for each i in N2 there is an open set V , such that - Fi C V , C cl(V,) C U,l, - Bx(V,) C Si. Finally let B’ = { V , : i E N2 U N3 }. Now suppose that a collection

+

+

-

+

a

+

is given. Then either n 2 indices out of { i l , . . .,in+m+2 } are in N2 or m 1 indices are in N3. In the first case, the intersection of D is empty by formula (3). In the second case the intersection of D is compact by the construction of B*. It follows that B* witnesses the fact that S k l X 5 n t m 1. By Kimura’s theorem, defX 5 n m t 1.

+

+

+

The inductive dimension ind in the formula of the last theorem cannot be replaced by def as the next example will show. 5.12. Example. In Example I.5.10.f we have studied the subspace X of lit2 given by X = Sl((0,O)) U { ( l , O ) } . It was proved there that defX = 1. For this example, consider Y = { ( 1 , O ) } and 2 = Sl((0,O)). Obviously d e f y = -1 and i n d Y = 0 hold. As 2 is locally compact, we have def 2 = 0.

VI. COMPACTIFICATIONS

300

6. T h e inequality K-dim

2 K-def

Smirnov characterized the compactness deficiency in the series of papers [1965], [1966] and [1966a]. In these papers the notion of border covers made its first appearance. The starting point of Smirnov’s work is the dimension theory of proximity spaces and its relationship to compactifications. A brief presentation of this approach to the compactness deficiency will be made. The inequality in the title of this section will come out of this endeavor. The first part of the section will be devoted to a brief outline of the more important facts about proximity spaces. Proximity spaces were introduced by Efremovich in [1952] and extensively studied by Smirnov in [1952] and [1954]. For more details the reader is referred to Engelking [1977] and Naimpally and Warrack [1970]. The reader is referred to Isbell [1964] for more details on proximity dimensions. 6.1. Definition. A proximity relation on a nonempty set X is a

relation 6 between subsets of X such that the following conditions are satisfied: For subsets A , B and C of X and for elements x and y of

x

( P l ) if ( A , B ) E 6 , then ( B , A ) E 6, (P2) ( A ,B U C ) E 6 if and only if ( A ,B ) E 6 or ( A ,C ) E 6 , (P3) ({ x},{ y}) E 6 if and only if x = y, (P4) ( X , @ @ 8, (P5) if ( A , B ) @ 6 , then there are subsets E and F of X such that E u F = X , ( E , B ) 4 6 a n d ( A , F ) @ 6 . is called a proxIf 6 is a proximity relation, then the pair (X,6) imity space. The sets A and B are near or proximal if ( A , B ) E 6. If ( A ,B ) 4 6 , then A and B are said to be far. We shall frequently use the following proposition: If A and B are far and if C C A and D C B , then C and D are far. The natural proximity relation on a metric space ( X ,d ) is defined by ( A ,B ) E 6 if and only if d ( A , B ) = 0. Suppose that (X,6) is a proximity space. We shall say that a subset A is deep inside B - denoted by A cz B - if A and X \ B are far. The following properties are easily verified: (1) If A g B , then X \ B G X \ A . ( 2 ) If A cz B , then A C B.

6 . T H E INEQUALITY K-dim

2 K-def

(3) If A1 c A G B c B I , then A1 G B I . (4) If Ai g Bi for i = 1,2, then A1 U A2 G B1 U B2. ( 5 ) If A G B , then there exists a C such that A @ C (6) 8 G 8.

301

@

B.

6.2. Definition. Let (X1,61) and (X2,62) be proximity spaces. A function f : X I t X2 is called a &function provided that f satisfies the condition ( f [ A ] , f [ B E] )6 2 , whenever ( A , B ) E 61. A &function is a &homeomorphism (or equimorphism) if f is bijective and f - l is also a &function.

Let ( X ,6) be a proximity space and let 2 be a subset of X . Then there is a natural proximity 62 defined on 2.Namely, for subsets A and B of 2 the relation SZ given by (A, B ) E SZ

if and only if

( A ,B ) E 6

is a proximity relation on 2. The proximity space ( 2 ,6 ~ is) called a subspace of (X, 6 ) . 6.3. Definition. The topology of the proximity space (X, 6) is defined by its closure operator: For a subset A of X ,

Equivalently, U is a neighborhood of z if and only if { z } G U . For a metric space ( X , d ) the topologies induced by the metric and the natural proximity coincide. Clearly, every &function is continuous and the topology of a proximity subspace (2,6 ~ of) ( X ,6) coincides with the relative topology of 2 in X. For subsets A and B of ( X ,6) we have

( A , B ) E 6 if and only if

(clx(A),clx(B)) E 6.

It then follows that A GB

if and only if

clX(A) C intxB.

The topology of a proximity space ( X , 6 ) is completely regular. Conversely, if X is a completely regular space, then there is a proximity relation 6 on X such that the topology of ( X , S ) agrees with

302

VI. COMPACTIFICATIONS

the topology of X . Such a S is called a proximity relation on the topological space X . These facts will follow easily from the relation between proximity spaces and compact Hausdorff spaces which will be discussed next. If X is a compact Hausdorff space, then there is one and only one proximity relation 6 on X such that the topologies of X and ( X , S ) agree, namely

(A, B ) E 6 if and only if

clx(A) fl clx(B) #

0.

Furthermore, a continuous mapping of a compact Hausdorff space into a proximity space is a 6-function. Two compact Hausdorff spaces are homeomorphic if and only if they are equimorphic. Therefore a proximity relation need not be specified for a compact Hausdorff space, 6.4. Definition. A proximity space (Y,6 1 ) is called a &extension of the proximity space ( X , 6) if ( X ,6) is a subspace of (Y,6 1 ) and X is dense in Y .

The fundamental results about compact extensions of proximity spaces, which were first proved by Smirnov in [1952], will be summarized next. 6.5. Theorem (Uniqueness theorem). Each proximity space has a unique compact 6-extension.

Here uniqueness means the following: If Y1 and Y2 are compact 6-extensions of the proximity space ( X ,S), then there is an equimorphism of Y1 onto Y2 whose restriction to X is the identity map on ( X , S ) . The unique compact Hausdorff extension of a proximity space ( X , S ) provided by the above theorem will be denoted by S X . The set 6X \ X is called the remainder of X in S X . For a fixed completely regular topological space X there is a natural one- to-one correspondence between the collection of all proximity relations on X and the collection of all classes of topologically equivalent Hausdorff compactifications of X . Each &function f from ( X I ,6 1 ) into ( X z , 6 2 ) can be uniquely extended to a continuous map 6f of 6x1into 6x2. Let us now turn to proximity dimension. The natural generalization of covering dimension to proximity spaces was defined by Smirnov in [1956].

6. THE INEQUALITY K-dim

2 K-def

303

6.6. Definition. Let ( X , S ) be a proximity space. A finite cover { v1,. .. ,v k } of X is called a &cover of X if there exists a cover { A1 , . . . ,Ak } of X such that Ai G V , for each i. The proximity dimension of X is minus one, Sdim X = -1, if and only if X = 8. For n in N, 6dim X 5 n if for each 6-cover V of X there exists a 6-cover W of X such that W is a refinement of V and ord W 5 n t 1. As usual, Sdim X = n means Sdim X 5 n holds but 6dim X 5 n - 1 fails.

Observe that if { V1, . . .,V,} is a 6-cover, then so is the collection { int V1, . . . ,int V, }. It will follow from Lemma 1.8.8 that every finite open cover of a compact space is a &cover. Consequently we have SdimX = d i m X for a compact spaces X . 6.7. Theorem. Let ( X ,S) be a proximity space. Then

Sdim X = dim SX

Proof. Suppose dim 6X 5 n. Let v = { v1,.. ., vk } be a &cover of the space X. Because V is a 6-cover and not just a cover, we may assume that each Vi is an open set. Then the open collection { exsx(Vl), . . .,exsx(Vk)} is also a cover of S X . This cover has an open refinement W of order a t most n 1. As W is a &cover, its trace on X is also a S-cover of order at most n -t 1. It follows that Sdim X 5 n. Suppose SdimX 5 n. Let U = { U 1 , . . . ,UI, } be an open cover of the space S X . We may assume further that the sets Ui are regularly open. The trace of U on X is a &cover. Let { W1,. . .,Wl } be a &cover of order at most n 1 that refines this trace. There is a cover A = { Al, . . .,Al} such that Ai Wj, i = 1,. . .,Z. For each i the open set ex(Wi) is contained in some Uj because each member of U is regularly open and Lemma V.2.10 (a) applies. As A covers X , it will follow that { ex(Wl), . . . ,ex(Wl) } is a cover of 6X that refines U. That the order of this cover is at most n + 1 will follow from Lemma V.2.10 (c). Hence, dim 6X 5 n.

+

+

We have as a corollary the following subspace theorem for Sdim. 6.8. Corollary (Subspace theorem). For any subspace 2 of a proximity space ( X ,S),

Sdim 2

5 Sdim X .

304

VI. COMPACTIFICATIONS

Proof. We have by the uniqueness theorem that c l ~ x ( 2 = ) 62. The closed subspace theorem for dim together with Theorem 6.7 yield Gdim 2 = dim 6 2 5 dim 6X = Sdim X . The following addition theorem can be proved in a similar fashion. Notice that the summands need not be closed sets. 6.9. Corollary (Addition theorem). Suppose X = Y U 2 for

a proximity space ( X , 6). Then

Sdim X = max { Sdim Y ,Sdim 2 }. With this brief introduction to proximity dimension we can now turn towards the characterization of the compactness deficiency. Following Smirnov [1966], we shall present first his characterization using proximity spaces. The key notion is that of an extendable K-border cover. 6.10. Definition. A family { U1,. . . , Uk’} of open sets of a proximity space (X, 6) is called an extendable K-border coueT of X if its enclosure li = X \ U{ Ui : i = 1,. . . ,k } is compact and if for each neighborhood U of I< the collection { U , U1,. . . , u k } is a &cover of

x.

The name “extendable K-border cover” is explained by the following lemma. 6.11. Lemma. A family { U1,. . .,Uk} of open sets of a proximity space (X,6) is an extendable K-border cover if and only if 6X \ X c ex 6~ ( U1) u - . u ex 6X ( Uk ) .

Proof. Suppose that 6X \ X C ex SX(U1) U and define

-

U exJx(Uk) holds

U Uk) holds, K is also The set K is compact. As K = X \ (U1 U a subset of X . Select an open neighborhood U of K in X . Then U = {ex6x(U),ex6x(Ul), ..., exbX(Uk)} is an open cover of the compact space 6 X . Since SX is compact, the collection U is also a 6-cover. Thereby the trace of U on X is a &cover of X . It follows that { U1 ,. . . , U k } is an extendable K-border cover.

6. THE INEQUALITY K-dim

2 K-def

305

Conversely, suppose that { U1 , . . . , uk } is an extendable K-border cover of (X, 6). Define K = X \ (171 U . - U U,). Let y be in 6X \ X and let V and U be disjoint open neighborhoods of y and K respectively. As the collection { U n X,U1 ,. . . ,U k } is a &cover of X , there is a shrinking { H , H I , . . . ,HI,} such that H G U and Hi G Ui, for i = 1, ..., k. Thus, 6X = clsx(H)UclsX(H1)U...UClsX(Hk) . As clsx(H) C clsx(U) holds, we have that y is not in c l s x ( H ) . It follows that y E clax(Hi) @ exax(U;) must hold for some i. Consequently, 6X \ X c ex sx(U l ) u - - - u exsX( U k ) . 6.12. Definition. The boundary dimension dim"X of a proximity space ( X , S ) is defined as follows. dim"X = -1 if and only if X is compact. For n in N, dim"X 5 n if for each extendable K-border cover U of X there exists an extendable K-border cover V of X such that V is a refinement of U and ord V 5 n 1.

+

The next theorem concerns the topological notion of compactness deficiency in the setting of proximity spaces. It should be no surprise that a topological condition such as Lindelof at infinity (see Definition 3.10) has been included in its hypothesis. The need for this hypothesis will become apparent in the proof of the theorem. 6.13. Theorem. Let (X,6) be a proximity space. that X is Lindelof a t infinity. Then

dim"X = dim ( 6 X

Suppose

\ X)

Consequently, dim"X 2 K-defX. Proof. Notice first that 6X \ X is a normal space because it is Lindelof and thus dim (SX \ X ) is well defined. Suppose dim (6X\ X ) 5 n. Let { U I ,. . ., uk } be an extendable K-border cover of X. By Lemma 6.11 we have

Since 6X \ X is normal and dim (6X \ X ) 5 n, the cover of 6X \ X consisting of the sets (6X\ X ) n exsx(Ui), i = 1 , . . .,k, has a closed shrinking H = { H I , . . . ,H k } in 6X \ X such that ord H 5 n 1. Let us observe that disjoint closed subsets of 6X \ X will have disjoint open neighborhoods in SX because X is Lindelof at infinity.

+

306

VI. COMPACTIFICATIONS

To see this, we consider disjoint sets G and H that are closed in the space 6 X \ X. Then F = c l 6 ~ ( Gr l) clax(H) is a compact subset X. The condition that X is Lindelof at infinity will yield an open neighborhood V of SX \ X such that V n F = 0 and V is a normal space. (See the proof of the converse in Lemma 3.11 where it is shown that V is an F,-set of SX.) The existence of the disjoint open neighborhoods of G and H will follow from the normality of V. We can now modify the proof of Proposition 11.4.4 t o get an open coflection w = { w1, . . .,wk } in 6x such that Hi C C ex 6X( U j ) for each i and such that W is combinatorially equivalent t o H . In particular ord W 5 n -I-1. Obviously, ( 6 X \ X ) C W1 U - . . U Wk. It will follow from Lemma 6.11 that { Wl n X,. . ., wk n X } is an extendable K-border cover of order at most n 1. Hence dim"X 5 n. Suppose that dim"X 5 n and let { V1, . . . ,v k } be an open cover of SX \ X. By Lemma 1.8.8 there is a closed cover { G I , . . . ,Gk } of SX \ X such that G; C V , for each i. From the observation made in the last paragraph there is an open collection { U1,. . . ,u k } in SX such that G; C Ui and c l ~ x ( U i n ) ( S X \ X ) c V , for each i. The collection { U1 f l X, . . . , u k n X } is an extendable K-border cover of X by Lemma 6.11. From dim"X 5 n we have an extendable K-border cover { W1,. . .,Wl } that refines this cover and has order at most n t 1. By Lemma V.2.10, { ex6x(Wl), . . . ,ex6x(W1)} is a refinement of { clax(Ul), . . . , c l s X ( ~ k ) }with order at most n t 1. So { ( e x ~ x ( W 1 )n) (SX \ X), . . . ,(ex6x(Wl)) n (SX \ X ) } has order at most n 4- 1 and refines { V1,. . .,v k }. Thereby dim ( S X \ X ) 5 n follows.

wi

+

We have need for two more lemmas. The first lemma has been included for our convenience because it is not readily available in the stated form. 6.14. Lemma. Every finite open cover of a normal space has a finite open star refinement.

Proof. Let v = { vl,.. ,,v k } be a finite open cover of the normal space X. Let { F1, . . . , F k } be a closed shrinking of V. Then there are continuous functions f;: X + 1, i = 1,.. .,k , such that f ; [ F i ]= -1 and fi[X \ K] = 1 for each i . It is clear that

6. THE INEQUALITY K-dim

2 K-def

307

is a cover that refines V.As the collection

is a star refinement of { (--m,l), (-1,m)}, it will follow that the meet of the collections

i = 1,.. . ,k, is a finite open star refinement of V and the lemma is proved. The second lemma will be proved by an argument that was used in the first part of the proof of Theorem 6.13. 6.15. Lemma. Let ( X , S ) be a proximity space and suppose that X is Lindelof a t infinity. Then every extendable K-border cover of X has a star refinement which is an extendable K-border cover.

Proof. Let U = { U l , . . . ,Ul } be an extendable K-border cover. By Lemma 6.11, the collection { exsx(U1), . . .,exax(Ul) } is a cover of 6X \ X . As the space SX \ X is normal, the open collection ( e x s x ( U i ) n ( S X \ X ) : i = l , ..., 1 ) has a finite open star refinement 0 by the last lemma. The cover 0 of SX \ X has a closed shrinking H = {HI,. . . , H I ,} in SX \ X . By the argument employed in the first part of the proof of Theorem 6.13, there is an open collection W = { W1,. . . , WI,} in SX that is combinatorially equivalent to H and satisfies the condition Hi c Wi for each i. For each Wj in W we define Vj by

Let V = { Vj n X : j = 1,. . . , k }. Suppose that j is in { 1,. . . ,k }. Then there exists an m such that the star of H j in H is contained in ex6X(Urn). It follows that Vj n X c Urn holds. Let us show that the star of Vj in V is contained in Urn. If V, n V, # 0 for some q , then H j n H , # 0, whence H , c Urn. Thus it follows that V, c Urn. The compactness deficiency is characterized by the existence of a special collection of K-border covers that will now be defined. Observe that the notion being defined is a topological one.

308

VI. COMPACTIFICATIONS

6.16. Definition. Let X be a topological space. A collection C of finite K-border covers is called a border structure of order n for X if the following conditions are satisfied. ( E l ) If U and V are in C, then there is a W i n C such that W is a star refinement of both U and V. (C2) For every point z of X and for every neighborhood U of z there exists a neighborhood V of z and an element W of C such that the star of V in W is contained in U , that is, S t W ( V ) c u. (C3) Every U in C has order at most n.

Smirnov’s topological characterization reads as follows.

6.17. Theorem. Suppose that the space X is Lindelof at infinity. Then K-def X 5 n if and only if there is a border structure for X of order a t most n f 1. Proof. Observe that the space Y \ X is normal for every compactification Y of X . We shall prove the necessity of the conditon first. Suppose K-defX 5 n. Then there exists a compactification Y of X such that dim (Y \ X ) 5 n. Let 6 be the unique proximity on X such that Y = SX. By Theorem 6.13 we have dimmX 5 n for this proximity. Let C be the collection of all extendable K-border covers of order at most n f 1. We shall show that C is a border structure of order at most n + 1. Obviously the elements of C are IC-border covers. Let U and V be in C. The meet UA V is an extendable border cover of X as is easily seen with the help of Lemma 6.1 1. By Lemma 6.15 there is a star refinement 0 that is an extendable K-border cover. As dimmX 5 n, there is a K-border cover refinement W of 0 of order at most n + 1. This shows that condition ( E l ) is satisfied. Let 2 be in X and U be an open neighborhood of 5. In view of axiom (P5)there is a closed neighborhood V of z such that V is deep inside U . The binary cover { U , X \ V } is a &cover of X and hence an extendable IC-border cover of X . By Lemma 6.15 there is a finite extendable K-border cover V that star refines the binary cover. Let W be an extendable K-border cover of order a t most 7~ 1 that refines V. It is easily seen that the condition (C2) is satisfied. Obviously condition (C3) is satisfied. Thereby the necessity part is proved.

+

6 . THE INEQUALITY K-dim

2 K-def

309

The proof of the sufficiency of the condition will be given in three steps. First, using the border structure, we shall define a proximity relation 6 on X that is compatible with the topology. Then we shall show that every element of C is an extendable IC-border cover of (X, 6). Finally dimmX 5 n will be proved. The theorem will then follow from Theorem 6.13. Suppose that C is a border structure for X of order at most n 1. We shall use the structure C to define a proximity relation 6 on X that is compatible with the topology of X .

+

Definition. Let A and B be subsets of X . Then 6 is defined as follows. (A, B ) 4 6 if clx(A) n clx(B) = 8 and there exists a U i n C such that B n S t U ( A ) = 8. Otherwise (A, B ) E 6. We shall verify that the axioms of Definition 6.1 are satisfied and that S is compatible with the topology of X . The axioms ( P l ) , (P4) and a part of (P3), namely that ({ x},{ y}) E 6 holds whenever 5 = y holds, are obviously satisfied. We shall verify a part of (P2) and leave the remainder of its proof to the reader. Let us prove that if (A, B ) 4 6 and (A, C) $! 6, then (A, B U C) 4 6. Suppose that ( A , B ) 4 6 and ( A , C ) 4 6. It is evident that cl (A) fl (cl ( B )U cl (C)) = 0. As ( A ,B ) 4 6, there is a U i n C such that B n St u ( A ) = 0. Similarly, there is a V i n C such that C n S t V ( A ) = 8. By ( E l ) there is a W i n C that is a common refinement of both U and V. Then we have ( B U C) n S t W ( A ) = 8. Thus it follows that (A, B U C) 4 6. Before we continue with the verification of the axioms, let us first show that the topology of X is compatible with 6. If z E cI(A), then obviously ( { z},A) E 6. Suppose that z 4 cl(A). Then we havecl({z})flcl(A)=~and,inview of(C2))thereisa W i n C such that S t W ( z ) c X \ cl(A). Consequently cl(A) n S t W ( z ) = 8. By the definition of S we have (A, { z }) 4 6. This completes the proof of the compatibility. Now the other part of (P3) will easily follow. That is to say, we have ({ z }, { y }) 4 6 whenever z # y because points are closed. Only axiom (P5) remains to be verified. First let us observe that if S is a compact set and T is a closed set such that S n T = 8, then there is a neighborhood U of S such that U and T are far. This can be shown in the following way. From (C2) it follows that for each point 5 of S there is an open neighborhood U, such that U, and T

310

VI. COMPACTIFICATIONS

are far, or (U,,T) 4 6. Using the compactness of S, we select a finite collection of such neighborhoods whose union is an open set U that contains S. By (P2), which has already been verified, we have ( U , T )4 6. Turning to the verification of (P5), we assume that A and B are far. Then cl ( A ) n cl ( B ) = 0 and there exists a V i n C such that B n S t V ( A ) = 0. As every set in V i s open, we have the equality cl(B) n S t v ( c 1 ( A ) ) = 0. It follows that we may assume A and B are also closed. There is a W i n C that is a star refinement of V. Then S t W ( A ) n S t W ( B ) = 0. Let C = X \ U W. By our observation there is a neighborhood U of C n A such that U and B are far. Similarly there is a neighborhood V of C n B such that V and A are far. Define F = X \ (U U S t W ( A ) ) and E = X \ (V U S t w ( B ) ) . As S t W ( A ) n F = 0, it follows by (P2) that F and A are far. Similarly E and B are far. Finally let F' = F U V and E' = E U U . Then (A, F') 4 6 and ( B ,E') 4 6. An easy computation will show E' U F' = X . It follows that axiom (P5) is satisfied. Thus we have completed the first step of the proof of the sufficiency. In the second step of the proof we shall show that every element U of C is an extendable K-border cover of the proximity space (X,6) that has just been defined. Suppose that U = { U 1 , . . . ,uk } is an element of C. Let V be an element of C that is a star refinement of U . Denote the enclosure of V by B . Let Vj be an element of V. Then there is an Uj in U such that Stv(v)c U j . So it follows that V, @ U j . For each j in { 1, . . . ,k } we define

Wj = U{V,: V , G U j , V , @ U1 for 1 < j } . Then Wj G Uj and U{ Wj : j = 1,.. . ,k } = X \ B . For each neighborhood W of B the sets X \ W and B are far and consequently { W,W1,. . .,Wk } is a &cover. It follows that U is extendable. For the final part of the proof we shall show dim"X 5 n holds. Suppose that U = { U 1 , . . .,u k } is an extendable K-border cover. Let B = X \ U and let Uo be a neighborhood of B . There are closed sets Ho, H I , . . . , Hk such that X = Ho U H I U - - U H k holds and H i G Uj hold for i = 0 , 1 , . . . ,Ic. There are VO,V1, . . . , vk in C such that StV, ( H i ) c Ui for i = 0,1,. . . ,k . Choose a member W of C that is a common refinement of all the Vi's. This W refines U and has order at most n 1. It follows that dim"X 5 n.

+

As a corollary to Smirnov's theorem we get the inequality in the title of this section.

6. T H E INEQUALITY K-dim

2 K-def

311

6.18. Corollary. Let X be a hereditarily normal space that is

Lindelof a t infinity. Then K-dim X 2 IC-def X.

Proof. Suppose K-dimX 5 n. As X is a hereditarily normal space, we infer from Lemma 6.14 that the family of all finite K-border covers of order at most n 1 is a border structure of order at most n t 1 for X. By the theorem, K-defX 5 n.

+

There are weight preserving compactifications with remainders of minimal dimension. 6.19. Corollary. Suppose that X is a space that is Lindelof at infinity and has infinite weight w(X) = r . Then the following conditions are equivalent. (a) K-defX 5 n. (b) A border structure C of order n t 1 with ICI 5 T exists. ( c ) A compactification Y of X with both dim (Y \ X ) 5 n and w(Y) = r exists.

Proof. We shall only sketch the proof. Providing the details will be left to the reader. That (c) implies (a) is obvious. For the proof that (a) implies (b) we let f? be a base for the open sets of X with IBI = r. By Theorem 6.17, there is a border structure C of order at most n -+ 1. We shall inductively define subfamilies Ci of C. For each pair (U,,Up) of elements of B for which there exists a V in C with St V ( U,) C Up we select precisely one such element of C. The family of all collections obtained in this way is denoted by CO. It can be verified that CO satisfies conditions (C2) and (C3). Suppose for i 1 that CO, . . . , &-1 have been defined. For each finite subfamily of Ci-1 we select a common star refinement W from C. The family Ci is defined to be the union of the family Ci-1 and the newly formed family of the common star refinements that were selected. Finally we let C, = U{ Ci : i E N}. It is clear that C, is a border structure. For the proof that (b) implies (c) we use the sufficiency part of proof of Theorem 6.17 to obtain the compactification Y = SX. Let B be a base witnessing the fact that w ( X ) 5 7 . One easily verifies that B* = {exax(U) : U E B } LJ (exax(V) : V E V E C,} is a base for the open sets of SX with cardinality at most T.

>

312

VI. COMPACTIFICATIONS

Remark. The outline of the proof gives the following stronger result. If 61 denotes the proximity relation induced by C and 6 denotes the proximity relation induced by C,, then the identity mapping of X can be extended to a continuous function of the compactification induced by 61 to the compactification induced by 6. 7. Historical comments and unsolved problems

The theory of the Wallman compactification was developed by Wallman in [1938]. The fact that the Wallman compactification preserves the covering dimension is in Wallman’s paper. The corresponding result for the large inductive dimension was first published in [1939] by Vedenisoff. Construction of compactifications of a space via bases for the closed sets have been discussed by Shanin in his papers [1943], [1943a] and [1943b]. The theorems in the first part of Section 2 are from the [1964] paper by Frink. Frink posed the question of whether every compactification was a Wallman compactification. A lot of effort was put into investigating this question; Ul’janov in [1977] finally answered the question in the negative. Sklyarenko [1958a] and Engelking and Sklyarenko [1963] present extensive studies of compactifications that allow extensions of mappings in the general situation as well as in the special case of rim-compact spaces. The thesis of de Vries [1962] also has many results along these lines. The first results on the extensions of families of mappings to compactifications were given by de Groot and McDowell [1960] and Engelking [1960]. See also de Groot [1961]. Almost all of the results in Section 3 originate from the work of Freudenthal in I19421 and [1951]. Freudenthal’s work is related to the problem of deciding whether a topological space is the underlying space of a topological group. In [1931] Freudenthal showed, for example, that a path connected topological group G has at most two ends, i.e., F X \ X consists of at most two points. This aspect of the theory is discussed by Peschke in [1990]. The publication of Freudenthal [1952] was unduly delayed because of the second world war; the manuscript was submitted in [1942]. That the Freudenthal compactification is a Wallman compactification was first proved by Njdstad in [1966]. An extensive discussion of the Freudenthal

7. HISTORICAL COMMENTS AND UNSOLVED PROBLEMS

313

compactification can be found in Dickman and McCoy [1988]. Theorem 3.6 is due to Alexandroff and Ponomarev [1959]. The existence of maximal compactifications of rim-compact spaces, Theorem 3.7, is due to Morita [1952]. The notion of a perfect compactification was introduced by Sklyarenko in [1962], but it was implicitly introduced in Freudenthal [1942]. In [1957] Henriksen and Isbell introduced the “properties at infinity”. Theorem 3.12 is due to Sklyarenko [1962]. An analysis of the fine distinction between zero-dimensional subsets and zero-dimensionally embedded subsets can be found in Diamond [1987] and Diamond, Hatzenbuhler and Mattson [1988]. The first examples like the ones presented in Example 3.5 are in Smirnov [1958]. Theorem 3.13 was first proved by Sklyarenko in [1958]. The Theorems 3.14 and 3.15 which generalize results of Freudenthal [1942] were proved by de Vries [1962]. Theorems 3.17 and 3.18 are due to Morita [1956], [1957] and [1959]. Example 3.19 was first published by Nishiura in [1964]. A metrizable Wallman compactification of a complete rim-compact metric space was constructed in 3.20; the construction also provided a proof of Zippin’s Theorem 1.5.2. The fact that Zippin’s compactification can be obtained as a Wallman compactification was first proved by Steiner in [1969]. In [1977] van Mill presented a more general result along these lines. Zippin’s result has also triggered research on the problem of characterizing spaces that can be compactified by adding at most countably many points. A characterization of such spaces is found in Charalambous [1980]. See also the papers Henriksen [1977], Hoshina [1977], [1979] and C r a d a [1977]. Most of the results in Section 4 were first published by de Vries in [1962]. But the notion of ( 5 n)-Inductionally embedded sets that is used in this section is slightly different from the one introduced by de Vries. Theorem 4.3 appears here for the first time. The main result of Section 5, Theorem 5.9, is due to Kimura [1988]. The lemmas leading up to Kimura’s theorem are essentially due to him. The proof of the existence of the required compactification by means of a Wallman-type constructiom appears here for the first time. Corollary 5.11 is an improvement of a result of Aarts, Bruijning and van Mill [1982] and appears here for the first time. The most important results from the papers [1956], [1965], [1965a], [1966] and [1966a] of Smirnov have been collected in Section 6.

314

VI. COMPACTIFICATIONS

Unsolved problems 1. Related to Corollary 3.16 is the following problem posed in Isbell [ 19641. If X is connected, metrizable and rim-compact, must X be separable? 2. Another question from Isbell [1964] that still remains open is:

Does every rim-compact space X have a compactification Y such that dim (Y \ X ) 5 O ? From the proof of Theorem 3.12 it will follow that the answer is yes for spaces X that are Lindelof at infinity. 3. In Theorem 4.10 we have presented a very limited result about the inequality K-Def 5 K-Ind. Prove or disprove the inequality K-DefX 5 K-IndX for every X in some universe that is contained in N and properly contains Mo.

4. In Section 3 we have studied the property that a subset Z is zero-dimensionally embedded in a space X . We have introduced in Section 4 the property Ind [ Z ;X] 5 n for subsets Z of a space X. One can also define ind [ Z ;XI 5 n in a similar way.

Is there a meaningful theory for ind[Z; XI ? 5 . Does Ind

[ Z ,X ] 5 0 imply that 2 is zero-dimensionally embedded

in X ? It will follow from Theorem 4.8 and the proof of Theorem 3.12 that the answer is yes when X is compact and 2 is Lindelof.

BIBLIOGRAPHY

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LIST OF SYMBOLS List of generalized dimension functions Bind Dim dim dimw ind Ind Odim

The proper dimension functions basic inductive dimension, 218 large covering dimension, 94 covering dimension, 43 boundary dimension, 305 small inductive dimension, 3 large inductive dimension, 9 order dimension, 47

P-Bind P-def P-Def P-dim P-dimG P-Dim P-ind P-Ind P-Mind P-Odim P-sur P-Sur ccd CmP CmP Comp def icd Icd loccom Skl

The generalized dimension functions basic inductive dimension modulo P,218 small P-deficiency, 108 large P-deficiency, 108 small covering dimension modulo P,94 relative dimension with respect to G, 199 large covering dimension modulo P,94 small inductive dimension modulo P, 76 large inductive dimension modulo P, 80 mixed inductive dimension modulo P ,241 order dimension modulo P , 234 P-surplus, 107 ?-Surplus, 85 covering completeness degree, 49 small inductive compactness degree, 15 large inductive compactness degree, 21 22 compactness deficiency, 16 small inductive completeness degree, 33 large inductive completeness degree, 33 210 23

List of other symbols Special compactifications of a space X are denoted by small Greek letters.

CUX PX FX

SX wx

one point compactification Cech-Stone compactification F’reudenthal compactification, 267 compact &extension, 302 Wallman compactification, 254

328

LIST OF SYMBOLS

w ( 3 , X ) Wallman compactification with respect to the base

3,260

Special subsets of Euclidean spaces are denoted by blackboard bold letters.

W il

P Q

Iw Iwn S"

natural numbers (starting with 0) interval [- 1,1] irrational numbers rational numbers real numbers n-dimensional Euclidean space n-dimensional sphere

With four exceptions, namely B,3, and 2, capital script letters denote classes of spaces. This convention is not in effect in Section VI.5 base for the open sets a class of complete metrizable spaces, 35 C Dowker universe, 156 D class of strongly hereditarily normal spaces, 150 E base for the closed sets 3 base for the closed sets G class of compact spaces x: W) set of metrizable compactifications of X , 16 class of locally compact spaces, 86 C class of metrizable spaces M Mo class of separable metrizable spaces class of normal spaces N NH class of hereditarily normal spaces, 87 NP class of perfectly normal spaces class of totally normal spaces, 147 NT a general class of spaces P a general class of spaces Q class of regular spaces R class of completely regular spaces Rc class of a-compact spaces, 54 S 7 class of topological spaces universe of discourse, 74 u class of subspaces of Iw" un class of subspaces of Euclidean spaces UU 2 family of zero-sets of a space, 255

INDEX absolute additive Borel class, 115 absolute Borel classes, 115 absolute multiplicative Borel class, 115 absolute G6-space, 31 absolutely closed-monotone, 78 absolutely monotone, 82 absolutely open-monotone, 110 addition theorem, 6 additive Borel class, 114 additive in U ,130 ambiguous, 231 axioms A, 178 Al, 180 H, 175 H extended, 176 M, 171 N, 172 S, 179 Sh, 180 base for the closed sets, 251 base-normal (base), 259 base-normal (space), 259 basic inductive dimension modulo P, 218 border structure of order n, 308 Borel set, 112 boundary dimension, 305 Brouwer fixed point theorem, 12 C-border cover, 49 C-kernel, 49 C-surplus, 58 class, 74 closed-additive in U ,130 closed-monotone, 77 closed-monotone in U ,77 coincidence theorem, 10 combinatorially equivalent, 88, 96 compactification problem, 16 compactness deficiency, 16 complementary, 231 complete, 28 completeness deficiency, 34 329

completeness order dimension, 40 cosmic family in U , 142 countable sum theorem, 6 countably closed-additive in U ,142 cover, 42 covering completeness degree, 49 covering dimension, 42

A, see Roy’s example &cover, 303 &extension, 302 &function, 301 6-homeomorphism, 301 decomposition theorem, 6 deep inside, 300 deficiency compactness, 16 completeness, 34 large P-, 109 small P-, 109 degree covering completeness, 49 large compactness, 21 large inductive compactness, 20 large inductive completeness, 33 large inductive a-compactness, 54 small inductive compactness, 15 small inductive completeness, 33 small inductive u-compactness, 54 diameter, 30 dimension basic inductive modulo P, 218 boundary, 305 completeness order, 40 covering, 42 large covering modulo P,94 large inductive, 9 large inductive modulo P, 80 mixed inductive, 241 order, 25 order modulo P, 234 proximity, 303 small covering modulo P, 94 small inductive, 3 small inductive modulo P , 76

330

INDEX

dimension of a mapping, 206 dimension offset, 208 Dimensionsgrad, 2 disjunction property, 251 disjunctive, 259 distributive, 248 Dowker space, 182 Dowker universe, 156 Dowker-open, 149 dual ideal, 249 €-mapping, 160 enclosure, 49, 94 equimorphism, 301 example Kimura’s, 68 Pol’s, 62 Roy’s, 9 excision theorem, 224 expansion, 226 extendable K-border cover, 304 extension, 28 F,-set, 6 far, 300 framework of a base, 219 Freudenthal base, 267 Freudenthal compactification, 267 Ga-set, 29 de Groot’s problem, 16 de Groot’s theorem, 15, 280 Hayashi sponge, 165 hereditary with respect to Ga-sets, 32 homomorphism, 249 ideal compactification, 266 ( 5 n)-Inductionally embedded, 282 inductive invariants, 126 isomorphic, 249 isomorphism, 249 join, 248

K C ( . ) , 288 Kimura’s example, 68

Kimura’s theorem, 68, 297 large compactness degree, 21 large covering dimension modulo P , 94 large inductive compactness degree, 20 large inductive completeness degree, 33 large inductive dimension, 9 large inductive dimension modulo P , 80 large inductive a-compactness degree, 54 large P-deficiency, 109 lattice, 248 Lavrentieff’s theorem, 31 Lindelof a t infinity, 272 locally finite sum theorem, 39 locally finitely closed-additive in U ,142 maximal, 249 meet, 248 mixed inductive dimension, 241 modulo notation, 50, 87 monotone, 82 monotone in U ,82 multiplicative Bore1 class, 114 near, 300 normal base, 259 normal family extension, 147 normal family in U ,142 normal universe modulo P , 99 open cover, 42 open-monotone, 110 open-monotone in U ,110 operation H, 140 M, 140

N, 147

s, 151

order dimension, 25 order dimension modulo P , 234 order of a cover, 41 order of a mapping, 206 order offset, 209 oscillation, 30 P-border cover, 94

INDEX P-hull, 109 P-kernel, 85 P-stable value, 100 p-surpius, a5 ?-surplus, 107 P-unstable, 100 partition, 3 perfect compactification, 270 point addition theorem, 8 point-additive in U ,130 Polish space, 117 Pol’s example, 62 product theorem, 7 proximal, 300 proximity dimension, 303 proximity relation, 300 proximity space, 300 quasi-component, 275 quasi-component space, 275 rational curve, 15 refinement, 42 regular curve, 15 regular family in U ,142 regularly closed, 267 regularly open, 267 relative dimension, 199 remainder, 302 rim-compact, 14 ring, 259 Roy’s example, 9 S-deficiency, 58 S-hull, 58 S-kernel, 55 S-Surplus, 55 a-locally finitely closed-additive in 2.4, 142 @-totally paracompact, 158 screening, 259 screening collection, 266 semi-normal family in u, 142 separated, 84 shrinking, 42 small covering dimension modulo P,94 small inductive compactness degree, 15

331

small inductive completeness degree, 33 small inductive dimension, 3 small inductive dimension modulo P, 76 small inductive a-compactness degree, 54 small P-deficiency, 109 stable value modulo P,89 strongly closed-additive in U ,130 strongly hereditarily normal, 150 strongly paracompact, 160 structure theorem, 40 subspace theorem, 5 super normal, 182 swelling, 88 theorem addition, 6 Brouwer fixed point, 12 coincidence, 10 countable sum, 6 decomposition, 6 excision, 224 de Groot’s, 15, 280 Kimura’s, 68, 297 Lavrentieff’s, 31 locally finite sum, 39 point addition, 8 product, 7 structure, 40 subspace, 5 Tumarkin’s extension, 32 topologically invariant class, 74 totally normal, 147 Tumarkin’s extension theorem, 32 universe of discourse, 74 unstable value modulo P,89 Wallman compactification, 254 Wallman compactification with respect to a base, 260 Wallman representation, 251 weight, 262 zero-dimensionally embedded, 268 zero-set, 255