Proximity Spaces (Cambridge Tracts in Mathematics 59)

Cambridge Tracts in Mathematics and Mathematical Physics GENERAL EDITORS

J. F. C. KINGMAN, F. SMITHIES, J. A. TODD, C. T. C. WALL, AND H. BASS

No. 59 PROXIMITY SPACES

PROXIMITY SPACES S.A.NAIMPALLY Professor of Mathematics Indian Institute of Technology Kanpur AND

B.D.WAERACK Department of Mathematics University of Alberta

CAMBRIDGE AT THE UNIVERSITY PRESS 1970

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521079358 © Cambridge University Press 1970 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1970 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 73-118858 ISBN 978-0-521-07935-8 hardback ISBN 978-0-521-09183-1 paperback

CONTENTS Preface

page vii

Index of Notations

1

2

3

ix

Historical Background

1

Basic properties

7

1

Introduction

7

2

Topology induced by a proximity

10

3

Alternate description of proximity

15

4

Subspaces and products of proximity spaces

19

Notes

26

Compactifications of proximity spaces

27

5

Clusters and ultrafilters

27

6

Duality in proximity spaces

34

7

Smirnov compactification

38

8

Proximity weight and compactification

47

9

Local proximity spaces

52

Notes

61

Proximity and uniformity

63

10

Proximity induced by a uniformity

63

11

Completion of a uniform space by Cauchy clusters

67

12

Proximity class of uniformities

71

13 Generalized uniform structures

78

14 Proximity and height

84

15 Hyperspace uniformities

87

Notes

92 [v]

vi

CONTENTS

4 Further developments 16 Proximal convergence 17 Unified theories of topology, proximity and uniformity

page 94 94 97

18 Sequential proximity

100

19 Generalized proximities

104

20 More on Lodato spaces

108

Notes

114

General References

116

Bibliography for proximity spaces

117

Index

125

PREFACE This tract aims at providing a compact introduction to the theory of proximity spaces and their generalizations. It is hoped that a study of the tract will better enable the reader to understand the current literature. In view of the fact that research material on proximity spaces is scattered and growing rapidly, the need for such a survey is apparent. The material herein is self-contained except for a basic knowledge of topological and uniform spaces, as can be found in standard texts such as the one by John L. Kelley; in fact, for the most part, we use Kelley's notation and terminology. The tract begins with a brief history of the subject. The first two chapters give the fundamentals and the pace of development is rather slow. We have tried to motivate definitions and theorems with the help of metric and uniform spaces; a knowledge of the latter is, however, not necessary in understanding the proofs. The main result in these two chapters is the existence of the Smirnov compactification, which is proved using clusters. Taking advantage of hindsight, several proofs have been considerably simplified. A reader not acquainted with uniform spaces will find it necessary to become familiar with such spaces before reading the third chapter. In this chapter, the interrelationships between proximity structures and uniform structures are considered and, since proximity spaces are intermediate between topological and uniform spaces, some of the most exciting results are to be found in this part of the tract. Various generalizations of uniform spaces find their way naturally into the theory presented here. The final chapter deals with several generalized forms of proximity structures, with one of them being studied in some detail. This chapter is rather sketchy and the interested reader is referred to the relevant literature for further information. In order to minimize the number of discontinuities occurring in the main body of the text, all references from which material is selected as well as those where further details can be found are [vii]

Vlll

PREFACE

collected together in the Notes at the end of each chapter. We have attempted to provide a reasonably complete bibliography of the literature on proximity spaces; to this end we were greatly aided by D. Bushaw's Bibliography on Uniform Topology (Washington State University, November, 1965). At the end of each item in the bibliography is found a reference to Mathematical Reviews. Appended separately is a list of general references used in the tract. An index of notations and another of terms are also included. With great pleasure we acknowledge our indebtedness to several colleagues. Dr K.M.Garg, Mr CM.Pareek, Professor A. J. Ward and Professor K. Iseki assisted with advice during the initial stages. Comments by Professor C. T. C. Wall on the first draft of the manuscript were useful during revision. Several mathematicians kindly sent us their unpublished manuscripts; we are especially grateful to Dr C. J. Mozzochi, who also made several suggestions. Mathematical manuscripts are difficult to type and we admire the skill and patience of our typists: Miss June Talpash, Mrs Vivian Spak and Mrs Georgina Smith. The first author would like to take this opportunity to express gratitude to his inspiring teachers: Professors D.S.Agashe, M. L. Chandratreya, D.P.Patravali, N. H.Phadke from India, and Professors J.G.Hocking and D.E.Sanderson from the U.S.A. This author was generously supported by operating grants from the National Research Council (Canada) and the Summer Research Institute of the Canadian Mathematical Congress (1967). We thank the staff of the Cambridge University Press for their help and cooperation. Miss M. Gagrat helped in the difficult task of proof correction. S.A.N. B.D.W. May 1969 The University of Alberta Edmonton, Canada

INDEX OF NOTATIONS Iff A <=B 8

If and only if xeA implies xeB proximity relation T(8) topology induced by proximity 8 {x:x8A} A* interior of A Int(A) JorCl(^) closure of A A<^B B is a ^-neighbourhood of A subspace proximity on Y 8T product of sets Xa, for ael,

page 8 8 11

15 23 23

ael

point cluster determined by x ultrafilter or cluster set of all clusters in X 2£ set of all clusters containing A ~srf W(T) topological weight \0$\ cardinal number of 3S proximity weight w(8) N set of natural numbers uniformity A diagonal, i.e. {{x,x):xeX} T{°U) topology induced by a uniformity % 8{°ll) proximity induced by a uniformity °ll subspace uniformity on Y induced by °M JK{cr) see paragraph following (11.4) proximity class of uniformities n(*) TT = iT{8) unique totally bounded member of II (8) n

u A(8) O}/
0

ForUczXxX,

n+l

28 27 27 39 39 47 48 63 63 64 64 68 68 71 72

n

U = £7 and U = UoU

for each neN supremum of uniformities °tt and y proximity class of A-N uniformities °ll is less than or equal in height to y height class of % [ix]

63 64 79 84 84

INDEX OF NOTATIONS

H(X) °li < 2 8% n Cj (J^) (x) Un] (fn:neD) oc ft £ Eo c2 y UAB T(8) Ac BA

) , where 8 = 8{<%) page 85 hyperspace of all closed subsets of X 87 hyperspace uniformity induced on H(X) 87 by uniformity t o n l topogenous order 98 generalized topological (or GT-) structure 98 proximity associated with GT-structure S 99 sequential proximity 100 first countable or first axiom of countability denotes a net or a sequence of functions constant sequence, each element of which is x range of (fn) net of functions fn on a directed set D 94 Leader (LE-) or Lodato (LO-) proximity 105 Pervin or P-proximity 105 arbitrary generalized proximity 105 separation axiom (x e y iff y e x) 106 quasi-uniformity 107 separation or S-proximity 108 IxI-[(ixB)U(Bxi)] 113 LO-proximity class of M-uniformities 113 complement of A the set of all functions from A to B 98

HISTORICAL BACKGROUND The germ of the theory of proximity spaces showed itself as early as 1908 at the mathematical congress in Bologna, when Riesz [95] discussed various ideas in his ' theory of enchainment' which have today become the basic concepts of the theory. The subject was essentially rediscovered in the early 1950's by Efremovic [18, 19] when he axiomatically characterized the proximity relation 'A is near B' for subsets A and B of any set X. The set X together with this relation was called an infinitesimal (proximity) space, and is a natural generalization of a metric space and of a topological group. A decade earlier a study was made by Krishna Murti [52], Wallace [116, 117] and Szymanski [113] concerning the use of 'separation of sets' as the primitive concept. In each case similar, but weaker, axioms than those of Efremovic were used. Efremovic later used proximity neighbourhoods to obtain an equivalent set of axioms for a proximity space and thereby an alternative approach to the theory. Defining the closure of a subset A of X to be the collection of all points of X 'near' A, Efremovic [19] showed that a topology can be introduced in a proximity space and that one thereby obtains, in fact, a completely regular (and hence uniformizable) space. He further showed that every completely regular space X can be turned into a proximity space with the help of Urysohn's function: namely, A $B iff there exists a continuous function / mapping X into [0,1] such that/(^l) = 0 and/(i?) = 1. Smirnov [98] subsequently proved that every completely regular space has a maximal associated proximity space, and that it has a minimal associated proximity space if and only if it is locally compact. More recently, Mrowka [75] has introduced a nearness relation on the set of all sequences from a proximity space. This provides a motivation for defining a notion, similar to that of proximity, in a Frechet L-space; such a notion was discovered independently by Goetz [28] to obtain what he terms a ^JSf-space. However, as opposed to the complete regularity of a proximity space, a need not even be a topological space. Poljakov [91] [ 1 ]

NPS

2

PROXIMITY SPACES

and Goetz [29] have since carried out further investigations in this area, and discuss the connection between this notion and the proximity of Efremovic. Svarc [20] had earlier introduced a nearness relation on the set of all nets from a given space. In order to study mappings from one proximity space to another, it was natural for Efremovic to introduce the concept of a proximity mapping. Defined to be a mapping which preserves the proximity of sets, a proximity mapping is a natural analogue of a continuous mapping in topological spaces and of a uniformly continuous mapping in uniform spaces. It is readily verified that a proximity mapping between two proximity spaces is continuous with respect to the induced topologies. Pervin [86] later revealed that the converse holds if the domain proximity space is equinormal. In 1952, Smirnov [98] pursued extensions of proximity spaces and in particular answered Alexandroff's query: 'which topological spaces admit a proximity relation compatible with the given topology?'. He discovered the connection between the Hausdorff compactification of a Tychonoff space and the compatible proximity relation, showing that 'a topological space admits a compatible proximity relation if and only if it is a subspace of a compact Hausdorff space'. Using the 'ends' of Alexandroff [A], Smirnov [98] obtained the compactification of a proximity space X by identifying each point xeX with the end consisting of all proximity neighbourhoods of x, and showing the compactification of X to be the set of all ends in X. The concept of a cluster, the analogue in a proximity space of an ultrafilter, was introduced by Leader [53] and provides an alternative approach to many proximity problems. In particular, Leader [53] obtained the compactification of a proximity space X to be the family of all clusters from X. Since the Smirnov compactification is unique, it is evident that a one-to-one correspondence exists between clusters and ends. In fact, Leader [55] proves (without resorting to compactification theory) that clusters and ends are dual classes. Csaszar and Mrowka [13] proved a stronger result regarding the compactification of a proximity space: namely, compactification can be effected preserving the proximity weight. This

HISTORICAL BACKGROUND

3

gave rise to the following interesting metrization theorem: a proximity space of proximity weight Ko is metrizable. Additional solutions to the metrization problem were offered earlier by Efremovic and Svarc [20], who used the 'sequence-uniformity' method, and by Ramm and Svarc [92] using uniform proximity covers. Smirnov [105] has used both the pseudo-metric and proximity neighbourhood approaches in deriving necessary and sufficient conditions for metrizability. More recently, Leader [61] has investigated this problem in a manner analogous to R. L. Moore's approach to the metrization of topological spaces. Using the alternate set of axioms for uniform spaces involving uniform coverings, Smirnov [98] showed that every proximity (orp-) equivalence class of uniform structures contains a coarsest member, which is also the unique totally bounded structure of the class. He also showed that there is an isomorphism between the partially ordered set of all proximities on a given completely regular space and the partially ordered set of all its compactifications, which reduces the theory of proximity spaces to that of compactifications. In 1959, Gal [27] continued the pursuit of the relationships between uniform structures and proximities, proving that there is a natural order-preserving one-to-one correspondence between totally bounded (precompact) structures and proximity relations. Gal also showed that there is a one-to-one correspondence between the compactifications of a uniformizable space and the totally bounded uniform structures which are compatible with its topology. Again, this yields a one-to-one map between the Hausdorff compactifications and the separated structures. In the same year, Alfsen and Fenstad [4] paralleled the work of Gal and treated many of the problems of Efremovic and Smirnov in the framework of Weil's uniform structures. Using maximal regular filters Alfsen and Fenstad perform completion and compactification, showing that there exists a maximal completion amongst all the completions determined by the structures of a given proximity equivalence class. This naturally raises the question as to whether there always exists a minimal completion, which is equivalent to asking if there exists a finest uniform structure in a given proximity equivalence class. This question

4

PROXIMITY SPACES

was answered affirmatively by Smirnov [98] for the case in which the proximity space is metrizable. That the answer is in general negative was first shown in 1961 by Dowker [17]. He unveiled an example of a proximity space (a product of two infinite spaces with the product proximity structure) which has no finest uniform structure inducing its proximity. The key concept in any completion theory for proximity spaces is that of 'small' sets, which can be introduced by means of pseudo-metrics, uniform structures or uniform coverings. Leader [54] has used the first device to define a local cluster, while the authors [121] have used the second to define a Cauchy cluster. They obtain a completion which consists of the family of all local (resp. Cauchy) clusters in X, a subspace of the family of all clusters, which Leader showed to be the Smirnov compactification. In [99, 100], Smirnov used uniform ^-coverings to introduce the concept of a complete uniform space and proved that every proximity space admits a minimal completion of this kind. Thus the general existence of a minimal completion was established, but with the sacrifice of the one-to-one correspondence between proximity structures and completions. This was regained in the work of Alfsen and Njastad [6], who in 1963 gave yet another example of a ^-equivalence class lacking a finest uniform structure. (Further examples are to be found in Leader [56], Fenstad [23] and Isbell [44].) This then led to the notion of a generalized uniform structure, obtained from Weil's uniform structure by replacing the 'intersection' axiom with a less restrictive one. They proved that for generalized uniform structures the answer to Smirnov's problem is affirmative, and gave an explicit characterization of those generalized uniform structures which occur as the finest member of their respective ^-equivalence classes. Such members are called total structures, and it was shown that the collection of all total generalized uniform structures embraces all ordinary metrizable (or pseudometrizable) uniform structures. Alfsen and Njastad also found a relation between proximal continuity and uniform continuity, obtaining as a particular consequence, Efremovic's result that metric uniform continuity is equivalent to metric proximal continuity. They further proved that every generalized uniform

HISTORICAL BACKGROUND

5

space can be completed, and established a one-to-one correspondence between generalized uniform structures and proximity space completions. In particular, the minimal completion of a proximity space is obtained by completion of the finest generalized structure compatible with the proximity structure. In the same year, Njastad [80] further developed the concept of a generalized uniform structure. He first noted that the collection of all generalized uniform structures on a set and the collection of all proximity structures on a set form complete lattices when ordered by the relations 'finer-coarser', and that lattice sums and products are compatible, unlike the case in which the usual uniform structures are considered. Moreover, he established the existence and compatibility of initial (final) generalized uniform structures and initial (final) proximity structures. He proved that if the uniformity is replaced by the associated proximity, then uniform convergence implies convergence in proximity, a notion introduced by Leader [54]. Convergence in proximity implies uniform convergence if the associated total uniformity is used in place of the proximity. Moreover, for a net of functions with a linearly ordered directed set, the two forms of convergence are equivalent. In 1965, Hursch [37] formulated a new concept, height, to help clarify the order structure of ^-equivalence classes of uniformities. As previously mentioned, some authors have worked with weaker axioms than those of Efremovic, enabling them to introduce an arbitrary topology on the underlying set. With such generalized proximities as quasi-proximity, paraproximity, pseudo-proximity and local proximity already existing in the literature, one almost wonders if a generalized proximity relation may be defined for each prefix which can possibly be attached to the word' proximity'! About 1963, both Pervin [84] and Leader [57] independently studied generalizations of Efremovic's original set of axioms. Pervin neglected the symmetry condition, obtaining what he called a quasi-proximity space. As well as omitting the symmetry condition, Leader used a weakened form of the 'Strong Axiom' to arrive at his topological eZ-space. I t was shown that every topological space gives rise to a generalized proximity space (X, S) of either form by defining the binary

6

PROXIMITY SPACES

relation S as follows: ASB iff Aft B #= 0 . Conversely, every quasi-proximity or topological d-space (X, S) becomes a topological space if the closure operator is defined by Cl(A) = {x: {x} 8 A}. Lodato [63] later added symmetry to Leader's set of axioms to obtain a symmetric binary relation which we shall refer to as a Lodato proximity. He proved that every set with a Lodato proximity defined on it satisfies the RQ axiom (i.e. every open set contains the closure of each of its points), and that given any U0-space we obtain a Lodato proximity compatible with the given topology if we define ASBiSA[]B + 0. Mozzochi [72] has since introduced the idea of a symmetric generalized uniform structure and has studied its relationship to a Lodato proximity structure, as well as extending results of Alfsen-Fenstad, Hursch and others to such a setting. In 1964, Hayashi [32] introduced the notion of' paraproximity' by replacing the word 'finite5 by 'arbitrary', and thereby strengthening Efremovic's 'union' axiom to read : for an arbitrary index set A, (\J Ax) SB iff A SB for some JLLEA. He AeA

showed that a paraproximity space X is completely normal if one defines 0 to be an open set if and only if 0 $ (X — G). A completely normal space becomes a paraproximity space if we define A SB iff A n B #= 0. Hayashi [33] also discussed a generalized proximity space, which he called a pseudo-proximity space, with even weaker axioms than those considered by Pervin or Leader. He proved that every such space can be topologized and that every topological space admits a pseudo-proximity. Recently, Leader [59] has defined a local proximity space, in which both' proximity' and' boundedness' are taken as primitive terms. The proximity spaces of Efremovic are the special cases in which all subsets are bounded. Just as every proximity space can be embedded as a dense subset of a compact Hausdorff space, it is shown that every local proximity space can be embedded as a dense subset of a locally compact Hausdorff space. Leader also showed that every proximity space (X, S) with its proximity relation localized with respect to any free regular filter from (X, S) gives rise to a local proximity space. Conversely, every local proximity space arises from the localization of some proximity relation.

CHAPTER 1

BASIC PROPERTIES 1. Introduction In a topological space X, the topology is determined by the closure axioms given by Kuratowski concerning the relation 'x is a closure point of A c: X'. When x is a closure point of A, we may say that 6x is wear A'. In terms of this nearness relation, a continuous function f:X -> Y may then be described as one exhibiting the property: if x is near A, ihenf(x) is near/(^l). This suggests axiomatizing the relation 6A is near J3' for subsets A and B of X. For the case in which X is a pseudo-metric space with pseudo-metric d, this nearness relation can be defined in a natural way. Let

D(A,B) = inf{d(a,b):aeA,beB}.

We may then define: A is near B if and only if D(A,B) = 0. In terms of D, the closure of a set A is A = {x:D(A,x) = 0}. But the nearness relation so defined goes a little further. Let (Y,e) be another pseudo-metric space, E be defined in a similar manner to D, and/be a function from X to Y. Then/is uniformly continuous if and only if D(A,B) = 0 implies E(f(A),f(B)) = 0 (see (4.8)). Thus the nearness relation between the subsets is somehow connected with uniformity. The above nearness relation (in a pseudo-metric space) satisfies the following properties, where we denote 'A is near B' by A SB: A SB implies BSA.

(1.1) (1.2)

(A{jB)SC

(1.3)

ASB

iff ASC or BSC.

implies

A + 0 , B 4= 0 .

(1.4) ^4 #i? implies there exists a subset E such that and(X-E)$B. (1.5) ^ 4 f l £ + 0 implies . 4 £ £ . [7]

8

PROXIMITY SPACES

In a metric space, the nearness relation also satisfies: (1.6) x8y implies x = y. (Strictly speaking one should use the notation {x}S{y}, but we shall simply write x8y.) All of the above properties, except perhaps (1.4), are immediate consequences of the definition of the nearness relation. To verify (1.4), we note that if A $B then D(A,B) = e > 0. Setting E = {xeX:D(x,B)

^ e/2} we obtain

D(A,E) ^ e/2

and D(X — E, B) ^ e/2, from which the desired property follows. The above discussion leads to the following definition of a proximity space: (1.7) D E F I N I T I O N . A binary relation 8 on the power set of X is called an (Efremovic) proximity on X iff 8 satisfies the axioms (1.1)—(1.5). The pair (X, 8) is called a proximity space. Proximity relations satisfying Axiom (1.6) will be referred to as separated (or Hausdorff) proximity relations. If a proximity is derived from a (pseudo-) metric, then it is called a (pseudo-) metric proximity. (1.8) R E M A R K S . The above axioms are different from, although equivalent to, the original axioms of Efremovic. The reason for writing the axioms in this way is to permit a smooth transition to generalized proximity spaces. It will be shown presently that a proximity 8 on X induces a topology r = T(8) on X if one defines the closure A of A to be the set {x: x8A). It will be seen that this topology is always completely regular: in fact it is always Tychonoff if Axiom (1.6), which is equivalent to the I^-axiom, is satisfied. Conversely, if (X,T) is any completely regular topological space, then there exists a proximity 8 on X such that T(S) = r. Actually it would be sufficient to concern ourselves solely with separated proximity spaces, as many authors do; for if a given space fails to satisfy condition (1.6), we can instead consider the separated quotient space formed from the equivalence classes consisting of all points near to one another. However, for the sake of generality, we shall not assume a proximity space to be separated.

BASIC PROPERTIES

9

Suppose 8' is a binary relation on the power set of X that satisfies (1.2)-(1.5) and (1.2') A8'(B[)C) iff A8'B or AS'C. Then 8' induces a topology r(8f) on X if one defines the closure A of A to be the set {x:x8r A). If in addition we require that 8' satisfy the Symmetry Axiom (1.1), then the induced topology will be completely regular. Thus we see that the Symmetry Axiom (1.1) is in a sense equivalent to the complete regularity of the induced topology. Axiom (1.4) plays an important role in the theory of proximity spaces, but is omitted or replaced by a weaker condition in some generalized proximity spaces. It will, therefore, be helpful to avoid the use of this axiom as far as possible. We shall refer to this axiom as 'the Strong Axiom.' It should be noted that the order of the sets in the Strong Axiom is important, particularly in generalized proximity spaces in which (1.1) is not satisfied. We shall strictly observe the order even in proximity spaces, so that the proofs can then be carried over to more general situations. Given below is an outline of a number of examples of proximity spaces in which the proximity is not constructed from a pseudometric. Since some of these will be carefully taken up in later sections, the details are not verified here. (1.9) E X A M P L E . Just as discrete and indiscrete topologies can be defined on any set, we have discrete and indiscrete proximities. If we define A 8XB iff A n B =(= 0 , then 81 is the discrete proximity on X. On the other hand, if A S2B for every pair of non-empty subsets A and B of X, then we obtain the indiscrete proximity onl (1.10) E X A M P L E . Given a completely regular space (X,r), we say that subsets A and B of X are functionally distinguishable iff there is a continuous function f:X -> [0,1] such that f(A) = 0 and/(B) = 1. We may then define a proximity 8 on X by (1.11)

A$ B iff A and B are functionally distinguishable.

For details, refer to Theorem (2.10) and Remarks (3.15).

10

PROXIMITY SPACES

(1.12) E x AMPLE. Given a uniform space (X,°U), one may define a proximity on X by A SB iff for every U E%, one (and hence all) of the three following equivalent conditions is satisfied: (i) U[A](]B+ 0; (ii) A n U[B] * 0 ; (iii) {AxB)f]U + 0. Several interesting results concerning the relationship between uniformities and proximities will be discussed in Chapter 3. (1.13) E X A M P L E . If (Z,-,r) is a topological group and the neighbourhood system of the identity, we may define A8XB

iff for every N eJf,

JV

is

NA n B 4= 0 .

A second proximity 82 may be defined by A82B

iff for every NeJf,

AN(]B+0.

In general the two proximities S1 and S2 differ. They coincide, however, if X is either commutative or compact. 2. Topology induced by a proximity In this section we consider the topology on X which is induced by a proximity on X, and study its elementary properties. Properties (i) and (ii) of the following lemma, which follow directly from Axioms (1.1), (1.2) and (1.4), are useful in several proofs. (2.1) LEMMA, (i) If A SB, A a C and B ^ D, then C SD. Hence X is near every non-empty subset. (ii) / / there exists an x such that A8x and x8B, then A SB. Note that the Strong Axiom is not used in the proof of the following theorem. (2.2) T H E O R E M . / / a subset A of a proximity space (X,8) is defined to be closed iffxSA implies xeA,then the collection of complements of all closed sets so defined yields a topology r = T(S) on X. Proof: Obviously 0 and X are closed sets. Let {Afiel} be an arbitrary collection of closed sets. If xS(~]Ai then by Lemma iel

BASIC PROPERTIES

11

(2.1), x8Ai for each iel, and so xeAi for e a c h i e / since is closed. Thus xe f| Ai9 which means f| ^% is closed. Finally, if iel

iel

Ax and A2 are closed and x8(A1 U A2) then by (1.2), either x8Ax or x SA2. But ^4X and A2 are closed, implying that x EA1 or a: e^42, i.e. x e f i j U i a ) . T h u s ^ U ^ * 1 3 closed. (2.3) T H E O R E M . Let (X,8) be a proximity space and r = Then the r-closure A of a set A is given by

T(S).

1 = {x:xSA}. Proof: If A denotes the intersection of all closed sets containing A and A8 = {x:xSA}, then we must show that A = A8. If xeA8 then xSA. By (2.1) this implies xS A and, since A is closed, xeA. Thus A8 c A. To prove the reverse inclusion it suffices to prove that A8 is closed, i.e. x8A8 implies xeA8. Assuming x^A8,then x$A so that, by the Strong Axiom, there is a set E such that Thus no point of (X-E) is near A, i.e. x$E and (X-E)SA. A8 c: E, which together with x $ E implies that x $ A8. (2.4) COROLLARY. If G is a subset of a proximity space (X, S), then G ET(S) iff x $ (X — G) for every xeG. (2.5) COROLLARY. If A and B are subsets of a proximity space (X,8), then A$B implies (i) B c (X-A)

and (ii) B c

Int(X-A),

where the closure and interior are taken with respect to r(8). Proof: Statement (i) follows directly from (2.1). To prove (ii), x^Int(X-A) we use the identity: Int (X - A) = X - A. Then implies xeA, so that x8A and hence x (2.6) R E M A R K S . Theorem (2.3) is true if we omit the Symmetry Axiom (1.1) and add (1.2'). An alternative method of introducing the same topology on a proximity space (X, 8) would be to define for each subset A of X, (2.7)

A8 = {x:x8A}

12

PROXIMITY SPACES 8

and show that is a Kuratowski closure operator as follows: (i) By (1.3), xS 0 implies 0 = 08. (ii) By (1.5), xeA implies xSA, so that A <= A8. (iii) By (1.2), xe{A[)B)8 iff xS(A[jB) iff xSA or ;r£J3 iff #e.4* or xeB8 iff ^ e (J.* U £*). Thus (A U J3)* = A8 u B*. (iv) To prove (A8)8 c J / , suppose x$A8,i.e.x $A. Then by the Strong Axiom, there exists an E such that x $ E and (X — E) S A. Now A8 czE and xSE, so that a<M* and x$ (A8)8. (2.8)

LEMMA.

For subsets A and B of a proximity space (X, S), A SB

iff A SB,

where the closure is taken with respect to r(S). Proof: Necessity is a trivial consequence of (2.1). To prove sufficiency, suppose A SB. Then by the Strong Axiom, there exists an E such that ASE and (X-E)SB. By Corollary (2.5), B c= E and by (2.1), A SE implies ASB.lt then follows from the Symmetry Axiom that ASB. (2.9) D E F I N I T I O N . If'onasetX there is a topologyTand a proximity S such that r = T(S), then r and S are said to be compatible. (2.10) T H E O R E M . If(X,r)is a completely regular space, then the proximity S defined by (1.11), namely ASB iff A and B are functionally distinguishable, is compatible with r. If (X, r) is a Tychonoff space, then 8 is separated. Proof: To show that 8 is a proximity it suffices to prove that the Strong Axiom is satisfied, since the other proximity axioms are easily verified. Suppose ASB and let/be a continuous function from X to [0,1] such that/(,4) - 0 and/(J5) = 1. Set E = {xeX: 1/2 ^f(x) ^ 1}. Then A SE and (X-E) SB. For instance, if g denotes the selfmapping of [0,1] defined by g(y) = 2 y = 1

O ^ y ^

1/2,

1/2 < y < 1

then g (and hence gf) is continuous, and gf: X -> [0,1] is such that gf{A) = 0 and qf(E) = l.

BASIC PROPERTIES

13

To see that 8 is separated if (X, r) is Tychonoff, we note that if x 4= y then x£y since (X, r) is Tv From the definition of a completely regular space, we are assured that x and y are functionally distinguishable, implying that x$y. We now show that r = T(S). Let GET and xeG. Then x is not in the closed set X — G, so that there exists a continuous function / : X - > [0,1] such thatf(x) = 0 andf(X-G) = 1, i.e. a#(X-G)Hence by (2.4), OET(S). Conversely, if GET(8) and XEG, then x $ (X — G). Hence there exists a r-continuous function

such that/(x) = 0 and/(X-G) = l. Then/^flX), 1/2)) is a r-open neighbourhood of x contained in G. We therefore have GET. Thus we know that on every completely regular (Tychonoff) space, we can define a compatible (separated) proximity. The converse of Theorem (2.10) is also true; that is, the topology r(8) induced by a (separated) proximity 8 is always (Tychonoff) completely regular (see Theorem (3.14)). We now show that in a T± (normal + T±) space, there is another way of defining a compatible proximity. (2.11) (2.12)

THEOREM.

In a T^-space (X,r). A8B

iff A(]B + 0

defines a compatible proximity. Proof: That (2.12) defines a proximity follows from Theorem (2.10) and the fact that, in a normal space, A n B = 0 iff if and i? are functionally distinguishable. (Note that for this part, the T^-axiom is unnecessary.) To show that r = r(S), we observe that a J^-space is Tychonoff and apply Theorem (2.10). (2.13) R E M A R K . In a normal space (X,r), the proximities defined by (1.11) and by (2.12) are equivalent. (2.14) T H E O R E M . / / a completely regular space (X, r) has a compatible proximity 8 defined by (2.12), then X is normal. Proof: If A and B are disjoint closed sets, then A$B. By the Strong Axiom, there exists an E such that A $E and (X — E)$B.

14

PROXIMITY SPACES

From 2.5 (ii), we then have A c Int(X-E) and B c IntE. Since Int i£ n Int (X - E) = 0 , X is normal. Just as the class of topologies on a given set can be partially ordered by inclusion, one can impose a partial order on the class of proximities denned on a set in the following manner: (2.15) D E F I N I T I O N . / / 8 1 and 82 are two proximities on a set X, we define (2.16)

8±>82

iff A81B

implies

A82B.

The above is expressed by saying that 81 is finer than 82, or 82 is coarser than 8V The following theorem shows that a finer proximity structure induces a finer topology: (2.17) T H E O R E M , (a) Let 8X and 82 be two proximities defined on a set X. Then 81 < 82 implies r(^x ) <= r(82). (b) Let rx and r2 be two completely regular topologies on X, and let S± and 82 be the proximities on X defined by (1.11) with respect to r x and r2 respectively. Then rx <= r 2 implies 8X < 82. Proof: (a) Suppose i e r ^ ) . Then by (2.4) x$1{X-A) for each XEA. Moreover, since 8± < 82, x$2{X — A) for each xeA. Thus A er(82), from which we conclude that r ^ ) c: T(82). (b) If A$XB, then there exists a Tycontinuous function / : (X,TX) -> [0,1] such that/(^4) = 0 and f(B) = 1. Since rx c r2, / is also a r2-continuous function, showing that A$2B. Hence 8X < 82. (2.18) R E M A R K S . For a slightly stronger form of Theorem 2.17 (6), the reader is referred to Remarks (3.15). It is sometimes said that a proximity is a finer structure on a set than a topology, since a topological space may have different compatible proximities. For example, let X be the real line with the usual topology and let 8± be defined bjA81B iff D(A, B) = 0, where D(A,B) = inf{|a-6| :aeA,beB}. Let 82 be defined by (2.12). Then both 8X and 82 are compatible with the topology of X. However, the sets A = {n: neN} and B = {n—ljn:neN} are such that A8XB but A $2B.

BASIC PROPERTIES

15

3. Alternate description of proximity Given a uniform space (X, °tt), a subset B may be said to be a uniform neighbourhood of A iff there is an entourage JJe°li such that U[A] c B. An analogous concept, that of a ^-neighbourhood, can be introduced in a proximity space and furnishes an alternative approach to the study of proximity spaces. This concept of a ^-neighbourhood is not only useful in the theory of proximity spaces, but is also, in a sense, dual to the concept of proximity. This duality will be studied in detail in Section 6. (3.1) D E F I N I T I O N . A subset B of a proximity space (X,d) is a (^-neighbourhood of A (in symbols A
IntB),

Equivalently,

i.e. A <§ Int B.

The following is an alternate way of stating the Strong Axiom and is employed by many authors: (3.3)

LEMMA.

The Strong Axiom (1.4) is equivalent to

(3.4) A $ B implies there exist subsets C and D such that A9(X-C),

{X-D)$B

and C$D.

Proof: To prove that (3.4) implies (1.4), we note that if C$D, we have A$E and then C c ( I - D ) . Setting E-=X-C, (X-E)$B. On the other hand, suppose (1.4) holds. Then

16

PROXIMITY SPACES

A $B implies there is a D such tht A $D and (X -D)$B. Moreover, there exists a C such that A $ {X — C) and C $D, completing the proof. (3.5) COROLLARY. A $B implies there exist subsets C and D such that A <4C, B <^ D and C$D. Consequently, the topology T(S) of a separated proximity space (X, 8) is Hausdorff. (3.6) LEMMA. Let 8 be a compatible proximity on a completely regular space (X, r). If A is compact, B is closed and Af]B= 0, then A SB. Proof: For each aeA, a$B. From (3.2) and (3.5), we know there exists an open neighbourhood Na of a such that Na$B. Now {Na: a e A} is an open cover of the compact set A, and so there is a finite subcover {JV^ : i= l,...,w}. By (1.2), N$B, where n

N = U Na.. But A cz N, implying that A $B. It is well known that a compact completely regular space has a unique compatible uniformity. A similar theorem is true in proximity spaces: (3.7) T H E O R E M . Every compact space which is completely regular (Tychonoff) has a unique compatible (separated) proximity, given hy

ASB

iff AnB±

0.

Proof: If 8 is any compatible proximity and A n B 4= 0 , then by (1.5) and (2.8), A 8B. Since closed subsets of a compact space are compact, the foregoing lemma implies the converse. (3.8) R E M A R K S . Contrary to a con j ecture made by Alexandroff, there also exist non-compact completely regular spaces with unique compatible proximities. (Such spaces are necessarily normal, as will be seen in Section 7.) The following is one such example: the space of all ordinals less than the first uncountable ordinal with the order topology is not compact, but has a unique compatible proximity. In a later section (Theorem (7.20)), we shall actually characterize those topological spaces which possess unique compatible proximities.

BASIC PROPERTIES

17

(3.9) THEOREM. Given a proximity space (X,8), the relation <4 satisfies the following properties: (i) X^X. (ii) A <^B implies A cz B. The converse holds iff (X, 8) is discrete. (iii) i c B ^ C c D implies A <4 D. (iv) A^BJori...l,...,niffA^

f] Bt.

(v) A <4 Bimplies {X-B) <4 (X-A). (vi) A <
+ y.

Proof: (i) Since X$ 0 by (1.3), X 4 X. (ii) If AS (X-B) then A(](X-B)= 0, implying A ^ B. The second part is an easy consequence of the definition of discrete proximity given in Example (1.9). This implies that BS(X-C) (iii) If A ^ D, then AS(X-D). or B ^ C, a contradiction. (iv) It suffices to consider n = 2. ^4 <^ 5 X and A <^ B2 iff iff 4 * [X - (#! n S2)] iff A
i.e.

(vi) A <^ B implies A$(X — B). By the Strong Axiom, there exists a set (X-G) such that ^ # ( X - C ) and CS(X-B); that is, ^4 4 G
COROLLARY. ^
[J A, ^ \J Bt.

All of the separated proximity axioms (1.1)—(1.6) are used in the above proofs. In particular we note that (1.4) is equivalent to 3.9 (vi), and (1.6) is equivalent to 3.9 (vii). The following is a converse of Theorem (3.9).

18

PROXIMITY SPACES

(3.11) T H E O R E M . If <4 is a binary relation on the power set of X satisfying 3.9 (i)-(vi) and 8 is defined by (3.12)

A$B

iff

A^(X-B),

then 8 is a proximity on X. B is a S-neighbourhood of A iff A <^ B. Moreover, if <^ also satisfies 3.9 (vii), then 8 is separated. Proof: (i) A $B implies A 4 (X-B). By 3.9 (v), B <4 (X-A),

and so B$A. (ii) (A[)B)$C implies (A\j B) <^ (X-C). Then by 3.9 (iii), A <^ {X-C) and B <4 (X-C); that is, A8C and B$C. Conversely, if {A[)B)dC then by part (i), C8(A\)B). Hence C £[X-(A[)B)l or C£[(X-A)n(X-B)]. Thus by 3.9(iv), C ^ (X - A) or C ^ (X - B). Hence C 8 A or C SB and it follows, since 8 is symmetric, that A 8C or BSO. (iii) (1.3) is a direct consequence of 3.9 (i). (iv) Suppose A $B, i.e. ^4 < (X — B). Then 3.9 (vi) assures the existence of a (7 such that A <^ (X-C) 4 (X-B). Thus there is a C such that ^1 j G and (X - C) 8B. (v) If ^ # 5 , then A <^ (X-B). From 3.9(ii) we have ,4 cz (X-B),

i.e. i n 5 =

0.

If 3.9 (vii) is satisfied, (1.6) follows immediately. That B is a 8neighbourhood of A iff A <^B follows easily from the definitions of the terms involved. In a uniform space (X, °ll), the closure of any subset A is given by

A= fl U[A]. Ue°//

The following result is an analogue of this in proximity spaces. (3.13)

THE OREM. / / (X, 8) is a proximity space and A cz Xjhen

A=

n B.

Proof: From 3.2 (i) and 3.9 (ii) we conclude that A <^ B implies A <= B, and hence l c f| B. To show the reverse inclusion, _ _ suppose that x$A. Then x$A and, by (3.5), A has a <J-neighbourhood Bx not containing x. Thus x$ f] B. A^B

BASIC PROPERTIES

19

In Theorem (2.10) it was shown that every completely regular space (X,T) has a compatible proximity S. The following is a converse. (3.14) T H E O R E M . If (X,§) is a (separated)proximity space, then T(S) is (Tychonoff) completely regular. Proof: That r(S) is Tx if 8 is separated follows easily from Axiom (1.6). We now indicate briefly why T(S) is completely regular. If A is a closed set and x$A, then x$ A. Hence we have a; <^ (X — A) and, after applying 3.9 (vi) twice, we find that there are sets B and C such that x [0,1] such that/(x) = 0 and/(^) = 1. (3.15) R E M A R K S . Actually, following the proof of Urysohn's Lemma one can prove that for a compatible proximity S defined on X, A SB implies the existence of a continuous function f:X^ [0,1] such that f(A) = 0 and f(B) = 1 (see Theorem (7.12)). Consequently, the proximity defined by (1.11), namely A SB iff A and B are functionally distinguishable, is the largest or finest compatible proximity which can be defined on a completely regular space. Or, as Smirnov proved: every completely regular space has a maximal associated proximity space. I t is useful to note that, as a result of Corollary (2.13), the proximity defined by (2.12) is the largest compatible proximity that can be defined on a normal I^-space. We can now strengthen 2.17(&) to read: Let r1 and r 2 be two completely regular topologies on X, Sx be any proximity on X compatible with r1? and S2 be defined by (1.11) with respect to r2 . Then rx <= r 2 implies Sx < S2. 4. Subspaces and products of proximity spaces In the study of general topological spaces, continuous functions play an important role. A similar role is played by uniformly continuous functions in uniform spaces. Their analogue in the

20

PROXIMITY SPACES

theory of proximity spaces is the concept of a proximity (or proximally continuous or equicontinuous or 8-) mapping. (4.1) D E F I N I T I O N . Let (X,^) and (Y,82) be two proximity spaces. A function/: X -> Y is said to be a proximity mapping iff A8XB

implies

f(A)S2f(B).

Equivalently, / is a proximity mapping iff C$2D implies or

G <^2D

t^SJ-^D),

implies f-\C)

<1f~1{D).

It is easy to see that the composition of two proximity mappings is a proximity mapping. The next theorem is similar to the well-known result: a uniformly continuous function is continuous with respect to the induced topologies. (4.2) T H E O R E M . A proximity mapping f:(X,8J -> (Y,82) is continuous with respect to r(^x) and T(82). Proof: This result follows easily from the fact that x 81A implies f(x)8J(A),i.e.f(A)czf(A). (4.3) R E M A R K S . The converse of the foregoing theorem is false. Consider the example of (2.18): the identity mapping on X is continuous with respect to r(<J1) and r(82), but is not a proximity mapping from (X, SJ to (X, 82). It is natural to inquire as to when the converse of Theorem (4.2) is true. Recalling that a continuous function on a compact space is uniformly continuous, we observe from the next theorem that a completely analogous result holds in proximity spaces. However, compactness is too strong and we shall prove a better result when we consider equinormal proximity spaces (see Theorem (7.22)). (4.4) T H E O R E M . / / (X, 8J and (Y, 82) are proximity spaces and X is compact, then every continuous function f from X to Y is a proximity mapping. Proof: If A and B are subsets of X such that A 8± B, then A n B #= 0 by Theorem (3.7). But this implies that f(A)nf(B)*

0,

i.e.

f(A)8J(B).

BASIC PROPERTIES

21

Since / is continuous, f(A) <=• f(A) and f(B) <=/(£), yielding f(A)S2f(B). From Lemma (2.8) it follows that f (A) S2f(B), and we conclude that/is a proximity mapping. (4.5) T H E O R E M . Given a function f:X -> (Y,82), the coarsest proximity 80 which may be assigned to X in order that f be proximally continuous is defined by (4.6)

A$0B

iff there exists a C <= Y such that

f(A)$2(Y-C)

and f-\C) c

(X-B).

Proof: We first verify that So is a proximity on X. (i) Suppose A $QB and let D = (Y-f(A)). Since f(B)c:(T-C)

and

f(A)$2(Y-C),

we have/(i?) $2 (Y — D). Moreover, Hence B$0A. (ii) (A U B) $Q C implies the existence of a D cz Y such that U(A)Uf{B)\h{J-D) and/- x (i)) cz ( Z - C ) , f r o m which A$0C and B$QC follow. If A $0 C and B $0 C, there exist D1 and Z>2 such

that f(A)$2(Y-D1),

f(B)$2(Y-D2),

f-\Dx) c (X-C) and

/-i(i) 2 ) c (X-C). Therefore [/(^)U/( J B)]* 2 [r-(D 1 ui) 2 )] and f-^D^DJ c (X-C),i.e. ( i u B ) ^ C . (iii) If ^ = 0 , then /(^!)£2 7 and /- x (0) <={X-B). Hence (iv) If A $0B, then there exists a C c 7 such that /-i(C)<= (X-2?)

and

f(A)S2(Y-C).

This latter relation and (1.4) together assure the existence of a set D such thatf(A)$2D and ( F - D ) # 2 ( r - C ) . Let E =f-\D). Since f(A)92D, A$0E. As f{X-E) a (Y-D)#2(Y-C) and /-i(C) c (X-B), we have (X-E)$0B. (v) ^4 (^0 J5 implies there exists a O c f such that f(A)92(Y-C)

and /-i(C) cz

(X-B).

Therefore f(A)n(Y-C)= 0 and f~1(f(A)){\f-\Y-C)= 1 Since ^ c f~\f(A)) and B<^f~ (Y- C), we have i n £ = 0 .

22

PROXIMITY SPACES

In order to show that / : (X, 80) -> (Y, 82) is proximally continuous, suppose that f(A)S2f(B). Sincef(A) <4 (Y-f(B)), there exists a C such that f(A)<^ C <£ (Y-f(B)) by 3.9(vi). Thus f(A)*2{7-C)*zdf-i(C) <= (X-tHf(B)) c (X-£),i.e.^ 0 £. It remains to show that if 81 is any proximity on X such that / : (X, SJ -> (F, 82) is proximally continuous, then #x is finer than #0. If A 80 B, then there exists a C c 7 such that f(A)S2(Y- C) and/- 1 (C) 80. (4.7) (4.6),

COROLLARY.

/ / / : {X,80) -> (7,5 2 ) wAere *0 is defined by

(4.8) R E M A R K . Perhaps we should mention at this point that in Chapter 3, it will be shown that every uniformly continuous function is proximally continuous with respect to the induced proximities. Moreover, a function/mapping a metric space (X, d) to a metric space (Y, e) is uniformly continuous if and only if it is proximally continuous, (a result due to Efremovic). We shall simply outline a proof here, as a more general result will be proved later (see (12.20)). Necessity follows trivially. To prove the converse, suppose that / is not uniformly continuous. Then for some e > 0 we can find sequences (an), (bn) in X such that d(an, bn) -> 0 while e(f(an),f(bn))>e. If we can find an infinite index set / such that E(A, B) > 0, where A = {f(an):nel}, B = {f(bn):nel}, and E(A,B) = inf {e(a,b):aeA,beB}, then it will follow t h a t / i s not proximally continuous. If

C = {f(an):neN}u{f(bn):neN} has an infinite subset of diameter at most e/2, then clearly we can find a subset consisting solely of/(a n )'s (or/(6n)'s). In the case that all elements of this subset of/(a n )'s (resp./(frn)'s) are within

BASIC PROPERTIES

23

e/2 distance of each other, every f(hn) (resp. f(an)) is at least e/2 distant from all/(a n )'s (resp. /(6w)'s) and the result follows. Otherwise, the e/4-neighbourhood of any of these points contains only finitely many others, and so by induction we can find an infinite set / such that E(A, B) ^ e/4, completing the proof. (4.9) D E F I N I T I O N . Two proximity spaces (X, Sx) and (Y, 82) are called proximally isomorphic (or 5-homeomorphic) iff there exists a one-to-one mapping ffrom X onto Y such that both f and / - 1 are proximity mappings. Such a mapping f is termed a proximity isomorphism or #-homeomorphism. The term equimorphism is also used. It follows from Theorem (4.2) that proximally isomorphic spaces are homeomorphic. However, as can be seen from the example of (2.18), homeomorphic spaces are not necessarily proximally isomorphic. A property is said to be a proximity invariant iff it is preserved under proximity isomorphisms. That every topological invariant is a proximity invariant is clear, again from Theorem (4.2). (4.10) D E F I N I T I O N . / / (X,S) is a proximity space and Y <= X, then subsets of Y are also subsets of X. For subsets A and B of Y, we define (4.11)

A8YB

iff

ASB.

It is easily verified that 8Y is a proximity on Y and that T(8Y) is the subspace topology induced on Y by r(8). We call 8Y the induced (or subspace) proximity. We next consider the product of a family {(Xa, Sa):aeA} of proximity spaces. Let X = X l a denote the Cartesian product aeA

of these spaces. A proximity 8 can be defined on X as follows: (4.12) Let A and B be subsets of X. Define A SB iff for each pair of finite covers {Afi = 1, ...,m} and {Bfj = 1, ...,n} of A and B respectively, there exists an Ai and a Bj such that Pa[A^\ 8aPa[Bj]

for each

(Pa denotes the projection of X onto Xa.)

aeA.

24

PROXIMITY SPACES

(4.13) T H E O R E M . The binary relation S defined in (4.12) is a proximity on the product space X. It is separated iff each 8a is separated. Proof: (i) Since each 8a is symmetric, so is S and (1.1) is satisfied. (ii) Let A, B and G be subsets of X and suppose A SG. Since every cover of (A U B) is a cover of A, it follows that {A [) B)SG. Conversely, suppose A$G and B $C. Then there are finite covers {Afi = 1, ...,m} and {Gfj = 1, ...,n} of A and C respectively such that Pa\Ai\$aPa\Gj\ for some a = S^EA, where i = 1, ...,m and j = 1, ...,n. Likewise, there are finite covers [Afi = m + 1 , ...,m+#>} and {Dk\k = 1, ...,#} of 5 and (7 respectively such that Pa[^4i]<JaPa[i)A.] for some , where i = m + 1 , ...,m+p and &=1,...,#. Now 1> ...j?&; & = 1, ••.,#} is a cover of C and

is a cover of ^4 u B. Since Pa [^lJ^a Pa[C^n -D/^J for a = stj or a = tik, we conclude that (A \}B)$G. (iii) That A SB implies that A and B are non-void follows easily. (iv) If A$B, then there exist covers {Afi = l,...,m} and {#. : j = 19... 9 n} of ^4 and JB respectively such that -P a [-4J* a ^[^]

for some « = ^ e A ,

where i = 1,..., m and j = l,...,n. Since each (X a, <Ja) is a proximity space, there exist E^ such that Ps|yl JfigEyand

Set ^ . = Pi K^ij]

and JS7 = U % Since P s [^.] c ^ , w e have

j] for 5 = ^.; that is, A $E. Let m

and

JJ = (Z-B i )= U 1

BASIC PROPERTIES

Then (X-E)=

25

f) F. Since 3= 1

(Xs-E^^Bj

and Dtj = P^(X8-Eij)9

BjfiDij for all i and j . This implies BJSFJ for all j ; thus (X-E)9BJ

for all j ,

showing that (X -E)$B. (v) If ^4 n B 4= 0 , then there exists an x = (xa)eA n 5 . For every pair of covers {Afi = l,...,m) and {^:j = l,...,w} of A and .S respectively, there exists an Ai and a 2^- containing x. Clearly xaGPa[Ai] n P a [ ^ ] for all ae A, which implies for ; Mi^aa^j] that is A SB. (vi) Suppose X is separated and xa 8a ya for points xa, ya in Xa. There are two points in X, x and y say, such that x = (xA) and V = (2/A)> where xx = yx for all AeA except A = a. Then ##£/, implying x = y. That is, xa = ya, showing Sa is separated. Conversely, x = (xa) Sy = (ya) implies xaSaya for all a e A. If each 8a is separated, then xa = ya for all as A, i.e. x = y. (4.14) COROLLARY.^4 mapping ffrom a proximity space (F, ^ ) to X = X, Xa is a proximity mapping if and only if the composition aeA

Paof :Y->Xa is a proximity mapping for each projection Pa. Consequently, S is the smallest proximity on X for which each projection Pa is proximally continuous. Proof: We need only prove that if each Pa of is proximally continuous, then so is/. Let A and B be subsets of Y such that A S± B, and let {Afi = 1, ...,m} and {By.j = 1, ...,n} be finite covers of f(A) and f(B) respectively. Then for each

Pa[f(A)]8aPa[fm

«O that

Hence by (1.2), Pa[Ai]SaPa[Bj] for some i,j. From (4.12) we then have/(^4) df(B), showing t h a t / i s proximally continuous. The second part of the above corollary reminds one of the manner in which the Tychonoff product topology is defined on the

26

PROXIMITY SPACES

Cartesian product of a collection of topological spaces. Similarly, an analogous definition to that of the quotient topology is used in the case of quotient proximity. We shall not give the definition here, however, as we do not make use of this concept. Notes 1. The axioms for a proximity space were originally given by Efremovic [18, 19], although they appeared in a slightly different but equivalent form to those presented in this section. The equivalence of the Symmetry Axiom for a proximity to the complete regularity of the associated topology has been shown by Pervin [84]. 2, 3. The theorems in these sections are mainly due to Efremovic [19], as is the concept of a (^-neighbourhood. They have been collectively presented by Smirnov [97, 98] in his early survey of proximity spaces. In the same survey, Smirnov announced that every completely regular space has a maximal associated proximity structure. The example referred to in Remarks (3.8) was contributed by Pervin [86]. For a clear proof of Urysohn's lemma, the reader is referred to page 100 of Thron [115]. It should be observed that, as with most order relations, there is no general agreement on the definition of a partial order on the set of all proximities. 4. The results concerning proximity mappings were first established by Smirnov [98]. A brief discussion of most of the topics covered in this section can be found in Dowker [17]. For an account of proximity on the product of proximity spaces, see Leader [57].

CHAPTER 2

COMPACTIFICATIONS OF PROXIMITY SPACES 5. Clusters and ultrafilters Ultrafilters play an important role in topological spaces inasmuch as such notions as convergence and compactness can be characterized in terms of ultrafilters. In this section we consider their counterparts, namely clusters, in proximity spaces. We show that ultrafilters and clusters are closely related, and use this relationship to derive several important results in the theory of proximity spaces. It is well known that a family S£ of subsets of a non-empty set X is an ultrafilter if and only if the following conditions are satisfied: (5.1)

If A and B belong to J§^, then A n B + 0 .

(5.2)

If A n C # 0 for every C e&, then

(5.3)

If (A U B) e£>, then A e& or

It is natural to expect that the collections of sets in a proximity space satisfying conditions similar to (5.1)—(5.3), with nearness replacing non-empty intersection, should be valuable in the theory of proximity spaces. This is indeed the case and we are led to the following definition: (5.4) D E F I N I T I O N . ^ collection cr of subsets of a proximity space (X, 8) is called a cluster iff the following conditions are satisfied: (i) If A and B belong to a, then A SB. (ii) If A8C for every Cecr, then Aecr. (iii) / / (A\J B)e a, then Aecr or Be a. (5.5)

REMARKS,

(i) For each xeX, the collection crx = {A ^X: [27]

28

PROXIMITY SPACES

is a cluster. We call such a cluster a point cluster and use the above notation. (ii) If {x} e a for some x e X, then or = crx. If (X, S) is separated, then no cluster can contain more than one point since that would contradict (1.6); consequently, x 4= y implies crx 4= ory. (iii) If crisany clusterin X, then X e or by 5.4 (ii). Hence for each subset E of X, either Eecr or (X-E)ea. Recall that an ultrafilter also has this property. (iv) If A e a and A c B, then B e or. This too is a property of an ultrafilter. (v) If or is any cluster in X, then A e cr iff A e or. This follows from (2.8), 5.4(ii) and 5.5(iv). (5.6) LEMMA. If cr1 and cr2 are two clusters in a proximity space {X,8) such that cr1 c: a2, then a1 = cr2. Proof. If Aecr2, then A8C for every (7GO*2. Since cr1 c: cr2, ^4 #1? for every Becr^ which shows that AEC^ Thus ,then there exists an ultrafilter J§? such that (a)

AOE^

and (b) ^ c 0>.

Proof. By Zorn's lemma, there exists a maximal collection ? (of subsets of X) satisfying (a) and (6')

AtE^

for i=l,...,n

implies

Obviously 0 £ JS?. If ^ and B belong to J^ then by (6'), Since J§? is maximal, we must have AOBEJJ?. If A e ££ and A <= D, then DESP and hence belongs toJ§? since JSf is maximal. Having shown that J§? is a filter, it remains to show that jSf is an ultrafilter. Supposing the contrary, there would exist a subset E of X such that neither i£ nor (X — E) belongs to J5f. Hence there are sets Ax and A2 in Jg? such that neither AX[\E nor ^42 n (X - 2£)

COMPACTIFICATIONS OF PROXIMITY SPACES

29

belongs to SP. If A = A1(]A2, then A e^ while neither A n E nor A n (X — E) belongs to SP, a contradiction. (5.8) T H E O R E M . A collection cr of subsets of a proximity space (X, S) is a cluster if and only if there exists an ultrafilter ££ in X such that (5.9)

cr = {Acz X.ASB

for every

Be^}.

Moreover, given a and Aoecr, there exists an ££ satisfying (5.9) which contains Ao. Proof. Let J? be an ultrafilter in X and let a* be defined by (5.9). We shall first show that a is a cluster. (i) Suppose A and B belong to cr. For every subset G, either G or (X — G) is in J§?. This means that both A and B are near to either G or (X — C). Hence for every subset C, either A8G or (X-G)SB which shows (by the Strong Axiom) that A SB. (ii) Since J§? c
and

B9(A'[\B').

Thus ( i u 5 ) < J ( 4 ' n 5 ' ) . Since (^'n£')eJ§?, it follows that {A u B) $ a, as required. Conversely, let or be a cluster and suppose A0eor. Taking ^ = cr in Lemma (5.7), we obtain an ultrafilter J§? c a* such that ^40GJSP. If cr' = £4 c Z : ^ * S for every Be&}, then cr c cr'. Thus by Lemma (5.6), cr = cr', and (5.9) is satisfied. (5.10) COROLLARY. If ££ is an ultrafilter such that 3?
30

PROXIMITY SPACES

by Theorem (2.11). Now consider the point cluster cr0. Let JS^ and J§?2 be ultrafilters containing the filter bases #\ = {(-a,0):a > 0} and J^ = {(0,a):a > 0} respectively. Then JSfx and j£?2 both generate cr0, but JSfx =t=«^V (5.12) LEMMA. If a cluster a in a proximity space (X, 8) is determined by an ultrafilter££, then a is a point cluster crx if and only if ££ converges to x. Proof, o* = CTX iff {x} e a iff x S A for every A e 3? iff x is a cluster point of J§? iff JS? converges to a?. A topological space is compact if and only if every ultrafilter in the space converges to a point. The following analogue of this result follows directly from the above lemma. (5.13) T H E O R E M . A proximity space is compact if and only if every cluster in the space is a point cluster. If A n B =(= 0 , then there exists an ultrafilter J§? which contains both A and B. A similar result holds for clusters in proximity spaces: (5.14) T H E O R E M . If A SB, then there exists a cluster a in (X, 8) such that A and B both belong to cr. Proof. Let 0> = {C a X:CSB}. From Lemma (5.7), there exists an ultrafilter JSf such that Ae<£ <=-3P. The cluster a determined by J? contains both A and B. (5.15) R E M A R K S , (i) Theorems (5.13) and (5.14) together yield the result: two subsets of a compact proximity space are near if and only if their closures intersect. Therefore, a compact completely regular space has a unique compatible proximity (cf. Theorem (3.7)). (ii) If A SB, then there may be different clusters or and ar such that A and B are members of both. For example, let X be the real line with the usual topology and define A SB iff I n 5 + 0 . If A = [0,1] and B = (0,1), then A SB, and both sets belong to the point clusters cr0 and alm

COMPACTIFICATIONS OF PROXIMITY SPACES

31

We now prove a result similar to: if J§? is an ultrafilter in Y and X a Y, then the trace of j£? on X is an ultrafilter in X iff (5.16) T H E O R E M . Let a be a cluster in a 'proximity space (Y, S) and let XECT. Then there exists a unique cluster in (X, Sx) contained in or, namely crf = {A c X: A e or}. Proof. By Theorem (5.8), or is determined by an ultrafilter & containing X. Then &x = {Ln X.LE^}, the trace of & on X, is an ultrafilter in X and so generates a cluster cr' in X. If Aecr', then i * ( i n l ) for each LE&. This implies ^ SL for each LE£? , i.e. ^4 ecr. Thus <x' c or, and clearly
for each

Proof. cr1 is determined by an ultrafilter J? in X. Now f(<3?) is an ultrafilter base in Y and generates a cluster a2 in Y. If A S2f(B) for every BECTV then A S2f(L) for every LEJ£, SO that A E cr2. To prove the reverse inclusion, we first note that This follows from the fact that if BECT-^, then BS±L for every LEJ?, and/being a proximity mapping implies f(B)S2f(L) for

32

PROXIMITY SPACES

each L eJ§?, i.e. f(B) E C 2 . Thus if A e o*2, then ^4 S2f(B) for every The following result corresponds to the well-known theorem: if J§? is an ultrafilter in X and J c J , then J§? is an ultrafilter base in Y and thus generates an ultrafilter in Y. (5.18) COROLLARY. If X is a subspace of a proximity space (Y, 8), then every cluster cr' in X is a subclass of a unique cluster cr in Y, and (r = {Acz Y:A8B for every Bea'}. We now present an axiomatic characterization of the family of all clusters in a separated proximity space. (5.19) D E F I N I T I O N . A semi-ultrafilter SP in a setX is a collection of non-empty subsets of X satisfying the following conditions: (i) (ii) (iii) (iv)

/ / (A [)B)eSr, then Ae^or / / A eS? and A cz B, then If A n S 4= 0 for every 8 e&, then / / {x} and {y} both belong to £f, then x = y.

Although every ultrafilter is clearly a semi-ultrafilter, the converse is not true. Every cluster in a separated proximity space is a semi-ultrafilter, showing that a semi-ultrafilter is not necessarily a filter. (5.20) T H E O R E M . Let <€ denote the family of all clusters in a separated proximity space (X, 8). Then the following conditions are satisfied: (i) Let A and B be subsets of X. If for every E c X there is a cre^S such that either A, Eea or (X — E), Be a, then there is a cr'G^7 such that A, Beo~f. (ii) Consider any cre^. If for each Beer there is a a' e<£? such that A, Beer', then Aecr. (iii) Each ultrafilter in X is contained in some member of <S. Proof, (i) is a consequence of the Strong Axiom and Theorem (5.14). Condition (ii) follows from the fact that any two sets in a cluster are near to one another, and (iii) is proved in Theorem (5.8).

COMPACTIFICATIONS OF PROXIMITY SPACES

33

(5.21) T H E O R E M . 7/ ^ is a family of semi-ultrafilters in X satisfying conditions 5.20 (i), (ii) and (iii), then there exists a separated proximity S on X such that *%> is the family of all clusters in (X,S). Proof. If ^ is a family of semi-ultrafilters satisfying 5.20 (i), (ii) and (iii), define 8 by: (5.22)

A SB iff there is a ore <€ such that A, Be a.

(Theorem (5.14) provides a motivation for this definition.) That 8 is a separated proximity on X will now be verified. (1) Symmetry is obvious from (5.22). (2) (AuB)8C iff there exists a cretf such that (A U B), Ceo-. This is equivalent to A, Cecr or B, Cecr, i.e. A 8G or B8C. (3) That A$ 0 follows from the fact that 0 does not belong to any semi-ultrafilter. (4) If for every subset E of X either A 8E or {X-E) 8B, then there exists a cre^ such that A,Eecr or (X — E), Bear. Since *€ satisfies 5.20(i) by hypothesis, there exists a cr' e^ such that A,Beer1, i.e. A SB. (5) A n B =(= 0 implies A and B both belong to some ultrafilter^Sf. Hence by 5.20 (iii), there exists a cre^ such that j£? <=• <x; that is, both A and B belong to
34

PROXIMITY SPACES

space to be a set X together with a family of semi-ultrafilters satisfying 5.20 (i), (ii) and (iii). Each member of this family will be called a cluster. This leads to an alternate treatment of proximity spaces and enables one to prove the compactification of a proximity space (see Section 7) without appealing to the axiom of choice. 6. Duality in proximity spaces In Section 3, we saw an alternate way of studying proximity spaces using (^-neighbourhoods. If 8 is a proximity on X, then 8 and <^ are dual relations on the power set of X in the following sense: (6.1) D E F I N I T I O N . TWO relations /? and /?* on the power set of X are dual iff Aft*B is equivalent to A/j!(X — B). Clearly y?** =ft,justifying the term dual. In this section, a study is made of this duality with special reference to clusters and their duals, 'ends'. (6.2) D E F I N I T I O N . Let & be a class of subsets of a proximity space {X,8). Define: (a) f ° = {£ c I : there exists an A e^such that A <| E). (b) <3' = {E aX:E 8A for every A e &}. (c) &* = {EczX:(X-E)$ #}, the dual of <$. (6.3) L E M M A . ^ ° and <&' are dual classes. Proof. Ee&°iff there exists an A e ^such that A <4 E. This is equivalent to the existence of an A e@ such that (X — E)$A,

(6.4) D E F I N I T I O N . An end J^ in a proximity space (X,8) is a collection of subsets of X satisfying the following conditions: (i) B,Ce^ implies the existence of a non-empty subset Ae^ such that A <4 B and A <^C. (X-A)e#rorBe^r. (ii) If A < B, then either (6.5) D E F I N I T I O N . A round filter J5" is a filter with the additional property that for each F1e^, there exists an F2e^ such

COMPACTIFICATIONS OF PROXIMITY SPACES

35

(6.6) R E M A R K S . Clearly if !F is an end in X, then 0 $!F and X e !F. An example of an end, and also of a round filter, is the system Jfx of (^-neighbourhoods of a point x e X. If !F is a maximal round filter converging to x, then J^ = JVX. Some authors use the term regular filter rather than round filter. A collection of sets fF from a proximity space is said to be a centred 8-system iff (a) A, Be^ implies A n B == j 0 , and (b) Ae
36

PROXIMITY SPACES

belong to ^ ° . Now by 3.9 (vi), there is an R such that (A n E) < R < (P n Q). By taking P = Q and noting that (A n E) e ^implies Re <&*, we see that the filter ^ ° is round. Since A <4 B, we have A n E <§ £ by 3.9(iii), so that Be@°. Finally, to show that ^ ° is finer than #~, suppose E e J^. Since #" is a round filter, there exists Fe 3F such that F
THEOREM.

^ is anendifand only if it is a maximal round

filter. Proof. In view of Theorem (6.7), it is sufficient to show that every maximal round filter J^ is an end. Condition 6.4 (i) is clearly satisfied by any round filter. In verifying 6.4 (ii), suppose A <^ B and J5^J^. Since J^ is maximal, Lemma (6.8) guarantees the existence of an E e SF such that A n E = 0. Thus E c (X-A) and (X-A)e^ since ^ is a filter, proving 6.4(ii). (6.10) C O R O L L A R Y . Every round filter is a subclass of some end. The following is the main result of this section, pointing out the duality between clusters and ends. (6.11) THEOREM. !F is an end if and only if J^* is a cluster. Proof. If J^* is a cluster, then: (a*) A, B E^* implies A SB. (&*) A SB for every Be J^* implies (c*) (^U^jG^^implies^GJ^^or The dual J^ of J^* satisfies the following conditions: (a) A<^B implies ( I - i ) G F o r £ e ^ . (b) Given 5 e J^, there is an E such that (X-E)$ & and JB7 < J5. (c) A,Be^ implies An Be 3*. Note that (a) is simply a restatement of 6.4 (ii). If SF is an end, it clearly satisfies (a) and (c). That (6) is satisfied follows from 6.4 (i), which states that given BetF, there exists an Ee^ such that E satisfies (a), (b) and (c), then 8F is an end. In doing so, we need only verify 6.4 (i). In view of (c) and 3.9 (iii), it is sufficient to show that given &Be J^, there

COMPACTIFICATIONS OF PROXIMITY SPACES

37

exists a non-empty AetF such that A <^B. If Be^, then (b) assures the existence of an E such that E <§ B and (X-E)$ &. By 3.9 (vi), there exists an A such that E <^A<^B. Since E <^ A and (X-E)$ &, we see from (a) that A e J^. Finally, (X-E)^^ implies i? 4= 0 , and so A 4= 0 . (6.12) C O R O L L A R Y . .For discrete proximity spaces, ends and clusters both coincide with ultrafilters. (6.13)

COROLLARY.

^° is an end if and only if <S' is a cluster.

(6.14) T H E O R E M . Every ultrafilter £F in a proximity space contains a unique end JF0. Proof. By Corollary (5.10), J^ is a subclass of a unique cluster, which in this case is 3F'. Taking duals, we obtain J ^ * c J^*. Using Lemma (6.3) and the self-duality of J^, this reduces to the statement that J^0 <= J^. That J^° is an end follows from Corollary (6.13). To prove uniqueness, suppose that 2? is any end contained in 3F. Then by (6.11) S?* is a cluster containing J^, and hence equals &'. Thus ^ = ^ / * = ^ ° , by (6.3). The following is the dual of Theorem (5.14): (6.15) T H E O R E M . A <^ Bin a proximity space (X, S) if and only if every end in X contains either (X — A) or B. Proof. Necessity is supplied by (6.4) (ii). To prove the converse, suppose A ^ B, i.e. AS(X-B). By Theorem (5.14), there exists a cluster J^* containing both A and (X — B). Hence both (X — A) and B are not in !F which, by Theorem (6.11), is an end. The axiomatic characterization of clusters (Theorems (5.20), (5.21)) has the following analogue regarding ends: (6.16) T H E O R E M . Let O be a family of filters (called -filter which does not contain C

38

PROXIMITY SPACES

contains B, and every <&-filier which does not contain D contains (X-B). (iv) Every ultrafilter in X contains some O-filter as a subclass. Proof: Let O be the family of all ends in X. By (6.7), every end is a filter. 6.16 (i) is a consequence of 6.4 (ii) and 3.9 (vii). In view of Theorem (6.15), 6.16(ii) follows from 6.4(i) while 6.16(iii) is equivalent to 3.9(vi). Statement 6.16(iv) is simply Theorem (6.14). To prove sufficiency, suppose O is a family of filters satisfying 6.16(i)-(iv). Motivated by Theorem (6.15), we define, A <^ B iff every O-filter contains either (X — A) or B. Verification that < satisfies 3.9(i)-(vii) is left to the reader. That every O-filter is an end follows from the definition of <^ and 6.16(ii). Finally, since every end is contained in some ultrafilter, (6.14) and 6.16 (iv) together imply that every end in X is a O-filter.

7. Smirnov compactification It should be mentioned at the outset that throughout this section, we shall work exclusively with separated proximity spaces. In Theorem (3.7), it was shown that every compact Hausdorff space has a unique compatible separated proximity. The problem considered in this section is to determine whether or not every separated proximity space (X, 8) can be embedded in a compact proximity space Y such that the proximity S is the subspace proximity induced by the unique proximity on Y. This can indeed be done; in fact, there is an intimate relationship between the proximities associated with a Tychonoff space and its compactifications. A one-to-one order-preserving mapping (i.e. an order-isomorphism) exists between the two, and consequently the study of proximities can be reduced to that of compactifications. We have already seen (Remarks (2.18)) that the real line has two distinct compatible proximities. These actually correspond to two compactifications of the real line, namely the Alexandroff one-point compactification and the two-point compactification. Using clusters, we shall now construct the Smirnov compactification of a separated proximity space. Let (X, S) be a separated

COMPACTIFICATIONS OF PROXIMITY SPACES

39

proximity space and let 2£ denote the set of all clusters in X. For icj let _ For xeX, letf(x) = o~x (the point cluster). Then it is easy to see that (i) / i s a one-to-one mapping, and (ii) f(A) cz ^ It is to obtain property (i) that we insist on the proximity 8 being separated, for we are then assured that each point xeX is a member of one and only one cluster in X. (7.1) D E F I N I T I O N . For SP <= <%?, we say that a subset A of X absorbs SP iff A ear for every aeSP, i.e. SP c srf. (7.2) L E M M A . Tine binary relation 8* on the power set of SC defined by (7.3) SPS*St iff A absorbs SP and B absorbs J2 implies A SB, is a separated proximity on SC. Proof, (i) Symmetry of 8* follows from that of 8. (ii) Suppose that £8* 01, that D absorbs {SP \}2), and that C absorbs St. Then D absorbs 1 and hence DSC. Thus {SP U 3) 8* St. Conversely, suppose that {SP\}£)8*01 and that &$*&. Let B absorb 3 and G absorb Si. Then we must show that B8G. Since 3P$*M, there are sets A and D absorbing SP and 3% respectively such that A$D. By the Strong Axiom, there is an E such that A$E and {X-E)$D. Because D absorbs St and (X-E)SD, (X — E) belongs to no cluster in Si. Consequently, (G — E) belongs to no cluster in Si. But G = (G — E)[) (G n E) absorbs Si, implying that G n E absorbs Si. Now (A\}B) absorbs {SP\)&), which shows that (A u B)8(G n E). Since A$E, we also have AS(C(]E) and hence BS(G(]E), implying that BSG. (iii) That SP 8* 21 implies SP and St are non-empty follows directly from (1.3). (iv) If SP$* St, there are sets A and B absorbing SP and St respectively such that A$B. By the Strong Axiom, there is an E such that A $E and (X -E)$B. Since (X-E)SB and B absorbs J2, (X - E) belongs to no cluster in St. Thus E absorbs St. Now let

40

PBOXIMITY SPACES

g$ = i. Then &$*0!t since A absorbs SP, E absorbs M and A SE. Since E belongs to no cluster in (2C — 3&), (X-E) absorbs (3T-31). Therefore (X-E)SB implies \sC-gi)$* J. (v) Suppose that 0> n £ * 0 and that ^4 and B absorb & and .2 respectively. Then both A and J? absorb 0> n =2, so that .4 £JB and hence &>8* 21. (vi) J. absorbs {a} where creS? iff Aecr. Hence cr18*o'2 iff every set in vx is in cr2 and vice versa iff cr1 and cr2 coincide. Thus 5* is a separated proximity on SC. NOTATION.

Let r* be the topology induced on iFby 5*.

(7.4) LEMMA. (X,S) is proximally isomorphic to f(X) with the subspace proximity induced by 8*, andf(X) is dense in SC. Proof. We first note that A absorbs f(B) iff B c A (r(8)closure of A). Hence if SI is any subset of SE, then MS*f(A) iff C absorbs J and D absorbs/(-4) implies C8D iff C absorbs M and i c 5 implies CSD iff (7 absorbs St implies G8A. Taking 3, to be the singleton {a}, we obtain {o-} **/(^4) iff C e o- implies C 8 A iff .4 GO*. That is, 88 is the r*-closure off(B). Therefore, since X belongs to every cluster,/(X) is dense in $£. Now/(.4)£*/(£) iSCSD whenever C and D absorb/(4) and f(B) respectively iff G8D whenever A ^C and B c D. But this last statement is equivalent to A SB, so that f(A) 8*f(B) iff A SB. Thus X is proximally isomorphic tof(X). (7.5) LEMMA. (3£,T*) is compact. Proof. By Theorem (5.13), it suffices to show that an arbitrary cluster or in 2£ is a point cluster. Since/(X) is dense in ^ 5.5 (v) implies that f(X)eo". From Theorem (5.16), there is a unique cluster a' inf(X) such that or' cz cr. But X is proximally isomorphic to/(X), so that there is a cluster or" in X which corresponds to a'.

COMPACTIFICATIONS OF PROXIMITY SPACES

41

To be specific, a' = {f(A )\AE(T"}. NOW from the proof of Lemma (7.4), we know that \cr"}8*f{A) iff A ea". Applying Corollary (5.18), we obtain er"' ecr' c cr, showing that cr is indeed a point cluster. An alternative proof of the next lemma, which states that the compactification (SC, 8*) of (X, 8) is essentially unique, is given in Remarks (7.9). (7.6) L E M M A . Any 8-homeomorphism g of (X,8) onto a dense subset of a compact proximity space (Y, 8X) extends to a 8-homeomorphism g of (SC, 8*) onto (Y, S±). Proof. Suppose (X, 8) is cMiomeomorphic to a dense subset of a compact proximity space (Y,Sj). To each cluster and 21 are subsets of 3C such that &> 8* 2,, then #f) J" + 0 . Hence there exists a crG^such that {a-} 8* 2P and {a} 8* 1. Let y = g{cr). With the help of (1.4) and 5.4(iii), we obtain {y]81g(^)) and {y}81g(^), whence g(^) 8^(3). Conversely, g{0*) 81g(^) implies the existence of a point yEg(^)(]g(M)9 since Y is compact. Let or = g^iy). Now if A ecr and B absorbs SP, then A8g{SP) and g(0>) a B (considering X as a subset of Y). But this implies A SB, so that {or} 8*SP. Similarly, {cr}8*J2, from which we conclude that Combining the above sequence of lemmas we obtain the main result of this section: (7.7) T H E O R E M . Every separated proximity space (X,S) is a dense subspace of a unique (up to 8-homeomorphism) compact Hausdorff space S£. Since 2£ has a unique compatible separated proximity, subsets A and B of X are near iff their closures in SC intersect [SE is called the Smirnov compactification of X).

42

PROXIMITY SPACES

(7.8) COROLLARY. The topology of a separated proximity space is Tychonoff. (7.9) R E M A R K S . In the statement of Theorem (7.7), we have identified X with/(X) as is usually done. The Smirnov compactification of (X, 8) can also be constructed by embedding (X, 8) in the proximity space (Y, 5**), where Y is the family of all ends in X and 5** is defined in either of the two following equivalent ways: (i) For subsets 77^ and n2 of Y, n18**7T2 iff for every finite collection of sets {Bfi = 1, ...,n} such that Bi^>Ai where n

U Ai = X, there exist Fen1 and Gen2 such that Bke(F ft G) for some 1 < h < n. (ii) Setting O(A) = {ot:Aea,aeY}, define TT1$**TT2 iff there exist sets A and B such that A SB, n1 c O(A) and n2 c O(B). Since 7.9 (ii) coincides with definition (7.3), it follows from (7.5) that (Y,T(8**)) is compact. Since the Smirnov compactification is essentially unique, the duality existing between clusters and ends is again brought to our attention. The following theorem was proved by Smirnov [136] in 1952: In order that two compactifications of the space X be distinct, it is necessary and sufficient that there exist a pair of closed subsets of the space X, the closures of which intersect in one compactification and do not intersect in the other compactification. This result may be used to provide a simple proof of the essential uniqueness of the compactification of a separated proximity space. For, suppose that a proximity space (X, 8) has two distinct compactifications (Y^S^) and (Y2,82). Then there does not exist a 5-homeomorphism between Y± and Y2 which is the identity on X. Hence, using Theorem (4.4), there is no homeomorphism from Yx onto Y2 which is the identity on X. Applying the above theorem together with Theorem (3.7), we then see that there must exist a pair of closed subsets of X which are near in one compactification but not near in the other. But this contradicts the fact that both Yx and Y2 are ^-extensions of the proximity space

COMPACTIFICATIONS OF PROXIMITY SPACES

43

(7.10) T H E O R E M . Every proximity mapping g of (X, S±) onto (Y, 82) has a unique extension to a continuous mapping g which maps the compactification of X onto the compactification of Y. Proof. It follows from Theorem (5.17) that if o'1 is a cluster in X, there corresponds a cluster cr2 in Y such that o-2 = {P a Y:P82g(C) for every Ceo-J. Let gio-^) = cr2. Then g is a mapping from 2E to <&. Clearly g maps the point cluster ) and B absorbs g(M) then A S2B. If A S2B, then by Lemma (3.3) there are subsets 0 and D of Y such that A92(Y-C), B$2(Y-D) and G92D. Since A absorbs g(0*), (Y — C) belongs to no cluster in g(SP). But g is a proximity mapping, so that g~\Y — C) — (X — g~1(G)) belongs to no cluster in SP. This shows that g~\C) absorbs 2P. Similarly, g~\B) absorbs J . Since 0*8* J£, we must have g-\C)8ig-\D). But g is a proximity mapping, yielding C82D, a contradiction. Therefore g must be a proximity mapping. That g{3T) = & follows from the facts that f(Y) c g(3C) c
44

PROXIMITY SPACES

We have already defined, in Section 2, a partial order on the set of all proximities on a set X. The following important result follows directly from Theorems (3.7) and (7.10). (7.11) T H E O R E M . Given any Tychonoff space X, the Smirnov compactification defines an order-isomorphism u of the partially ordered set {Sfiel} of compatible proximities onto the partially ordered set {Xfiiel} of Smirnov compactifications. That is, 8t > Sj iffXf ^ Xf where u(8k) = Xt for he I. The above theorem is actually valid for a completely regular space as well. Following is an analogue of Urysohn's lemma for normal spaces: (7.12) T H E O R E M . In a proximity space (X,S), A$B implies that there exists a proximity mapping g:X->[0,l] such that g(A) = 0andg(B) = 1. _ Proof. IfA$B, then A n B = 0 in 9£. But % is normal so that by Urysohn's lemma, there exists a continuous function [0,1] such that g(A) = 0and^(5) = 1. By Theorem (4.4), g is a proximity mapping. Setting g = g\f(X) and identifying X with/(X) where f(x) = crx, we obtain the required mapping. The next result follows directly from Tietze's extension theorem and Theorem (7.10). (7.13) T H E O R E M . Let A be any subspace of a proximity space X and let g be a proximity mapping from A to [0,1]. Then g can be extended to a proximity mapping g: X -> [0,1]. The compactification of a proximity space, which we have considered above, is a special case of the general theory of proximal extensions. We shall now consider briefly this general theory and show its relationship to the theory of compactifications. (7.14) D E F I N I T I O N . A proximal (or S-) extension of a proximity space (X,S) is a proximity space (Y,8') such that X= Y and 8 = S'x- A proximity space is maximal (or absolutely closed) iff it has no proper 8-extension. (7.15) T H E O R E M . A separated proximity space (X, 8)ismaximal if and only if every cluster in X is a point cluster.

COMPACTIFICATIONS OF PROXIMITY SPACES

45

Proof. If X is not maximal, there is a proper ^-extension Y; let £e Y — X. Then from Theorem (5.16), there exists a unique cluster a in X which is a subclass of the point cluster cr^ in Y. Obviously a is not a point cluster in X. Conversely, if there are clusters in X which are not point clusters, then the proximal extension of X given by (7.4) is proper. Thus X is not maximal. (7.16) C O R O L L A R Y . A separated proximity space is maximal if and only if it is compact. Every Tychonoff space has a maximal compactification, namely the Stone-Cech compactification. The following theorem characterizes those separated proximity spaces which have minimal compactifications, (7.17) T H E O R E M . A Tychonoff space X has a minimal compactification if and only if it is locally compact. Proof. If X is locally compact, we may resort to the Alexandroff one-point compactification, which is the minimal compactification of X. (Recall that the one-point compactification is Hausdorff iff X is locally compact Hausdorff.) Conversely, suppose that X has a minimal compactification X*. In order to show that X* is the Alexandroff one-point compactification, suppose there are two different points £ and y in (X* — X). We can then construct a smaller compactification, Xf, than X* by 'pasting' £ and y together. Open sets in X* are then those which do not contain £ or y, and those which contain both £ and y. (7.18) C O R O L L A R Y . Every Tychonoff space X has a maximal compatible proximity. It has a minimal compatible proximity if and only if it is locally compact. A separated proximity relation 8 may be defined on a locally compact Hausdorff space by (7.19) A$B\RA(\B= 0 and either A or B is compact. Indeed, (7.11) and (7.17) show that a space has a smallest compatible proximity only if it is locally compact Hausdorff, and in this case the proximity is given by (7.19). We have previously

40

PROXIMITY SPACES

seen that the largest proximity which can be denned on a Tychonoff space is A SB iff A and B are functionally distinguishable. If a Tychonoff space possesses a unique proximity, the maximal and minimal compactifications must clearly coincide. These considerations together with the preceding results yield the following important theorem: (7.20) T H E O R E M . A Tychonoff space X has a unique compatible proximity if and only if it is locally compact and every pair of non-compact closed subsets A and B of X are functionally indistinguishable. We now consider those topological spaces which have a unique compatible proximity given by A SB

iff AnB 4= 0 .

It has already been shown (Theorem (3.7)) that compact completely regular spaces are of this form. (7.21) D E F I N I T I O N . A separated proximity space (X,S) is equinormal iff A [) B = 0 implies A SB. It follows from Theorem (7.12) that every equinormal proximity space is normal. The converse is not true, however, since in (2.18) we have AnB = 0 although AS1B. The following is a generalization of Theorem (4.4). (7.22) T H E O R E M . A normal separated proximity space (X,S) is equinormal if and only if every real-valued continuous function on X is a proximity mapping. Proof. The proof of necessity is exactly the same as that for Theorem (4.4). To prove the converse, let X be a normal proximity space such that every real-valued continuous function on X is a proximity mapping. If A n B = 0, then by Urysohn's lemma there exists a continuous function /:X->[0,1] such that f(A) = 0 and f(B) = 1. By hypothesis, / is also a proximity mapping and so ASB, i.e. ASB. (7.23) COROLLARY. A normal separated proximity space (X, S) is equinormal iff the proximity S is induced by the Stone-Cech compactification of X.

COMPACTIFICATIONS OF PROXIMITY SPACES

47

(7.24) R E M A R K . If/: X -> Y is continuous, 8 is the proximity on X corresponding to the Stone-Cech compactification 2E, and 8' is any proximity on Y, then/is a proximity map with respect to * and 8'. 8. Proximity weight and compactification In the previous section we studied compactifications of proximity spaces and the various implications of this theory. In this section we shall define topological weight and proximity weight, and prove stronger results concerning compactifications. These developments will lead to some interesting metrization theorems concerning proximity spaces. (8.1) D E F I N I T I O N . The topological weight w(r) of a topological space (X, r) is the smallest cardinal number a ^ Ko such that r has a topological base 38 with \£#\ ^ a. It is easy to see that w(r) is a topological invariant. (8.2) D E F I N I T I O N . A proximity base 2P for a proximity space (X, 8) is a subset of the power set of X such that A$B implies there are members U and V of SP such that A cz U, B cz V and U$V. It is obvious from Lemma (2.8) that the collection of all closed sets, in a proximity space, forms a proximity base. (8.3)

ILEMMA.

If & is a proximity base, then so is

Proof. IfA$B, then by (3.5) there are sets C and D such that A <^ C, B <^ D and G $D. This in turn implies the existence of members U and V of & such that C a U, D c V and US V. But A cz IntO cz Int U <= U and B a I n t D cz Int V e V. Hence Int US Int F, and the conclusion follows. (8.4) COROLLARY. Given a proximity space (X,S), r(8) is a proximity base. (8.5) LEMMA. If & is a proximity base of open sets, then & is a topological base. Proof. If G is an open set and xeG, then x$(X — G). Hence there are sets U and V in 3P such that x e U, (X — G) cz V and

48

PROXIMITY SPACES

F. Also U n V = 0, which shows that xe U <= (X- V) c: G. Thus SP is a topological base. (8.6) T H E O R E M . Let 3% be a topological base for a compact proximity space (X, 8). Then 3#* = {B\B = \J Bi9 where Bi e38 and I is iel

finite} is a proximity base. Proof. From Lemma (2.8), A SB implies A(]B = 0. Since X is regular, for each x e A there is a Ux E3§ such t h a t xeUxc:Uxc:(X-B). _

_

r

r

Now A is compact, so that A <= U Ux. <= (J Ux. <=

_

(X-B).

Using a similar argument, we obtain _

Let

s

s _

U* - U C/c. and F* - U >;..

Then U*G&*, V*e@*, AczU*, B c F^ and [ 7 * n F * = 0 . Therefore J7* (J F*, and we conclude that 33* is a proximity base. The following result is obvious. (8.7) L E M M A . Let SP be a proximity base of X and let 7 c: X. Then 3PY = [U'nT:ZJ' <E3P} is a proximity base for the sub space 7 . (8.8) LEMMA. Let Ybea dense subspace of a proximity space X and let Slbea proximity base for 7 . Then 3P — {U :U e J2} is a proximity base for X. Proof. If A and B are subsets of X such that A SB, then by (3.5) or (8.4) there are open sets 0 and H such that A c G, B <= H and GSH. Hence (6rfi Y)SY(H[\ 7), which implies there are members U and F of J such that (Gn 7) <= U c 7, (J^n 7) c F c 7 and USY V. Since 7 = X, G c= (Gn 7) and ^ c (fin 7). Thus

(8.9) D E F I N I T I O N . TAe proximity weight w(<J) o/ a proximity space {X,8) is the smallest cardinal number a ^ Ko a proximity base SP with \3P\ < a.

COMPACTIFICATIONS OF PROXIMITY SPACES

49

Clearly w(S) is a proximity invariant. Applying Corollary (8.4) and Lemma (8.5), we obtain the following result: (8.10)

THEOREM.

In a proximity space (X,S) withr = T(8), W(T)

^ w(S) < 2W^\

Using this theorem together with Theorem (8.6), we obtain: (8.11) COROLLARY. For a compact proximity space (X9S), = w(S).

W(T)

(8.12) R E M A R K . It should be pointed out that the converse of (8.11) is false. For an infinite discrete space, w(r) = w(S) although the space is not compact. The next lemma follows easily from Lemmas (8.7) and (8.8). (8.13)

LEMMA.

/ / (Y, 8) is a subspace of (X, 8*), then

w(8) < w(8*). If Y is dense in X, their proximity weights are equal. We shall now consider a strengthened form of Theorem (7.7): namely, that the compactification of a separated proximity space can be effected so as to preserve the proximity weight. (8.14) T H E O R E M . Every separated proximity space (X, 8) with proximity weight d can be embedded 8-homeomorphically as a dense subspace of a compact Hausdorff space X* with the same proximity weight 6. X* is unique up to 8-homeomorphism. This theorem follows from (7.7) and (8.13), but will be proved alternatively below in a sequence of lemmas. The construction used here reminds one of a similar one used in constructing the Stone-Cech compactification of a Tychonoff space. Let 3P be a proximity base for X such that \0*\ < 6, and let [0,1] such that fy(U) = O and f (V) = 1. Now let K = X [0,1] be the Tychonoff cube of |^|-copies of [0,1]. Then K is a compact Hausdorff space and so, by Theorem (3.7), has a unique proximity 8* given by A8*B iff If] B+ 0. T>e&nef:X-^Kbyf(x) = (fy(x)).

50

PBOX1MITY SPACES

(8.15) LEMMA. The function f is one-to-one. Proof. If x 4= y, then x$y. Therefore there are members V and F of V such that x e U, y e F, fy(U) = 0 and / r (F) = 1, where y = (tf,F). Hence/(*) (8.16) LEMMA. / - 1 : / ( X ) -+ X is a proximity mapping Proof. It A and B are subsets of Z such that A$B, then there are members U and F of SP such that ^ cz U,B c F and £/# F. Let y0 = (?7, F). Now / ( ^ ) c X ^ and f(B) c X F^, where [7ro = {o}, F?« = {1} and W = V? = [0,1] for all other y. The sets containing/^) and/(J5) are closed and disjoint, showing that f(A)<$*f(B). Thus/" 1 is a proximity mapping. is a proximity (8.17) LEMMA. The function f:X->f(X) mapping. Proof. Since fy: X -> [0,1] is a proximity mapping, Theorem (7.10) assures the existence of a unique extension fy \2£-> [0,1] where fy is a proximity mapping on the Smirnov compactification^of X. Since fy is continuous for each ye %?,f: 3£ -^ if is also continuous, wheref(x) = (fy{x)). But «S"is compact, so t h a t / i s a proximity mapping. This, then, implies that / t o o is a proximity mapping, a s / = / | X . (8.18) D E F I N I T I O N . A proximity space is said to be metrizable iff it is 8-homeomorphic to a metric proximity space. (8.19) T H E O R E M . A proximity space (X,8) is proximally homeomorphic to a totally bounded metric space if and only if its proximity weight w(8) = No. Proof. Suppose w{8) = No. By Theorem (8.14), there exists a compactification SE of X with proximity weight Ko, which equals the topological weight w(r). But a Tychonoff topological space with W(T) =tf0is metrizable. Now a compact metric space is totally bounded and total boundedness is hereditary, showing that X is a totally bounded metric space. Conversely, if (X, 8) is proximally homeomorphic to a totally bounded metric space (Y,d) then Y*, the completion of Y, is compact and hence separable. Now a separable metric space has a countable base. Therefore by (8.11) the topological weight,

COMPACTIFICATIONS OF PROXIMITY SPACES

51

and hence the proximity weight, of both Y and T* is Ko. Thus the proximity weight of X is Ko as well. (8.20) COROLLARY. Every proximity space with proximity weight Ko is metrizable. (8.21) T H E O R E M . The proximity weight of a metric proximity space is Ko if and only if it is totally bounded. Proof. Sufficiency follows from Theorem (8.19). On the other hand, if a metric space (Y, d) has proximity weight Ko then Y is cMiomeomorphic to a totally bounded metric space (Y',d'). By (4.8) Y and Y' are uniformly isomorphic, so that Y too is totally bounded. The above theorem is a particular case of a more general result concerning the proximity weight of a metric space. (8.22) D E F I N I T I O N . A subset E of a metric space (Y,d) is e-discrete iff d(x, y) ^ efor every pair of distinct points x, y of E. Given e > 0, there exists, by Zorn's lemma, a maximal e-discrete subset Ee of Y; that is, i?e <= E c Y and E being e-discrete implies E = Ee. (8.23) T H E O R E M . Let {Y,d) be a metric space of topological weight i/r and proximity weight 6. For neN, let En be a maximal I/n- discrete subset of Y and let \En\ = crn. Then GO

f = 2 vn

00

and

71=1

0 = S 2 °^ 71 = 1

Proof. The family consists of pairwise disjoint non-empty open sets in Y. The cardinality of this family is crn, so that cn ^ i/r for each neN. Hence n=\

On the other hand,

covers Y and forms a topological base. The cardinality of this 00

CO

00

family is ^ 2 °"n> s o t h a t i/r ^ S crn. Thus \[r = 2 o*wn=l

n=l

7i = l 4-2

52

PROXIMITY SPACES

Every pair of disjoint subsets of En are not near, so that every proximity base of En must contain all subsets of En. Thus the proximity weight of En is ^ 2°« for each neN, showing that 00

0 ^ 2 2°". On the other hand, if two subsets A and B of Y are not 71=1

near, then D(A,B) = e > 0. Choose w such that \\n < e/5. Let [7= U s(s,-) and F = U Then A a U, B a V and D(£/, F) ^ 3e/5 > 0, implying that US* V. It is therefore clear that a proximity base for Y can be constructed by forming, for all possible subsets ofEn and for each ne N, unions of spheres

4-1) with points of subsets of En as centres. This proximity base has 00

00

00

c a r d i n a l i t y ^ 2 2°"», i . e . 0 ^ 2 2 ° " w - T h u s 0 = 2 2
w= l

n=l

(8.24) C O R O L L A R Y . A metric proximity space has proximity weight Ko or ^ 2^o according as it is totally bounded or not. The proofs of the next two theorems are omitted, as they involve techniques similar to those used above. (8.25) T H E O R E M . If crn = \[r for at least one neN, then 6 = 2&. This is the particular case when \Jr = r/^o? where r) is an arbitrary cardinal number. (8.26) T H E O R E M . For a metric proximity space, 6 = 2* orijr according as there exists or does not exist an neN such that an = \Jr. (The generalized continuum hypothesis is needed to prove this.)

9. Local proximity spaces In Section 7 it was shown that a compact Hausdorff space 3C is (uniquely) determined by a dense proximity space (X, 8), and that A8B in X iff A 0 B + 0 where the closures are taken in 2€. It will be shown in this section that a locally compact Hausdorff

COMPACTIFICATIONS OF PROXIMITY SPACES

53

space Y is determined by a dense subspace X when we know not only the proximity of X, but also which sets in X have compact closures in Y. (Such sets may be called bounded.) Boundedness in topological spaces has been studied axiomatically by various authors since 1939. This section is devoted to a treatment of the interrelationships existing between proximities and bounded sets, and eventually leads to the concept of a local proximity space. It will be seen that such spaces can be embedded as dense subspaces of locally compact Hausdorff spaces. First, however, we shall pursue a brief study of boundedness in topological spaces. (9.1) D E F I N I T I O N . A non-empty collection 38 of subsets of a topological space X is called a boundedness in X iff (i) A e3% and B <= A implies B e£8, and n

(ii) {At:i = l,...,n}

<= & implies (J i=l

Elements of 3$ are called bounded sets. (9.2) R E M A R K S , (i) If (X,d) is a metric space, we may define a metric boundedness ld ={ i c l

:

sup d(x,y) < 00}. x, yeA

It is clear that a metric boundedness is a boundedness. (ii) Given any topological space X, we may define a boundedness J o n I in the following manner: SS = {A c X : A is finite}. (iii) If 3$ is any boundedness, then 0 (iv) The intersection of a non-empty collection of bounded sets is bounded. (v) If X is bounded, then every subset of X is bounded. (9.3) then

LEMMA.

If 3$ is a boundedness in X and X is unbounded, & =

is a filter.

{F:(X-F)e&}

54

PROXIMITY SPACES

Proof. Since X is unbounded, each F e ^ is non-empty. If F± and F2 both belong to J*\ then (X-Fx) and (X-.F 2 ) both belong to 8$. Then by 9.1 (ii), in other words,

F1 [\

It Fe ^ and FczG, then (X - (?) c ( X - F ) . Since (X 9.1 (i) demands that G e &. Thus & is a filter. (9.4) COROLLARY, / / / o r e^er?/ <m&se£ E of X either EedS or (X - E) e0g, then & is an ultrafilter. Throughout the sequel, we shall always suppose that 3$ is a boundedness in a topological space X. (9.5)

DEFINITION.

A topological space is locally bounded iff

each point of the space has a bounded neighbourhood.

The next result is easily proved using standard techniques. (9.6) T H E O R E M . A compact subset of a locally bounded topological space is bounded. (9.7) D E F I N I T I O N . A topological space with a boundedness 38 is compactly bounded iff £8 = {B c= X:Bis

compact).

We then have the following obvious theorem: (9.8) T H E O R E M . A Hausdorff space X is locally compact if and only if X being compactly bounded implies X is locally bounded. (9.9)

DEFINITION.

X is boundedly compact (' MonteV space)

iff every closed bounded subset of X is compact.

In proving the following theorem and corollary, we use two well-known characterizations of compactness: A subset E of a topological space X is compact iff every filter base in E has a cluster point in E iff every ultrafilter in E converges to a point in E. (9.10) T H E O R E M . X is boundedly compact if and only if every closed bounded filter base (i.e. a filter base consisting of closed bounded sets) has a cluster point.

COMPACTIFICATIONS OF PROXIMITY SPACES

55

Proof. Suppose that X is boundedly compact and let £F be any closed bounded filter base in X. Given any Fo e ^, consider the filter base
56

PROXIMITY SPACES

(v) B e& implies the existence of a C e& such that

(vi) The following is equivalent to 9.12 (a), where <^ is defined with respect to oc. and A 4, C, there exists a B e J such that

(9.14)

DEFINITION.

A filter^ is free iff f] F = 0.

(9.15) D E F I N I T I O N . Given a local proximity space (X,oc,&), a binary relation J3 on the power set of X is said to agree locally with oc iff/3 and oc agree whenever either of the sets involved is bounded. (9.16) T H E O R E M . Let 3? be a free round filter in a proximity space {X,8). Define (a) & = {BaX:(X-B)e^r},and (b) AocB iff (E n A)8{E n B)for some Ee3S. Then (X,oc,&) is a local proximity space and oc agrees with 8 locally. Proof. That 38 is a boundedness is immediate from (a). To prove that oc agrees with 8 locally, suppose A SB where BeSS. Since J^ is a round filter, the duality between !F and 88 yields an Ee& such that B <$ E relative to 8. Now B8[(X-E)(] A], so that BS{E(]A). Since (E n B) = B, (E n A)8(E oB). Then by definition, A a B. That AocB implies A 8 Bis obvious. The definition of oc and the fact that {x}e& for every xeX together imply that a satisfies (1.5). The remainder of the axioms for a local proximity space, with the possible exception of 9.12 (a), follow easily. We now prove the contrapositive of 9.12 (a), namely 5 G J and B <^ D implies there is a 0e£% such that

relative to oc. Suppose J S G J 1 and B <4 D relative to oc. Then B<^D relative to 8 also, for oc and 8 agree locally. Since 8 is a proximity, there exists an A such that B <§ A < D relative to 8. As before, choose an E e ^ s u c h that B <^E. Setting C = E n A, we obtain the required result since C G J 1 and 8 agrees locally with oc.

COMPACT1FICATIONS OF PROXIMITY SPACES

57

The next result indicates how a proximity may be defined on a local proximity space. This is valuable since it enables one to derive properties of local proximity spaces from known properties of proximity spaces. The proof is trivial. (9.17) T H E O R E M . Let (X,a,&) be a local proximity space, and define S by A SB iff AOLB, or A$38 and B^SS. Then 8, called the Alexandroff extension of a, is a proximity on X. The next corollary follows from the fact that, due to 9.13 (iii), xSA iffxocA. (9.18) COROLLARY, (i) If for A <^ X,wedefine A = \X\XOLA\, then X becomes a completely regular topological space. (ii) AocB iff AOLB. Observe that the proximity 8 defined in the above theorem is the smallest binary relation on the power set of X which agrees locally with a. (9.19) LEMMA. Every local proximity space (X,a,&) is locally bounded. Proof. This lemma follows directly from 9.13 (iii) and the contrapositive of 9.12 (a), namely B e& and B <4 D implies there is a C G J 1 such that B <4 C <^ D relative to a. (9.20) LEMMA. In a local proximity space (X, CL,8%), the closure of a bounded set is bounded. Proof. If Be@, then by 9.13(vi) there exists a C e£§ with B<^C <4X. Hence BaC. (9.21) LEMMA. Every compact subset of a local proximity space is bounded. Proof. Since every local proximity space is locally bounded, we simply apply Theorem (9.6). (9.22) LEMMA. Let (X, a, 8%) be a local proximity space and define S as in Theorem (9.17). / / X is unbounded, then o- = {A is a cluster in (X,8).

58

PROXIMITY SPACES

Proof. By the definition of S, any two sets in <x are near. From 9.13(v), it follows that if ASC for every Ceor, then A GC. If A$ o- and B$cr, then A e ^ and EeJ*. Hence (A[)B)e@, i.e. (^4 U 5) <£ o". Thus, a* is a cluster. We now turn our attention to the local compactification of a local proximity space. This could be accomplished by considering the space of all bounded clusters (i.e. clusters determined by bounded ultrafilters, the latter being those ultrafilters containing at least one bounded set) and defining a suitable topology on this space. However, since a compactification of a proximity space has already been constructed in Section 7, and by Theorem (9.17) a proximity space is always associated with a local proximity space, this local compactification can be performed more easily. (9.23) T H E O R E M . Given a separated local proximity space (X,ot,&), there exists a locally compact Hausdorff space L and a one-to-one mapf:X -> L satisfying the following conditions:

(i) AaBifff(A)(]J(BJ*

0%nL.

(ii) B e88 ifff(B) is compact in L. (iii) K Such a local compactification is unique; that is, given satisfying the above conditions, there exists a homeomorphism h from Lx onto L2 such that f2 = h o fv The function f is onto iff X is boundedly compact. L is compact iff X is bounded. Conversely, if we are given an injective mapping f:X->L with L locally compact Hausdorff and cc, 8% are defined by statements (i) and (ii) then (X,a,&) is a local proximity space. Proof. Let S be the Alexandroff extension of oc (Theorem (9.17)). Let (/,#*) be the Smirnov compactification of (X,S) (Theorem (7.8)); i.e. 2£ is a compact Hausdorff space, A SB iff f{A) n/CB) * 0 , and f(X) =&.IfX is bounded, then S = a and we take L = 3C. So we need only consider the case in which . Now 2£ consists of all clusters in X and so, by Lemma (9.22), cr = {A:A^^}E^. Clearly
OOMPACTIFICATIONS OF PROXIMITY SPACES

59

words, B e@ ifff(B) ^(3C- {&}). LetL = &- {&}. ThenL is locally compact and Hausdorff. Since {x)e£% for each xeX,f(X) <= L. One can now readily see that 9.23 (ii) and (iii) follow. Moreover, a and 8 are locally equivalent and, by 9.12(6), if AOLB we may assume A eSS and B e&$. As a result, we obtain 9.23 (i), which in turn implies t h a t / i s one-to-one. In order to complete the proof, we need the following result: (9.24) For every qeL, there exists a Be& such that qef(B). This becomes apparent by noting that, since L is locally compact, q has a compact neighbourhood Q. Hence by 9.23 (ii), B = f-\Q) is bounded and q ef(B). We now prove uniqueness. Let 9Ci = Li U {cr^ be the one-point compactification of Lt. Then (/$,^) is the Smirnov compactification of (X, S). By Theorem (7.8), the Smirnov compactification is unique and there is a homeomorphism h from SCX onto «$"2 such that h o/ x = / 2 . It suffices to show that h{(T^) = cr2. If B is a bounded set in X,fx(B) is a compact subset of Lx which does not contain crv Since h is a homeomorphism, A ^ ) ^hif^B)) = f2(B). Hence by (9.24), A(o"1)£.L2, i.e. ^(crj) = X2 be a function. Then f is a local proximity mapping iff: (i) ACLXB implies f(A)oc2f(B), and (ii) Be&± implies

60

PROXIMITY SPACES

It is easy to see that the composition of two local proximity mappings is a local proximity mapping. Moreover, a local proximity mapping carries bounded clusters into bounded clusters. (9.26) T H E O R E M . Let (X^c^,^) (i = 1,2) be two local proxIf imity spaces with local compactifications (fi,Li)(i=l,2). f: Xx -> X2isa local proximity mapping, then there exists a unique continuous mapping h:L1->L2 such that h ofx — f2 of. Conversely, if h\Lx-^ L2 is continuous and h of^X^ a f2(X2), then the mapping f = f^1 o h o/ x : Xx -> X2 is a local proximity mapping. Proof. Let g = f2 of. Then g: Xx -> L 2 satisfies (9.27)

AOLXB

implies g(A) n g{B) + 0, and

(9.28) B e^± implies g(B) is compact. Given peLv let J^ = {gf^M) c: L2\M a neighbourhood of p}. Since f^X-^ is dense in Ll9 no member of J^ is empty. Moreover, the intersection of finitely many neighbourhoods of p is a neighbourhood ofp. This together with the fact that some member of J^ is compact (by 9.23 (ii), (9.28) and the local compactness of L±) shows that (9.29) J^ has a non-void intersection. To see that (9.30) qegf^M) for every neighbourhood M of p implies for every neighbourhood N of q, suppose q has a neighbourhood N such that p^f-ig~\N). Since L± is regular, there exists a neighbourhood M of p such that

^W

T h e r e f o r e fi\M) n g~\N) = 0 a n d h e n c e gfi1{M)

.23 (i). (]N = 0, s o

If s and t are distinct points in L2, they have disjoint compact neighbourhoods S and T respectively. Now (9.27) implies

that g-\S)
COMPACTIEICATIONS OF PROXIMITY SPACES

61

Let N be any open neighbourhood of q. Now we know that J^, a class of closed sets directed by inclusion, has a compact member C and intersects in a unique point q e C. Hence some member of J^ is disjoint from the compact set C — N, since C — N^lF. But C belongs to J^, so that there exists an Fe^ such that F c C (]N c JV. That is, for some neighbourhood i f of p, iV.

This, together with the fact that h(M) c: gf~11(M) for every open M c l j (by definition of A), implies the continuity of A. If P = /i(#) f° r some # e X, then g(x)egfj1(M) for every set i f containing£>. Thus g(x) = hf^x), and we have hofx = g = / 2 o / . To show uniqueness, suppose h': Lx -> i 2 is a continuous mapping such that A' o/i = / 2 of For ^p G £ l9 let q be defined as above. Since L2 is Hausdorff, h'(p) = q will follow if we can show that h'(p)eN for every neighbourhood N oiq. If N is any open neighbourhood of q, then from the second last paragraph, there exists an open neighbourhood M of p such that h'fxfi^M) <= N. Since M is open in i x and/ 1 (X 1 ) = i l s 2/x = Mof^X^) u ( ^ - if). But fifiHM) = Mnf1(X1), so that M c / ^ i f f ) , Thus by the continuity of A', h\p)eh\M) c h'(fJ^(M)) c h'f±f?(M) c iV. Since Z^1, A a n d ^ are local proximity mappings and the local proximity of mappings is preserved under composition, the converse follows easily. Notes 5. A thorough treatment of ultrafilters may be found, in Gaal [C], for example. Clusters were introduced by Leader [53]. Relationships between clusters and ultrafilters are given in Leader [54] and Mrowka [76], the latter presenting the axiomatic characterization of the family of all clusters in a proximity space. The treatment presented here is due to the authors [120]. A weak form of Theorem (5.8) is implicitly contained in the aforementioned paper of Mrowka. 6. This section owes its existence to the work of Leader [55]. An excellent survey of centred systems in topological spaces has recently been published by Iliadis and Fomin [H]. The centred system is a subbase for a filter, and is generally used by the Soviet school in place of the filter. The concept of an end was originated by Alexandroff [A], while both Freudenthal [B] and Alexandroff [A] defined a round filter. Smirnov [98]

62

PROXIMITY SPACES

then used these devices in his proximal extension theory. Alexandroff originally denned an end in a Tychonoff space X using such terms as 4 centred system' and ' completely regular system'. In filter terminology, he would define an end to be a maximal completely regular filter, the latter being a maximal filter with an open base 86 such that for each B e 86, there exists an A &86 with A <= B and a continuous function/: X -» [0,1] such that f(A) = 0 and f(X — B) = 1. Clearly, this definition of an end coincides with that of Smirnov (see (6.4)) when the proximity in question is that defined by (1.11). 7. A general treatment of compactifications can be found in Kelley [J]. For an account of n-point compactifications, the reader is referred to Magill [K]. The results on proximal extensions are due to Smirnov [98], who was the first to expose the relationship between proximities and compactifications (which he effected using ends). The construction of the Smirnov compactification presented here is a modification of the method of Leader [53]. Equinormal proximity spaces were first investigated by Pervin [86]. For a discussion of uniformizable spaces with a unique uniformity (and hence a unique proximity), refer to Gal [D]. 8. The material in this section was taken from Csaszar and Mrowka [13]. Taking advantage of the Smirnov compactification constructed in the previous section, a few proofs have been simplified. The topological theorem used in the proof of (8.17) can be found on page 91 of Kelley [J]. The definition of a proximity base given in this section is due to Csaszar and Mrowka [13]. It is somewhat unsatisfactory, however, in that a unique proximity is not necessarily associated with a given proximity base; for instance, the family of all closed subsets of X is a base for every compatible proximity on X. Njastad [83] has eliminated this problem by defining a proximity base for (X, 8) to be a collection 86 of subsets of X satisfying: (i) A $ B implies that there exist C,De8tf such that A c C,B <= D and C$D. (ii) If A and B are disjoint members of 86, then A SB. 9. A comprehensive study of boundedness, on which the first part of this section is based, was carried out by Hu [F], Taking proximity and boundedness as primitive concepts, Leader [59] defined and studied local proximity spaces. Recently, Leader [130] has considered the problem of finding general conditions under which a topological space is determined by a dense subspace possessing a local proximity structure, and has generalized Theorem (9.26).

CHAPTER 3

PROXIMITY AND UNIFORMITY 10. Proximity induced by a uniformity In this chapter, we shall study questions concerning the relationship between uniform structures and proximity structures. Proximity structures lie between topological structures and uniform structures in the sense that all topological invariants are proximity invariants and all proximity invariants are uniform invariants; however, some uniform invariants, such as total boundedness and completeness, are not proximity invariants. A uniform structure on X was first defined by Weil in terms of subsets of X x X. Tukey later provided an alternate description of a uniform structure using covers of X. Although we shall adhere to the Weil approach here, it should be noted that Tukey's method is widely used by the Soviet school. If U cz X x X then C/"1 = {{z, y): (y, x) e U}. Whenever U = C/"1 U is called symmetric. For subsets U,V ofXx X, Uo V = {(x, z): there exists a yeX such that (x,y)eV and (y,Z)GU}. We inn

1

n+1

n

ductively define V for neN by V = V, V = Vo V. Let A = {(x,x):xeX}. It A c= X, then U[A] = {y:(x,y)eU U[x] = U[{x}].

for some xeA}. For xeX,

(10.1) D E F I N I T I O N . A uniform structure (or uniformity) °ll on a set X is a collection of subsets (called entourages) of X x X satisfying the following conditions: (i) Every entourage contains the diagonal A = {(x,x): xeX}. (ii) IfUcWandVeWjhenUnVeW. (iii) Given Ue°ll, there exists a Ve°U such that Vo V <= U (triangle inequality). (iv) IfUeWandU^VciXxX, then F e f . (v) If Ue^, then U^e^. The pair (X,°ll) is called a uniform space. [63]

64

PROXIMITY SPACES

A subfamily 38 of a uniformity °ll is afrasefor ^ iff each entourage in °U contains a member of 3%. A family Sf is a subbase for ^ iff the family of finite intersections of members of ^ is a base for °tt'. A uniformity ^ is totally bounded iff for each U E&, n

there exists a family of sets {^: 1 ^ i ^ w} such that X = (J At i=l

and AixAi^ U for each i = 1,..., n. \i°U, Y* are two uniformities on X such that °ll <=. ir9 then we say that y isfinerthan °tt or that ^ is coarser than y . The supremum of two uniformities °ll and y , denoted by ^ V ^> is that uniformity which has as a base {f7n ViUeWtVe-r}. Just as we defined a (pseudo-) metric proximity in Section 1, it is possible to define a (pseudo-) metric uniformity. Given a (pseudo-) metric d on X, let Ue = {(x,y)eXxX:d(x,y) < e}. Then the collection {Ue: e > 0} is a base for the (pseudo-) metric uniformity induced by d. It can be shown that for each xeX, {U[x]: JJe°ll) is a neighbourhood filter. Thus °tt generates a topology r = r(^) on X. As is well known, this topology is always completely regular. If °ti satisfies the additional condition (vi) n u = A, then °li is called a Hausdorff or separated uniformity. In this case, r(^) is Tychonoff. Conversely, every (Tychonoff) completely regular space (X, r) has a compatible (separated) uniformity, i.e. a uniformity ^ such that r = r(^). Every uniformity has a base consisting of open (closed) symmetric members, and it is frequently more convenient to work with such a base for °U rather than with °ti itself. (10.2) T H E O R E M . Every uniform space (X,°H) has an associated proximity 8 = §($/) defined by (10.3)

ASBiff(AxB)(]U+

0 for every

Furthermore, T(<%) = r(8). If°tt is separated, then so is Proof. All the axioms for a proximity, except perhaps (1.4), are easily verified. To verify (1.4), suppose A SB. Then there exists an entourage U such that (A x B) n U = 0 . Now by

PROXIMITY AND UNIFORMITY

65

10.1 (iii), there exists an entourage V such that F o F Let E = V-\Bl Then (Ax E)(] V = 0 and ((X-E)xB)[)

V= 0,

i.e. A$E and

c

U.

{X-E)$B.

To show that T(S) = r(^), we observe that x is in the T(%)closure of A iff xe U[A] for every entourage UiE(xxA)[\ ?7#= 0 for every entourage U iff ##^4, i.e. x is in the r(#)-closure of A. Finally, suppose that ^ i s separated. If xSy, then (x, y) n U + 0 for every entourage U. This implies (x, y) n A 4= 0 , so that x = y. Thus 5 is separated. (10.4) R E M A R K S , (i) Instead of using (10.3), S could equivalently be defined by A SB iff U[A] n Z7[£] * 0 A SB iff £7|>4] n B #= 0

for every

for every

Ue<%, or

C/ e^r.

Definition (10.3) has been used here and care has been taken in writing the above proof, because the proof in this form will essentially go through if we are dealing with quasi-uniformities and quasi-proximities. (ii) Two different uniformities on X may induce the same proximity S on X. For example, let X be the real line and °llx be the usual metric uniformity. Let °ll^ be the subspace uniformity on X induced by the (unique) uniformity of its Smirnov compactification corresponding to the usual metric proximity. Clearly, °llx and ^ 2 induce the same (metric) proximity on X. However, since °UX is not totally bounded whereas ^ 2 is totally bounded, °UX and °U^ are different. (iii) It is possible for different uniformities on X to induce the same topology and yet induce different proximities on X. This becomes immediately clear if one recalls the proximities defined in Remarks (2.18) and considers the subspace uniformities induced by the Smirnov compactifications of X corresponding to Sx and S2. (iv) Since every (Tychonoff) completely regular space has a compatible (separated) uniformity, the above theorem provides an alternate proof of the fact that every (Tychonoff) completely regular space has a compatible (separated) proximity. The reader

66

PROXIMITY SPACES

should note the incompatible use of the term ' compatible'! It is hoped that in each case the meaning will be clear from the context. (v) It should be observed that the triangle inequality 10.1 (iii) corresponds to the Strong Axiom. Given a subset A of X and an entourage U, U[A] may be called a 'uniform neighbourhood' of A. The following result justifies the application of this term to a subset B when A <^ B: (10.5) T H E O R E M . Let (X,%) be a uniform spaceandlet S = 8(W). Then A <^ B if and only if there is an entourage U such that U[A] c B. Proof A^B iff A${X~B) iff (Ax(X-B))[) U = 0 for some U e%. But the last statement is equivalent to U[A] c: B. (10.6) R E M A R K . The above result suggests an alternate definition of the associated proximity relation: A <^ B iff U[A] <= B for some TJe°tt. (10.7) T H E O R E M . / / ^ <= ^ 2 , then S± < S2 where 8i = S(%) fori= 1,2. Proof. A$XB implies the existence of a Ux£°llx such that (A x B) n Ux = 0. But Uxe°ti^ and thus A $2B. (10.8) T H E O R E M . If /-.(X,^)-* (Y,<%2) is uniformly continuous, then f: (X, Sj) -> (Y, S2) is proximally continuous where Si = S(%) for i= 1,2. Proof. Suppose on the contrary that AS1B, but f(A)S2f(B). Then there exists a U2 eW2 such that (f(A) xf(B)) ()U2= 0. Since / is uniformly continuous, there exists a U1e^/1 such that (x, y) e Ux implies (f(x)J(y)) e U2. But A St B, so that (A x B) n U± 4= 0 which implies (f(A) xf(B)) (]U2 + 0, a contradiction. (10.9) R E M A R K S . The converse of the above theorem is not true. Consider the identity mapping i: (X,°U2) -> {X,^x) where

PROXIMITY AND UNIFORMITY

67

(

X, %1 and °tt2 are defined as in 10.4 (ii). Then i is a proximity mapping from (X, 8) onto itself, but i is not uniformly continuous. However, as will be shown later, a proximity mapping is uniformly continuous under certain conditions: for example, when the uniformity of the domain space is pseudo-metrizable (12.20), or that of the range space is totally bounded (12.12). Moreover, given a proximity mapping / : (X, 8J -> (Y, 82) and a uniformity ^ 2 on Y such that 82 = 8(ffl2), there exists a uniformity °MX on X such that 8± = 8(°ll^ and / : (X, <%x) -> (Y, °U2) is uniformly continuous. 11. Completion of a uniform space by Cauchy clusters Let (X,°ll) be a uniform space and 8 = 8(^11). We have seen (Theorem (5.8)) that every ultrafilter in a proximity space generates a cluster and that given a set A in a cluster cr, there exists an ultrafilter containing A which generates a. It therefore seems natural to call a cluster Cauchy if it is generated by a Cauchy ultrafilter (recall that a filter #~is Cauchy iff for every U e% there is an FG^ such that FxF c= JJ). One can readily convince oneself that every Cauchy cluster can be considered to be a point cluster, determined by a point of the completion of X (in which X is embedded). The neighbourhood system of this point will contain arbitrarily ' small' sets which intersect every member of the cluster. This leads to the following definition of a Cauchy cluster, which is equivalent to the above (as will be seen later), but is easier to work with. Throughout this section, we shall suppose that °ll is an open symmetric base for a separated uniformity. (11.1) D E F I N I T I O N . A cluster cr in (X,<%,8) is Cauchy iff there exists a round Cauchyfilter*J( cz cr such that M n C 4= 0 for each Me^ and Cea. (11.2) R E M A R K S , (i) Every point cluster in X is Cauchy. In fact, in the above definition we may take (as we shall always do in this section) *J? to be the neighbourhood filter of the point. (ii) If a cluster is Cauchy, then every ultrafilter which generates it is Cauchy. 5-2

68

PROXIMITY SPACES

(iii) Given a round Cauchy filter ^#, Ue% and neN, there n

exists an MeJ?

n

and Ve°ll such that V[M]x V[M] <= U and

n

Fc= U. This can be seen from the following argument. Since J( is Cauchy and round, there exist sets M', MeJ( such that M' x M'
Then F may be chosen to be that entourage satisfying F <= C7 n Vx. The following definition is easily seen to be equivalent to the usual one (see Remark (11.9)): (11.3) D E F I N I T I O N . A uniform space (X,°ll) is complete iff every Cauchy cluster in (X, °ll, 8) is a point cluster ax for some xeX. (11.4) L E M M A . A closed subspace Y of a complete uniform space (X,°ll) is complete. Proof. The trace <%T = {U (\{YxY): U e°U) of ^ on 7 is a base

for the subspace uniformity on Y. If cr1 is any Cauchy cluster in Y, then a slight modification of the proof of Theorem (5.17) shows that cr1 is a subclass of a unique Cauchy cluster o*2 in X. Since X is complete, a2 = crx for some xeX. But xSB for every Becr1 and, since Y is closed, xeY. Therefore {x]ecr1. Let / be a mapping which associates with each point xeX, the point cluster crx. Then / is a one-to-one mapping of X onto the space/(X) of all point clusters. Let X* denote the set of all Cauchy clusters in X. From 11.2 (i), it follows that f(X) aX*^3C. For each Cauchy cluster a, let ^£(a) be one of the filters given by (11.1). For each Ue%, define [7* = {(o-l9
MeJV(crJ,

M x N c U).

To see that U* is independent of the choice of JKi^cr^ and suppose (o-^CgjeP*. Then by 11.2 (iii), there exists an ), Ne^(o-2) and a F e * satisfying V[M] x V[N] c U. Given any J(\(T^) #= ^#(0^), we can find an M' e^\o-^) such that M'xM' a F. Now MnM' 4= 0 , so that M' a V[M]. Hence

PROXIMITY AND UNIFORMITY

69

) and V[M]x V[N] a U, showing that U* is well defined. (11.5) LEMMA. °ll* = {£/*: UeW) is a uniformity base on X*. Proof. Every U* obviously contains the diagonal, and (U(\ F ) * c (7*n F*. Given ? 7 * G ^ * , there exists a F e ^ such that F o F c f / . That J7* 0 y* d J7*followsfrom the following argument: if

then there exists a c 3 e l * such that (o^, O" 3 )GP and (o*3, c2) e F*. Hencethereexistsan^L G^((T 1),5G^(o'2)and(7/ , Cf//G^#( that 4 x C ' c F a n d C " x 5 c F. Setting C = C n C/r we have i x C c f and 0 x B c F. Therefore

which implies (o^, o*2) e ?7*. Since/(Z) c X* cz ^ and/(Z) is dense in^* (by (7.4)), we have the following result: (11.6)

LEMMA./(X) is a dense subset o/X*.

Let 8* be the proximity induced by °U* on X*. The restriction %* oi°ll* to/(X) is a uniformity base on/(X) and so induces the proximity 8* on/(X). (11.7) LEMMA. (X9<%,8) and (f(X),Wf,8?) are proximally isomorphic. Proof. Clearly/is one-to-one and onto. Suppose A SB. Given UeW, let Uf = U*()(f(X)xf(X)). Then we must show that (f(A)xf(B))n Uf * 0. Let VeW be such that F a U. Since A SB, there exist aeA, beB such that (a,b)eV. Therefore V[a] x V[b] c: JJ, and the point clusters cra, crb satisfy the condition (o-^a^eUf. Conversely, if f(A)8*f(B), then for each Uf (corresponding to an arbitrary U e%) there exists a (cra9
where


orbef{B).

Hence (a, b)eU and we have (A x B) n U 4= 0 for arbitrary , showing that ^4 SB.

70

PROXIMITY SPACES

It can similarly be proved that (X,^T) and (f(X),<%*) are uniformly isomorphic. (11.8) L E M M A . Every Cauchy cluster in (X*,<%*9 8*) is a point cluster. Proof. Let cr* be any Cauchy cluster in X*. Since/(X) is dense in X*, a slight modification of the proof of Theorem (5.16) shows that cr* determines a unique Cauchy cluster cr' inf(X) such that cr' c= cr*. But cr' is isomorphic to a Cauchy cluster cr in X. In order to show that cr e a"*, it is sufficient to verify that for each ?7*e^* and each Mecr', ((rxM)nU* =t= 0. Given U*e<%*, 3

there exists a F e t and CeJ?(cr) such that F <= U and C x C c F. Then V[C]x V[G] a U. Setting MQ - FtC^n/^tflf) we have Moecr since F[C]e^#(
PROXIMITY AND UNIFORMITY

71

12. Proximity class of uniformities If (X, T) is a Tychonoff space, then it is known that (i) every compatible uniformity on X contains (as a subset) a compatible totally bounded uniformity; (ii) there exists a largest compatible uniformity (called the universal uniformity)] (iii) it is locally compact iff there exists a smallest compatible uniformity on X. In this section, we consider similar problems concerning a proximity space. Specifically, for a proximity space (X, 8), let H(8) (called the proximity class of uniformities) denote the set of all uniformities f o n l such that 8 = 8{^ll). We are interested in the problems: (a) Does U(S) have a smallest member? (b) Does U(8) have a largest member? We shall see that U(S) always has a smallest member, which is in fact the unique totally bounded member of H(8). I t will also be shown, by means of an example, that Tl(8) need not have a largest member. For the special case in which 8 is a pseudometric proximity, however, the corresponding pseudo-metric uniformity is the largest member of II (8). Necessary and sufficient conditions for H(8) to have a largest member will be discussed in the next section. The following result is obvious and we omit the proof. (12.1) LEMMA. / / W is a uniformity on X and 7 c l , then 8Y(iT) = 8(1Tr). (12.2) T H E O R E M . Given a separated proximity space (X, 8), there exists a totally bounded member of Tl{8). Proof. By identification of #-homeomorphic spaces, we may consider X to be a dense subspace of its Smirnov compactification 3C. SC, being a compact Hausdorff space, has a unique compatible separated uniformity #/**. From (12.1) and the fact that a compact uniform space is totally bounded (a hereditary property), it follows that the subspace uniformity # ^ = i^* *s a member of H(8) and is totally bounded. We now give an explicit construction for H^\

72

PROXIMITY SPACES

(12.3) T H E O R E M . Let (X, 8) be a separated proximity space and let iT be any totally bounded uniformity belonging to U{8). Then the family i^ of sets of the form V = []{AkxAk:k

= l,...,n},

n

where Ak > Bk and \J Bk = X, constitutes a uniformity base

for or. Proof. Recall that by Theorem (10.5), Ak > Bk(k = 1, ...,n) iff there exist WkeiT such that Wk[Bk] cz Ak(k = 1, ...,n). Defining W = n Wkei^,wehaveW[Bk]c:

Ak(k=

l,...,n).B.ence

for every Vei^ there is a If e iTsuch that W cz F, so that "P cz HT, 3

Conversely, given T F e ^ let Wx = T f ^ G ^ satisfy M^ cz Tf. Since iT is totally bounded, there exists a family of sets m

{Cf i = 1, ...,m} such that

U C^ = Z

and Q x Q cz W1(i = l,...,m). Clearly, 3

- wa w for each i, so that Vx = U "^[CJ x W^C^ei^ and is a subset of If. Thus y is a uniformity base for iT. (12.4) COROLLARY. ^ is the smallest member of U(S). Proof. Let We U{8) and WeiT. By the above theorem, there w

n

exists a F = IJ ^

x

Ak cz If, where <4fc > ^ and \J Bk = X.

Since ^ e l l ( * ) , there exists a Ue<% such that U[Bk]cAk(k=l,...,n). Clearly, ^7 c (J U[Bk] x f/[BJ e F cz W, i.e. k

(12.5) ofU{8).

l

COROLLARY.

# " is the unique totally bounded member

PROXIMITY AND UNIFORMITY

73

(12.6) R E M A R K S . The family {Ak\1c = l,...,n} described in the preceding theorem, i.e. such t h a t Ak > Bk(k = 1, ...,n) n

where U Bk = X, is called a S-cover of X. k= l

In the above discussion we only considered a separated proximity. This restriction is, however, not necessary and it is possible to prove directly that i^ as defined in the statement of Theorem (12.3) is a base for a unique totally bounded uniformity which is compatible with a given proximity. It was in this way, in fact, that the result was first proved in the literature. However, in this approach the triangle inequality presents a little difficulty; our use of the Smirnov compactification avoids the manipulations. Throughout the remainder of this section, #^= W(8) denotes the unique totally bounded member of (12.7) T H E O R E M . Let 8{ (i = 1, 2) be two proximities on X and let W: = UTidi). If 8± < 82, then Wx c HT%. Proof. Given T^e#^, there exists a symmetric member W'x 3

of H\ such that W[ c Wv Since H\ is totally bounded, there n

exists a family of sets {Bk :1c = 1,..., n} such that X = \J Bk and k= l

BkxBkc: W[ for each h. Set Ak = W^B^. Then Bk <1Ak and, since S± < 82, Bk <^2^/r Consequently, W2 = U AkxAkei^2, by (12.3). Moreover, W2 c W[ ^ TFl5 showing that W±e 1T2. (12.8) LEMMA. Let (X, S) be aproximity space and let W= If V is a subset of X x X such that (12.9)

A <4 V[A] for all A c X,

then we also have (12.10)

A<4(V(]W)[A]

for all

WeiT,A<=:X.

Consequently, if 8* < S and % e II(**), Proof. In view of Theorem (12.3), we may assume that

74

PROXIMITY SPACES n

W = U AtxAi,

where {Afi = 1, ...,n} is a #-cover of X. We

shall first verify (12.10) for n = 2, i.e. for W = Since

(A1xA1)[)(A2xA2).

A = (A - A2) U (A - A±) U (A n A1 n 4 2 ),

(Fn T f ) M = (Fn W0D4-4JU (Fn = (V[A -A2] n ^ x ) U (V[A -A{\ n u F[-4n-4in^2]Using the distributive property, we obtain (V(]W)[A]= Now A-A2c: X-A2<4 W[X-A2] c ^4X and, by hypothesis, ^4 n ^42 <^ F[^4 n ^42]. With the help of (3.9) and (3.10), we obtain A = (A -A2) u (A n A2) 4 Ax u F[^ n ^42] and similarly, ^
ViAnAil

We also know that A <^ V[A]9 so that A <^(V(]W) [A]. For w > 2, we observe that W = n Tf (/, J) where

corresponding to a partition {/;J}

of {1,...,»},

AI=[jAi i

G

and /

i ^ U i ; , j eJ

with the intersection being taken over all partitions. The result then follows by induction on n. (12.11) THEOREM. Let f be a proximity mapping from (X, #x) to (Y,S2). Given an arbitrary member %2eU(S2), there exists a uniformity ^ e l l ^ ) such that f: (X,^) -> (J,°li2) is uniformly continuous. If °U2 is totally bounded', then °ilx may be chosen to be Proof. (/x/)~1[^r2] is a base for a uniformity °ll* on X. It is known that / is uniformly continuous with respect to °tt\ and °tt2, and if °U2 is totally bounded then so is <&?. Let 8? = S(^f). If

PROXIMITY AND UNIFORMITY

75

1

A if B, there exists a U* (= ( / x / ) - ^ ] , where Z72 G ^ 2 ) belonging to <%? such that (J. x B) n £7? = 0 , i.e. (f(A) xf(B)) ()U2= 0. This shows that/(^4) i2f(B). But/: (X, 8±)-+ (7, <J2) is a proximity mapping, so that Ai±B. Thus £f < 8V Set ^ = ^ Then by Lemma (12.8), ^ G I I ^ ) . Clearly, the mapping

is uniformly continuous. If ^ 2 is totally bounded, then so is <Wf. Thus by (12.5), =

implying that

(12.12) C O R O L L A R Y . Let °tti{% = 1,2)feeuniformities on X and Y respectively, W2 be totally bounded and St = S(%ri). Then / : X -> Y is a proximity mapping if and only if it is uniformly continuous. (12.13) LEMMA. / / ^ e n ( # ) and TT is a uniformity such that iT{8) c= - ^ c: ^ then i^ell{8). Proof. This result follows immediately from (10.7), as 8 < 8("T) < 8. (12.14) T H E O R E M . If We 11(8), then iT= iT(8) is the largest totally bounded uniformity contained in °tl. Proof. If °ll* is any totally bounded uniformity contained in 91, then W* V ^ is totally bounded and TT Cfy*\J ifr c (%. By (12.13), W*\ZiTeU(8) and by (12.5), W*\/^=iT; that is, fy* c if/\ (12.15) R E M A R K S . The above discussion establishes a one-toone correspondence between all proximity structures and all totally bounded uniform structures. Since a completion of a totally bounded uniform space is a compact completely regular space, this then provides an alternate approach to the construction of the Smirnov compactification of a proximity space. The following example shows that, in general, 11(5) does not possess a largest member. (12.16) E X A M P L E . Let X be a set admitting two countable partitions {A^.ieN} and {Bj-.jeN} such that A^Bj =(= 0 for

76

PROXIMITY SPACES

i, j e N. Let $lx and ^ 2 be the uniform structures generated by the single entourages TJX = \J (Ai x A^ and U2 = U {B$ x Bj) respecieN

jeN

tively. As usual, let # ^ and # ^ denote the respective totally bounded uniform structures in the proximity classes of °UX and ^r2. If 8X = £ ( ^ ^ 2 ) a n d #2 = < W V ^ ) , then we shall show that 82%8V Clearly £2 < 8V Now °Ux\l°k^ is defined by the single entourage

and W*! V ^ 2 n a s W=

a

^ a s e °f entourages of the form U

[(^ni^)x(^mni^)],

where {Em} and {Fn} are finite partitions of X such that each En is a union of sets Ai9 and each Fn is a union of sets By Define G = \J Then Oi^X-O). However, G82{X-G) as the following argument shows. Let W e iV*x \j W2. Among the sets Em, there exists at least one EmQ which contains more than one A^ Suppose Ai± U Ah c Emo,

where

ix < i2.

Among the sets Fn, there exists a unique Fno containing B^. Let xGAti n Bix and ^/G^4^2 n Bti. Then xeG, ye(X-G) and

(», ^/) G [(J5mo n FJ x (^mo n J^)] c TT. Since IF is arbitrary, this shows that G82(X-G). Thus £2 J *1# We next show that II (82) does not have a largest member. Suppose on the contrary that W is the largest member of ). Consider the uniform structures

and

7^2 = ^ 2 V ^ l = ^ 2

Clearly 5(^) < #2 and 5(1^) < 52 (as can be seen from the fact

that 8(%x) = 8^)

< 8(irx\JiT^ = 82, and by (12.8),

PROXIMITY AND UNIFORMITY

77

Hence i^czir and ^ 2 c #"'. Therefore ^ V ^ <= ^ ' , i.e. y ^ V ^ e n ^ ) . On the other hand, ^ V ^ 2 = ^ i V ^ ^ n ^ ) , a contradiction. If 8 is a pseudo-metric proximity, then 11(5) does in fact have a largest member. In order to prove this, we first need a lemma: (12.17)

LEMMA.

Let U and W be symmetric subsets of XxX

4

such that W <= U. Then for every sequence ((xn,yn)) from XxX such that (xn,yn)<£U for each neN, there exists a subsequence (fe P y ni )) suchthat (xnk,yni)$ Wfor every k,leN. Proof. For each neN, let Bn = {m: (xn,ym)e W) and Cn = {m:(xm,yn)eW}. 2

2

If p,qeBn, then (yp,yq)E W. Hence {xq,yp)$ W, since otherwise 4

(xq, yq)eW c: U, contrary to hypothesis. Thus if Bn is infinite for any n, then ((xm,ym))meBn is the required subsequence. The case when Gn is infinite is similarly disposed of. The only remaining case is that for which both Bn and Gn are finite for each neN. Let (f)(n) denote thefirstnatural number greater than both n and all the elements of Bm and Cm (m = 1, ...,n). Define nx = 1 and nk+1 =
"/* such that W ^ V. Then by the previous lemma, there is a subsequence ((xnk,ynk)) such that (xnk,yni)£W for every

78

PROXIMITY SPACES

k, IEN. Set A = U {xnj} and B = U {?/nJ. Then A SB, since given any neN there is a k > n such that xnke^4,2/nfce.B and But (AxB)(]W=0

and therefore .4 # B, a contradiction.

(12.19) COROLLARY. H(8) contains at most one pseudo-metric uniformity. If it does contain one, then it is the largest element of the proximity class. (12.20) C O R O L L A R Y . Let °llx he a pseudo-metric uniformity on X and let S± = #(^ x ). Let S2 be a proximity on Y induced by a uniformity °ll^. Thenf: X -> Y is a proximity mapping if and only if it is uniformly continuous. 13. Generalized uniform structures In the previous section it was shown that if S is a pseudometric proximity, then the corresponding pseudo-metric uniformity is the largest member of H(S). This naturally suggests the problem of characterizing those proximities S which enjoy the property: U(8) has a largest member. In the present section this problem is studied using a generalized notion of uniform structure, which differs from the usual uniform structure in that a weaker intersection axiom is used (see 13.1 (*) below). Total uniform structures, whose role in the theory is dual to that of totally bounded structures, are also considered. Finally, correct structures are introduced, with the help of which a characterization of those filters from X x X which induce a separated proximity is obtained. (13.1) D E F I N I T I O N . An Alfsen-Njastad (or A-N) uniform structure % on a set X has all the properties of a uniform structure (10.1) with the possible exception of 10.1 (ii), which is replaced by: (*) / / {Af 1 ^ i ^ n) is a family of subsets of X and U^^ for 1 ^ i ^ n, then there is a V E% such that Members of °tt are called entourages, and the pair (X,°ll) is called an A-N uniform space. °tt is said to be separated if it also satisfies 10.1 (vi).

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79

Although it is evident that every uniform structure is an A-N uniform structure, it will be shown later that there exist A-N uniform structures which are not uniform structures. Many of the results of the previous sections concerning uniformities hold for A-N uniformities, and are summarized in the following theorem: (13.2) T H E O R E M . Every A-N uniform structure °ll induces a proximity structure 8 = 8{%), and hence a topology r = r(8) =

T(<&)

also, in the same way as a uniform structure, r(tf/) is completely regular and, in fact, Tychonoff if °U is separated. If A(8) denotes the proximity class of A-N uniform structures, then A(S) contains a unique totally bounded member #^ and *W is the smallest member of the class. Every totally bounded A-N uniform structure is a uniform structure. (13.3) D E F I N I T I O N . A subset V of Xx X is called entouragelike with respect to a proximity 8 on X iff there exists a sequence (Ki)neN °f symmetric subsets of X x X such that Vn+l^Vn, and

where V0=V,

n = 0,l,...,

A 4 Vn[A] for all i c j .

Equivalently, we sometimes say that V is entourage-like with respect to A(8) or with respect to We A(8). Clearly, every entourage of an A-N uniform structure is entourage-like. (13.4) D E F I N I T I O N . An A-N uniform structure °U is said to be total iff every entourage-like set with respect to 8{W) is an entourage. In contrast to the situation with uniform structures, we have the following result concerning A-N uniform structures: (13.5) T H E O R E M . Every proximity equivalence class A(S) contains a largest member s/9 which consists of all entourage-like sets with respect to 8. Proof. Let Hr be the smallest member of A(8) and let s/ be the class of all entourage-like sets relative to 8. Clearly # ^ c s/. To prove that s/ is an A-N uniform structure it suffices to verify 13.1(*), since the other axioms follow easily. Let

80

PROXIMITY SPACES

{Af 1 ^ i ^ n} be a family of subsets of X and let l^ssi for I ^i ^n. Now by (13.3), ^ <4 U^A^ for 1 < i ^ n. Since iTeA{8), there exist T^eiT such that WlA^^U^A^ for n

1 < i ^ %. Since #^is a uniform structure, W =

^

and W[At~\ <= [ ^ J for 1 < i < n. Clearly 8 < 8{si) and, by (13.3), 8{si) < S; i.e. Since every entourage of % e A(#) is entourage-like relative to 8, s0 is the largest member of A(8). (13.6) C O R O L L A R Y . In the above theorem, srf is the only total structure of A(8). (13.7) T H E O R E M . The largest A-N uniform structure stfeA(8) is equal to each of the following: (i) the union °UX of all members of A(8). (ii) the union W2 of all members of the classes A(8') for all 8' < 8. (iii) the union °llz of all members of 11(5). (iv) the union °tt± of all members of the classes U{8') for all 8' <8. (v) the union W5 of all pseudo-metric members of the classes U(8f),forall8f < 8. Proof. Since j / e A ( i ) , si <= <%v We also have ^ 2 <= tf since every entourage of a member of A(8'), with 8' < 8, is entouragelike relative to 8. Thus s/ <= ^ c <%2a ^ , from which (i) and (ii) follow. Statements (iv) and (v) are implied by the relations

The non-trivial part si <= °llh is due to the fact that, if V is an entourage-like set with a denning sequence (Vn) as in (13.3), then {Vn} is a base for a pseudo-metric uniformity which contains V. (Recall that a uniformity with a countable base is pseudometrizable.) Finally, to prove (iii), it suffices to show that si c ^ 3 since the reverse inclusion is immediate. We know from (iv) that if

PROXIMITY AND UNIFORMITY

81

, then there exists a "TeTL(8') with 8' < 8 and such that . Clearly ,

and by (12.8),

(13.8) COROLLARY. / / 11(5) has a largest member, then such a member is precisely stf. The next theorem, which follows from the above discussion, characterizes those II(#)'s which possess a largest member. (13.9) T H E O R E M . 11(5) has a largest member if and only if the entourage-like sets with respect to 8 form a filter. (13.10) R E M A R K S , (i) Example (12.16) illustrates the existence of a total A-N uniform structure which is not a uniform structure. (ii) Theorem (12.18) illustrates that every pseudo-metric uniformity is total. (13.11) D E F I N I T I O N . A contiguity % on X is a family of symmetric and reflexive relations such that if U etf/ and U <= V where A c V = V~\ then F e f . Given a contiguity °ll on X, one can define a binary relation S = 8(%) on the power set of X as follows: (13.12)

A$B

iff there is a C/G^T such that U[A](]B=

0.

A natural question arises as to what conditions are necessary and sufficient to ensure that 8 — 8(tf/) is a separated proximity onl. (13.13) D E F I N I T I O N . A contiguity °ll on X is called correct iff 8 = 8(^1) is a separated proximity on X. We have seen that separated uniform and A-N uniform structures are correct. The following theorem gives a characterization of correct spaces: (13.14) T H E O R E M . A contiguity °li onX is correct if and only if each of the following conditions is satisfied: (i) Ifx,yeX, then x 4= yiff there is a U e°U such that

82

PROXIMITY SPACES

(ii) For each i c j and U,Ve^, that

there exists a We°ll such

W[A]cz U[A]nV[A].

(iii) For A c X and Ue%, there exist V, We°ti such that W\V\A~]\ c U[A], Proof. Let us first suppose °tt to be correct and let S = 8(<%). We need only prove (ii) and (iii). If (ii) is not satisfied, there exists a n i c j and U, Y e°tt such that for each We °ll, we can find an x = x(W) e W[A] ~(U[A] n V[A]). Consider Then B = Bx u B2 where U[A] n B1 = 0 and V[A] (]B2= 0. As this implies A$BX and .4 $B2, we have A SB, contradicting the fact that W[A] n £ # 0 for each W e°ll. To prove (iii), let A c: X and E/e^. Since = 0, At{X-U[A]). By (3.5) (which is equivalent to the Strong Axiom), there exist disjoint ^-neighbourhoods E and F of A and (X— U[A]) respectively. Then A$(X-E) and (X - U[A]) t(X- F), implying the existence of V, We°ll such that V[A] (] (X- E) = 0 and W[X-U[A]]n(X-F)= 0. Hence V[A] c= E and W[X - U[A]] a F. But E n F = 0 , so that = 0. Therefore W[F[^4]] n ( X - E7|>4]) = 0 , since Tf = W~\ which proves (iii). To prove the converse, it is sufficient to verify the Strong Axiom. If A SB, then there is a U e°U such that U[A] n B = 0. By 13.14(iii), there exist F, WeW such that W[V[A]] c U[A]. We first verify that V[A] n ^ [ 5 ] = 0 . Suppose instead that there exists an xe V[A] n W[B]. Then TF[x] c TT[F[-4]] c J7[^l] and there exists aye W[x] 0 B c: U[A]. But this contradicts the fact that U[A] o B = 0. Clearly V[A] and W[B] are disjoint ^-neighbourhoods of A and I? respectively. An example is now given of a correct space which is not a uniform space:

PROXIMITY AND UNIFORMITY

83

(13.15) E X A M P L E . Let an increasing sequence (an) of natural numbers be called rapidly increasing iffajn -> oo. Let X be the set of all natural numbers and °U' be the collection of all sequences which either have a finite range or are rapidly increasing. For each U'e°ll' define JJ = {(x, y) eX x X: either x and y are both in or both not in the range of U'}. Then °ll = {U: V e °tt'} is a contiguity on X. For M c: X and UeW, let us first determine U[M]. Let N denote the range of V. We are then faced with one of the three following situations: (a) I f M ( \ N = 0 , t h e n U[M] = X-N. (b) If M c N, then U[M] = N. (c) If M n N 4= 0 and M * N, then U[M] = X. The correctness of ^ will now be verified. If x 4= y, we take U corresponding to N = {x}. Then y$ U[x]. Conversely, if y$ U[x], then x 4= y, verifying 13.14(i). For each A c X and TJ<E°ll, U[U[A]] = J7[^l] and 13.14(iii) is verified. Finally, to show 13.14(ii), let A c X and [/, F G ^ . Let JVi and N2 denote the respective ranges of the sequences V and V. We now consider the three possible situations separately: (I) If A(\NX = 0 = A niV2, then U[A] = X-N± and V[A] = X-N2. Let N0=N1[j N2. Then iV0 e W and, if Tf is the corresponding member of ^ , then W[A] ^X-Noa

U[A]n V[A].

(II) If A c: j ^ l 9 then ^4 is the range of some W'. In this case, W[A] = A c Z7[>4]n F ^ ] (III) If ^1 n-^i 4= 0 and ^4 4= JVi, then J7[^4] = X and we may choose W = V. Finally, °U is shown not to be a uniformity by observing that 10.1 (ii) is not satisfied. Suppose x 4= y and let U, V be members of °ll corresponding to Uf = (x) and V = (y). We then clearly haveC/n %

6-2

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PROXIMITY SPACES

14. Proximity and height In this section, a new partial order for uniform structures is introduced through the concept of height. In some respects, this concept is dual to that of proximity and will clarify the order structure of (14.1)

D E F I N I T I O N . A subset U of XxX

is totally bounded n

iff there exists a family of sets {At: 1 ^ i ^ n} such that X = \J Ai and AixAi^ U for 1 ^ i ^ n. A uniform structure is totally bounded iff every entourage is totally bounded. This is equivalent to the condition: for each U e °ll there exists afiniteset {x1,...,xn}aX n

such that X = \J U^] (sometimes called precompactness). (14.2)

D E F I N I T I O N . Let °ll and Y* be two uniformities on X.

°U is said to be less than or equal in height (^ ) to Y* iff for each UeW, there exists a F e f such that U[)VC is totally bounded (where Vc = X x X - V). Equivalently, we may define °ti ^ "T iff for each V e°ll, there exists a F e f and a finite family {Af 1 ^ i ^ n} with n

U Ai = X

such that

V n (Ai x At) c U for each i.

i=l

The following results are immediate consequences of this second form of the definition: (14.3) LEMMA. The relation ^ as defined above is a preorder (i.e. reflexive and transitive) on the set of all uniformities on X. (14.4) LEMMA. If°ll c Y\then% < iT. Clearly ^ induces an equivalence relation on the set of all uniformities on X, and so gives rise to equivalence classes. We shall denote by H(W), called the height class ofW, the collection of all uniformities on X which are equal in height to °ll'. The following result is obvious.

PROXIMITY AND UNIFORMITY

85

(14.5) L E M M A . The smallest height class is the class of all totally bounded uniformities on X. The next theorem provides the first step towards showing that the family of all height classes of uniformities on a set X is a complete lattice. The corresponding result for the family of all uniformities, assigned the partial order induced by set inclusion, is well known. (14.6). T H E O R E M . / / ^ < i^andiT^ -T, then {<% V'TT) ^ -T. Proof. A typical member of % V ^ is U 0 W, where U and W are members of °ll and ^respectively. By hypothesis, there exist entourages V1,V2ieir and finite covers {Af 1 < i ^ n) and {By. 1 ^ j < m) of X such that AixAi^U{]Vc1 and

Clearly {At n By. I ^ i ^ n, 1 ^ j ^ m} is a finite cover of X and ( ^ n Bj) x (^4^ n ^ ) c ([/ n TF) U (Pi (11^)0, proving the theorem. The following corollary is an immediate consequence of the above theorem together with (14.4). (14.7)

COROLLARY.

If°tt ^ ^ , then

(14.8)

COROLLARY.

If0)/ is totally bounded, then

for every i r . If °tt is a uniformity on X and 8 = S(W), we shall denote by °UW the unique totally bounded member of 11(5). Recall that such a member exists by Corollaries (12.4) and (12.5).

NOTATION.

(14.9)

THEOREM.//^ ^ i

Proof. To simplify the proof, we suppose that all entourages throughout the discussion are symmetric. 3

3

Given UeW, let Ul9 U2eW be such that U± c U and U2 c Uv Since ^ ^ y , there exists a F e f and a finite cover {Af 1 ^ i ^ n) n

of X such that

U^xi^c^y V°\

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PROXIMITY SPACES 3

Let V±ei^ be such that Vx <= V, and define Bi = (U20 Vx) [A^. Then {Bf 1 ^ i ^ n) is a £(^ V ^)-cover of X, so that n

Uw= U ( ^ x ^ ) The theorem is now proved, since UW(]V1^ U as verified below. If (t, u) e Uw n Pi, then both £ and w belong to 1?^ for some i; that is, there exist x,yeAi such that (x, t), (y, U)GU20 VV Hence (x, y)eV

and, since Ai x Ai c= Ux U Fc,

(x, ^/) G Cj_. Since (x, ^), (y, u)eU2 c: [7X and all entourages are symmetric, it follows that (t, u) e U, as required. (14.10) COROLLARY. / / H(8) has a least upper bound, then W ^-TforW^e U(d) implies that fy c i^, Proof. If #"is the least upper bound of 11(5), then Hence ^ = \T V ( ^ V ^ ) J and, by (14.9), No confusion should arise from the fact that the same notation ^ is used for the preorder on the set of all uniformities on X, as for the preorder which it induces on the height classes. (14.11) COROLLARY. If H1 and H2 are two height classes, then i < H2 iff there exist a(%1eH1 and aW2EH2 such that %1 c $r2. Proof. In view of Lemma (14.4), it is sufficient to prove necessity. Let H± ^ H2, ^1EH1 and i^eH2. Then ^x^i^ and, by (14.5) and (14.7),

H

According to Theorem (14.9), °UX <= ^r2. In Corollaries (12.4) and (12.5), we have seen that every has a smallest member. The following result concerning the height classes is, in a sense, dual to this. (14.12) T H E O R E M . Every height class H has a largest member %h. If°tt < <%h, then fy <= <%h.

PROXIMITY AND UNIFORMITY

87

Proof. Let °ttA be the largest uniformity on X: namely, that generated by the diagonal A. Then every finite cover of X is a <J(«rA)-cover of X. For each WeH, define

Then ^ E # and, if ^ < ^ , Theorem (14.9) implies that

(14.13) COROLLARY. The family of all height classes of uniformities on a set X forms a complete lattice. Proof. In view of (14.5), it suffices to show that every subfamily {Ha: aeA} of height classes has a least upper bound. Let Wa be the largest member of Ha for each aeA, and let °tt = V %a- Then oceA

H(W) is an upper bound for {Ha:aeA}. HH1 is any upper bound for {Ha: aeA} and ^ is the largest member of H', then (14.12) implies that Wa <= iTh for each aeA. Hence °U <= ^ and, by

15. Hyperspace uniformities If (X, r) is a uniformizable space, then any compatible uniformity f o n l induces a uniformity °U on the hyperspace H(X) of all non-empty closed subsets of X as follows: for each TJe°U, an element U e ^ is defined by

U = {(A,B)eH(X)xH(X):A

c C7[JS] and 5 cz U[A]}.

Two compatible uniformities on X which induce compatible uniformities on H(X) will be referred to as H -equivalent, examples of which can be found in the literature. The following questions naturally arise: (a) Under what conditions are two uniformities ^-equivalent ? (b) Under what conditions does ^-equivalence of uniformities imply identity? There is an ever-growing literature centred on these problems (see Notes for references); in this section, we merely study the role played by proximity in such problems.

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PROXIMITY SPACES

(15.1) T H E O R E M . / / two uniformities on X are H-equivalent, then they are in the same proximity class. Proof. Suppose °ttx and °tt% are two uniformities on X which are not in the same proximity class; that is, if Si = $(%i) for i = 1,2, then 8± =# S2. Then by (2.8), there are closed subsets A and B of X such that AS1B but A $2B. In other words, U±[B] n A =t= 0 for each Ux G ^ 1 ? while there exists a[/ 2 G^ 2 such that J72[JB] n -4 = 0. Let A = {FeH(X):F(]A # 0 } . Given C ^ G ^ , there exists a such that ^ o ^ cz Uv Choose a e VX[B] n ^4. Then

so that i u j f l j c U^B]. Clearly JS c C/^5 u {a}] and £ U Ja}e A. Hence U x [5] n A + 0 , and we conclude that Ber^^-closure of A. On the other hand, since U2[B] contains no set intersecting A, we know that B<£T(W2)-closure of A. Thus °ttx and ^ 2 are not ^-equivalent, establishing the theorem. (15.2) COROLLARY. TWO different uniformities on X, at least one of which is totally bounded, are not H-equivalent. Proof. This follows from the two facts: (i) every proximity class contains exactly one totally bounded uniformity, by Corollary (12.5), and (ii) the family of closures of finite subsets of X is dense in (H(X)), T{°U)) iff ^ is totally bounded. To see the validity of statement (ii), suppose Ue^ and let B be a finite subset of X such thatBe\J[X]. Then clearly X = U[B], so that °ll is totally bounded. Conversely, if °ll is totally bounded and U = TJ~xe°ll, then by (14.1) there exists a finite subset {xlf ...,xn} of X such that X = U Ufa],

Clearly,

{a-J c Ufa] for 1 ^ i ^ n. Given

i=i

AeH(X), let 5 =_{^:^. n Ufa] 4= 0 } . Then .4 cz ?7[J5] and 5 c= U[A], so that 5 G U [ - 4 ] . Hence the closures of finite subsets of X are dense in H(X). (15.3) COROLLARY. TWO different uniformities on X, each having a countable base, are not H-equivalent.

PROXIMITY AND UNIFORMITY

89

Proof. This follows from Theorem (12.18) and the fact that every uniform structure which has a countable base is pseudometrizable. (15.4) D E F I N I T I O N . Let °ll and 'V be two uniformities on X and let A c: X.°ttis said to be uniformly finer than "V on A over X iff given any V ei^, there exists aU e% such that U f) (A x X) c= V. (15.5) H(X).

DEFINITION.

°U is H-finer than -T iff T(°U) •=>

T("T)

in

(15.6) D E F I N I T I O N . A subset E of X is [/-discrete for UeW iff for each xeE, U[x] n E = {x}. E is ^-discrete if it is V-discrete for some Ue%. The next result answers question (a) of the introductory paragraph. (15.7) T H E O R E M . Let °tt and 1^ he two uniformities on X. Then °U is H-finer than *V if and only if it is both (i) proximallyfiner,and (ii) uniformly finer over X on every ^-discrete set. Proof. Let °ll be i7-finer than i^. Statement (i) is implicit in the proof of Theorem (15.1). In proving (ii), we again work solely with symmetric entourages. Suppose Eo c X is pQ-discrete for some Vo G V. Let Yx e V and F2 = V± n Vo. Since °ll is #-finer than ^ , there is a Ue<% such that EeU[E0] implies EeY2[E0]. In particular, let E = {y} U (i?o - {^o}) where x0 s Eo and (x0, y) e U. Then Eo cz V2[E]; that is, there exists a y' e E such that (x0, yf)eV2^ Vo. Since Eo is 1^-discrete, we have y' = y and hence

To establish the converse, suppose that (i) and (ii) are satisfied, and let Eo <= X and Vo e i^. Now Eo and X - V0[E0] are not near relative to S(i^), and hence not near relative to 8(%); that is, there is a UoeW such that U0[E0] c TJ[^0]. Let VxeY° be such that 2

T^ c: ]^ and let i ^ be a maximal ^-discrete subset of Eo, so that Jgf0 cz V^E-L]. By (ii), there exists a C^e^ such that

90

PROXIMITY SPACES

and also, Ux c Uo. If E
aeA,

then ^ is J^-finer than rK (15.9) T H E O R E M . Let °ll and "V be two uniformities on X^which are equal in height and are in the same proximity class, i.e.

Then °ll and *V are H-equivalent. Proof. Let Sei^ and Dv ...,Dn be a finite cover of A c X. Since °tt and i^ are in the same proximity class, there exists a UeW such that C7[Z>J c 8[Dt] for all i=l,...,n. If and (a, x) e U, then

(a,a:)e U (A* ^ M c U Thus, in view of (15.8), it is sufficient to prove that given and a F-discrete subsets of X, there exist TE°U, Sei^ and a finite cover Dl9..., D^ of ^4 such that

(a,x)e\J ( 1=1

implies (a, a?) G F .

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91

3

Let W e Y* be such that W c F. Since if ^ °U, there exists a finite cover iT 1? ..., Kr of X and a T e f such that

U ( r n ^ x ^ c w. This in turn (since we also have °U ^ V) yields a finite cover El9...,E8 of X and an tfe^such that 8 c Tf and U (iSn(^x^))c!T.

(i) Given an x e X there are at most r elements aeA such that (a, x) G 7. For if there were more than r, then at least two of them, say a± and a2, would be members of the same Ki (for some i). 2

Therefore (al9 a2) e(T n {Ki x j?^)) c: }f c 7 ? a contradiction since A is F-discrete. (ii) Given aoeA, there exist at most r elements ak #= a0 of ^4 satisfying: for each k there exists an xkeX such that (ao,^)<ET

and

(ak,xk)e8.

2

If ^fc = Xj for i =(= j , then (ak,aj)eS c F, a contradiction. Hence the xk's are distinct. If there were more than r of them, at least two of them, say xx and x2, would fall in the same Ki (for some i). 2

In this case (xl9 x2) G (T n ( ^ x ^ ) ) <= TF, which in turn implies 3

(al9 a2) e W c F, a contradiction. (iii) Given aoeA, there exist at most sr2 elements ak (=)= a0) of ^4 satisfying: for each /^, there exists a.nxkGX such that (ak,xk)eT and ( O O ^ J G S . If this were not true, then by (i), more than sr of the xks would be distinct. Consequently, more than r of them, say x1,...,xr+1 would fall in the same Ei (for some i) Hence

for all j,k between 1 and r +1. Thus for the same J's and

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PROXIMITY SPACES 3

(ak, a,j) e T. But at least two of these, say ax and a2, are in the same 3

Kt. Thus (al9 a2) e(Tn (Ki x K^) <= W <= F, a contradiction. Combining (ii) and (iii) we obtain: (iv) Given aoeA, there exist at most w = r + sr2 members ak + a0ofA satisfying: for each k there exists an xk e X such that either (a0, xk) e T and (a^, xk) e S, or (a0, xk) e $ and (afc, xk) e T. Now let Dl9..., Dn+1 be maximal disjoint subsets of A such that for a, a1 eZ^ and any a i e l , (a, x)eT and (a', a;) e 8 implies a = a'. By (iv), A =n\J Dt.Nowaei,

x e X a n d (a,x) E(T n ( A x

implies there exists an a1 e Z)^ such that (a', x) e $. But then a = a' and (a, a : ) e S c 75 completing the proof.

Notes 10. The definition of a uniformity as given in this section is due to Weil [N]. That every such uniformity has a base consisting of open symmetric members is proved on p. 179 of Kelley [J]. Smirnov [98] was the first to investigate the relationships between proximity spaces and uniform spaces, although he worked with uniform structures defined by systems of uniform coverings. This equivalent method of defining a uniformity was earlier employed by Tukey [L]. Further work concerning the connection between uniformity and proximity was carried out by Gal [27] and by Alfsen and Fenstad [4] using Weil's uniform structures. Ramm gave the first example of two different uniform structures inducing the same proximity. 11. This section owes its existence to the work of the authors [121]. Definition (11.3) is equivalent to the usual one: namely, a uniform space is complete iff every Cauchy filter in the space converges to some point of the space. 12. A uniform structure ^ on X is precompact iff for every Ue%, there n

are finitely many points xt, ...,xn

in X such that X = U U\x^\. This i=l

concept is equivalent to that of total boundedness (see Gaal [C], p. 279) which is defined as follows: (X,tf/) is totally bounded iff for every Ue^, there is a finite cover {A^. 1 < i ^ n} of X such that A t x At c JJ for each i. For more information regarding statements (i) and (ii) of the opening paragraph, the reader is referred to pp. 234-5 of Gillman and Jerison [E]. Statement (iii) was proved by Gal [D]. The equivalence of proximities

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93

and totally bounded uniform structures has been shown by Smirnov [98], Alfsen and Fenstad [4] and Gal [27]. Our presentation of the theory in this section follows closely that of Thron [115] and of Alfsen and Njastad [6]. Theorem (12.11) was independently proved by Alfsen-Njastad [6] and Hursch (see Thron [115]). Example (12.16) was provided by Alfsen and Njastad [6]. Another example illustrating the same point has been given by Leader [56]. Lemma (12.17) and Theorem (12.18) are due to Efremovic [19]. 13. The first part of this section is based on the work of Alfsen and Njastad [6]. Further interesting results concerning lattice operations, completions, etc., of generalized uniform structures are also given in this paper. In proving Theorem 13.7 (v), we make use of the fact that a uniform space with a countable base is pseudo-metrizable. The material on correct structures is due to Mordkovic [70] and Efremovic, Mordkovic and Sandberg [21]. The term' contiguity' was denned by Lubkin in a somewhat different manner. 14. The concept of height was introduced by Hursch [37], who used the notation <* for the height relation. We have simply used the notation ^ , as there is no danger of confusion. This section is based on the papers of Hursch [37, 38]. Thron initially conjectured that there exists at most one uniformity in the intersection of a height class and a proximity class. This, however, was later disproved by Hursch. For details of this example along with other interesting results and examples, the reader is referred to the original papers. 15. It was first conjectured by Isbell [44] (p. 35, Ex. 17) that different uniformities on X are not iJ-equivalent; in fact, he suggested that they induce non-equivalent families of neighbourhoods of the element XeH(X). Smith [109] gave a counter-example to show the latter statement to be false, but nevertheless proved several results supporting Isbell's conjecture. Ward [M], however, disproved this conjecture. Theorem (15.1), along with Corollaries (15.2) and (15.3), is due to Smith [109]. Ward [118] proved Theorem (15.7), while Theorem (15.9) is due to Hursch [G]. The statement' tfl is proximally finer than i^' used in Theorem (15.7) means, of course, that 8(ir) < §(°ll). More material concerning this fascinating problem can be found in Isbell [I] and Ward [119].

CHAPTER 4

FURTHER DEVELOPMENTS 16. Proximal convergence In this section we introduce the concept of proximal convergence, or convergence in proximity, which is analogous to uniform convergence. Proximal convergence preserves (proximal) continuity and enjoys interesting relationships with both uniform convergence and continuous convergence. Recall that a net is a function on a directed set (D, ^ ) . A subset E of D is a cofinal subset of (D, > ) iff for every neD, there exists anmeE such that m ^ n. (16.1) D E F I N I T I O N . Let (Y,S) be a proximity space and let (fn: neD) be a net of functions mapping X to Y. Then (fn) is said to converge proximally tofe Yx ifff(A)$Bfor A <= X and B e f implies fn(A) $B eventually. That uniform convergence is stronger than proximal convergence follows as a corollary to the next theorem. (16.2) T H E O R E M . Let (Y,ir) be an A-N uniform space and let S = 8(i^). If a net (fn) of mappings from X to Y converges uniformly to f, then the convergence is proximal. Proof. Let A ^ X, B <= Y and f(A)$B. Then there exists an 2

entourage Ve^ such that V[f(A)](]B= 0. However, since (fn) converges uniformly to f,fn(x) e V[f(x)] eventually for each xeA; that is, fn(A) cz V[f(A)] eventually. We therefore eventually have V[fn(A)]oB = 0 and hence fn{A)$B, showing that (fn) converges to / proximally. We now prove a near converse of the above result: (16.3) T H E O R E M . If a net (fn:neD) of mappings from X to a totally bounded uniform space (Y,ir) converges proximally to fe Yx, then the convergence is uniform. Proof. Suppose the convergence is not uniform with respect [94]

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95

to y. Then there exists a F e ^ a cofinal subset Do of D and, for each neD0, dLnxneX such that (fn{xn),f(xn)) $ V. Choose We^ 2

m

such that W a V and W = \J (Atx At)9 where {^: 1 ^ i ^ w} i= l

is a #-cover (see 12.6) of X. Now for some i0 ^ m, there is a cofinal subset 5 of Do such that fp(xp) cAio for each /?e J3. Set 0 = {xp'./leB}

and let

xp,xqeG.

If (fP(xP), f(xq)) E W, then (fQ(xq),fp(xp))eAio

x ^, o c: W implies

2

that (fq(xq),f(xq))e W c= F, a contradiction. Hence (/P(^)JK))^^

or

/p(

1

Therefore /^(C?) 4 TT[/((r)] for every j3eB, contradicting the fact that ( / ^ : / ? G £ ) , and hence (fn)9 converges proximally t o / . (16.4) COROLLARY. If Y is a uniformizable space with a unique uniformity, then proximal and uniform convergence are equivalent. The following results show that proximal convergence preserves both continuity and proximal continuity. (16.5) T H E O R E M . Let (fn) be a net of proximity mappings from (X, 8-L) to (Y, 82). If (fn) converges proximally to f, then f is a proximity mapping. Proof. Suppose A S^, but f(A) <]S2f(B). Then there is a subset E of Y such that f(A)S2E and (Y-E)92f(B). We therefore eventually have/ n (^.) $2 E and (Y-E)$2 fn(B); that is, eventually fn(A)S2fn(B), contradicting the hypothesis that each fn is proximally continuous. The following result is proved similarly, using the characterization that a mapping/is continuous iff/(if) ^ f(A) for each subset A ofX. (16.6) T H E O R E M . If a net (fn) of continuous mappings from a topological space X to a proximity space Y converges proximally to f, then f is continuous. We next consider continuous convergence. A subnet (xn.: ieE) of a net (xn: n eD) is said to be isotone iff the mapping i -> ni of E into D is order preserving.

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(16.7) D E F I N I T I O N . Let X and Y be topological spaces. A net (fn:neD) of mappings from X to Y converges continuously to fe Yx iff for every isotone subnet (fn.:ieE) of (fn) and every net (xf ieE) in X converging to x, (/ ni (^)) converges tof(x). (16.8) T H E O R E M . Let Xbea compact topological space, (Y,S) a separated proximity space and (fn:neD) a net of continuous mappings from X to Y. If (fn) converges continuously to a continuous function fe Yx, then (fn) converges proximally tof. Proof. Let <& be the Smirnov compactification of Y and let the closure of subsets of Y be taken in (W. Suppose A a X, B a Fand fn(A) SB, where n is an arbitrary member of a cofinal subset H of D. Now/n(^4) SB implies/w(-4) n 5 + 0 . Since/^ is continuous and A is compact, fn(A) is compact and hence closed (<@f is Hausdorff). Thus/w(.Z) contains fn(A), and so meets B. Pick xneA with fn{xn) = yneB. Now the net (xn:neH) has a subnet (xn. :ieE) converging to some x e A and, without loss of generality, we may assume this subnet to be isotone. Ity continuous convergence, {fni{xni))-+f{x) and, since fn.{xn.)eB for all ieE, f(x) eB. Thus f(A) n B + 0, implying that f(A) n B #= 0 . We therefore have/(^4) S B, and (fn) converges to/proximally. (16.9) T H E O R E M . Let X be a topological space and (Y,S) a separated proximity space. If a net (fn:neD) of continuous mappings from X to Y converges proximally to fe Yx, then (fn) converges continuously to f. Proof. Let & be the Smirnov compactification of Y, and let the closures of subsets of Y be taken in &. If the conclusion is not true, then there exists an isotone subnet (fn.:ieE) of (fn:neD) and a net (xfieE) converging to xeX such that (/ ni (^)) does not converge tof(x). Because Y is regular, there are neighbourhoods V and W of f(x) such that W <= F, and a cofinal subset C of E such t h a t / n i ( ^ ) ^ V for all ieC. Since/is continuous by (16.6), there exists an i0 such that f{xi)eW for all i ^ i0. Set A = {xf i eC, i ^ Q and B = {/7l.(^): i eC}. Then and

f(A)aW.

Hence f(A)n B — 0 , so that/(^4)#i?. However, for each ieC,

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97

fn.(A) n B 4= 0 ; that is, it is false that eventually/J.4) <2> B. Thus (fn) does not converge to / proximally. 17. Unified theories of topology, proximity and uniformity E.H.Moore, in his New Haven Colloquium lectures in 1906, said ' the existence of analogies between the central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features'. Since several features are common to topology, proximity and uniformity, one would expect the existence of general structures which include these as special cases. In this section we briefly outline, without proof, two such general structures. In one such general theory, the idea of a transitive relation between subsets of a set X is taken as the basis. If (X, r) is a topological space, one can define a transitive order between the subsets of X as follows: (17.1)

A
iff ^ c I n t ( j B ) .

It is possible to give precise axioms for this order which will determine the topology in a unique way, namely (17.2)

GET iff G < G.

One such set of axioms is the following: (i) 0 < 0 and X < X. (ii) A < B and A' < B' implies Af]Af < Bf]Bf. (iii) Ai < B^iel) implies (J Ai < \J Bt for any index set I. iel

iel

For the case in which (X, S) is a proximity space, we have already seen in Section 3 that the transitive relation
A
iff U[A]czB

for some UeW.

In order to find the axioms for a general structure encompassing topology, proximity and uniformity, one has merely to

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PROXIMITY SPACES

note the significant common properties satisfied by (17.1), < and (17.3). The following definition arose from such considerations: (17.4) D E F I N I T I O N . A topogenous order on a set X is a relation on the power set ofX, denoted by < , satisfying: (i) (ii) (iii) (iv)

0 < 0 and X < X. A < B implies A c= B. A c A' < B' c B implies A < B. A < B and A' < B' together imply An A' < B(]Bf and AvA'


(17.5) D E F I N I T I O N . A syntopogenous structure on a set X is a family SP of topogenous orders on X satisfying: (S±) If
U[A] n V[A] for each A e Jt.

(17.6) D E F I N I T I O N . 2 <= ^ ^ is a GT-structure (GT standing for 'generalized topologicaV) on X with respect to Jt iff the following conditions are satisfied: (i) IfUeZ, then A c U[A] for each AeJP. (ii) IfU,VeX,then(UnV)e?:. (iii) / / U e 2, then for each A e ^# there is a V e 2 (depending on A) such that B e ^ and B c V[A] together imply the existence of a (depending on B) for which W[B] c U[A].

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J

(iv) IfUe 3P * and, for every A e Ji, there is a F e 2 (depending onA)suchthat V[A] a U[A],then £7e2. The pair (X, 2) is called a GT-space. With every GT-space (X,2) is associated a 'proximity 7 relation <JS in the following manner: (17.7)

DEFINITION.

A S^B

If A E ^ and B E^, define

iff U[A] n B #= 0 /or ever?/

£7e2.

2 is said to be symmetric iff A Sz B implies BS^A whenever It is easy to see that a GT-structure is a generalization of a topological structure; for if ^#= «^", then 17.6(i)-(iv) are precisely the usual neighbourhood axioms of Hausdorff, where the fundamental neighbourhood system at x is {£/[#]: ?7e2}. It can also be verified that for A <= X, (17.8)

Z = {a::a:*£^}

defines a Kuratowski closure operator. Thus every GT-structure on X induces a topology on X. Conversely, given a topological space (X, r), we may define a 6rT-structure 2 on JT by 2 = {Ue^^:xeInt{U[x\)

for each xeX).

Let us now consider a symmetric trT-space (X, 2) with respect to ^ — 3P. In this case, conditions 17.6 (i)-(iv) and the symmetry condition (17.7) are equivalent to the following: (17.9)

(i) Ue 2 implies A c U[A] for each A a X. (ii) U, V e 2 implies (C7 n V) e 2. (iii) C/G2 implies that for each i c j 5 there is a F e 2 (depending on A) such that F[F|\4]] c U[A]. (iv) IfUe&& and, for each i c j , there exists a F e 2 (depending on A) such that F[^4]
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PROXIMITY SPACES

tions Ue0& such that A$(X- U[A]) for every i c j yields a symmetric G^F-structure on X. Finally, we consider uniform structures. For Ue^x, define (17.10)

U[A]= U U[x]. xeA

If, in (17.6), we set Jt = ^ and require that the mappings involved be independent of x, y etc., we obtain the following set of conditions: (17.11)

(i) If * 7 G 2 , then xeC/[£] for each £ e Z . (ii) If £7, F G 2 , then([7n F ) G 2 . (iii) If £ / G 2 , then there exists a F G 2 such that V[V[x]] a U[x] for each xeX. (iv) If, for UE^X, there exists a F G 2 such that V[x] c= U[x] for all xeX, then C7G2.

In this case, 2 is a quasi-uniform base. If in addition 2 is symmetric i.e. (v) UeE assures the existence of a VeH such that ye V[x] implies xe U[y], then 2 is a uniformity base. Conversely, given a uniform structure °ll on X, one can define a symmetric GT-structure 2 on X in the following manner: for TJ'e°ll and # G X , define Ue0>x by tf[*] = {*/GX: (*,?/)G £7'}.

Definition (17.10) then supplies the desired element of £P^, with 2 consisting of all such elements. 18. Sequential proximity In this section, we briefly consider nearness relations that can be introduced in Frechet spaces, i.e. spaces in which convergence of sequences is axiomatized. We motivate the discussion by considering sequences in a separated proximity space (X, 8). If (xn) and (yn) are sequences of elements in X, define (18.1)

(xn) n (yn)

iff {xkJ 8 {ykn} for every increasing sequence (Jcn) of natural numbers.

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101

Using Greek letters £, rj etc. to denote sequences in X, the nearness relation n of (18.1) can easily be shown to satisfy the following properties, the first three of which imply that n is an equivalence relation: (18.2)

(i) £ng for all g. (ii) gnv implies vnE,. (iii) £ n £' and £' n £" together imply £ n £". (iv) (x) n (y) iff a; = y. (v) (z n )n(yj implies ( ^ J n ^ J for every subsequence (kn) of JV. (vi) If (xn)xi(yn), then there exists an increasing sequence (&n) in N such that for every subsequence

(kj

of (fcj,

fejj*^).

(18.3) T H E O R E M . If(X,d) is a metric space and Sis the induced metric proximity on X, then (xn)n(yn)

iff

Proof. Clearly, if limd(xn,yn) = 0 then (xn)n(yn). Conversely, n—>co

suppose lim d(xn9 yn) 4= 0. Without loss of generality, we may n—>oo

suppose that d(xn, yn) ^ e > 0 for all n e N. Then one of two cases can arise, which we shall consider separately: (i) Suppose there is a finite set {xv ..., xk} such that min {d(xn,xi)} < el4: for neN, lik

or similarly a finite set of y's. Then there exists an xio and an increasing sequence of natural numbers (pn) such that But d(yPm,xPn) > d(yPm,xPm)-d{xPm,xPn)

>e-e/2

= e/2, so that

}n (ii) Suppose there exists an increasing sequence (rn) of natural numbers such that d(xrn,xrjn) ^ e/4 and d(yrn,yrm) ^ e/4 for n, meN, n #= m. Let &x = 1. Define kno+1 inductively such that d(xk.,yk.) ^ e/8 for i =\=j, where i,j ^ n0. There exist at most finitely many x e {xn: n ^ &nJ such that

d(xn,{{xki}[){yk}:i^no})

< e/8,

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PROXIMITY SPACES

and likewise for [yn\ n ^ kno}. Thus there is an n' > Jcno such that the distances between each of xn,, yn, and the set {xk^ U {y^}* where i ^ n0, are greater than e/8. Set kno+1 = n'. Then, for the sequence (kn) in N, defined inductively in this way, (xkn)fi(ykn). (18.4) D E F I N I T I O N . A sequence (xn) in a separated proximity space (X, S) is Cauchy iff (xn) n (xkn) for every increasing sequence (JcJinN. Every convergent sequence is Cauchy, although the converse is not true. If every Cauchy sequence in a metric space converges, then the metric space is complete in the usual sense. (18.5) T H E O R E M . Let Xbea complete metric space and letSC he its Smirnov compactification. Then the first axiom of countability Cz is not satisfied at points of3C — X. Proof. If Cj is satisfied at xe3? — X, there exists a sequence (xn) in X converging to x. But (xn) is Cauchy and, since X is complete, we have that xeX, a contradiction. Having seen the usefulness of the concept of nearness of sequences in a proximity space, we now turn our attention to the axiomatization of the nearness relation. Properties in (18.2) suggest the following definition: (18.6) D E F I N I T I O N . A nearness relation n, between sequences of elements of a set X, which satisfies 18.2(i)-(v) is called a °U3?structure. The pair (X, n) is called a °U££-space. / / in addition n satisfies 18.2 (vi), then (X, n) is a <^i?*-space. Theorem (18.3), a special case of (18.1), gives an example of a °ll££*-space. We now present two further examples: (18.7) E X A M P L E S , (i) A trivial ^J§?*-structure can be defined on X by letting (xn) n (x'n) iff xn = x'n eventually. (ii) Let (F,n) be a ^^-(resp.^J^*-) space and let & be a collection of functions from a set X to Y which separates points; that is, xx #= x2 for xvx2eX implies the existence of an such that/(^ x ) =h/(#2)- Define a nearness relation N on X by (xJTX{x'n)iK(f{xn))n{f(x'n)) for all feF. Then (X, N) is a <&S?-(resp. <%&*-) space.

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(18.8) D E F I N I T I O N . A sequence (xn) in a °U3?-space (X,n) converges toxeX iff (xn) n (x). (18.9) D E F I N I T I O N . A Frechet or 3?-space is a pair (X,j£?) where X is any set and ^ is a collection of sequences, converging to certain points called limits, satisfying the following conditions: (i) Ifxn = xfor all neN, then (xn)e££ and Urnxn = x. (ii) limxn = x implies (#n.)eJ£? and limxn. = x, for every subsequence (xn.) of (xn). (X, J§?) is called an <J§?*-space iff whenever (xn) eJ? does not converge to x, there exists a subsequence (xni) e J§? such that no subsequence of it converges to x. The following result is obvious. (18.10) T H E O R E M . A °k3?-(tyl<£*-) space with the convergence relation of (18.8) defined on it is an J§?-(J§?*-) space, where j£? is the set of all convergent sequences. We now turn to the problem of defining a nearness relation in an JS?-(J2?*-) space. (18.11) D E F I N I T I O N . A nearness relation n in an JS?-(JS?*-) space is compatible iff the convergence induced by n coincides with the original. There may exist several compatible °ll3?-(!%<£*-) structures on a given j£?-(J2?*-) space. A partial order can be defined on these structures, but we shall not go into the details. Suffice it to say that given an JS?-(J§?*-) space X, we may introduce two extreme compatible ^oSf-structures: (18.12) limit.

(a) (xn) n0 (x'n) iff both sequences converge to the same

(b) (xn) iii {x'n) iff for each increasing sequence (in) in N, either (xin) and (x'in) are both divergent or both converge to the same limit. Given a ^J^-space (X, n), we may define (18.13) (xn)n* (yn) iff each increasing sequence (in) in N contains a subsequence (jn) such that (xjn) n (yjn).

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Then (X, n*) is a °l/<£*-space. If (X,«j£?*) is compact (i.e. every sequence in X has a convergent subsequence), then n* = n*; that is, it has a unique compatible °U3?*-structure. Finally, we consider i?-uniform convergence. (18.14) D E F I N I T I O N . Given a °USe-{pr °ll£e*-) space (Y, n), a sequence (fn) in Yx converges J?-uniformly to fe Yx iff

(/>„)) n OK)) for every sequence (xn) in X. (18.15) T H E O R E M . Let X be a topological space and (Y, n) a °ttJ£-space, where n is induced by the proximity S on Y. If fn:X -> Y is sequentially continuous at xoeX for each neN and (fn) converges J§?-uniformly tofe Yx, then f is sequentially continuous at x0. Proof. Suppose (xn) converges to x0, but (f(xn)) does not converge to f(x0). We may then suppose, without loss of generality, that there is a neighbourhood U of f(x0) such that f(xn) ^ U for every neN. Now there exists a neighbourhood U1 of f(x0) such that Ux <^ U. Since (fn(x0)) converges to f(x0) and/^ is sequentially continuous at x0, for each neN there exists a subsequence (kn) of N such thatfn(xkn)eUv Thus (fn{xkn))jii{f{xkn)),

contrary to the hypothesis that (fn) converges uniformly to/. 19. Generalized proximities Several generalized forms of proximity structures are known in the literature, some of which were introduced even before the appearance of Efremovic proximity spaces. In this section we briefly study some of these generalized proximities and their interrelationships. One of these is studied in some detail in the next section. Since a compatible proximity can be introduced only in completely regular topological spaces, one wonders whether the axioms of proximity can be relaxed so as to embrace more general topological spaces. Indeed, generalized proximities can in fact be introduced in any topological space, as we shall see presently.

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(19.1) D E F I N I T I O N . A binary relation a defined on the power set of X is called a Leader or LE-proximity on X iff it satisfies the following conditions: (i)

Aa(B[)C)iffAocBorAocC,and (A[jB)aCiffAocCorBocC. (ii) AaB implies A 4= 0 , B 4= 0. (iii) AocB and bocC for each beB together imply AocC. (iv) A n B 4= 0 implies AocB. If in addition a satisfies (v) AaBiffBotA, then a is called a Lodato or LO-proximity. The pair (X, 8), where Sis a LO -proximity, is referred to as a Lodato space. (19.2) D E F I N I T I O N . A binary relation ft defined on the power set of X is called a Pervin or P-proximity on X iff fl satisfies 19.1 (i), (ii), (iv)and (iii') A fjtB implies there exists an E c X such that AfitE and

(X-E)fiB. It should be noted that symmetry is demanded neither for a LE-proximity nor a P-proximity, and thus care must be taken in writing proofs involving either of these. Throughout this section we shall use the letter £ to denote an arbitrary generalized proximity on X. £ is said to be separated iff it satisfies the additional condition x£y implies x = y. A partial order may be defined on the collection of all generalized proximities on a set X by (2.16), namely g x <£ 2 iff A^B implies A^B. We now compare a with p. First note that 2.1 (i), namely A E,B, A c C and B a D together imply C g D, holds for £ = a or j3. (19.3) T H E O R E M . Every Y-proximity p on X is also a LEproximity on X. Proof. It is sufficient to show that /? satisfies 19.1 (iii). Suppose ApB and bj3C for every beB. If A0C, then by 19.2(iii') there exists an E
106

PROXIMITY SPACES

A / E and A J3 B together imply B cj: E, i.e. B n (X - E) 4= 0 . If 6 G 5 n ( I - £ ) , then 6/?0 implies (X-E)/3C,a contradiction. Given £ = a or /3, define (19.4) AZ = {xeX:x£A}. As in (2.7), one can prove that if £ = a or /?, then (19.4) defines a Kuratowski closure operator on X. In the case that £ = a, the result (A£)£ = A% follows from 19.1 (iii). Actually a weaker axiom, namely (19.5) for XGX, xaB and bocG for all beB implies xaC, is sufficient to guarantee the idempotence of the operator £. Thus either £ = a or /? yields a topology r = r(£), and we say that r and £ are compatible. Moreover, it is easy to show that (19.6) AgB iff AiB*>. In contrast to (Efremovic) proximity structures, which are compatible with completely regular spaces, we have the following result: (19.7) T H E O R E M . Every topological space (X,T) has a compatible LE- or P-proximity £0 given by (19.8) A^B iff AnB* 0. Moreover, £0 is the largest compatible LE- or ~P-proximity. Proof. In view of Theorem (19.3) we need only verify 19.2 (iii'), since the other axioms follow readily. If A $0 B, then A n B = 0. Set E = B. Then i n l = 0 and (X-E)(]B= 0 , i.e. A$oE and (X - E) g0 5 . That r = r(£0) follows from the fact that a; £0 J. iff xeA. Finally, if £ is any LE- or P-proximity, then from (19.6) we have that AfcB implies A n B = 0, i.e. A %0 B. A topological space (X, r) is Ro iff either of the following equivalent conditions is satisfied: (19.9)

(i) xey iff yex. (ii) XEGGT implies x c ( J .

(19.10) T H E O R E M . / / a is any LO-proximity, then r(oc) is necessarily Ro. Conversely, a compatible LO -proximity OLX can be defined on every R0-space by (19.11) A(xxB iff AnB+0. Furthermore, ax is the largest compatible LO-proximity.

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Proof. That r(oc) is Ro follows from the fact that xey iff xocy iff y oc x iff y e x. To prove that oc± is a LO-proximity, it suffices to verify 19.1 (iii). Suppose AOLXB and boc1G for each beB. Then An B + 0 and 6 n C 4= 0 for each 6 G 5 , i.e. there exists a c e C such that ceb. Since X is Ro, bee c: C and hence ^4 n C 4= 0 , showing that ^ o ^ C Since xeA"* iff #0^4 4= 0 iff #e^4, it follows that r = T(OC1). For every LO-proximity a, A oc B iff ^4 a i?, and thus ax is the largest compatible LO-proximity. (19.12) COROLLARY. There exists a LO -proximity which is not a P-proximity. Proof. There exist i?0-spaces which are not regular and (19.11) shows that if OLX were a P-proximity, then it would also be an Efremovic proximity, which is impossible. A quasi-uniformity i o n a set X is a family of subsets of X x X which satisfies all the conditions for a uniformity with the possible exception of the symmetry axiom. The topology r(£>) induced by £} is the family {6? <= X: for each xeG, there exists a P e i such that U[x] <= G}. It is known that every topological space (X,r) has a compatible Pervin quasiuniformity SP consisting of all sets which contain finite intersections of sets of the form: (19.13)

SG = [GxG][)[(X-G)xX]

where GET.

The proof of the following theorem is similar to that of Theorem (10.2). (19.14) T H E O R E M . Every quasi-uniformity 21 on X induces a P- (or LE-) proximity £ on X defined by (19.15)

A£B

iff

(AxB)f)U*0

for every

Uel.

Moreover, r(£) = T(£). If 2, = 3P (the Pervin quasi-uniformity), then £ = £0, as defined by (19.8). Let X be a 7^-topological space and A, B be non-empty subsets ofZ.If (19.16)

(Af]B)[)(A[]B)= 0,

108

PROXIMITY SPACES

then (A u B) is separated, i.e. not connected in the HausdorffLennes sense. If we write AtyBto denote this separation, then y obviously satisfies the axioms given in the following definition: (19.17) D E F I N I T I O N . A binary relation y defined on the power set ofX is called a Separation or S-proximity iff the following conditions are satisfied: (i) (ii) (iii) (iv) (v) (vi)

Ay B implies By A. (A[jB)yCiffAyCorByC. AyB implies A 4= 0,5=)= 0 . xyB and byC for every beB together imply xyC. A n B =f= 0 implies AyB. xyy implies x = y.

Clearly 19.1 (iii) is stronger than 19.17(iv), and the following result is immediate: (19.18) T H E O R E M . Every separated LO -proximity on X is also an S-proximity on X. We now give an example of an S-proximity which is not a LO-proximity: (19.19) E X A M P L E . If on the set of real numbers with the usual topology we define y by (19.16), then y is strictly larger than cc1 as defined by (19.11). But OLX is the largest compatible LO-proximity, as shown in Theorem (19.10), so that the S-proximity y cannot be a LO-proximity. 20. More on Lodato spaces In the theory of (Efremovic) proximity spaces, one of the most important results (Theorem (7.7)) is the following: every separated proximity space is 5-homeomorphic to a dense subspace of a compact Hausdorff space in which the unique compatible proximity 80 is given by A80B

iff A(]B+ 0 .

A natural question arises which, in fact, provides a motivation for Lodato spaces: does there exist a set of axioms for a binary relation 8 on the power set of X such that 8 satisfies these

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109

axioms iff there is a topological space Y in which X can be topologically embedded so that A SB in X i& AnB =£ 0 in 7? Granting that such an embedding exists, one can easily show that S must necessarily satisfy the axioms of a LO-proximity as given in Definition (19.1). The converse question was first handled in the literature using a technique involving clusters which is similar to the construction of the Smirnov compactification (Section 7), the difference being that instead of defining a proximity on the set of all clusters, one merely defines a topology via a Kuratowski closure operator. This approach provided an affirmative solution under the restriction that Y be a T^-space and X be regularly dense in Y. The latter condition was subsequently relaxed, when the notion of a bunch was introduced. Recently the theory of Lodato spaces has been extended with the introduction of 'symmetric generalized uniform structures', which we shall call M-uniform structures. In this section, we outline the theory of Lodato spaces only briefly and refer the interested reader to the literature for a more detailed account. (20.1) D E F I N I T I O N . A subset X of a topological space (Y,T) is regularly dense in Y iff given Uer and peU, there exists a subset A of X withpeA <= U. Use of the term 'regularly dense5 is justified by the following readily-verified facts: (i) A regularly dense set is dense. (ii) If Y is regular, then dense and regularly dense sets coincide. Definition (5.4) is used to define a cluster in a Lodato space exactly as it is in a proximity space. One can prove without difficulty that Remarks (5.5) and Lemma (5.6) remain valid in the generalized setting. We are now in a position to present the first solution to the question posed in the opening paragraph. (20.2) T H E O R E M . If Sis a binary relation on the power set ofX, then the following statements are equivalent : (a) There exists a T±-space Y in which X can be topologically embedded as a regularly dense subset such that (20.3)

A SB

in

X

iff A{\B * 0

in

Y.

l]0

PROXIMITY SPACES

(b) 8 is a separated LO -proximity satisfying the additional axiom: (20.4) A SB implies the existence of a cluster to which both A and B belong. Proof. The symbol ~ will be used to denote closure in Y. In showing that (a) implies (6), wefirstnote that by Theorem (19.10), S is a LO-proximity on Y, and so induces one on the subspace X. Furthermore, the ^-axiom is equivalent to the condition that xSy implies x = y. To prove that (20.4) is satisfied, suppose A SB in X, and hence Ap\ B 4= 0 in Y. Let ye An B and set or = {E c: X:yeE}. That cr is a cluster in X will be clear if we can show that C 8 E for every E e cr implies Cecr. Suppose on the contrary that C
Cl {0>) = {cr e3C\ E C X absorbing SP implies E e cr}.

Observing that (20.6)

Cl(f(A)) = J / for i c l ,

it becomes a routine verification to show that Cl is a Kuratowski closure operator on 2£. If cr'GCl(cr), then clearly cr c cr1 and, since clusters are maximal (Lemma (5.6)), cr = cr'. Consequently X is Tv We next prove (20.3). If A SB, then by (20.4) there exists a (ie9C such that A,Beer. Hence
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111

Finally, to show that X is topologically embedded in SC, we need only show t h a t / i s bicontinuous; that is, XGA8

iff

But clearly, xeA

8

f(x)eCl(f(A))

= tf for

iff xSA iff Aeax ifff(x)

all

A c X.

= crx

(20.7) D E F I N I T I O N . ^4 non-empty collection er of subsets of a Lodato space (X, S) is called a bunch iff the following conditions are satisfied: (i) IfA.BeaJhenASB. (ii) If (A[j B)eor,thenAecr or BGCT. (iii) If A eor and a$B for every aeA, then Bea. (20.8) R E M A R K S , (a) Although every cluster is a bunch, a bunch need not be a cluster. To illustrate the latter statement, take the family of all infinite subsets of an infinite set X with cofinite topology and LO-proximity ax (19-11). (b) XGGT for every bunch a, as a consequence of 20.7 (iii). Although redundant, this condition was originally included in the definition of a bunch. Consider a family 8ft of bunches in (X, S) which satisfies the two conditions: A,Bee (20.9) (i) A SB implies there is a cre& such that (ii) If or, a'' e& and either A e cr or B e cr' for all subsets A, B of X such that A U B = X, then a = &'. By introducing a Kuratowski closure operator o n ^ using (20.5), one can prove the following result in a manner similar to the proof of Theorem (20.2). (20.10) T H E O R E M . Given a set X and a binary relation 8 on the power set of X, the following statements are equivalent: (a) There exists a Hausdorff space Y in which X can be topologically embedded so that (20.3) holds. (b) (X, S) is a separated LO-proximity space possessing a family 38 satisfying (20.9). In the theory of proximity spaces, interesting relationships exist between uniform structures and proximity structures, as we saw in Chapter 3. The notion of a Mozzochi or M-uniform

112

PROXIMITY SPACES

structure is now introduced, playing a role in Lodato spaces similar to that played by uniform structures in proximity spaces. (20.11) D E F I N I T I O N . A non-empty subset °ll of the power set ofXxXisa Mozzochi or M-uniform structure on X iff the following axioms are satisfied: (i) A c U for each UeW. (ii) UG^implies U = TJ-1. (iii) For every A <= X and U, VeW, there exists aWe°U such that W[A] cz U[A] n V[A]. (iv) For every pair of subsets A,B of X and every U e%, V[A] n 5 + 0

for every F e f

implies the existence ofanxe Band W e% suchthatW[x] <= U[A]. (v) UeWandU c V = F" 1 <= XxX implies Y<E°U. The pair (X,°ll) is called an M-uniform space. It is separated iff (vi)

n f / = A.

A subfamily 3% of an ^.-uniformity °llis a base for °tt iff each member of°ll contains a member of £%. Clearly, the family of all symmetric entourages of a uniformity (and, in fact, of an A-N uniformity) forms an M-uniform structure. The existence of M-uniformities which are not uniformities will be demonstrated shortly (see 20.19). The proof of the following theorem, being reasonably straightforward, is omitted. (20.12) T H E O R E M . Every M-uniform space (X,°U) has an associated topology r = r(^) defined by (20.13)

GET iff for each XGG, there exists a VG°ll such that U[x] c G.

Alternatively, the operator Cl defined by (20.14)

Cl(^) = f| U[A]

is a Kuratowski closure operator yielding r. The topology is necessarily Bo; T($/) is T± if and only if °U is separated.

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Just as every uniform structure has an associated proximity relation, it can be shown that every M-uniform structure °ti induces a LO-proximity 8 = 8(%) defined by (20.15)

A SB iff iff

U[A]nB+0 (A x B) n U # 0

for every

Ue<%

for every

= 5(^),then5and^aresaidtobecompatible.Itisclearthat whenever one is separated, so is the other.) In fact, an even stronger result is valid, the proof of which is similar to that of Theorem (13.14) and is thus omitted. (20.16) T H E O R E M . Let 8 be a binary relation on the power set of X and % be a collection of symmetric subsets of X x X such that 8 and & satisfy (20.15). Then 8 is a LO -proximity if and only if °ll is a base for an M-uniform structure. Furthermore, r(8) = (20.17) T H E O R E M . Every Lodato space (X, 8) has a compatible M-uniform structure °ll (i.e. 8 = 8(%)). Proof. For every pair of subsets A and B of X, define UAB =

XxX-[(AxB){j(BxA)].

Set -T = {UAiB\A$B}. Then each member of -T is clearly symmetric, so that 20.11 (ii) is satisfied. Now if A SB, then UA,B\A\ fl B = 0. Conversely, if UcfD[A] (]B = 0 for some pair G,D such that C$D, then either A a C and B a D or A a D and B <= C. In either case, A SB. Hence by the previous theorem, i^ is a base for an M-uniform structure °ll l0 0 and 8 = (20.18) COROLLARY. The topology r on X is the topology induced by some M-uniform structure if and only ifr is Ro. Proof. This result follows from Theorems (19.10) and (20.17). (20.19) R E M A R K . Since there are i?0-spaces which are not completely regular, the above corollary shows the existence of Muniformities which are not uniformities. (20.20) D E F I N I T I O N . Given a Lodato space (X,8), the collection of all compatible M-uniform structures on X is a LO-proximity class of M-uniformities and is denoted by

114 PROXIMITY SPACES (20.21) T H E O R E M . Let (X,8) be a Lodato space. Then °ll^ as defined in (20.17), is the smallest member of T(8) (under the usual partial order of set inclusion). Proof. Suppose UA}Be<%0 and i^eY(8). Since A$B, we know from (20.15) that there exists a F e f such that (AxB)nV = 0. But V is symmetric, so that (BxA)n V = 0 also. We therefore have V <= UA B, showing that UA}BE^In contrast to the situation with a proximity class of uniformities, every LO-proximity class of M-uniform structures possesses a largest element and hence forms a complete lattice. That such is the case is evident from the next theorem. (20.22) T H E O R E M . The union of an arbitrary family of Muniform structures belonging to the same proximity class Y(8) forms a base for a member of Y(8). Proof. If 0$ denotes the union of such a family, then A8B iff U[A](]B 4= 0 for every Ue&. Consequently, by Theorem (20.16), 38 is a base for some M-uniform structure which is a member of T(8).

Notes 16. The concept of proximal convergence and Theorems (16.2), (16.3) and (16.5) are due to Leader [54]. The proof of Theorem (16.3) appearing in this section, however, is that of Njastad [80]. Leader has also shown that if a net (fn) is in fact a sequence, then proximal convergence and uniform convergence coincide. Njastad has generalized this result to the case in which the directed set of the net is linear and the range space possesses a generalized uniform structure. Theorems (16.8) and (16.9) were proved by Wolk [122]. The definition of continuous convergence given here is stronger than that given in Kelley [J] (p. 241, Problem M). 17. As pointed out in the text, we have presented only bare outlines of two general theories. Csaszar [12] first gave his theory of syntopogenous structures. Subsequently, Dolcinov [15] used the neighbourhood approach. 18. Mrowka [75] introduced sequential proximity in a proximity space and proved Theorems (18.3), (18.5) and (18.15). An axiomatic treatment of sequential proximity was first given by Goetz [28]. For an up-to-date account of this subject see Goetz [29], where a summary of the work of Poljakov [91] is also included.

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19. We have taken the liberty to deviate from the names originally used by authors to describe their generalized proximities, in that we have used the terms LE-proximity and LO-proximity space rather than ' topological d-space' and ' symmetric generalized proximity space' respectively. The definition of LE-proximity was first given by Leader [57], his motivation being an explicit formulation of the proximity relation induced on the product of proximity spaces by the proximity relations on the co-ordinate spaces. Lodato [63, 64, 65] introduced and developed the LO-proximity relation. Mozzochi [72] has since carried out an extensive study of LO-proximity, a brief account of which is given in Section 20. For information and references concerning the i?0-axiom, consult Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 10, 53-4 (1967). Pervin [84] introduced the P -proximity relation, calling it ' quasiproximity'; Steiner [110] later corrected an error in this paper. The S-proximity relation was conceived independently by Krishna Murti [52], Szymanski [113] and Wallace [116, 117]. For an interesting discussion of some of the generalizations of proximity, see Pervin [85]. Interrelationships between various generalized proximities discussed in this section are due to the authors. Recently, Mattson [67] has studied S-proximity spaces in relation to the extended topologies of Hammer. Around 1964, Hayashi [32, 33] introduced two new proximities: (i) paraproximity, and (ii) pseudo-proximity. Given a paraproximity he defines a topology by calling a set U open iff US(X — U), while in the case of pseudo-proximity U is defined to be open iff x$(X — U) for each xeU (cf. (2.4)). The topology of a paraproximity is necessarily completely normal. Only two axioms are used to define pseudo-proximity: (i) A $ 0 for every A <= X, and (ii) C8{A[)B)iff CdA or CdB. Recently, Fedorcuk [22] has defined ^-proximity in regular topological spaces. 20. Theorems (20.2) and (20.10) were proved by Lodato [63, 64], with the notion of a bunch being introduced in the second paper. In his dissertation, Mozzochi [72] generalized many results of Chapter 3 to Lodato spaces with the introduction of' symmetric generalized uniform structures', which we have called M-uniform structures. Only a small part of his work is outlined in this section. LO-spaces have recently been used to derive a rather general theorem concerning the extensions of continuous functions from dense subspaces (see [138]). Example (20.8) was discovered by P. L. Sharma.

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P. (1939). On bicompact extensions of topological spaces, Mat. Sb. 5 (47), 403-423 (In Russian; German summary). FEEUDENTHAL, H. (1942). Neuaubau der Endentheorie, Ann. of Math. 43 (2), 261-279. GAAL, S. A. (1964). Point Set Topology, Academic Press, New York. GAL, I. S. (1959). Uniformizable spaces with a unique structure, Pacif. J. Math. 9, 1053-1060. GILLMAN, L. and M. JERISON (1960). Rings of Continuous Functions, Van Nostrand, Princeton. Hu, S.-T. (1949). Boundedness in a topological space, J. Math, pures appl. 28, 287-320. HTJRSCH, J. L. (1967). A theorem about hyperspaces, Proc. Camb. phil. Soc. 63, 597-599. ILIADIS, S. and S. FOMIN (1966). The method of centred systems in the theory of topological spaces, Usp. math. Nauk, 21, 47-76 (in Russian); English translation in Russ. Math. Survs. 21, 37-62. ISBELL, J. R. (1966). Insufficiency of the hyperspace, Proc. Camb. phil. Soc. 62, 685-686. KELLEY, J. L. (1961). General Topology, Van Nostrand, Princeton. MAGILL, K. D., Jr. (1965). iV-point compactifications, Am. math. Mon.72, 1075-1081. TUKEY J. W. (1940). Convergence and Uniformity in Topology, Ann. Math. Stud. 2, Princeton. WARD, A. J. (1966). A counter-example in uniformity theory, Proc.

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10 11 12 13

4

AARTS, J. M. (1966). Dimension and deficiency in general topology, Druk. V.R.B. Kleine der A 3-4 Groningen. ALEXANDROFF, P. (1954). Aus der Mengentheoretischen Topologie der letzten zwanzig Jahren, Proc. of the International Congress of Mathematicians, Amsterdam, I, 177-196; MR 20 # 2697.* ALEXANDROFF, P. (1956). On two theorems of Yu. Smirnov in the theory of bicompact extensions, Fundam. Math, 43, 394-398 (in Russian); MR 18, 813. ALFSEN, E. M. and J. E. FENSTAD (1959). On the equivalence be-

tween proximity structures and totally bounded uniform structures, Math. Scand. 7, 353-360 (Correction (1961), Ibid. 9, 258); MR 22 # 5958. ALFSEN, E. M. and J. E. FENSTAD (1960). A note on completion and compactification, Math. Scand. 8, 97-104; MR23# A 3543. ALFSEN, E. M. and O. NJASTAD (1963). Proximity and generalized uniformity, Fundam. Math. 52, 235-252; MR 27 # 4207a. ALFSEN, E. M. and O. NJASTAD (1963). Totality of uniform structures with linearly ordered base, Fundam. Math. 52, 253-256; MR 27 # 42076. BANASCHEWSKI, BERNHARD and

JEAN-MARIE MARANDA (1961).

Proximity functions, Math. Nachr. 23, 1-37; MR 29 # 2768. BOGNAR, M. (1962). Bemerkungen zum Kongressvortrag 'Stetigkeitsbegriff und abstrakte Mengenlehre', von F. Riesz, General Topology and Its Relations to Modern Analysis and Algebra, (Proc. Sympos., Prague, 1961), Academic Press, New York, Publ. House Czech. Acad. Sci., Prague, 96-105; MR 21r# 2953. CECH, EDUARD (1966). Topological Spaces, Interscience Publishers, John Wiley and Sons. CSASZAR, A. (1957). Sur une classe de structures topologiques generates, Rev. Math, pures appl. 2, 399-407; MR 20 # 1289. CSASZAR, A. (1963). Foundations of General Topology, Macmillan, New York. CSASZAR, A. et S. Mrowka (1959). Sur la compactification des espaces de proximite, Fundam. Math. 46, 195-207; MR 20 # 7255. DOICINOV, D. (1964). A method of introducing the concept of proximity. C.R. Acad. Bulgare Sci. 17, 349-351 (in Russian); MR 29 # 4031. * MR denotes Mathematical Reviews [ 117]

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22

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BIBLIOGRAPHY D. (1964). A unified theory of topological spaces, proximity spaces and uniform spaces, Bold. Akad. Nauk SSSR, 156, 21-24 (in Russian); MR 29 # 5221. DOOHER, T. E. (1966). Proximity relations on an abstract lattice, Ph.D. dissertation, University of Colorado. DOWKER, C. H. (1961). Mappings of proximity structures, Proc. Symp. Gen. Top. Prague, 139-141; MR 26 # 4312. EFREMOVIC, V. A. (1951). Infinitesimal spaces, Dokl. Akad. Nauk SSSR, 76, 341-343 (in Russian); MR 12, 744. EFREMOVIC, V. A. (1952). The geometry of proximity I, Mat. Sb. 31 (73), 189-200 (in Russian); MR 14, 1106. EFREMOVIC, V. A. and A. S. SVARC (1953). A new definition of uniform spaces. Metrization of proximity spaces, Dokl, Akad. Nauk SSSR, 89, 393-396 (in Russian); MR 15, 815. EFREMOVIC, V. A., MORDKOVIC, A. G. and V. J. SANDBERG, (1967). Correct spaces, Dokl. Akad. Nauk SSSR, 172, 1254-1257 (in Russian); English translation in Soviet Math. Dokl. 8, no. 1, 254-258; MR 35 # 977. FEDORCUK, V. (1967). #-space and perfect irreducible mappings of topological spaces, Dokl. Akad. Nauk SSSR, 174, 757-759 (in Russian); English translation in Soviet Math. Dokl., 8, 684686; MR 35 # 7288. FENSTAD, J. E. (1964). On Z-groups of uniformly continuous functions. III. Proximity spaces, Math. Z. 83, 133-139; MR 30 # 1494. FOMIN, S. V. (1958). On the connection between proximity spaces and the bicompact extension of completely regular spaces, Dokl Akad. Nauk SSSR, 121, 236-238 (in Russian); MR 20# 4255. GACSALYI, SANDOR (1964). On proximity functions and symmetrical topogenous structures, Publ. Math. Debrecen, 11, 165-174; Mi?30# 5271. GACSALYI SANDOR (1965). Skew proximity functions, Publ. Math. Debrecen, 12, 271-280; MR 33 # 1836. GAL, I. S. (1959). Proximity relations and precompact structures, Indag. Math. 21, 304-326; MR 21 # 5944. GOETZ, A. (1962). On a notion of uniformity for L-spaces of Frechet, Colloq. Math. 9, 223-231; MR 25 # 5491. GOETZ, A. (1967). Proximity-like structures in convergence spaces, (Presented at conference on General Topology at Arizona State Univ.); Preprint, Univ. of Notre Dame. HACQUE,M. (1962). Sur les ^-structures, C.R. Acad. Sci. Paris, 254, 1905-1907; MR 26 # 1848a. HADDAD, L. (1962). Sur la notion de tramail, C.R. Acad. Sci. Paris, 255, 2880-2882; MR 26 # 1849. HAYASHI, E. (1964). On some properties of a proximity, J. math. Soc. Japan 16, 375-378; MR 31 # 2708. HAYASHI, E. (1965). A note on proximity spaces, Bull. Aichi Gakngei Univ. 14, 1-4.

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Supplement 124

DOICINOV, D. (1969). A generalization of topological spaces, Contributions to Extension Theory of Topological Structures (Proc. Sympos. Berlin 1967), VEB Deutscher Verlag Der Wissenschaften, Berlin. 125 ENGELKING, R. (1968). Outline of General Topology, NorthHolland Publishing Company, Amsterdam. 126 FEDORCTJK, V. (1968). ^-proximities and ^-absolutes, Soviet Math. Dokl. 9, 661-664. 127 HADZIIVANOV, N. (1969). Pseudometrics of proximity spaces, Contributions to Extension Theory of Topological Structures (Proc. Sympos. Berlin 1967), VEB Deutscher Verlag Der Wissenschaften, Berlin. 128 HAYASHI, E. (1967). Closures and neighbourhoods in certain proximity spaces, Proc. Japan Acad. 43, 619-623. 129 HUSEK, M. (1969). Categorical connections between generalized proximity spaces and compactifications, Contributions to Extension Theory of Topological Structures (Proc. Sympos. Berlin 1967), VEB Deutscher Verlag Der Wissenschaften, Berlin.

124 130 131 132 133 134 135

136 137 138

BIBLIOGRAPHY LEADEB, S. (1969). Extensions based on proximity and bounded ness, Math. Z. 108, 137-144. MOZZOCHI, C. J. (1968). A partial generalization of a theorem of Hursch, Comment. Math. Univ. Carolinae, 9, 157-159. MOZZOCHI, C. J. (1969). Symmetric generalized uniform and proximity spaces, Trinity College, Hartford, Connecticut. NACHMAN, L. J. (1968). Weak and strong constructions in proximity spaces, Ph.D. dissertation, The Ohio State University. NAGATA, J. (1968). Modern General Topology, North-Holland. POLJAKOV, V. Z. (1969). On some proximity properties determined only by the topology of the compactification, Contributions to Extension Theory of Topological Structures (Proc. Sympos. Berlin 1967), VEB Deutseher Verlag Der Wissenschaften, Berlin. SMIRNOV, Y. M. (1952). Mappings of systems of open sets, Mat. Sb. 31 (73), 152-166 (in Russian); MR 14, 303. SMIRNOV, Y. M. (1960). Generalization of the Stone-Weierstrass theorem and proximity spaces, Czech. Math. J. 10 (85), 493-500 (in Russian, English summary); MR 23 # A 3474. GAGRAT, M. S. and S. A. NAIMPALLY (1970). Proximity approach to extension problems, Fundam Math, (to appear).

INDEX absolutely closed, 44 absorbs, 39 agree locally, 56 Alexandroff extension, 57 Alfsen-Njastad (or A-N) uniform structure, 78 separated, 78 total, 79 axiomatic characterization of clusters, 32 base proximity, 47, 62 bounded cluster, 58 bounded compactly, 54 bounded locally, 54 bounded sets, 53 bounded totally, 64 bounded ultrafilter, 58 boundedly compact, 54 boundedness, 53 metric, 53 bunch, 111 Cauchy cluster, 67 Cauchy filter, 67 Cauchy sequence, 102 centred £-system, 35 cluster, 27 as dual of end, 36 bounded, 58 Cauchy, 67 determined by ultrafilter, 29 cluster point, 28 clusters and ultrafilters, 27 axiomatic characterization of, 32 coarser proximity, 14 coarser uniformity, 64 cofinal, 94 compact, boundedly, 54 compact proximity space, 16, 30 compactification local, 58 maximal, 45 minimal, 45 one-point, 45

partial order, 43 Smirnov, 41, 44 Stone-Cech, 45, 46 compactly bounded, 54 compatible nearness relation in .£?spaces, 103 compatible proximity and uniformity 65 compatible topology and proximity, 12 complete uniform space, 68 contiguity, 81 correct, 81 continuous convergence, 96 converges, continuously, 96 in proximity, 94 in^r^f-space, 103 -^-uniformly, 104 proximally, 94 uniformly, 94 correct contiguity, 81 £-closure in terms of, 11, 18 8-cover, 73 ^-extension, 44 £-homeomorphic, 23 ^-isomorphism, 23 ^-mapping, 20 ^-neighbourhood, 15 dense, regularly, 109 discrete proximity, 9 dual classes, 34 dual relations, 34 e-discrete, 51 Efremovi<5 proximity, 8 ends, 34 and compactification, 42 axiomatic characterization of, 37 characterization of, 36 entourage, 63, 78 symmetric, 63 entourage-like, 79 equicontinuous, 20 equimorphism, 23

[ 125 ]

126

INDEX

equinormal, 46 extension Alexandroff, 57 8-, 44 proximal, 44 filter Cauchy, 67 free, 56 regular, 35 round, 34 finer proximity, 14 finer topology, 14 finer uniformity, 64 Frechet (orJS?-) space, 103 j£?*-space, 103 free filter, 56 function equicontinuous, 20 proximally continuous, 20 functionally distinguishable, 9 generalized proximities, 104 generalized topological (GT-) structure, 98 associated proximity, 99 generalized uniform spaces, 78 H-equivalent, 87 H-finer, 89 Hausdorff (or separated) proximity, height, 84 class of % 84 less than or equal in, 84 proximity and, 84 hyperspace uniformity, 87 indiscrete proximity, 9 induced proximity, 23 induced topology, 8, 10 invariant, proximity, 23 isomorphism, proximity, 23 isotone subnet, 95 Kuratowski closure operator, 12 103 J^*-space, 103 ££-uniform convergence, 104 Leader (LE-) proximity, 105 compatible, 106 induced by quasi-uniformity, 107

LO-proximity class of M-uniformities, 113 local compactification, 58 local proximity mapping, 59 local proximity space, 55 locally agree, 56 locally bounded, 54 Lodato (LO-) proximity, 105 Lodato (LO-) space, 105, 108 mapping, 8-, 20 equicontinuous, 20 local proximity, 59 proximally continuous, 20 proximity, 20 metric proximity, 8 metric uniformity, 64 metrizable proximity space, 50 Montel space, 54 Mozzochi (M-) uniform structure, 112 associated topology, 112 Lodato space and, 113 near, 7 nearness relation n, 101 neighbourhood, S-, 15 uniform, 15, 66 net, 94 order isomorphism, 44 topogenous, 98 paraproximity, 115 partial order compactifications, 43 partial order proximities, 14 Pervin (P-) proximity, 105 induced by quasi-uniformity, 107 point cluster, 28 precompact uniform structure, 92 product proximity, 24 proximal convergence, 94 proximal extension, 44 proximally continuous, 20 proximally isomorphic, 23 proximity, 8 associated with a uniformity, 10, 64 coarser, 14 coarsest, 21 compatible with a topology, 12

127 INDEX proximity (cont.) round filter, 34 discrete, 9 semi-ultrafilter, 32 Efremovi<3, 8 separated (or Haudsdorff) A-N unifiner, 14 formity, 78 generalized, 104 separated(or Hausdorff) proximity, 8 Hausdorff (separated), 8 separated (or Hausdorff) uniformity, indiscrete, 9 64 induced, 23 separation (S-) proximity, 108 invariant, 23 sequential proximity, 100 Leader, (LE-), 105 Smirnov compactification, 41, 44 local, 55 strong axiom, 9 subspace proximity, 23 Lodato (LO-), 105 symmetric entourage, 63 maximal, 44 symmetry axiom, 9 minimal, 45 syntopogenous structure, 98 of a compact space, 16, 30 para-, 115 Pervin (P-), 105 Tietz's extension theorem, analogue, pseudo-, 115 44 (pseudo-) metric, 8 topogenous order, 98 quasi-, 115 topological group, 10 topological weight, 47 separation (S-), 108 topology, induced by a proximity, 8, sequential, 100 0-, 115 12, 13, 19 topology induced by, 8, 12, 13, 19 total A-N uniform structure, 79 totally bounded, 64 proximity base, 47, 62 proximity class of uniformities, 71 ^-discrete, 89 proximity extension, 43 ^J^-space, 102 proximity isomorphism, 23 proximity mapping, 20 convergence in, 103 proximity product, 24 ^J^-structure, 102 proximity space, 8 ultrafilters, 27 and clusters, 27, 29 absolutely closed, 44 bounded, 58 compact, 16, 30 clusters generated, by, 29 duality in, 34 semi-, 32 equinormal, 46 uniform neighbourhood, 15, 66 local, 55 uniform space, 63 maximal, 44 uniform structure, 63 metrizable, 50 uniformities, proximity class of, 71 proximity subspace, 23 uniformity (uniform structure), 63 proximity weight, 48 pseudo-metric proximity, 8 Alfsen-Njastad, (A-N), 78 pseudo-metric uniformity, 64 coarser, 64 compatible, 64, 66 quasi-uniformity, 107 finer, 64 Pervin, 107 Hausdorff (separated), 64 P-proximity induced by, 107 hyperspace, 87 topology induced by, 107 Mozzochi (M-), 112 precompact (totally bounded), 92 1^-axiom, 106 proximity associated with, 64 rapidly increasing, 83 pseudo-metric, 64 regular filter, 35 quasi-, 107 regularly dense, 109 symmetric generalized, 115

128 uniformity (cont.) total, A-N, 79 universal, 71 uniformity base, 64 uniformity subbase, 64 uniformly continuous, 22, 66 uniformly finer, 89

INDEX universal uniformity, 71 Urysohn's lemma, analogue of, 44 weight proximity, 48 topological, 47