Beyond Topology

CONTEMPORARY MATHEMATICS 486 Beyond Topology Frédéric Mynard Elliott Pearl Editors American Mathematical Society Thi...

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CONTEMPORARY MATHEMATICS 486

Beyond Topology Frédéric Mynard Elliott Pearl Editors

American Mathematical Society

This page intentionally left blank

Beyond Topology

This page intentionally left blank

CONTEMPORARY MATHEMATICS 486

Beyond Topology Frédéric Mynard Elliott Pearl Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor George Andrews

Abel Klein

Martin J. Strauss

2000 Mathematics Subject Classiﬁcation. Primary 54Axx, 54–02.

Library of Congress Cataloging-in-Publication Data Beyond topology / Fr´ed´eric Mynard, Elliott Pearl, editors. p. cm. — (Contemporary mathematics ; v. 486) Includes bibliographical references and index. ISBN 978-0-8218-4279-9 (alk. paper) 1. Topological spaces. 2. Topology. I. Mynard, Fr´ed´eric, 1973– QA611.3.B49 514—dc22

II. Pearl, Elliott.

2009 2008050812

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

14 13 12 11 10 09

Contents Preface

vii

Categorical Topology Robert Lowen, Mark Sioen and Stijn Verwulgen

1

A Convenient Setting for Completions and Function Spaces H. Lamar Bentley, Eva Colebunders and Eva Vandersmissen

37

Proximity: a powerful tool in extension theory, function spaces, hyperspaces, boolean algebras and point-free geometry Anna Di Concilio

89

An Initiation into Convergence Theory Szymon Dolecki

115

Closure Marcel Ern´ e

163

An Introduction to Quasi-uniform Spaces ¨nzi Hans-Peter A. Ku

239

Approach Theory Robert Lowen and Christophe Van Olmen

305

Semiuniform Convergence Spaces and Filter Spaces Gerhard Preuß

333

Index

375

v

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Preface Often in mathematics, the context in which a problem is ﬁrst studied is not the most adapted to the problem at hand. For instance, diﬃcult questions in the context of real numbers become easy to tackle in the realm of complex numbers, problems of planar geometry requiring intricate arguments become trivial when considered as projections of a three dimensional situation, etc. Topological questions are no exception, even though this fact is not widely recognized yet. After various attempts at axiomatizing the notions of nearness and convergence in the early 20th century, the concept of topology introduced by Felix Hausdorﬀ in 1914 was relatively quickly accepted as the answer to the problem of ﬁnding solid foundations for analysis and geometry. There are, of course, reasons why topology has been widely accepted as the standard structure to describe nearness, convergence and continuity. Not the least of them is the fact that topologies can be introduced in so many equivalent ways: system of open sets, of closed sets, of neighborhoods at each point, closure operator or interior operator, in terms of covers, of convergent ﬁlters, to name a few. However, working with topological spaces has its shortcomings, many of which will be presented in this volume, together with various approaches to remedy them. Let me just mention two examples that the reader will repeatedly encounter in this book. Firstly, while a quotient set can be canonically endowed with a quotient topology, this operation does not yield very satisfactory results. To be more speciﬁc, consider an equivalence relation ∼ on a topological space X and denote by q : X → X/∼ the map associating to each element of X its equivalence class: The quotient topology on X/∼ is the ﬁnest topology on X/∼ that makes q continuous. This construction is not hereditary in the following sense: if B is a subset of the quotient X/∼ the induced topology by X/∼ does not necessarily coincide with the quotient topology induced by q|q−1 (B) : q −1 (B) → B. A second fundamental problem is the lack of a canonical function space topology that would yield as nice a duality as the usual algebraic duality. If X and Y and Z are sets and Z X denotes the set of all functions from X to Z, the sets of functions are well-behaved in the sense that Y Z X×Y = Z X , where the equality represents the bijection f → t f where t f (y)(x) = f (x, y). But if X, Y and Z are topological spaces and C(X, Z) denotes the set of continuous functions from X to Z, there is in general no topology on C(X, Z) that ensures that f : X × Y → Z is continuous if and only if the companion map t f : Y → C(X, Z) of f is continuous. This situation can be viewed in two ways: either you consider that the class of topological spaces is too large and leaves room for too much pathology, in which vii

viii

PREFACE

case you will try to remedy the above problems by ﬁnding a subclass of topological spaces behaving better, or you realize that the class of topological spaces is too small to perform certain operations in a natural way, just like the ﬁeld of real numbers is too small to factor any polynomial into linear factors. The former approach led among others to the theory of k-spaces, which became reasonably popular, notably in homotopy theory, even though this solution suﬀers from obvious problems, like the necessity of using a product that is diﬀerent from the usual topological product. The present book is about the latter, less widely known, approach. It presents to a reader with only a basic knowledge of point-set topology various generalizations of topologies, each addressing one or several particular shortcomings of topologies. Written with the graduate student in mind, this volume should also be an eyeopener for the working mathematician, the day-to-day user of topology: there is sometimes much to gain in looking beyond topology. Any reference to category theory has been carefully avoided so far, because the present volume is not a book on category theory. However, the book focuses on several related structures on a set and the natural way to describe structured sets and their relationships is in the language of category theory. Roughly speaking and restricting ourselves to the present context, a category (of structured sets) is composed of objects that are sets with a structure (think of groups, rings, vector spaces, topological spaces, topological vector spaces, etc.) and morphisms between these objects, usually maps preserving the structure in some sense (for the previous examples: group homomorphisms, ring homomorphisms, linear maps, continuous maps, continuous linear maps, etc.). The categories to be considered in this book all contain the category Top of topological spaces and continuous maps. The language of category will be necessary to describe how Top sits in the category at hand, how these categories relate to each other, and some qualities that Top lacks but are enjoyed by these categories. Therefore, an introductory chapter on categorical topology presents the necessary categorical background. This chapter can be seen as an appendix to refer to when running into an unknown categorical notions while reading another chapter. Taking this into account, each chapter is self contained and can be read independently of the others, despite ocasional overlaps. Each one of them should be seen as an introduction to a ﬁeld, and a guide for the interested reader who wants to go further. Finally, I wish to express my deep appreciation to all of the authors who contributed to this book, and to my co-editor Elliott Pearl whose tremendous work and technical skills made it possible to ﬁnish this book. Fr´ed´eric Mynard August 2008

Contemporary Mathematics Volume 486, 2009

Categorical Topology Robert Lowen, Mark Sioen, and Stijn Verwulgen Abstract. It is the aim of this chapter to give a basic introduction to the theory of topological constructs, i.e., topological categories over Set, together with their main categorical features. Also procedures to embed such constructs into larger ones satisfying some convenience properties that are lacking are discussed.

Contents 1. Introduction 2. Topological constructs 3. Limits and topological constructs 4. Special morphisms 5. Fiber-small topological constructs 6. Reﬂective and coreﬂective subcategories 7. Convenience properties 8. Convenient hulls 9. Topological hull = MacNeille completion References

1 2 7 8 12 14 19 22 25 26

1. Introduction At present there are many books which contain a detailed account of the basic notions of categorical topology [5, 198]. A revised online version of [5] dating from 2006 has been reposted in the electronic journal Reprints in Theory and Applications of Categories. Hence, in the following text many proofs are either not given or are very short. The purpose of this chapter is to make the present book suﬃciently self-contained. In particular, the deﬁnitions of most notions and the basic results on categorical topology used in the other chapters can be found here. 2000 Mathematics Subject Classiﬁcation. 18A20, 18A30, 18A32, 18A35, 18A99, 18B30, 18B99, 54B30, 54E. Key words and phrases. category, categorical topology, morphism, source, sink, factorization structure, topological construct, cartesian closed, extensional, exponential, convergence, approach space, metric space, topological space. c 2009 2008 American Mathematical Society

1

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R. LOWEN, M. SIOEN, AND S. VERWULGEN

2. Topological constructs The following are well-known and often used categories in topology and analysis. 2.1. Examples. Top topological spaces and continuous functions, Unif uniform spaces and uniformly continuous, functions, QU quasi-uniform spaces and quasi-uniformly continuous functions. App approach spaces and contractions, Cap convergence approach spaces and contractions, UG uniform gauge spaces and uniform contractions, Near nearness spaces and uniformly continuous functions PrTop pretopological spaces and continuous functions, PsTop pseudotopological spaces and continuous functions, Conv convergence spaces and continuous functions, Lim limit spaces and continuous functions, Cls closure spaces and continuous functions, pMet pseudometric spaces and contractions, Prost preordered sets and order preserving functions, Pos partially ordered sets and order preserving functions, Metc metric spaces and continuous functions, Top0 T0 spaces and continuous functions, Top1 T1 spaces and continuous functions, Haus Hausdorﬀ spaces and continuous functions, Tych Tychonoﬀ spaces and continuous functions, Comp2 compact Hausdorﬀ spaces and continuous functions, Seq sequential topological spaces and continuous functions, Unif 2 Hausdorﬀ uniform spaces and uniformly continuous functions, CompUnif 2 complete Hausdorﬀ uniform spaces and uniformly continuous functions, (25) Set sets and functions. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

In what follows we will see that some of these categories, as well as many others which we did not mention, share a number of characteristic properties which in a precise sense can be qualiﬁed as being topological. Some will have supplementary good properties which we explain in the section on convenience properties. The category Top is of course a main source of inspiration. For our purpose the following properties are noteworthy (where some concepts and terms used will be deﬁned later on). (1) The forgetful functor U : Top → Set is faithful. (2) There are discrete and indiscrete topologies on each set. (3) The ﬁber of a set X, i.e., the class of all topological spaces (X, τ ), is a set. Moreover, ordered by (X, τ1 ) ≤ (X, τ2 ) ⇔ τ2 ⊂ τ1 , it is a complete lattice with largest element X endowed with the indiscrete topology and smallest element X endowed with the discrete topology. (4) Limits are constructed as limits in Set provided with the initial lift and likewise colimits are constructed as colimits in Set provided with the ﬁnal lift.

CATEGORICAL TOPOLOGY

3

(5) The monomorphisms in Top are precisely the injective continuous functions and the epimorphisms are precisely the surjective continuous functions. (6) Structured sources have a unique initial lift in Top and structured sinks have a unique ﬁnal lift in Top. (7) The homeomorphisms are precisely the surjective embeddings or, equivalently, the injective quotients. For the sake of contrast with more algebraic categories, we will also refer to the categories Grp, resp. VectR , and Rng, of groups, resp. real vector spaces and rings, each with their resp. structure preserving maps as morphisms. Topological constructs, which we deﬁne below, will possess all of these and several other properties. We suppose that the basic notions of general category theory are known and refer the reader to the literature for any concepts which we do not deﬁne in this text [121, 5]. 2.2. Definition. A construct is a pair (A, U ), where A is a category and U : A → Set is a faithful functor. Sometimes U is called the forgetful or underlying set functor. Although we mainly focus on constructs, it should be noted that many deﬁnitions and results can also be considered in the more general setting where Set is replaced by another base category X . A category (A, U ) equipped with a faithful functor U : A → X is called a concrete category (over X ). Note that every category A is a concrete category (A, idA ) over itself and that all examples of Example 2.1 are constructs. 2.3. Definition. Let (A, U ) be a construct. If for A-objects A and B and a function f : U A → U B there exists a (necessarily unique) A-morphism f : A → B with U f = f , then we say that f is an A-morphism. Often we denote f simply by f and often too the functor U is suppressed in notation. 2.4. Definition. An object A in a construct (A, U ) is called discrete whenever, for each A-object B, every function f : U A → U B is an A-morphism and it is called indiscrete whenever, for each A-object B, every function f : U B → U A is an Amorphism. 2.5. Definition. Let (A, U ) be a construct. The ﬁber of a set X is the class of A-objects A with U A = X. (A, U ) is said to be ﬁber-small provided that the ﬁber of each set is a set. The ﬁber of a set X can be ordered by: A ≤ B ⇔ idX : U A → U B is an A-morphism. 2.6. Example. This order is obviously reﬂexive and transitive, but not always antisymmetric. For instance, in the category Metc of metric spaces and continuous maps, consider two diﬀerent topologically equivalent metrics d1 , d2 on the same set X. Then both idX : (X, d1 ) → (X, d2 ) and idX : (X, d2 ) → (X, d1 ) are continuous but (X, d1 ) = (X, d2 ). 2.7. Definition. A construct (A, U ) is said to be amnestic provided that the order ≤ is antisymmetric, i.e., A ≤ B and B ≤ A imply A = B.

4

R. LOWEN, M. SIOEN, AND S. VERWULGEN

A source in a category consists of an object A together with a (class) indexed collection of morphisms (fi : A → Ai )i∈I and dually a sink consists of an object A together with a (class) indexed collection of morphisms (fi : Ai → A)i∈I . 2.8. Definition. Let (A, U ) be a construct. A source (fi : A → Ai )i∈I in A is called initial provided that a function f : U B → U A is an A-morphism whenever each composite fi ◦f : U B → U Ai is an A-morphism. Dually, a sink (fi : Ai → A)i∈I in A is called ﬁnal provided that a function f : U A → U B is an A-morphism whenever each composite f ◦ fi : U Ai → U B is an A-morphism. If I is a singleton, the initial source (resp. ﬁnal sink) is called an initial (resp. ﬁnal) morphism. 2.9. Proposition. If (fi : A → Ai )i∈I is an initial source and the sources (fij : Ai → Aij )j∈Ji are initial for each i ∈ I, then the source (fij ◦ fi : A → Aij )i∈I,j∈Ji is initial. Dually if (fi : Ai → A)i∈I is a ﬁnal sink and the sinks (fij : Aij → Ai )j∈Ji are ﬁnal for each i ∈ I, then the sink (fi ◦fij : Aij → A)i∈I,j∈Ji is ﬁnal. Proof. Immediate from the deﬁnitions.

2.10. Proposition. If (fi : A → Ai )i∈J is an initial source for some J ⊂ I, then so is (fi : A → Ai )i∈I and if (fi : Ai → A)i∈J is a ﬁnal sink for some J ⊂ I, then so is (fi : Ai → A)i∈I . Proof. Immediate from the deﬁnitions.

2.11. Definition. Let (A, U ) be a construct. A U -structured source is a source in Set of the form (fi : X → U Ai )i∈I . An initial lift for a U -structured source is an initial source (fi : A → Ai )i∈I in A such that U A = X and U fi = fi for all i ∈ I. A U -structured sink and a ﬁnal lift are deﬁned dually. Often fi is again simply denoted fi . 2.12. Definition. A construct (A, U ) is called topological provided that every U -structured source has a unique initial lift. The functor U is referred to as a topological functor. Note that existence of unique initial lifts for all class indexed sources is required in the deﬁnition. The reason for this will become clear in the proof of the Topological Duality Theorem below. 2.13. Examples. All constructs of Example 2.1 (1)–(13) are topological. Although we will only work with topological constructs in this text, it should be noted that many results hold in the more general case of topological categories, i.e., when working over an arbitrary base category X rather than over Set, if need be modulo extra conditions on the base category. The forgetful functor U is often not emphasised, and, for topological constructs, usually even omitted. This however is no problem as it was shown by Hoﬀmann in 1975 [124] that any two topological functors from A into Set are naturally isomorphic. 2.14. Proposition. If (A, U ) is a topological construct then it is amnestic. Proof. Let A and B be objects in A with U (A) = U (B) = X and A ≤ B, B ≤ A. The morphisms idX : B → A and idX : A → A are initial morphisms. By the uniqueness of initial structures it follows that A = B.

CATEGORICAL TOPOLOGY

5

2.15. Proposition. If U : A → Set is a faithful functor such that every U structured source has an initial lift then the following are equivalent: (1) (A, U ) is topological. (2) (A, U ) is uniquely transportable, i.e., for every A-object A and every X isomorphism k : U A → X, there exists a unique A-object B with U B = X and k : A → B an A-isomorphism. (3) (A, U ) is amnestic. Proof. [5].

2.16. Theorem (Topological Duality Theorem). If (A, U ) is a topological construct, then each structured sink has a unique ﬁnal lift. Proof. Let (fi : U Ai → X)i∈I be a structured sink. Consider the structured source T = (gj : X → U Bj )j∈J consisting of all structured arrows (gj , Bj ) with the property that gj ◦ fi : U Ai → U Bj is an A-morphism for each i ∈ I. Let (gj : A → Bj )j∈J be the initial lift of T in A. Then, since all compositions gj ◦ fi are A-morphisms, it follows that fi : Ai → A is a morphism for each i ∈ I. Let g : U A → U B be a function such that g ◦ fi is an A-morphism for each i ∈ I. This means that there must be a j ∈ J with g = gj and thus g : U A → U B is an A-morphism. Therefore the sink (fi : Ai → A)i∈I is ﬁnal. If there is another ﬁnal lift (fi : Ai → A )i∈I , then idX : A → A and idX : A → A are morphisms. Thus by amnesticity: A = A . Note that this entails that the dual or opposite category (Aop , U op ) of a topological construct (A, U ) is topological over Setop and hence not a construct, making the property of being a topological construct not self-dual. 2.17. Proposition. Let X be a set, then the initial structure of the empty source on X is indiscrete and dually the ﬁnal structure of the empty sink is discrete. These objects are the largest (coarsest) and respectively smallest (ﬁnest) objects of the ﬁber of X. Proof. Immediate from the deﬁnitions.

2.18. Proposition. In a topological construct (A, U ) the ﬁber of any set is a complete lattice. Proof. By Proposition 2.14 it follows that the ﬁber of a set X is a partially ordered class. Let (Ai )i∈I be a family of A-objects with U Ai = X. The initial lift of (idX : X → U Ai )i∈I is inf Ai . The ﬁnal lift of (idX : U Ai → X)i∈I is sup Ai . 2.19. Proposition. If (fi : A → Ai )i∈I is an initial source, then A = sup{B ∈ Ob A | U A = U B, fi : U B → U Ai are A-morphisms} and if (fi : Ai → A)i∈I is a ﬁnal sink, then A = inf{B ∈ Ob A | U A = U B, fi : U Ai → U B are A-morphisms}. Proof. Immediate from the deﬁnitions.

2.20. Proposition. Let (A, U ) be a topological construct. Then U has a left and right adjoint R and S that both satisfy U ◦ R = U ◦ S = idSet .

6

R. LOWEN, M. SIOEN, AND S. VERWULGEN

Proof. For a set X, put RX the discrete object with U RX = X. Let a : X → U A be a function. Then, since RX is discrete and U is faithful, there exists a unique A-morphism b : RX → A such that a = U b ◦ idX . In an analogous way (SX , idX ) where SX is the indiscrete object with U SX = X is a co-universal arrow of X. 2.21. Proposition. For ﬁber-small concrete categories (A, U ), the following conditions are equivalent: (1) (A, U ) is topological. (2) Every small structured source (fi : X → U Ai )i∈I has a unique initial lift. (3) Every small structured sink (fi : U Ai → X)i∈I has a unique ﬁnal lift. Proof. (1) ⇒ (2) follows immediately from the deﬁnition of a topological construct. (2) ⇒ (1). Let (fi : X → U Ai )i∈I be a structured source indexed by a class I. For each i ∈ I there exists an initial lift fi : Bi → Ai of fi : X → U Ai . By ﬁber-smallness {Bi | i ∈ I} is a set. Thus there exists a set J ⊂ I with {Bj | j ∈ J} = {Bi | i ∈ I}. Let (fj : A → Aj )j∈J be the initial lift of (fj : X → U Aj )j∈J . Then A ≤ Bj for each j ∈ J (Proposition 2.19). Hence A ≤ Bi for each i ∈ I. Thus fi : A → Ai is a morphism for each i ∈ I. Consequently, by Proposition 2.10, the source (fi : A → Ai )i∈I is initial. (1) ⇔ (3) follows by duality. 2.22. Definition. A topological construct (A, U ) is said to be well-ﬁbered if (A, U ) is ﬁber-small and if the ﬁber of a set with at most one element has exactly one element. 2.23. Examples. All constructs of Example 2.1 (1)–(13) are well-ﬁbered topological. Nowadays diﬀerent terminologies are used: sometimes the term topological construct already is meant to include well-ﬁberedness. Almost all interesting topological constructs are well-ﬁbered. It is important to note that well-ﬁberedness in a topological construct guarantees that constant maps are, i.e., can be uniquely lifted to, morphisms. 2.24. Remark. Note that there is a small redundancy in our deﬁnition of a topological construct (A, U ) is the sense that faithfulness of the functor U : A → Set follows automatically from the existence of unique U -initial lifts which can be seen as follows. r / / B be a pair of A-morphisms such that U r = U s. Let Let A s

(fh : Aˆ → A )h∈Mor(A) be the initial lift of the U -structured source (fh : U A → A )h∈Mor(A) , with fh := U r. For each A morphism h deﬁne gh : A → A by r if fh ◦ h = s, gh = s otherwise. Then U gh = fh ◦ idUA for all h ∈ Mor(A). By initiality there exists a morphism k : A → Aˆ such that U k = idUA and hence such that gh = fh ◦ k for each Amorphism h. In particular we obtain gk = fk ◦ k. From the deﬁnition of gk thus r = s follows.

CATEGORICAL TOPOLOGY

7

3. Limits and topological constructs A topological functor is faithful and (co-)adjoint, and hence it preserves (co)limits in the sense that if D : I → A is a functor (called a diagram) and L = (li : L → Di ) is a limit of D (resp. C = (li : Di → is a colimit of D) in A, then U L = (U li : U L → U Di ) is a limit (resp. U C = (U li : U Di → U C) is a colimit of U ◦ D in Set. One rephrases this fact by saying that (co)limits in topological constructs are concrete (co)limits. This means that a product and an equalizer in A are constructed respectively on the cartesian product of the underlying sets and on a subset of the domain of the parallel pair. Initiality and limits, moreover, are closely related in a general way. 3.1. Theorem. Suppose that (A, U ) is a construct and that U preserves limits. Let D : I → A be a functor and let S = (fi : A → Di )i∈I be a source. Then the following are equivalent. (1) S is a limit of D in A. (2) U S is a limit of U ◦ D in Set and S is initial. Proof. Take an A-object B and a function f : U B → U A such that any U fi ◦f can be lifted to an A-morphism U fi ◦ f . Then, by the assumption that S is a limit of D, there exists a unique A-morphism g : B → A such that the diagram: AO g

B

/ Di }> } } }} }} Ufi ◦f fi

commutes. Since U S is a limit of U ◦ D we deduce that U g = f , which proves initiality of S. Conversely, if U S is a limit of U ◦ D then automatically, by the fact that U is faithful, S is a natural source or cone (see [5]) for D. Suppose (Di : B → Di )i∈I is another cone of D. Then (U gi : U B → U Di )i∈I is a cone of U ◦ D. Since U S is a limit of U ◦ D, there exists a unique function f : U BU A such that, for any i ∈ I, U fi ◦ f = U gi . Since S is initial, f is an A-morphism. 3.2. Proposition. Let (A, U ) be a construct, where U preserves colimits. Let D : I → A be a functor and let L = (li : Di → L)i∈I be a sink in A. Then the following conditions are equivalent: (1) L is a colimit of D in A. (2) U L is a colimit of U ◦ D in Set and L is ﬁnal. Proof. This follows by duality.

So, in a topological construct limits (resp. colimits) are constructed in two steps: ﬁrst form the limit (resp. colimit) in Set, and then take the initial lift of this limit (resp. ﬁnal lift of this colimit). 3.3. Corollary. A topological construct is complete and co-complete. Proof. This follows immediately from previous result and the fact that Set is complete and co-complete.

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Proposition 2.20 can easily be generalized to topological functors over an arbitrary base category. Hence such functors are adjoint and co-adjoint. From Theorem 3.1 it follows that they uniquely lift limits, via initiality, and colimits, via ﬁnality. In particular we note that (co-)completeness is determined by the base category: if we replace Set by an arbitrary base category X then a topological category over X is (co)-complete if and only if X is (co)-complete. Also note that the result below is formulated for topological constructs but that the same characterization holds for general topological categories. 3.4. Theorem. Let (A, U ) be a concrete category. Then the following are equivalent. (1) (A, U ) is topological. (2) (a) U lifts limits uniquely. (b) Any ﬁber has an indiscrete object.

Proof. [5]. 4. Special morphisms

4.1. Proposition. Let (A, U ) be a construct and suppose that U preserves limits. Then for a morphism f : A → B the following conditions are equivalent: (1) f is a monomorphism. (2) U f is an injective function. Proof. (2) ⇒ (1) follows from the fact that injective functions are monomorphisms in Set. (1) ⇒ (2). Recall that f : A → B is a monomorphism if and only if the following square is a pullback: A

idA

/A

f

/ B.

f

idA

A

Since U preserves limits the proposition holds. We also have the dual of this statement.

4.2. Proposition. Let (A, U ) be a construct and suppose that U preserves colimits. Then for a morphism f : A → B the following conditions are equivalent: (1) f is an epimorphism. (2) U f is surjective. 4.3. Corollary. In a topological construct monomorphisms are precisely the injective morphisms and epimorphisms are precisely the surjective morphisms. 4.4. Definition. A morphism m : M → A is a regular monomorphism prof

vided that there is a pair of morphisms A

g

//

B such that m is the equalizer of

f and g. The dual concept is that of a regular epimorphism.

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CATEGORICAL TOPOLOGY

4.5. Example. Note that in Set, regular monomorphisms and monomorphisms are the same since they coincide with injective functions. Indeed, any injective r / / {0, 1} with r the constant map 1 function f : A → B is an equalizer of B s 1 b ∈ f (A), and s : b → 0 b∈ / f (A). Likewise the regular epimorphisms in Set are precisely the surjective functions: p1 // if e : A → B is a surjective function, then e is a coequalizer of the pair D A p 2

with D = {(a, a ) ∈ A × A | e(a) = e(a )} and p1 , p2 the projections.

It is clear from the deﬁnition that regular monomorphisms (resp. regular epimorphisms) are monomorphisms (resp. epimorphisms). 4.6. Proposition. In a topological construct, for any morphism f the following conditions are equivalent: (1) f is an isomorphism. (2) f is a regular monomorphism and an epimorphism. (3) f is a monomorphism and a regular epimorphism.

Proof. See [5].

4.7. Proposition. If (A, U ) is a construct and U preserves equalizers, then a regular monomorphism is injective and initial. Proof. Suppose that m : M → A is an equalizer of a pair of morphisms r / A s / B . Since U preserves equalizers, U m is also a regular monomorphism in Set, i.e., injective. Let f : U C → U M be a function such that U m ◦ f is an A-morphism. Then, since U r ◦ (U m ◦ f ) = U s ◦ (U m ◦ f ), there exists an A-morphism g : C → M with U m ◦ U g = U m ◦ f . Since U m is injective, this implies that f = U g. Hence m is initial. 4.8. Proposition. If (A, U ) is a construct and U preserves coequalizers, then a regular epimorphism is surjective and ﬁnal.

Proof. Dual to Proposition 4.7.

4.9. Definition. An initial and injective morphism is called an embedding and a ﬁnal and surjective morphism is called a quotient. 4.10. Proposition. In a topological construct a morphism f : A → B is a regular monomorphism if and only if it is an embedding. Proof. The only if part follows from Proposition 4.7. To show the converse part, let f : A → B be an embedding. Then, since U f is injective, U f is a regular monomorphism in Set. There exists a pair of functions r r / / / Y be U B s / X such that U f is an equalizer of r and s in Set. Let B s

the ﬁnal lift of the U -structured sink U B

r s

//

X in A. Then for each morphism

g : C → B such that r ◦ g = s ◦ g, there exists a function h : U C → U A such that

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U g = U f ◦ h. Since f : A → B is initial and g is a morphism in A, it follows that h is an A-morphism. Hence, f is an equalizer of r and s in A. By dualization we also have the following. 4.11. Proposition. In a topological construct a morphism f : A → B is a regular epimorphism if and only if f is a quotient. 4.12. Corollary. In a topological construct (A, U ), the following assertions are equivalent for a morphism f : (1) f is an isomorphism, (2) f is an embedding and U f is surjective, (3) f is a quotient and U f is injective. Proof. This follows immediately from Propositions 4.6, 4.10 and 4.11.

The following alternative characterization of embeddings and quotients often is very useful. 4.13. Definition. A monomorphism m is called an extremal monomorphism if, whenever m = f ◦ g with g an epimorphism, g is an isomorphism. Dually, an epimorphism e is called an extremal epimorphism if, whenever e = f ◦ g with f a monomorphism, f is an isomorphism. 4.14. Proposition. In a topological construct a morphism is an embedding (resp. a quotient) if and only if it is an extremal monomorphism (resp. extremal epimorphism). Proof. Straightforward.

4.15. Examples. (1) As mentioned before, the regular = extremal monomorphisms (resp. regular = extremal epimorphisms) in Set are precisely the injective (resp. surjective) functions. (2) In Top the regular = extremal epimorphisms correspond to topological quotients and the regular = extremal monomorphisms are (up to isomorphism) precisely the inclusions of subspaces. (3) In Haus the regular monomorphisms correspond (up to isomorphism) to the inclusions of closed subspaces. (4) In Comp2 the regular epimorphisms correspond (again up to isomorphism) to topological quotients associated with equivalence relations which are closed with respect to the product topology. (5) In Pos regular = extremal monomorphisms correspond to inclusions of sub-posets. (6) Whereas in Grp and VectR all monomorphisms are regular, this is not true for Rng where the inclusion of Z in Q can be proven to be a counterexample. A similar remark holds for regular epimorphisms ans epimorphisms. (7) In Grp, VectR and Rng, regular = extremal epimorphisms correspond (up to isomorphism) to quotients with respect to so-called congruences (i.e., equivalence relations which are compatible with the algebraic operations). An important concept, not only in topological category theory but in category theory in general, is that of factorization structures [5].

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4.16. Definition. Let E and M be classes of morphisms in A that are closed under composition with isomorphisms. We say that A is an (E,M)-category if every morphism is essentially uniquely (E,M)-factorizable. That is, for every morphism f there exist e ∈ E and m ∈ M with f = m ◦ e and whenever f = m ◦ e with m ∈ M and e ∈ E, there exists a unique isomorpism j with m ◦ j = m and j ◦ e = e . If E and M are classes of morphisms in A which are closed under composition and under composition with isomorphisms and if, moreover, E ∩ M contains the class of all isomorphisms of A, then A is an (E,M)-category if and only if it satisﬁes the following (E,M)-diagonalization property: for any commuting square A

e

/B

m

/D

g

f

C

in A with e ∈ E and m ∈ M, there exists a unique A-morphism d making the diagram e / A B ~ d ~~ g f ~~ ~~~ m /D C commutative. One of the main reasons why factorization strucures are so important is the fact that E-reﬂective (resp. M-coreﬂective) subcategories and hulls in an (E,M)-can be so elegantly described (see section 6. If A is an (E,M)-category with E the class of epimorphisms and M the class of extremal monomorphisms then we say A is an (epi, extremal mono)-category. Analogously, notations like (extremal epi, mono)-category are self explanatory. 4.17. Examples. (1) VectR , Grp, Rng are (regular epi, mono)-categories. (2) Top is a (epi, regular mono)-, resp. (regular epi, mono)- and (dense, closed embedding)-category. (3) Tych is a (dense C ∗ -embedding, perfect map)-category. Two of the factorization structures we have in Set are lifted to topological constructs. 4.18. Theorem. Let (A, U ) be a topological construct. Then (1) A is an (epi, extremal mono)-category and (2) A is an (extremal epi, mono)-category. Proof. (1). For a morphism f : A → B the desired factorization is given by f = i ◦ f where i : B → B is the initial lift of the U -structured source U f (U A) → U B and f : A → B the unique A-morphism such that U f = (U A → U f (U A) : x → U f (x)) which exists by initiality of i. (2). Let f : A → B be a morphism in A. Let π : U A → U A/R denote the quotient in Set corresponding to the equivalence relation x R x ⇐⇒ U f (x) = U f (x ) and let fR : U A/R → U B be the unique (injective) map such that U f = fR ◦π. Then, with π : A → A the ﬁnal lift of the U -structured sink π : U A → U A/R

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and f : A → B the unique A-morphism such that U f = fR , which exists by ﬁnality of π, we get the desired factorization f = f ◦ π. 4.19. Definition. An object A in a topological construct is called a subobject of an object B if there exists an extremal monomorphism A → B and dually it is called a quotient of B if there exists an extremal epimorphism B → A. There are important categorical and structural diﬀerences between the topological constructs (and categories) which we have deﬁned in the foregoing sections and constructs of an algebraic nature like Grp, VectR and Rng of groups, resp. real vector spaces and rings with the corresponding operation preserving maps as morphisms. We list some of their features: (1) Structured sinks do not always have a ﬁnal lift. (Note that mono-sources are always initial!) (2) There are no discrete objects. (3) These categories are co-complete. (4) The forgetful functor U does not preserve colimits. (5) U has a left adjoint. (cf. the existence of free groups, free vectorspaces and free rings) (6) U need not have a right adjoint (the empty set is the initial object for Set, while every one-element group is an initial object for Grp and VectR and Z is an initial object for Rng). (7) U need not preserve epimorphisms (the inclusion i : Z → Q is an epimorphism in Rng, but is not surjective). (8) In none of the categories Grp, VectR and Rng the forgetful functor U preserves coequalizers. However, we do have that regular epimorphism have surjective underlying maps. So the forgetful functor does preserve regular epimorphisms. It moreover reﬂects them. In a topological category this doesn’t hold (Proposition 4.11). (9) In VectR , Grp and Rng the forgetful functor also reﬂects isomorphisms. In a topological category, this property does not hold. In [5] the several notions of algebraicity of a construct and their properties are discussed, capturing and pinpointing these diﬀerences. For completeness’ sake we include the following and refer to [5] for the categorical terminology needed. 4.20. Definition. A construct (A, U ) is called algebraic provided that it satisﬁes the following three conditions: (1) A is (epi, mono source)-factorizable, (2) U has a left adjoint, (3) U is uniquely transportable, (4) U preserves and reﬂects extremal epimorphisms. 5. Fiber-small topological constructs 5.1. Definition. A category A is called wellpowered provided that for each A-object A there is a set-indexed family (mi : Ai → A)i∈I of monomorphisms which is representative in the sense that for each monomorphism m : B → A there is an i ∈ I and an isomorphism hm : B → Ai satisfying m = mi ◦ hm . If in the deﬁnition above “monomorphism” is replaced with “morphism in M for to a particular class M of monomorphisms then we say A is M-wellpowered.

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Dually, a category A is called co-wellpowered provided that for each A-object A there is a set-indexed family (ei : A → Ai )i∈I of epimorphisms which is representative in the sense that for each epimorphism e : A → B there is an i ∈ I and an isomorphism he : Ai → B satisfying e = he ◦ ei . If in the deﬁnition above ‘epimorphism” is replaced with “morphism in E for to a particular class E of epimorphisms then we say A is E-co-wellpowered. 5.2. Proposition. Let (A, U ) be a topological construct. Then A is both wellpowered and co-wellpowered if and only if A is ﬁber-small. Proof. Suppose A is ﬁber-small and take an A-object A. For every X in the powerset P(U A) the ﬁber U −1 X is a set. Then P(U A) × X∈P(UA) U −1 X is also a set. Let m : B → A be a monomorphism, then U m : U B → U A is an injective map (Proposition 4.1) and hence there is a bijection f between U B and a subset X of U A and there is an injection g : X → U A, such that U m = g ◦ f . Let f : B → Y be the ﬁnal lift of f : U B → X, then f is an isomorphism (Corollary 4.12) and g : U Y → U A is an A-monomorphism. Hence, (A, U ) is wellpowered. Cowellpoweredness follows analogously. Conversely, assume that for some set X the ﬁber U −1 X is a proper class. Then, with Xind the indiscrete object overlying X, (idX : A → Xind )A∈U −1 X provides us with a proper class-indexed family of A-monomorphisms that can’t be represented by a set of A-monomorphisms. Hence A cannot be wellpowered. The examples Grp, VectR , Rng considered in the foregoing are ﬁber-small. Since monomorphisms are injective in each of these three categories, we can prove in an analogous way as for topological constructs that Grp, VectR and Rng are wellpowered. In the categories Grp and VectR epimorphisms are surjective. So again by an analogous reasoning we can conclude that Grp and VectR are co-wellpowered. Even though epimorphisms are not surjective in Rng, it can be proved that Rng is co-wellpowered. For ﬁber-small topological categories the above introduced concepts are characterized in terms of the base category. Actually, if A is a ﬁber-small topological category over X then A is (co-)wellpowered if and only if X is (co-)wellpowered. One can recast the nice characterizing conditions of Theorem 3.4 in an even more convenient form. 5.3. Theorem. For ﬁber-small constructs (A, U ) the following conditions are equivalent: (1) (A, U ) is topological. (2) (a) A has products and U preserves them. (b) Every structured injection m : X → U A has a unique initial lift. (c) (A, U ) has indiscrete objects, i.e., every ﬁber contains an indiscrete A-object. Proof. (1) ⇒ (2) is obvious. (2) ⇒ (1). Suppose that (fi : X → U Ai )i∈I is a small structured source. Choose an element j0 ∈ I, let Aj0 be an indiscrete A-object with U Aj0 = X, let fj0 be the U -structured morphism idX : X → U Aj0 let J := I ∪ {j0 } and let (pj : P → Aj )j∈J be the product of the family (Aj )j∈J in A. Since U preserves products, there exists a unique function f : X → U P such that U pj ◦ f = fj for each j ∈ J. Hence

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fj0 = idX = pj0 ◦ f . Thus f : X → U P is a structured injection and so has an initial lift f : A → P . Then, by Proposition 2.9, the source (fj : A → Aj )j∈J is initial. Since Aj0 is indiscrete, (fi : A → Ai )i∈I is initial too. Uniqueness follows from the uniqueness requirement in condition (b). Thus (A, U ) is topological. 5.4. Definition. A ﬁber-small construct is called monotopological if it satisﬁes the conditions (2 a) and (2 b) of the previous theorem. Monotopological constructs behave in many respects very analogously to topological ones as can be seen from their treatment in [5]. The nice thing is that they capture very nicely subconstructs of topological categories determined by separation axioms (see e.g., [175]). Such an example of a monotopological construct is for instance the category Haus of Hausdorﬀ spaces. Note that the epimorphisms and embeddings in Haus are very diﬀerent form the ones in Top, since they are the dense maps, resp. the closed topological embeddings. It can be shown that Haus is also a (co-)wellpowered (epi, extremal mono)- and (extremal epi, mono)-category. Also examples (17),(18) and (23) from Example 2.1 are monotopological. 6. Reﬂective and coreﬂective subcategories 6.1. Definition. Let E be a class of morphisms in a category mathcalC. We say that a category A is an E-reﬂective subcategory of C provided that A is an isomorphism closed full subcategory of C such that each C-object has an E-reﬂection arrow into an A-object. This means that for each object B of C there is a morphism r : B → A in E with A an A-object such that for any morphism f : B → A with A an A-object there exists a unique morphism f : A → A such that the diagram r /A B ~ ~~ f ~~f ~ ~~ A commutes. The dual notion is that of an an E-coreﬂective subcategory. In particular we use the terms epireﬂective (resp. monoreﬂective, bireﬂective, reﬂective) subcategory whenever E is the class of epimorphisms (resp. monomorphisms, bimorphisms, all morphisms). It can be shown (see [5, 198]) that E-reﬂectivity (resp. M-coreﬂectivity) of A in C is equivalent to the property that the embedding functor E : A → B has a left (resp. right) adjoint R : C → A (resp. C : C → A) and that the co-unit (resp. the unit) of this adjunction, which are in eﬀect the reﬂection (resp. coreﬂection) arrows, consists of E- (resp. M-) morphisms. The functor R (resp. L) is called the reﬂector (resp. coreﬂector). We say that (A, U ) is a full concrete subconstruct of a construct (C, V ) if A is a full subcategory of C and U factors through the embedding functor and V . Often we therefore also denote the forgetful functor U by V . 6.2. Examples. (1) Top0 , Top1 and Haus of T0 , T1 and Hausdorﬀ spaces are all reﬂective subconstructs of Top.

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(2) The construct Comp2 of compact Hausdorﬀ spaces considered as a full subconstruct of Tych , the construct of all Tychonoﬀ spaces, is an example of an epireﬂective subconstruct. The epireﬂection arrow of a Tych-object ˇ X is given by its Cech–Stone compactiﬁcation X → βX. (3) In Unif 2 , the construct of all separated uniform spaces, the full subconstruct ComplUnif 2 of all complete objects is epireﬂective, the epireﬂection arrow of a separated uniform space given by the embedding into its completion. (4) In Top, the full subconstruct Seq of sequential spaces (i.e., spaces in which all sequentially closed subsets are closed) is bicoreﬂective and for a topological space (X, τ ), its sequential modiﬁcation idX : (X, τseq ) → (X, τ ) with τseq the topology of all sequentially closed subsets of X, is the corresponding coreﬂection arrow. (5) Unif is a bicoreﬂective subconstruct of QU, the construct of quasi-uniform spaces and quasi-uniformly continuous maps. For (X, U) a uniform space, the bicoreﬂection arrow is idX : (X, U ∨ U −1 ) → (X, U). 6.3. Proposition. Every monoreﬂective subcategory is bireﬂective and dually every epicoreﬂective subcategory is bicoreﬂective. Proof. Again we only prove the ﬁrst statement, the second one following by dualization. Let A be a monoreﬂective subcategory of a category C and ﬁx a C-object B together with a corresponding monic reﬂection arrow m : B → A r / / B with with A an A-object. Consider a parallel pair of C-morphisms A s

r ◦ m = s ◦ m and let m : B → A be a, automatically monic, A-reﬂection arrow for B . Then (m ◦ r) ◦ m = (m ◦ s) ◦ m, hence by unicity in the deﬁnition of reﬂectivity, m ◦ r = m ◦ s, which entails r = s since m is a monomorphism. This shows that m is an epimorphism. 6.4. Definition. An object S is called a separator provided that for every f

pair of distinct morphisms A

g

//

B there exists a morphism h : S → A such

that f ◦ h = g ◦ h. A coseparator is deﬁned dually. 6.5. Proposition. In a well-ﬁbered topological construct any object with a nonempty underlying set is a separator, and any indiscrete object with an underlying set with at least two elements is a coseparator. Proof. Straightforward.

6.6. Proposition. Every coreﬂective subcategory that contains a separator is bicoreﬂective and dually every reﬂective subcategory that contains a coseparator is bireﬂective. Proof. To prove the ﬁrst one of the dual statements, consider a category C with separator S and a coreﬂective subcategory A of C containing S. It suﬃces to prove that C is epicoreﬂective, so ﬁx a C-object B, together with an A-coreﬂection r / / B with r ◦ c = s ◦ c. arrow c : A → B and a parallel pair of C-morphisms B s Because S belongs to A, we know that for every C-morphism f : S → B, there exists a unique C-morphism f : S → A with f = c ◦ f . Therefore r ◦ f = s ◦ f and since S is a C-separator, r = s.

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6.7. Corollary. Let (A, U ) be a full and isomorphism-closed concrete subconstruct of a well-ﬁbered topological construct (C, V ). Then: (1) If A is coreﬂective and contains an object with nonempty underlying set then A is bicoreﬂective. (2) If A is reﬂective and contains an indiscrete C-object with at least 2 elements then A is bireﬂective. Proof. Direct consequence of Propositions 6.6 and 6.5.

6.8. Definition. Let C be an (E,M)-category. If a C-morphism f : A → B belongs to M (resp. E), then f is called an M-subobject (resp. E-quotient) and A is called an M-subobject of B (resp. B is called an E-quotient of A). 6.9. Theorem. Let C be an E-co-wellpowered (E,M)-category that has products and let A be a full and isomorphism closed subcategory of C. Then the following are equivalent. (1) A is E-reﬂective in C. (2) A is closed under the formation of products and M-subobjects in C. Proof. For a full proof see [5]. To give an idea of how things work, we will instead limit ourselves to the case where (C, V ) is a topological construct, (A, U ) is a full concrete subconstruct, E the class of all epimorphisms and M the class of all embeddings, which by Proposition 4.14 coincides with the class of extremal monomorphisms. (1) ⇒ (2). Let (Ai )i∈I be a set indexed family of A-objects and let (pi : P → Ai )i∈I be its product in C. Let r : P → A be the A-epireﬂection arrow for P then by deﬁnition, for every i ∈ I there exists a unique morphism pi : A → Ai with pi = pi ◦ r. By deﬁnition of a product, there exists a unique morphism p : A → P such that pi = pi ◦ p for all i ∈ I. But then pi ◦ (p ◦ r) = pi = pi ◦ idp for all i ∈ I so again by deﬁnition of a product p ◦ r = idP . Because idp is an extremal monomorphism and r is an epimorphism, r has to be an isomorphism and hence P belongs to A. Now let f : B → A be an extremal monomorphism in C with A an A-object. With r : B → A the epireﬂection arrow, there exists a unique morphism f : A → A with f ◦ r = f . Since r is an epimorphism it follows from the deﬁnition of extremal monomorphism that r has to be an isomorphism and hence B is an A-object. (2) ⇒ (1). Fix a C-object B together with a set indexed family (ei : B → Ai )i∈I of epimorphisms with all Ai A-objects which is representative for the class of all epimorphisms with domain B and codomain an A-object. Let (pi : P → Ai )i∈I be the C-product of (Ai )i∈I . Then by (2), P is an A-object. By deﬁnition of a product, there exist a unique morphism f : B → P such that pi ◦ f = ei for all i ∈ I. If f = m ◦ e is an (epi-extremal, mono)-factorization of f it follows from (2) that the domain of m is an A-object A. We are done if we show that e : B → A is an epireﬂection arrow, so take a morphism g : B → A with A an A-object. If g = m ◦ e an (epi, extremal mono)-factorization then again by (2) the domain of m is an A-object and by representativity of the chosen family of epimorphisms we can, without loss of generality, assume this domain to be Aj and e = ej for some j ∈ I. Let g := m ◦ pj ◦ m. Obviously g ◦ e = g and since e is an epimorphism g is unique with this property.

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This helps reformulating the second statement in Corollary 6.7 in the following equivalent but more elegant form. 6.10. Corollary. Let (A, U ) be a full and isomorphism-closed concrete subconstruct of a well-ﬁbered topological construct (C, V ). If A is reﬂective and contains all indiscrete C-objects then A is bireﬂective. The statement below is obtained by dualisation of Theorem 6.9. 6.11. Theorem. Let C be an M-wellpowered (E,M)-category that has coproducts and let A be a full and isomorphism closed subcategory of C. Then the following are equivalent. (1) A is M-coreﬂective in C. (2) A is closed under the formation of coproducts and E-quotient objects in C. 6.12. Definition. A full concrete subconstruct A of a construct (C, V ) is called initially closed provided every V -initial source whose codomains are A-objects has an A-object as a domain. The dual notion, a ﬁnally closed subcategory, is deﬁned accordingly. 6.13. Definition. We say that a full concrete subconstruct (A, U ) of (C, V ) is concretely reﬂective if for each C-object there exists an identity carried A-reﬂection arrow. Again the dual notion of concrete coreﬂectivity is deﬁned in the obvious way. If (A, U ) and (C, V ) are constructs, a functor E : A → C is called concrete if it commutes with the forgetful functors. If (A, U ) is a concretely (co)reﬂective subconstruct of (C, V ) and E the corresponding embedding, then the associated (co)reﬂector is a concrete functor but the inverse implication does not hold however (see [5]). In the concrete case Theorems 6.9 and 6.11 can be strengthened to 6.14. Theorem. Let (A, U ) be a full concrete subconstruct of a topological construct (C, V ). Then the following are equivalent. (1) (A, U ) is initially closed (resp. ﬁnally closed) in (C, V ). (2) (A, U ) is concretely reﬂective (resp. concretely coreﬂective) in (C, V ). Proof. [5].

Note that in the context of topological structures, concrete (co)reﬂectors have, as is easy to see from their deﬁnition, an elegant interpretation as modiﬁcations in the following way (for a proof see [198]). If (A, U ) is a concretely reﬂective (resp. concretely coreﬂective) full concrete subconstruct of a topological construct (C, V ), then for each C-object B the A-bireﬂection (resp. A-bicoreﬂection) arrow is given by idVB : B → A (resp. idVB : A → B) with A = min{A ∈ U −1 (V B) | B ≤ A } (resp. A = max{A ∈ U −1 (V B) | A ≤ B}). In the following results we see how factorization properties are related to the existence and construction of so-called (co)reﬂective hulls. 6.15. Theorem. Let A be a full subcategory of an E-co-wellpowered (E,M)category C. Then the full subcategory determined by all objects that are isomorphic to M-subobjects of products of A objects (performed in C) is the smallest E-reﬂective full concrete subcategory of C that contains A. Proof. [198].

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6.16. Theorem. Let A be a full subcategory of an M-wellpowered (E,M)category C. Then the full subcategory determined by all objects that are isomorphic to E-quotients of coproducts of A objects (performed in C) is the smallest M-coreﬂective full and isomorphism-closed subcategory of C that contains A. Proof. Follows from the foregoing theorem by dualization.

6.17. Examples. (1) Tych is the epireﬂective hull of the full subconstruct of all metrizable topological spaces in Top. The epireﬂective hull of {[0, 1]} in Tych equals Comp2 . (2) For every ordinal α and every ultraﬁlter U on α, make α + 1 into a topological space by putting the discrete neighbourhood ﬁlter at each element of α and deﬁning the neighbourhood ﬁlter at α to be {U ∪ {α} | U ∈ U}. The monocoreﬂective hull in Top of the class of all spaces obtained this way is Top. These examples also motivate the following deﬁnition. 6.18. Definition. A full concrete subcategory A of a concrete category (C, V ) is ﬁnally dense in (C, V ) provided that for any C-object C there exists a V -ﬁnal sink fi : (Ai → C)i∈I with each domain Ai in A. The dual notion, an initially dense subcategory, is deﬁned likewise. In the following statement we see that initiality is preserved by ﬁnally dense subcategories. 6.19. Theorem. If (A, U ) is a ﬁnally dense full concrete subcategory of (C, V ) then any U -initial source is also V -initial. Proof. The proof can be found in [5] but is basically straightforward.

We conclude this section with a theorem which proves to be very useful in determining full concrete subconstructs of topological constructs that are themselves topological. 6.20. Theorem. For a full concrete subcategory (A, U ) of a topological construct (C, V ) the following are equivalent: (1) (A, U ) is topological (2) There exists a concretely coreﬂective full concrete subcategory (B, W ) of (C, V ) such that (A, U ) is concretely reﬂective in (B, W ) (3) There exists a concretely reﬂective full concrete subcategory (B, W ) of (C, V ) such that (A, U ) is concretely coreﬂective in (B, W ), (4) There exists a concrete functor from (C, V ) to (A, U ) acting as the identity on A-objects. Proof. See [5].

From Theorem 6.14 and the proof of the previous theorem, one can make the following interesting observations: if (A, U ) is a concretely reﬂective (resp. concretely coreﬂective) concrete full subcategory of a topological construct (C, V ) then the following hold:

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• the U -initial lift of a U -structured source (resp. the U -ﬁnal lift of a U structured sink) is obtained by performing the V -initial lift (resp. the V -ﬁnal lift) of the same source (resp. sink) but now considered as a V -structured source (resp. a V -structured sink), • the U -ﬁnal lift of a U -structured sink (resp. the U -initial lift of a U structured source) is obtained by ﬁrst forming the V -ﬁnal lift (resp. the V -initial lift) of the same sink (resp. source) but now considered as a V -structured sink (resp. a V -structured source) and then applying the concrete reﬂector (resp. concrete coreﬂector) to this lifted sink (resp. source). 7. Convenience properties Some topological constructs are better behaved than e.g., Top in the sense that they satisfy some so-called “convenience property” like having nice function spaces or representable partial morphisms via one-point extensions. In the following paragraph we will start from a general construct-free deﬁnition of the notions of cartesian closedness and then discuss how it appears in the setting of topological constructs. Cartesian closed topological constructs. 7.1. Definition. A category C is called cartesian closed if it has ﬁnite products and for each C-object A, the endofunctor A × − : C → C has a right adjoint, which is often denoted as (−)A . More generally, if C is a construct with ﬁnite products, an object A for which Atimes− has a right adjoint is called exponential . A category with ﬁnite products therefore is cartesian closed if all objects are exponential. As a consequence of the “adjoint functor theorem” (see e.g., [5]) one has the following characterization of cartesian closedness as a preservation property. 7.2. Theorem. A co-wellpowered, co-complete category C with a separator is cartesian closed if and only if it has ﬁnite products and for each C-object A, A × − preserves colimits. Proof. See [5].

7.3. Theorem. If C is a cartesian closed category, A, B, C are C-objects and (Ai )i∈I , (Bi )i∈I are set indexed families of C-objects, then the following properties hold. (1) First exponential law: AB×C ∼ (AB )C . = B ∼ (2) Second exponential law: (‘ i Ai ) = i (Ai B ). Bi (3) Third exponential law:A i Bi ∼ = i (A ). (4) Distributive law: A × i Bi ∼ = i (A × Bi ). (5) Finite products of regular epimorphisms are regular epimorphisms. Proof. Follows from Theorem 7.2 and can be found in [5, 198].

This already enables a new characterization of cartesian closedness in wellﬁbered topological constructs in the style of Theorem 7.2.

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7.4. Definition. A sink (fi : Bi → B)i∈I of morphisms in a category C is called an episink if it is cancellable from the right, i.e., if for every parallel pair of r / / C such that r ◦ fi = s ◦ fi for all i ∈ I, it follows that r = s. morphisms B s In a topological construct (C, V ), the episinks are exactly the jointly surjective ones, i.e., the sinks for which i∈I V fi (U Bi ) = V B. 7.5. Theorem. If (C, V ) is a well-ﬁbered topological construct, then (C, V ) is cartesian closed if and only if for each C-object A the functor A × − preserves ﬁnal episinks meaning that for every V -ﬁnal episink (fi : Bi → B)i∈I , the sink (idA ×fi : A × Bi → A × B)i∈I is a V -ﬁnal episink.

Proof. See [198].

However, in the case of a well-ﬁbered topological construct, it is often more informative to describe cartesian closedness as the concrete property of having nice function spaces, in the sense of the statement below. 7.6. Theorem. A well-ﬁbered topological construct (C, V ) is cartesian closed if and only if for every pair A, B of C-objects the set hom(A, B) can be equipped with the structure of a C-object, denoted [A, B] such that the following properties are fulﬁlled: (CC1) The evaluation map ev : A × [A, B] → B : (x, f ) → f (x) is a C-morphism. (CC2) For each C-object C and C-morphism f : A × C → B, the map f ∗ : C → [A, B] deﬁned by f ∗ (c)(a) := f (a, c) which renders the following diagram commutative / B v: v v vv idA ×f ∗ vv f v vv A×C A × [A, B] O

ev

is a C-morphism.

Proof. See [5] and [198].

Important to remember from the previous theorem is that in this case the right adjoint of A × − is given by the functor [A, −] : C → C deﬁned by h

h◦−

[A, −](B → C) := ([A, B] → [A, C] : g → h ◦ g). 7.7. Examples. Examples of well-ﬁbered cartesian closed topological constructs are PsTop and Conv. In both cases the structure on function spaces providing the cartesian closedness is given as follows. Let A and B be two objects, Ψ a ﬁlter on hom(A, B) and f ∈ hom(A, B) then Ψ → f if and only if for any ﬁlter F on A and for all x ∈ A, F → x ⇒ ev(F × Ψ) → f (x). Also the construct CAp was shown to be cartesian closed in [158].

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21

Extensional topological constructs. For certain purposes there is another convenience property which is more interesting than cartesian closedness. It was shown by Herrlich, Salicrup and V´ azquez in 1979 [118] that the investigation of connectedness in a topological construct beneﬁts from the following property which is independent from cartesian closedness. 7.8. Definition. A construct (C, V ) is called extensional if and only if it has pullbacks and ﬁnal episinks are hereditary, i.e., are preserved by pullbacks along regular monomorphisms. Precisely this means that if (fi : Yi → Y )i∈I is a ﬁnal episink, m : X → Y is a regular monomorphism and mi : Xi → Yi are morphisms such that for each i ∈ I, the diagram Xi

mi

/ Yi

m

/ Y

gi

X

fi

is a pullback (ensuring that the mi automatically have to be regular monomorphisms), then (gi : Xi → X)i∈I is a ﬁnal episink. In a topological construct (C, V ) the regular monomorphisms are as we know the embeddings. Also the pullback squares above get a more familiar look: in the case that V m is a subset inclusion, we can take mi to be the V -initial lift of subset inclusion of (V fi )−1 (V X) in V Yi and gi is determined by saying that V gi is the restriction of V fi to (V fi )−1 (V X). In well-ﬁbered topological constructs, extensionality can be characterized via the following theorem as the property of having good one-point extensions, making up for the terminology. 7.9. Theorem. A well-ﬁbered topological construct (C, V ) is extensional if and only if partial morphisms are representable. More precisely, if and only if the following condition is fulﬁlled. (ET) Every object B can be embedded in a one-point extension B # , say B # = B ∪ {∞B }, ∞B ∈ B, such that for every object A, for every subobject C of A, and for every morphism f : C → B, the extension, f # : A → B # , ﬁxed by f # (A \ C) := {∞B } is a morphism. / A C f

B

f#

/ B#

Extensional topological constructs too have nice properties. We recall the notion of an injective hull. Given an embedding i : A → B in a concrete category, this embedding is called essential if a morphism f : B → C is an embedding if (and of course only if) f ◦ i : A → C is an embedding. Given an object A, an injective hull of A is an essential embedding i : A → B whereby B is an injective object in C. Recall that B is called injective if and only if for every morphism f : A → B and every embedding m : A → A , there exists a morphism g : A → B with g ◦ m = f . 7.10. Theorem. If (C, V ) is an extensional topological construct then the following properties hold.

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(1) Every object has an injective hull. (2) Final sinks are hereditary. (3) Quotients and coproducts are hereditary. The last two properties are in fact equivalent to being extensional as was shown by Herrlich in 1988 [109]. 7.11. Examples. An example of an extensional topological construct is given by PrTop. Given an object B the structure on B # is determined as follows. For any ﬁlter F on B # , if F has a trace on B then it converges as that trace and it also converges to ∞B . The ﬁlter generated by {∞B } converges to every point of B # . Also PsTop and CAp are extensional. Topological universes. Topological universes, the deﬁnition of which follows below, are also known under the name concrete quasi-topoi. However in the work of Dubuc of 1979 [70], where this term was introduced, no ﬁber-smallness was required. Later in 1983 these constructs were called strongly topological by Herrlich [104]. The term topological universe is due to Nel, and it ﬁrst appeared in his paper [185] from 1984. 7.12. Definition. A well-ﬁbered topological construct (C, V ) is a topological universe if it is both cartesian closed and extensional. One of the reasons that topological universes are important and interesting is given in the following equivalent characterisation from Herrlich [108]. 7.13. Theorem. A well-ﬁbered topological construct (C, V ) is a topological universe if and only if ﬁnal episinks are preserved by pullback along arbitrary morphisms. Precisely, if (fi : Yi → Y )i∈I is a ﬁnal episink, f : X → Y , and hi : Xi → Yi are morphisms such that for each i ∈ I, the diagram Xi

hi

/ Yi

f

/ Y

gi

X

fi

is a pullback, then (gi : Xi → X)i∈I is a ﬁnal episink. 7.14. Examples. Examples of topological universes are the categories Conv, Lim and CAp. 8. Convenient hulls It is well known that in Top the functor Qtimes− does not preserve all quotients (where Q denothes the space of rationals with the Euclidean topology) and it is also easy to construct an example using only ﬁnite spaces showing that in Top qouotients in general fail to be hereditary, so Top is neither cartesian closed, nor extensional, hence not a topological universe. One is therefore often interested in ﬁnding extensions, i.e., larger topological constructs into which the given one can be embedded which have additional convenience properties and which are minimal with respect to this property. This motivates the following deﬁnition. 8.1. Definition. Let (A, U ) and (C, V ) be a topological constructs. Then (C, V ) is called a cartesian closed topological (resp. extensional topological, topological universe) hull of (A, U ) if and only if the following conditions are satisﬁed:

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23

• there exists a concrete embedding E : (A, U ) → (C, V ) as a ﬁnally dense concrete full subcategory, • (C, V ) is a topological construct which is moreover cartesian closed (resp. extensional, a topological universe), • If E : (A, U ) → (C , V ) is another full concrete ﬁnally dense embedding into a topological construct which is cartesian closed (resp. extensional, a topological universe), then there exists a unique concrete F : (C, V ) → (C , V ) such that F ◦ E = E . If such a cartesian closed topological (resp. extensional topological, topological universe) hull, which is then by deﬁnition essentially unique, exists it is denoted by CCTH(A, U ) (resp. ETH(A, U ), TUH(A, U )). 8.2. Definition. If (C, V ) is a topological construct and A is a class of Cobjects, the initial (resp. bireﬂective) hull of A in (C, V ) is the smallest full concrete initially closed (resp. bireﬂective) subcategory of (C, V ) the object class of which contains A. 8.3. Proposition. If (C, V ) is a topological construct and A is a class of Cobjects, the initial and bireﬂective hulls of A in (C, V ) coincide and are determined by the object class {B ∈ Ob(C) | there exists a V -initial source (fi : B → Ai )i∈I with all Ai ∈ A}. Proof. See [215].

The next theorem guarantees the existence of the convenient hulls CCTH(A, U ), ETH(A, U ) and TUH(A, U ) as long as one can prove the existence of a concrete topological ”super-construct” of (A, U ) with the desired convenience property in which A is ﬁnally dense. 8.4. Theorem. Assume that (A, U ) is a full concrete and ﬁnally dense subcategory of a topological construct (C, V ) which is moreover cartesian closed (resp. extensional, a topological universe). Then the cartesian closed topological (resp. extensional topological, topological universe) hull of (A, U ) exists and is realised within (C, V ) as the initial = bireﬂective hull of {[A, B] | A, B ∈ Ob(A)} (resp. {A# | A ∈ Ob(A)}, {[A, B # ] | A, B ∈ Ob(A)}). Proof. See [215].

The ﬁrst problem at hand is thus to ﬁnd such a convenient topological supercategory which is not too large in the sense that the ﬁnal density criterion needs to be satisﬁed. But even then, although the previous theorem, tells us how to ﬁnd the desired hull, it still is often a hard problem to give an internal description of exactly which C-objects belong to the hull. We start with a topological construct (A, U ). To simplify the language, we will call a topological construct (C, V ) into which (A, U ) is embedded as a full concrete ﬁnally dense subcategory a ﬁnally tight extension of (A, U ). We give a procedure to enlarge a concrete category until any of the above discussed convenience requirement is fulﬁlled. All of these hulls rely upon the presence of a largest ﬁnally tight extension, which objects constitute a proper conglomerate. So the largest ﬁnally tight extension is a quasi-construct. And only when it is isomorphic to some proper construct there are no foundational problems concerning the hulls. Note that indeed constructs exist for which the topological hull (see next

24

R. LOWEN, M. SIOEN, AND S. VERWULGEN

paragraph) is a proper quasi-category [4] , as well as constructs for which the cartesian closed topological hull is a proper quasi-category [3]. So, for a given construct, even if the legitimacy condition is fulﬁlled, it is often a challenge to determine the convenience hulls in a more concrete way than below, i.e., to ﬁnd internal realisations of them in more familiar convenient topological constructs. The examples we give are legitimate in the sense that the description of the hull is a settled problem, that is, a proper and known category. Conditions for an construct to have a legitimate hull can be found in [4, 2]. So, with some precautions in mind, we give the following deﬁnition. 8.5. Definition. Let (A, U ) be a construct. The largest ﬁnally tight extension, denoted Max(A, U ), has as objects all pairs (X, S) consisting of a set X and a U structured sink S = (ai : U Ai → X)i∈I subject to the following conditions: (1) S is composition closed, in the sense that for all i ∈ I and all A-morphisms b : B → A, it follows that ai ◦ U b : U B → X is in S, (2) S contains all constant maps, i.e., for any A-object A, any constant map x : U A → X is in S. A morphism in Max(A, U ) between (X, S) and (Y, T ) is a function f : X → Y such that f ◦ a : U A → Y is in T whenever a : U A → X is in S. The construct (A, U ) can be regarded as a full and ﬁnally dense subconstruct of Max(A, U ) with the obvious forgetful functor, via the full embedding E : A → Max(A, U ), where EA := (U A, {U b | b : A → A ∈ Mor(A)}). It can be shown moreover that Max(A, U ) is a topological universe. The hulls CCTH(A, U ), ETH(A, U ) and TUH(A, U ) can now be realised as full concrete sub-quasi-categories of Max(A, U ) via the procedure outlined in Theorem 8.4. Internal descriptions of them can be found in [215]. Note however that they also at this stage are only quasi-categories. The topological universe hull of a construct can also be obtained by a two-step process. First one takes the extensional topological hull and then one makes the cartesian closed topological hull, precisely: TUH(C) = CCTH(ETH(C)) This result is due to Schwarz [214]. It was observed by Schwarz in 1989 that the order of taking hulls on the right-hand side can not be interchanged [214]. In general ETH(CCTH(C)) is strictly smaller than CCTH(ETH(C)) and need not be cartesian closed. The relation among the various hulls of a topological construct C (provided they exist) is given in the following diagram, see the paper by Schwarz [214]. 8.6. Examples. One of the ﬁrst constructed such hulls was the cartesian closed topological hull of Top. This was achieved in a series of papers by Antoine, Machado and Bourdaud in the period 1966–1976, [12, 10, 172, 36, 37]. Antoine gave the start with his description of the objects of the cartesian closed topological hull as those which are initial for a particular source, but he did not give an internal description of these objects. Machado made a ﬁrst step towards this internal description, especially in the case of Hausdorﬀ spaces. Bourdaud ﬁnally rounded the internal description oﬀ with the elegant characterisation given below. The ideas that came out of these papers can not be overestimated, they were the source for

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25

TUH(C) = CCTH(ETH(C)) VVVV VVVV VVVV VVVV VV ETH(CCTH(C)) hh h h h hhhh hhhh h h h hhhh ETH(C) CCTH(C) PPP u u PPP uu PPP u u PPP uu PP uu C Figure 1. The relation among the various hulls of a topological construct C many generalisations and for developing techniques to ﬁnd cartesian closed hulls of other constructs. The cartesian closed topological hull of Top is the construct EpiTop of so-called epi-topological spaces (also called Antoine spaces). That means it is the full subconstruct of PsTop having as objects those spaces (X, q) which satisfy the two supplementary conditions (EpiTop1) and (EpiTop2) below. We denote by clq the closure operator of the Top-bireﬂection of q. Further, for a ﬁlter F on X we denote by Lim F the set of all points x ∈ X such that (F, x) ∈ q and by F ∗ the ﬁlter generated by all the sets {x ∈ X | clq (x) ∩ F = ∅}, F ∈ F. (EpiTop1) For any ﬁlter F on X, Lim F is closed in the Top-bireﬂection. (EpiTop2) For any ﬁlter F on X, Lim F = Lim F ∗ . It was also Bourdaud who proved that PsTop is the cartesian closed topological hull of PrTop. In [109] Herrlich proved, using earlier results from Machado and Bourdaud, that PrTop is the extensional topological hull of Top. Combining these facts yields that PsTop is the topological universe hull of Top. 9. Topological hull = MacNeille completion If (A, U ) is a concrete category with subspaces and ﬁnite concrete products the foregoing constructions can be used to determine a smallest ﬁnally tight topological extension as well. 9.1. Definition. Let (A, U ) be a concrete category with subspaces and ﬁnite concrete products and (C, V ) be a topological construct. Then (C, V ) is called a topological hull or MacNeille-completion of (A, U ) if and only if the following conditions are satisﬁed: • there exists a concrete embedding E : (A, U ) → (C, V ) as a ﬁnally dense concrete full subcategory, • (C, V ) is a topological construct,

26

R. LOWEN, M. SIOEN, AND S. VERWULGEN

• If E : (A, U ) → (C , V ) is another full concrete ﬁnally dense embedding into a topological construct, then there exists a unique concrete F : (C, V ) → (C , V ) such that F ◦ E = E . If such a topological hull exists it is denoted by TH(A, U ). 9.2. Theorem. Assume that (A, U ) is a concrete category with subobjects and ﬁnite concrete products which is a full concrete and ﬁnally dense subcategory of a topological construct (C, V ). Then the topological hull of (A, U ) exists and is realised within (C, V ) as the initial = bireﬂective hull of Ob(A).

Proof. See [215].

It follows from [215] that under these weaker conditions on (A, U ), the quasicategory Max(A, U ) still remains a ﬁnally tight extension and hence TH(A, U ) is the bireﬂective = initial hull of the class of all A-objects within Max(A, U ). Then TH(A, U ) can be characterized (see [215]) as the full subcategory of Max(A, U ) whose objects are those (X, S) with S a closed sink in the sense that g : U A → X belongs to S if h ◦ g : A → B is an A-morphism for every map h : X → U B having the property that h ◦ f is an A-morphism for all f in S. The idea about the topological hull is captured in the following meta statement, for which the proof consists of a technical veriﬁcation. 9.3. Theorem. Let A ⊂ B be full subcategories of a concrete category C. Then the following are equivalent (1) B is the smallest topological extension of A. (2) B is the largest initially and ﬁnally tight extension of A. (3) B is an initially and ﬁnally topological extension of A. (4) B is concretely isomorphic to TH(A). Proof. The statement is found in [108], without proof.

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Contemporary Mathematics Volume 486, 2009

A Convenient Setting for Completions and Function Spaces H. Lamar Bentley, Eva Colebunders, and Eva Vandersmissen Abstract. We develop the completion theories for regular nearness spaces as well as for regular Cauchy spaces and we describe the respective suitable classes of maps such that uniqueness of completion is obtained. We moreover give basic references and some historical background on these topics. Both completion theories provide tools for studying extensions of topological spaces. We illustrate the fact that although the context we are basically interested in is the setting of topological spaces, the theory of extensions can beneﬁt a lot from leaving the topological framework and going beyond Top. In this chapter we add an important aspect to these completion theories giving a formulation of both theories by embedding them in the common setting Mer of merotopic spaces and uniformly continuous maps. This enables us to make a comparison between them. Also for the second topic treated in this chapter, dealing with function spaces, we show that by going beyond Top to certain constructs we encounter as subconstructs of Mer, the theory is put into its right context. We prove that by enlarging Top canonical function spaces do exist, thus showing that Mer contains several cartesian closed subcategories.

Contents 1. Introduction: Why go beyond Top 2. The category of merotopic spaces and some of its subcategories 3. Smallness and nearness 4. Filter spaces and limit spaces 5. Separation and regularity for nearness spaces 6. Completion theory for nearness spaces 7. Separation and regularity for Cauchy spaces 8. Completion theory for Cauchy spaces 9. A comparison of the completion theories for subtopological spaces 10. Function spaces 11. Relations to other constructs 12. Where to ﬁnd more information References

38 40 46 50 55 60 66 69 73 77 79 81 84

2000 Mathematics Subject Classiﬁcation. Primary: 54A20, 54B30, 54D35, 54E17 Secondary: 54A05, 54E15. Key words and phrases. Merotopic space, uniform cover, micromeric collection, near collection, nearness space, ﬁlter space, limit space, Cauchy space, subtopological space, regularity, completeness, completion, function space, cartesian closedness, uniform limit space. c 2009 2008 American Mathematical Society

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H. L. BENTLEY, E. COLEBUNDERS, AND E. VANDERSMISSEN

1. Introduction: Why go beyond Top In this chapter our main motivation is the study of extensions of topological spaces and the investigation of function spaces. The point we want to make in both of these areas is that, although the context we are basically interested in is the setting of topological spaces, the theory itself can beneﬁt a lot from leaving the topological framework and going beyond Top. The well known completion theory of uniform spaces is the right tool for constructing all kinds of extensions of completely regular topological spaces. Given a T0 -uniform space X, it is well known that the space X can be densely embedded ˜ This extension is built on the set of all minimal into a complete T0 -uniform space X. Cauchy ﬁlters and is called the completion of X. It has some interesting features. First of all it is a reﬂection. Secondly the completion is unique, meaning that any complete T0 -uniform space which contains X as a dense subspace is isomorphic to ˜ The behaviour of the completion for uniform spaces will be exemplary for the X. completion theories we will develop in other categories. In an attempt of constructing extensions of topological spaces (not just of the completely regular ones), one needs a completion theory in a more general setting. Seeking such a more general setting where completeness and completions can be reasonably deﬁned, we are ﬁrst led to nearness spaces and uniformly continuous maps. The intuitive concept underlying nearness spaces is a generalization of the proximity space concept of the nearness of a pair of sets—the generalization involves allowing a general collection of sets, not just a pair. Completeness in nearness spaces is deﬁned using clusters (maximal near collections) or equivalently using minimal micromeric collections and these are the points of a completion. In the slightly restricted case of separated nearness spaces the theory of completions can be developed and if one restricts even further to regular nearness spaces (regular nearness spaces being a still more general class than uniform spaces) uniqueness of the completion can be obtained with respect to all dense embeddings. In fact the completion functor on regular nearness spaces restricts to the usual one for uniform spaces. When applied to the so-called subtopological separated nearness spaces, the completion technique provides an important tool for generating Hausdorﬀ topological extensions. Other attempts for building completion theories originated in the setting of convergence. In that theory, the term “Cauchy” has evolved as a synonym to being ‘potentially convergent’. The following quote from Bushaw expresses this idea perfectly: “A Cauchy ﬁlter without a limit can be regarded as a ﬁlter that has the attributes of a convergent ﬁlter, except that there is no point in the space to which it converges. By deﬁnition, a complete space is one in which this phenomenon of the missing limit cannot occur.” Of course, in the classical theory of uniform spaces, completions serve the purpose of providing the missing limit points and one has the result that a ﬁlter on a uniform space is a Cauchy ﬁlter if and only if there is a point in the completion to which the ﬁlter converges. One of the reasons why the completion theory for uniform spaces works out nicely is that the natural relation between two Cauchy ﬁlters, expressing the fact that their intersection is again Cauchy, is an equivalence relation. Formalizing these ideas leads us to the introduction of the category of Cauchy spaces and Cauchy continuous maps. This is an alternative framework in which completion theory will be developed. Cauchy spaces play an important role when studying extensions of spaces in a broad subclass

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of limit spaces (containing, e.g., the class of all Hausdorﬀ limit spaces), since their completion theory inherits the result from uniform space theory, that a ﬁlter on a Cauchy space is a Cauchy ﬁlter if and only if there is a point in the completion to which the ﬁlter converges. The completion obtained for Cauchy spaces is a reﬂection. Restricting further to regular Cauchy spaces uniqueness of completion can be obtained with respect to the class of all strictly dense maps. In this chapter we add an important aspect to the completion theories mentioned so far. We give a formulation of both theories by embedding them in the common setting Mer of merotopic spaces and uniformly continuous maps. This will enable us to make a comparison between them. Merotopic spaces are the generalization of uniform spaces that results when the star reﬁnement axiom is dropped. Merotopic space theory has a richness in the sense that the structures can be given in at least three diﬀerent ways: via uniform covers, via near collections, or via micromeric collections. The near collections intuitively are those that contain sets that are near and this aspect will enable us to fully embed the category of nearness spaces. On the other hand a merotopic space can be equivalently described using micromeric collections, intuitively, those that contain arbitrarily small sets. Through this description we can fully embed the category of Cauchy spaces as well. Having obtained the common supercategory Mer, it becomes possible to investigate the behaviour of both completion theories, nearness completion and Cauchy completion, on spaces to which both techniques are applicable. Going back to our initial concern, the construction of a Hausdorﬀ topological extension of a given topological space, both completion techniques can be used, based on the induced merotopic structure. In the ﬁrst setting the induced structure is handled as a separated subtopological nearness structure. The second line of thoughts apparently uses diﬀerent aspects of the induced structure, namely the fact that it is also a Cauchy structure in which every Cauchy ﬁlter contains a smallest Cauchy ﬁlter having an open base. As one can expect, the separated subtopological nearness spaces coincide with Cauchy spaces in which every Cauchy ﬁlter contains a smallest Cauchy ﬁlter with an open base and for regular subtopological spaces both completion theories coincide. Another important topic in topology is the construction of function spaces. Although in the setting of topological spaces several constructions such as pointwise convergence or compact-open topology do exist making the hom sets into topological spaces, none of these is quite satisfactory. The problem is that these function spaces are not canonical in the sense that their hom sets do not become power objects. Using the terminology of the chapter on Categorical Topology in this book, the problem is that Top is not cartesian closed. The lack of canonical function spaces in a topological construct which is not cartesian closed, such as Top, has long been recognized as a disadvantage for various applications. Steenrod suggested to replace Top by the subcategory of all compactly generated Hausdorﬀ spaces for use in homotopy theory and topological algebra, Dubuc and Porta showed the importance of cartesian closedness in the setting of topological algebra, in particular Gelfand duality theory, and in inﬁnite dimensional diﬀerential calculus the advantage of working in a cartesian closed setting has convincingly been demonstrated by several authors.

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In the “Function Spaces” section 10 we will show that by going beyond Top to certain constructs we encounter as subconstructs of Mer, a lot of cartesian closed candidates are available. We prove that the full subconstructs Fil consisting of ﬁlter spaces, Chy consisting of Cauchy spaces and Lim consisting of limit spaces are cartesian closed, by providing the explicit canonical function spaces in each case. Throughout the chapter we use categorical terminology as developed in the chapter on Categorical Topology in this book. In particular we use the language on topological constructs and on (concretely) reﬂective and coreﬂective subconstructs. 2. The category of merotopic spaces and some of its subcategories In this section we introduce the construct Mer of merotopic spaces and uniformly continuous maps. We describe the objects in terms of uniform covers and we prove that the construct of merotopic spaces is topological over Set. Further we describe the full embeddings of the constructs Tops , of symmetric topological spaces and continuous maps and of Unif, of uniform spaces and uniformly continuous maps. 2.1. Notations. We use the following notations. If A and B are collections of subsets of X then we use A ∩ B for the set theoretical intersection and A ∪ B for the union. Further we put A ∧ B = {A ∩ B | A ∈ A, B ∈ B}, A ∨ B = {A ∪ B | A ∈ A, B ∈ B}. When X and Y are sets and A and B are collections of subsets of X and Y respectively then we put A ⊗ B = {A × B | A ∈ A, B ∈ B}. If f : X → Y is a function, A is a collection of subsets of X and B is a collection of subsets of Y , then we use the notation f A = {f A | A ∈ A} and

f −1 B = f −1 B | B ∈ B . We call a collection A of subsets of X a stack if it satisﬁes A ∈ A and A ⊂ B ⇒ B ∈ A.

Further, for any collection A ⊂ P(X) we put stack A = {B ⊂ X | ∃A ∈ A, A ⊂ B}. In particular we denote x˙ = stack{{x}}. For A ⊂ P(X) let sec A = {B ⊂ X | ∀A ∈ A, B ∩ A = ∅}. Note that the sec operator is idempotent on stacks and reverses the order (A ⊂ B ⇒ sec B ⊂ sec A). We say that collections of subsets A and B on X mesh if A ∩ B = ∅ whenever A ∈ A and B ∈ B. In this case we use the short notation A # B.

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This property can also be expressed in terms of the sec operation as A ⊂ sec B (or B ⊂ sec A.) The set of all ﬁlters on X is denoted by F(X). A grill is a non-empty collection G of non-empty subsets of X such that ∃F ∈ F(X) : G = sec F. Note that every grill is a stack and that a stack G is a grill if and only if whenever A ∪ B ∈ G, then A ∈ G or B ∈ G. Since for a ﬁlter F, F = sec sec F, we have that sec F is a grill and if G is a grill, sec G is a ﬁlter. If U and V are covers of X then we write U < V if U is a reﬁnement of V, that is, ∀U ∈ U, ∃V ∈ V : U ⊂ V. Further we deﬁne the star of x with respect to U as St(x, U) = {U ∈ U | x ∈ U }. For A ⊂ X we further let St(A, U) =

{U ∈ U | U ∩ A = ∅}.

U is called a weak star reﬁnement of V, in symbols, U <∗ V, if {St(x, U) | x ∈ X} < V, and U is called a star reﬁnement of V, in symbols, U ∗ V, if {St(U, U) | U ∈ U} < V. ∗

∗

Recall that U < V and V < W implies U ∗ W. 2.2. Uniform covers. 2.2.1. Definition. A collection µ of collections of subsets of X is called a merotopy on X if the following conditions are fulﬁlled: (U1) If U ∈ µ then U = X. (U2) {X} ∈ µ. (U3) If U ∈ µ and U < V then V ∈ µ. (U4) If U ∈ µ and V ∈ µ then U ∧ V ∈ µ. The pair (X, µ) is called a merotopic space and the covers belonging to µ are called uniform covers. When no confusion can occur we simply write X to denote the merotopic space. A map f : X → Y between merotopic spaces is said to be uniformly continuous whenever the following condition is fulﬁlled: whenever U is a uniform cover of Y then f −1 U is a uniform cover of X. Mer denotes the concrete category whose objects are all merotopic spaces and whose morphisms are all uniformly continuous maps between such spaces. Mer is a well ﬁbered topological construct in the sense of [1]. 2.2.2. Proposition. Let X be a set and (fi : X → (Xi , µi ))i∈I a source indexed by some class I, where (Xi , µi ) are merotopic spaces for i ∈ I.

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Deﬁne µ as follows: U ∈ µ ⇔ {X} < U or U is a cover of X for which ∃{ij }nj=1 ⊂ I : Uj ∈ µij , ∀j = 1, . . . , n,

n

fi−1 (Uj ) < U. j

j=1

Then (X, µ) is the unique initial lift determined by the given source. In view of this result and by application of the Topological Duality Theorem in [1] we also have unique ﬁnal lifts for arbitrary sinks. 2.2.3. Proposition. Let X be a set and (fi : (Xi , µi ) → X)i∈I a sink indexed by some class I, where (Xi , µi ) are merotopic spaces for i ∈ I. Deﬁne µ as follows: µ = {U | U cover of X, fi−1 U ∈ µi , ∀i ∈ I}. Then (X, µ) is the unique ﬁnal lift determined by the given sink. For a given set X the ﬁber of X is a set ordered in the usual way by putting (X, ν) ≤ (X, µ) ⇔ 1X : (X, µ) → (X, ν) is uniformly continuous. The ﬁber is a complete lattice with indiscrete object X endowed with {U | X ∈ U} and discrete object X endowed with {U | U a cover}. Using the terminology explained in the chapter on Categorical Topology, Mer is complete and cocomplete. Limits (colimits) are determined by the corresponding construction in Set and by forming the appropriate initial (ﬁnal) lift. For later use we explicitly formulate the construction of subobjects and products. 2.2.4. Proposition. (1) For a subset A ⊂ X and (X, µ) a merotopic space, the subspace (A, µA ) is given by U ∈ µA ⇔ ∃V ∈ µ : U = {V ∩ A | V ∈ V}. (2) Given merotopic spaces (Xi , µi ) for i ∈ I, the cartesian product is endowed with the initial lift determined by the source (prk : ( i∈I Xi , µ) → (Xk , µk ))k∈I .

i∈I

Xi

In particular, for I = {1, 2} we have U ∈ µ ⇔ ∃U1 ∈ µ1 , U2 ∈ µ2 : U1 ⊗ U2 < U. 2.3. Nearness spaces. In any merotopic space (X, µ) there is an underlying interior operator intµ : P(X) → P(X) deﬁned by intµ A = {x ∈ X | {A, X \ {x}} ∈ µ} for A ⊂ X. Equivalently we have intµ A = {x ∈ X | ∃U ∈ µ, St(x, U) ⊂ A}. When U is a cover of X we denote intµ U = {intµ U | U ∈ U}. The operator satisﬁes the following properties: (1) intµ X = X,

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(2) intµ A ⊂ A whenever A ⊂ X, (3) intµ A ⊂ intµ B whenever A ⊂ B ⊂ X, (4) intµ (A ∩ B) = intµ A ∩ intµ B whenever A, B ⊂ X. In general however the operator is not idempotent. 2.3.1. Definition. A merotopic space (X, µ) is said to be a nearness space if on top of the conditions (U1), (U2), (U3) and (U4) also the following condition is fulﬁlled: (N) U ∈ µ ⇒ intµ U ∈ µ. Near is the full subconstruct of Mer with nearness spaces as objects. 2.3.2. Proposition. In any nearness space (X, µ) the underlying interior operator intµ : P(X) → P(X) is idempotent. Proof. Suppose x ∈ intµ A and take U ∈ µ with St(x, U) ⊂ A. Then we have St(x, intµ U) ⊂ intµ A. So we can conclude that intµ A ⊂ intµ (intµ A). The other inclusion follows from (2). In case (X, µ) is a nearness space the interior operator deﬁnes a topology on the underlying set X, to which we come back in the next subsection. In the next proposition we use the terminology concretely reﬂective. By this we mean reﬂective, where all the reﬂection arrows are identity carried. For the notion of reﬂectivity we refer the reader to [1] or to the chapter on Categorical Topology in this book. 2.3.3. Proposition. Near is concretely reﬂective in Mer. Proof. We prove that Near is initially closed. Let (fi : (X, µ) → (Xi , µi ))i∈I be an initial source in Mer, where (Xi , µi ) are nearness spaces for i ∈ I. For U ∈ µ there are i1 , . . . , in in I and Uj ∈ µij for j = 1, . . . , n such that n −1 j=1 fij (Uj ) < U. For B ∈ Uj we have fi−1 (intµij B) ⊂ intµ fi−1 B j j and hence we have

fi−1 (intµij Uj ) < intµ fi−1 Uj j j

for j = 1, . . . , n. From this we now conclude n

fi−1 (intµij Uj ) < j

j=1

n

intµ fi−1 Uj j

j=1

< intµ

n

fi−1 (Uj ) j

j=1

< intµ U which means that (X, µ) also satisﬁes (N).

The explicit description of the reﬂector Mer → Near is obtained through an iterative process. Given a merotopic space (X, µ) we put µ0 = µ, µα+1 = {U ∈ µα | intµα (U) ∈ µα } and µα = β<α µβ whenever α is a limit ordinal. This transﬁnite sequence (µα )α eventually becomes stationary at some ordinal γ in the sense that µγ = µγ+1 . The reﬂection then is given by 1X : (X, µ) → (X, µγ ).

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2.4. Embedding Tops and Unif. First we deal with the construct Unif of uniform spaces and uniformly continuous maps. We use the Tukey description of a uniform space in terms of covers, which is known to be isomorphic to the Weil description in terms of entourages or to the description in terms of gauges. The basic axiom of a uniform space in terms of covers is the star reﬁnement axiom. 2.4.1. Definition. A merotopic space (X, µ) is said to be a uniform space if on top of the conditions (U1), (U2), (U3), and (U4) also the following condition is fulﬁlled: (U) ∀U ∈ µ, ∃V ∈ µ : V ∗ U. Unif is the full subconstruct of Mer with uniform spaces as objects. 2.4.2. Proposition. Every uniform space (X, µ) is a nearness space. Proof. Let U ∈ µ and determine some cover V ∈ µ such that V ∗ U. For V ∈ V we take U ∈ U such that St(V, V) ⊂ U . For x ∈ V we have V < {U, X \ {x}} and hence x ∈ intµ U . Finally we conclude that V < intµ U. The previous proposition means that Unif is fully embedded in Near. 2.4.3. Proposition. Unif is concretely reﬂective in Mer (and in Near). Proof. Let (X, µ) be a merotopic space and put µu the collection of all U ∈ µ for which ∃(Un )n : U0 = U, Un ∈ µ, Un+1 <∗ Un , ∀n. It is easy to check that µu satisﬁes (U1), (U2), (U3) and (U4) and by deﬁnition also (U). Then 1X : (X, µ) → (X, µu ) is the reﬂection of (X, µ). In order to see this, let f : (X, µ) → (Z, ν) be uniformly continuous, with (Z, ν) a uniform space. For U ∈ ν, let (Un )n be such that U0 = U, Un ∈ ν, and Un+1 <∗ Un for all n. Then (f −1 Un )n is a sequence satisfying the conditions in the deﬁnition of the reﬂection and so we have f −1 U ∈ µu . Finally we can conclude that f : (X, µu ) → (Z, ν) is uniformly continuous too. Next we turn to topological spaces. Covers play also an important role in this setting. For instance for the study of compactness and paracompactness open covers are essential. We will succeed in embedding a large part of Top, the construct of topological spaces and continuous maps, into Mer. For this we need the following symmetry axiom. 2.4.4. Definition. A topological space X with Kuratowski closure cl is called symmetric (or equivalently R0 ) if for any x ∈ X and y ∈ X we have: x ∈ cl{y} ⇔ y ∈ cl{x}. We denote by Tops the full subconstruct of Top whose objects are the symmetric topological spaces. 2.4.5. Definition. A cover U of a topological space X with interior operator intX is called an interior cover if {intX U | U ∈ U} is still a cover. Using the same terminology as before we can also say that U is an interior cover of X if and only if it is reﬁned by some open cover. 2.4.6. Proposition. In a symmetric topological space X the following are equivalent:

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(1) x ∈ intX A. (2) {X \ {x}, A} is an interior cover of X. 2.4.7. Proposition. Let X be a symmetric topological space and let µ be its collection of all interior covers. Then µ satisﬁes the conditions (U1), (U2), (U3), (U4) and (N). Moreover the associated nearness space (X, µ) satisﬁes the following axiom: (T) If intµ U is a cover of X then U ∈ µ. A nearness space satisfying the supplementary axiom (T), is called a topological nearness space. We let TNear be the subconstruct of Near whose objects are the topological nearness spaces. 2.4.8. Proposition. Let (X, µ) be a topological nearness space. Then there exists exactly one symmetric topology T on X such that the collection of all interior covers coincides with µ. Proof. Put A ∈ T ⇔ ∀x ∈ A : x ∈ intµ A.

2.4.9. Theorem. The construct Tops is concretely isomorphic to the construct TNear. Proof. In view of the previous propositions it suﬃces to check that given a map f : X → Y between symmetric topological spaces X and Y and their associated topological nearness structures µX and µY , the following expressions are equivalent: (1) f : X → Y is continuous. (2) U ∈ µY ⇒ f −1 U ∈ µX . The only non-trivial implication is (2) ⇒ (1). In order to see that this is indeed the case let B ⊂ X such that x ∈ clX B for the Kuratowski closure of X. We have {X \ B, X \ {x}} ∈ µX . Since {f −1 (Y \ f (B)), f −1 (Y \ f (x))} reﬁnes {X \ B, X \ {x}}, we have that {Y \ f (B), Y \ f (x)} ∈ µY . Finally we can conclude that f (x) ∈ clY f (B). From now on we will identify Tops and TNear and we will use the notation Tops . 2.4.10. Proposition. Tops is concretely coreﬂective in Near. For a given nearness space (X, µ) the coreﬂection is given by (X, µT ) where µT = {U ⊂ P(X) | intµ U = X}. U∈U

Proof. Let (X, µ) be a nearness space. The interior operator intµ deﬁnes a symmetric topological structure T on X with collection µT of interior covers. We show that T : Near → Tops mapping (X, µ) to (X, µT ) deﬁnes the coreﬂector. In order to see this, we ﬁrst remark that intµ U = X}. µT = {U ⊂ P(X) | U∈U

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Let f : (Z, ν) → (X, µ) be a uniformly continuous map with (X, µ) a nearness space and (Z, ν) a topological nearness space. Then we have U ∈ µT ⇒ intµ U = X U∈U

⇒

f −1 intµ U = Z

U∈U

⇒

intν f −1 U = Z.

U∈U

By application of (T) we get {intν (f −1 U ) | U ∈ U} ∈ ν. So ﬁnally we can conclude that f : (Z, ν) → (X, µT ) is (uniformly) continuous.

We can summarize the results obtained so far by means of the diagram in Figure 1. Tops o Unif @@ ~ Oo ~ @@ ~~ @ c @@ ~~ r ~ ~ Near _ r

Mer Figure 1. The relationships among Mer, Near, Tops , and Unif Now that both Tops and Unif are fully embedded in a common superconstruct Near we can consider their intersection. 2.4.11. Proposition. Tops ∩Unif equals the class of all paracompact symmetric topological spaces. Proof. The objects in Tops ∩ Unif are those symmetric topological spaces the open covers of which satisfy the axiom (U). Using the characterization of paracompactness used in [72] we are done. The previous diagram already gives an indication about the ﬁrst step we take in going beyond Top. So far we described the superconstruct Near that will play a prominent role when generalizing the completion theory in Unif. 3. Smallness and nearness We make free use of the results of Herrlich [67] who showed that merotopic spaces can also be given in other natural ways: by means of collections which contain arbitrary small members, i.e., the micromeric collections of Katˇetov, or by means of collections that are near, a concept intuitively similar to the one which is involved in proximity spaces. These other approaches will prove to be most fruitful when considering completions later on in this chapter. The micromeric approach is the most directly applicable to ﬁlters. It deﬁnes a notion of Cauchyness for ﬁlters and that will in fact be our starting point for this section.

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3.1. Micromeric collections. Before discussing the general situation of merotopic spaces with respect to smallness axioms, we turn to the subconstructs we embedded in Mer in the previous section. For a symmetric topological space it is well known that the topology is completely determined once the convergent ﬁlters (and their limits) are known. Conversely, once the topology is given its class of convergent ﬁlters is ﬁxed. This result is expressed in the next proposition. 3.1.1. Proposition. Let (X, T ) be a symmetric topological space and µT the corresponding collection of interior covers. For a ﬁlter F on X we have F converges ⇔ ∀U ∈ µT : U ∩ F = ∅. Proof. If F → x and U ∈ µT then also intµ (U) is a cover of X. Take U ∈ U with x ∈ intµ (U ), then U ∈ U ∩ F. Conversely, if F is not convergent then for x ∈ X choose Gx open such that x ∈ Gx and Gx ∈ F. Then G = {Gx | x ∈ X} is an open cover and hence G ∈ µT . However G ∩ F = ∅. Next we turn to uniform spaces and we express the corresponding condition on ﬁlters. 3.1.2. Definition. Let (X, µ) be a uniform space and let F be a ﬁlter on X. The ﬁlter F is said to be Cauchy if ∀U ∈ µ : U ∩ F = ∅. Whereas in a topological merotopic space the ﬁlters satisfying the condition ∀U ∈ µT : U ∩ F = ∅ completely determine the structure, the Cauchy ﬁlters in a uniform space in general do not determine the structure. 3.1.3. Example. Let X = R and consider the usual topology T , its collection of interior covers µT , the usual uniform structure D and its associated collection of uniform covers µD . Since µD is not topological, µT and µD are diﬀerent merotopic structures. With N0 the set of strictly positive natural numbers, consider the cover {R \ N0 } ∪ n − n1 , n + n1 | n ∈ N0 which belongs to µT but not to µD . However for a ﬁlter F we have ∀U ∈ µT : U ∩ F = ∅ ⇔ F is T -convergent ⇔ F is D-Cauchy ⇔ ∀U ∈ µD : U ∩ F = ∅. In order to be able to have a smallness notion from which it is possible to recover the given merotopic structure, we will work with collections of subsets rather than with ﬁlters. 3.1.4. Definition. Let (X, µ) be a merotopic space and A ⊂ P(X). The collection A is said to be micromeric if ∀U ∈ µ : U ∩ stack A = ∅. Other terminology that is used to express the same property is “A is small” or “A is Cauchy”. The collection of all micromeric collections is denoted by γµ .

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It follows from Propositions 3.1.1 and 3.1.2 that for ﬁlters in a topological space “micromeric” means “convergent” and in a uniform space “micromeric ﬁlter” coincides with the usual notion of “Cauchy ﬁlter”. It follows from the next proposition that the merotopic structure can now be recovered from the micromeric collections. 3.1.5. Proposition. Let (X, µ) be a merotopic space and γµ its set of micromeric collections. Then we have U ∈ µ ⇔ ∀A ∈ γµ : U ∩ stack A = ∅. Proof. The only non-trivial implication is suﬃciency. Suppose U ∈ µ. For every V ∈ µ we can choose a subset AV ∈ V such that AV ⊂ U for every U ∈ U. Let A = {AV | V ∈ µ}. Then A ∈ γµ . However U ∩ stack A = ∅. 3.1.6. Proposition. Let (X, µ) be a merotopic space. Then γ = γµ , its collection of micromeric collections, satisﬁes the following properties: (M1) {∅} ∈ γ, ∅ ∈ γ. (M2) x˙ ∈ γ whenever x ∈ X. (M3) If A, B are collections of subsets and A ⊂ stack B and A ∈ γ then also B ∈ γ. (M4) If A and B are collections of subsets such that A∪B ∈ γ then either A ∈ γ or B ∈ γ. 3.1.7. Proposition. Let (X, γ) be given by γ ⊂ P(P(X)) satisfying (M1), (M2), (M3) and (M4). Then there exists exactly one merotopic structure µ on X such that the structure γµ = γ. Proof. As before let µ = {U | ∀A ∈ γ : U ∩ stack A = ∅}.

3.1.8. Proposition. Let f : X → Y be a function. Then f : (X, µ) → (Y, ν) is uniformly continuous if and only if f : (X, γµ ) → (Y, γν ) preserves micromeric collections in the sense that A ∈ γµ ⇒ f (A) ∈ γν . Combining the previous propositions yields: 3.1.9. Theorem. The construct Mer is concretely isomorphic to the construct having as objects (X, γ), with γ ⊂ P(P(X)) satisfying (M1), (M2), (M3) and (M4), and as morphisms functions preserving micromeric collections. As usual we will identify the isomorphic descriptions and in the sequel we will use the notation Mer for both of them. Also, if we are mainly concerned with the structure using micromeric collections then we will use the notation (X, γ), but if our main concern is with uniform covers, we will use (X, µ). 3.2. Near collections. 3.2.1. Definition. Let (X, γ) be a merotopic space. Then A ⊂ P(X) is called near if sec A is micromeric. The relationship between the various concepts we encountered so far is given by the following.

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3.2.2. Proposition. Let (X, γ) be a merotopic space then we have: A is near ⇔ sec A is micromeric. A is micromeric ⇔ sec A is near. A is a uniform cover ⇔ {X \ A | A ∈ A} is not near. A is near ⇔ {X \ A | A ∈ A} is not a uniform cover. A is near ⇔ U ∩ sec A = ∅ for every uniform cover U. 3.2.3. Proposition. Let (X, γ) be a merotopic space. Then ξ = ξγ , its structure of near collections, satisﬁes the following properties: (N1) {∅} ∈ ξ, ∅ ∈ ξ. (N2) A = ∅ ⇒ A ∈ ξ. (N3) If A, B are collections of subsets and A ⊂ stack B and B ∈ ξ then also A ∈ ξ. (N4) If A and B are collections of subsets such that A ∨ B ∈ ξ then either A ∈ ξ or B ∈ ξ. 3.2.4. Proposition. Let (X, ξ) be given with ξ ⊂ P(P(X)) satisfying (N1), (N2), (N3) and (N4). Then there exists exactly one merotopic structure γ on X such that the structure ξγ = ξ. Proof. As before put γ = {A | sec A ∈ ξ}.

3.2.5. Proposition. Let f : X → Y be a function. Then f : (X, γ) → (Y, γ ) is uniformly continuous if and only if f : (X, ξγ ) → (Y, ξγ ) preserves near collections in the sense that f (A) ∈ ξγ whenever A ∈ ξγ . Combining the previous propositions yields: 3.2.6. Theorem. The construct Mer is concretely isomorphic to the construct having as objects (X, ξ), with ξ ⊂ P(P(X)) satisfying (N1), (N2), (N3) and (N4), and as morphisms functions preserving near collections. The three isomorphic descriptions are identiﬁed and we use Mer to denote all of them. If we denote a merotopic space by (X, γ) (respectively (X, ξ), resp. (X, µ)) then it should be clear that we are using the presentation in terms of micromeric collections (respectively, near collections, resp. uniform covers). We have seen in any merotopic space (X, γ) there is an underlying interior operator intγ . The associated closure operator deﬁned by x ∈ clγ M ⇔ x ∈ intγ (X \ M ) for M ⊂ X, can now be described directly from the near collections. 3.2.7. Proposition. Let (X, γ) be a merotopic space. Then for any subset M of X clγ M = {x ∈ X | {{x}, M } is near}. Sometimes we will use the notation clX (M ) or simply cl(M ) for the closure on the merotopic space X = (X, γ). For a subobject m : M → (X, γ) in Mer let clX m : clX M → X be the corresponding subobject. 3.2.8. Definition. A closure operator C of a topological construct X is given by a family C = (cX )X∈X of maps cX : P(X) → P(X) such that for every X ∈ X the following properties hold:

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(1) (Extension) M ⊂ cX (M ) for all M ⊂ X. (2) (Monotonicity) M ⊂ M for subsets M and M of X, then cX (M ) ⊂ cX (M ). (3) (Continuity) f (cX (M )) ⊂ cY (f (M )) for every morphism f : X → Y and M subset of X. The closure operator is hereditary if moreover (4) (Hereditariness) cY (M ) = cX (M )∩Y for Y a subobject of X and M ⊂ Y . The closure operator is idempotent if on top of (1), (2) and (3) we have (5) (Idempotency) cX (cX (M )) = cX (M ) for all M ⊂ X. More information on closure operators can be found in [50]. 3.2.9. Proposition. (clX )X is a hereditary closure operator on Mer. Proof. Given a merotopic space X = (X, γ) applying (N1), (N2), (N3) and (N4), the closure operator clX is extensive and monotone. Moreover if f : X → Y is a morphism in Mer then applying 3.2.5 gives that f : (X, clX ) → (Y, clY ) satisﬁes the condition for continuity. To see that the closure operator is hereditary let Y be a subobject of X and M a subset of Y . That clY M = Y ∩ clX M follows from the description in 2.2.2 of the initial structures in Mer. Note that (clX )X needs not be idempotent. Through the isomorphisms described in Theorems 3.1.9 and 3.2.6, Near has isomorphic copies too. The objects of Near are characterized as follows in terms of micromeric collections or near collections respectively. 3.2.10. Proposition. A merotopic space (X, γ) is a nearness space if and only if one of the following equivalent conditions are fulﬁlled: (N1) sec{cl A | A ∈ A} ∈ γ ⇒ sec A ∈ γ. (N2) {cl A | A ∈ A} ∈ ξγ ⇒ A ∈ ξγ . 3.2.11. Proposition. The restriction of the closure operator (clX )X to Near is an idempotent hereditary closure operator on Near. In any nearness space X the closure operator (clX )X describes the closure in the topological coreﬂection. 4. Filter spaces and limit spaces In the section on micromeric collections we gave an example (Example 3.1.3) showing that in general for a merotopic space the class of all Cauchy ﬁlters is not big enough to determine the merotopic structure. In this section we investigate merotopic spaces for which the class of all Cauchy ﬁlters determines all micromeric collections and hence the merotopic structure itself. Those spaces are called ﬁlter spaces and the full subconstruct of Mer whose objects are the ﬁlter spaces is denoted by Fil. When developing a completion theory, the relation on Cauchy ﬁlters deﬁned by “F is related to G if the intersection of the two ﬁlters is again a Cauchy ﬁlter” appears to be very important. In general, on a ﬁlter space this relation need not be an equivalence relation. Sorting out exactly those ﬁlter spaces for which the relation becomes an equivalence leads to the deﬁnition of the full subconstruct Chy consisting of Cauchy spaces. In the ﬁnal subsection, we investigate how they relate to limit spaces.

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4.1. Filter spaces and Cauchy spaces. 4.1.1. Definition. A collection ϕ ⊂ F(X) is said to satisfy the ﬁlter axioms if (F1) x˙ ∈ ϕ, whenever x ∈ X, (F2) F ∈ ϕ, F ⊂ G ⇒ G ∈ ϕ. 4.1.2. Definition. Fil is the full subconstruct of Mer whose objects (X, γ) satisfy the condition (F) ∀A ∈ γ, ∃F ∈ F(X) : F ∈ γ, F ⊂ stack A. The objects of Fil are called ﬁlter spaces. Those ﬁlters which are members of γ are called Cauchy ﬁlters. 4.1.3. Proposition. Given a merotopic space (X, γ) let ϕγ = F(X) ∩ γ. This collection ϕγ satisﬁes the ﬁlter axioms. Moreover given any collection ϕ of ﬁlters on X satisfying the ﬁlter axioms, there is a unique ﬁlter space (X, γ) such that ϕγ = ϕ. Proof. The ﬁrst assertion is clear. In order to construct a merotopic structure out of the given ϕ let γϕ = {A | A ⊂ P(X), ∃F ∈ ϕ : F ⊂ stack A}. That this collection satisﬁes (M1), (M2) and (M3) is clear. In order to see that (M4) is also fulﬁlled observe that for a ﬁlter F and for collections A and B one has F ⊂ stack(A ∪ B) ⇒ F ⊂ stack A or F ⊂ stack B. 4.1.4. Proposition. Given ﬁlter spaces (X, γ) and (X, γ ) and a function f : X → Y , then f : (X, γ) → (Y, γ ) is uniformly continuous if and only if f preserves Cauchy ﬁlters, that is F ∈ ϕγ ⇒ stack f (F) ∈ ϕγ . 4.1.5. Theorem. The construct Fil of ﬁlter spaces is concretely isomorphic to the construct with objects spaces (X, ϕ) with ϕ a collection of ﬁlters on X satisfying the ﬁlter axioms (F1) and (F2) and with morphisms functions f : (X, ϕ) → (Y, ϕ ) satisfying F ∈ ϕ ⇒ stack f (F) ∈ ϕ . 4.1.6. Theorem. The construct Fil of ﬁlter spaces is concretely coreﬂective in Mer. Proof. Given a ﬁlter space (X, γ) with ϕ as collection of Cauchy ﬁlters, as in the proof of Proposition 4.1.3 let γϕ = {A | A ⊂ P(X), ∃F ∈ ϕ : F ⊂ stack A}. We show that 1X : (X, γϕ ) → (X, γ) is the coreﬂection arrow. Let (Z, γ ) be a ﬁlter space with collection ϕ of Cauchy ﬁlters and let f : (Z, γ ) → (X, γ) be uniformly continuous. For F ∈ ϕ we then have stack f (F) ∈ γ and hence stack f (F) ∈ ϕ. It follows that also f : (Z, γ ) → (X, γϕ ) is uniformly continuous.

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Therefore the ﬁnal lift of a sink in Fil is constructed as in Mer. Moreover the coreﬂector preserves limits, in particular products and equalizers. We remark that Fil is closed under subspaces in Mer. Products in Filare particularly easy to describe. Let (Xi ) Xi be their product in i∈I be a family of ﬁlter spaces, and let Fil. A ﬁlter F on Xi is Cauchy if and only if for each i ∈ I, the projection of F on Xi is Cauchy on Xi . In the case of two factors we use the following notation (based on the notation introduced in section 2.1: If F is a ﬁlter on X and G is a ﬁlter on Y then F ⊗ G generates a ﬁlter stack(F ⊗ G) on X × Y . 4.1.7. Proposition. Given ﬁlter spaces (X, γ ) and (Y, γ ) with ϕ and ϕ as corresponding collections of Cauchy ﬁlters, then F ∈ ϕ ⇔ ∃G ∈ ϕ , ∃H ∈ ϕ : G ⊗ H ⊂ F deﬁnes a ﬁlter space structure γϕ on X × Y which is the product structure in Fil. Proof. If (X × Y, γ) is the product space and F is a Cauchy ﬁlter on X × Y then putting G = pr1 F and H = pr2 F we obtain the right hand side by simple application of uniform continuity of the projections. To prove the other direction we have to show that when F satisﬁes the condition on the right hand side, it is a Cauchy ﬁlter for the product deﬁned in Proposition 2.2.4. Suppose U and U are uniform covers for γ and γ respectively. Choose G ∈ G ∩ U and H ∈ H ∩ U . Then G × H ∈ (U ⊗ U ) ∩ F. For further use in completion theory we next introduce a full subconstruct of Fil. We remark that for ﬁlters F and G the operation F ∨ G produces the set theoretical intersection F ∩ G of the ﬁlters. 4.1.8. Definition. Chy is the full subconstruct of Fil whose objects are ﬁlter spaces satisfying the following extra axiom: (CH) If F is a Cauchy ﬁlter and G is a Cauchy ﬁlter and F # G then F ∩ G is a Cauchy ﬁlter. The objects in Chy are called Cauchy spaces. Another way of putting this deﬁnition is the following. 4.1.9. Proposition. A ﬁlter space is a Cauchy space if and only if the relation between Cauchy ﬁlters deﬁned by F ∼ G ⇔ F ∩ G is a Cauchy ﬁlter is an equivalence relation. 4.1.10. Theorem. Chy is concretely reﬂective in Fil. Proof. If (X, γ) is a ﬁlter space then the reﬂection is 1X : (X, γ) → (X, δ), where A is micromeric in (X, δ) if and only if there exists a ﬁnite sequence F1 , . . . , Fn of Cauchy ﬁlters in (X, γ) such that Fi # Fi+1 for all i = 1, . . . , n and such that n i=1 Fi ⊂ stack A. In Chy subspaces are formed as in Mer or as in Fil. Moreover in Chy products are formed as in Fil. Chy moreover is closed with respect to taking coproducts in Mer. We remark that in particular a Cauchy space (X, γ) satisﬁes the condition that if A and B are micromeric and if for some point x ∈ X the collections A ∨ x˙ and B ∨ x˙ are micromeric, then also A ∨ B is micromeric.

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4.1.11. Definition. Let C be the full subconstruct of Fil whose objects are those ﬁlter spaces satisfying the condition that if A and B are micromeric and if for some point x ∈ X, both A ∨ x˙ and B ∨ x˙ are micromeric, then also A ∨ B is micromeric. We omit the proof of the following result and we refer to [19] for details. 4.1.12. Proposition. C is concretely reﬂective in Fil. We remark that C is closed with respect to taking coproducts in Mer, hence also in Fil. 4.2. Limit spaces. 4.2.1. Definition. The objects of Lim are pairs (X, q), where X is a set and q ⊂ F(X) × X. The expression (F, x) ∈ q is also written as F → x and means that F converges to x. The axioms for convergence are (L1) x˙ → x. (L2) F → x and F ⊂ G ⇒ G → x. (L3) F → x and G → x ⇒ F ∩ G → x. The objects of Lim are called limit spaces and morphisms are called continuous maps. If f : (X, q) → (Y, p) is a function between limit spaces then f is continuous iﬀ stack f (F) → f (x) whenever F → x. An object of Lim is called a pretopological space if the convergence moreover satisﬁes the following axiom: (L4) Fi → x for all i belonging to some arbitrary index set I ⇒ i∈I Fi → x. Pretop is the full subconstruct of Lim with objects the pretopological limit spaces. An object of Lim is called a symmetric limit space if the convergence satisﬁes the following symmetry condition: (S) x˙ → y ⇒ x and y have the same convergent ﬁlters. Given a space (X, γ) in C and x ∈ X, in order to deﬁne F → x one can select those ﬁlters F such that F ∩ x˙ is Cauchy. This selection deﬁnes a limit space in the sense of the previous deﬁnition. Several objects in C could generate the same limit space. For this reason we now restrict to a subconstruct of C. 4.2.2. Definition. A merotopic space (X, γ) in C is called a limit space if it satisﬁes: for every Cauchy ﬁlter F in (X, γ) there exists x ∈ X such that F ∩ x˙ is Cauchy. The full subconstruct of Mer whose objects are limit spaces is denoted by Lims . 4.2.3. Theorem. The construct Lims of limit spaces is concretely isomorphic to the full subconstruct of Lim consisting of symmetric limit spaces with continuous maps. Proof. Given a space (X, γ) in Lims , its isomorphic image is deﬁned as follows. For a ﬁlter F and x ∈ X, we put F → x ⇔ F ∩ x˙ ∈ γ.

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As usual we will identify the isomorphic descriptions and in the sequel we will use the notation Lims for both of them. 4.2.4. Theorem. The construct Lims of limit spaces is concretely coreﬂective in C. Proof. Given a ﬁlter space (X, γ) in C the coreﬂection 1X : (X, γ ) → (X, γ) is given by A ∈ γ ⇔ ∃F ∈ F(X), ∃x ∈ X with F ∩ x˙ ∈ γ such that F ∩ x˙ ⊂ stack A. 4.2.5. Proposition. The construct Lims of limit spaces is productive in C and hence also in Fil. If we restrict the previous coreﬂector to the subconstruct Chy then the limit spaces in the image satisfy an even stronger symmetry axiom, to be deﬁned next. 4.2.6. Definition. A limit space (X, q) is said to be reciprocal if the following condition is fulﬁlled: (∃F : F → x, F → y) ⇒ (∀H : H → x ⇔ H → y). Let Limr be the full subconstruct of Lims consisting of those objects for which F → x ⇔ F ∩ x˙ ∈ γ deﬁnes a reciprocal limit space. In particular T2 -limit spaces (those for which F → x and F → y ⇒ x = y) are reciprocal. 4.2.7. Proposition. (1) Limr is concretely reﬂective in Lims . (2) Limr is a concretely coreﬂective subconstruct of Chy. Proof. (1). The reﬂector Lims → Limr is the restriction of the reﬂector C → Chy. (2). Let (X, γ) ∈ Limr and suppose F and G are Cauchy ﬁlters such that F # G. There exist x and y such that F ∩ x˙ ∈ γ and G ∩ y˙ ∈ γ. It follows that stack(F ∧G) converges to x as well as to y. Since the limit structure is reciprocal we can conclude that x and y have the same convergent ﬁlters. In particular F ∩G ∈ γ. So Limr is a subconstruct of Chy. To see that it is a concretely coreﬂective subconstruct it suﬃces to restrict the coreﬂector C → Lims to Chy. 4.2.8. Proposition. The construct Limr of reciprocal limit spaces is productive in Chy and hence also in Fil.

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The results encountered in section 4 are summarized in the diagram in Figure 2.

Chy

m Limr Fq FF yM y FFr c yy FF y y y F y qEE EE E r EEE E C _

Lims x Mm x x xx xx c x x

r

Fil _ c

Mer Figure 2. The relationships among categories of ﬁlter spaces and Cauchy spaces In section 2, we described the embedding of Tops in Near. It can also be embedded in Fil. A topological space (X, T ) is said to be a reciprocal topological space if for every x, y ∈ X with neighborhood ﬁlters V(x) and V(y), resp., such that V(x) # V(y) then one has V(x) = V(y). 4.2.9. Proposition. (1) Every symmetric topological space is a symmetric convergence space. (2) Every reciprocal topological space is a reciprocal convergence space. Proof. A ﬁlter F in a topological space converges to some point x if and only if V(x) ⊂ F and clearly this convergence satisﬁes the limit axioms (L1), (L2) and (L3) described in Theorem 4.2.3. The R0 axiom for topological spaces deﬁned in 2.4.4 is equivalent to the symmetry deﬁned in 4.2.3. Deﬁnitions of reciprocal given in 4.2.6 and above coincide as well. The following theorem summarizes these results. 4.2.10. Theorem. (1) Topr = Limr ∩ Near. (2) Tops = Lims ∩ Near. 5. Separation and regularity for nearness spaces In order to be able to develop a completion theory in the more general setting of nearness spaces that behaves comparably to the completion theory of uniform spaces, we need to restrict to suitable subconstructs of Near. Separatedness and regularity play an important role in this respect and these properties deﬁne concretely reﬂective subconstructs of Near.

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5.1. Regularity for nearness spaces. Regularity can be introduced for a merotopic space but as we will see the property implies that the space is a nearness space. 5.1.1. Definition. Let (X, µ) be a merotopic space. A uniform cover U is called a regular reﬁnement of a uniform cover V provided for any U ∈ U there exists a member V of V, which is a uniform neighbourhood of U (i.e., {V, X \ U } is a uniform cover). A merotopic space (X, µ) is regular if every uniform cover has a regular reﬁnement. 5.1.2. Proposition. A regular merotopic space is a nearness space. Proof. Let (X, µ) be a regular merotopic space and let U ∈ µ. Choose a regular reﬁnement V of U. Then clearly V < {intµ U | U ∈ U}. Hence (X, µ) is a nearness space. The full subcategory of regular nearness spaces is denoted by RegNear. We use RegTop to denote the construct of regular topological spaces. 5.1.3. Proposition. (1) Every regular nearness space has a regular topological coreﬂection. (2) Every uniform space is regular. Proof. (1). Consider the restriction to RegNear of the coreﬂector Near → Tops . We prove that this restriction deﬁnes a coreﬂector RegNear → RegTop. As before, given a regular nearness space (X, µ), its topological coreﬂection is (X, µT ). In order to prove that this space is regular, let x ∈ X and let U ⊂ X be a neighbourhood of x. So {X \ {x}, U } is a uniform cover of (X, µ) and due to the regularity of (X, µ) there exists a regular reﬁnement V of {X \ {x}, U }. Let V ∈ V, such that x ∈ intµ V . It now follows that U is a uniform neighbourhood of V , i.e., {U, X \ V } is a uniform cover. By applying (N), we get that {intµ U, X \ clµ V } is a uniform cover too. clµ V is a neighbourhood of x which is contained in U . (2). Let (X, µ) be a uniform space and U a uniform cover. We choose V ∈ µ, a star reﬁnement of U. For any V ∈ V, there exists a member U of U such that St(V, V) ⊂ U . Therefore V < {U, X \ V } and so it follows that {U, X \ V } is a uniform cover. We conclude that V is a regular reﬁnement of U. 5.1.4. Proposition. RegNear is concretely reﬂective in Near. Proof. We prove that RegNear is initially closed in Near. Let (fi : (X, µ) → (Xi , µi ))i∈I be an initial source in Near, where (Xi , µi ) are regular nearness spaces for i ∈ I. For every uniform cover U in µ we can determine a ﬁnite subset J of I and Uj ∈ µj for j ∈ J such that j∈J fj−1 (Uj ) < U. Due to the regularity of (Xj , µj ), there exists a regular reﬁnement Vj ∈ µj of Uj for every j in J. Hence n V := j=1 fj−1 (Vj ) is a regular reﬁnement of U. Indeed let V ∈ V, then for any j ∈ J there exists Vj ∈ Vj such that V = j∈J fj−1 (Vj ). For every Vj , there exists a member Uj of Uj which is a uniform neighbourhood of Vj . Choose U ∈ U such that j∈J fj (Uj ) ⊂ U . It easily follows that j∈J fj−1 {Uj , X \ Vj } < {U, X \ V } and hence U is a uniform neighbourhood of V .

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RegTop s Unif l KK vv L KK v v KK v c KKK vv r vv RegNear _ r

Near Figure 3. The relationships among constructs of regular nearness spaces 5.2. Separatedness for nearness spaces. A condition somewhat weaker than regularity but nevertheless a very useful condition is separatedness. Again the deﬁnition can be formulated in the setting of merotopic spaces. 5.2.1. Definition. Let (X, µ) be a merotopic space. A collection of subsets of X is called concentrated when it is both near and micromeric. The merotopic space (X, µ) is called separated provided that whenever a collection A is concentrated then the collection B := {B ⊂ X | A ∪ {B} is near in (X, µ)} is near. We denote by Sep the full subconstruct of Mer consisting of all separated spaces and SepNear = Sep ∩ Near. We remark that the collection B in the previous deﬁnition is in fact a grill. 5.2.2. Proposition. Every regular merotopic space is separated. Proof. Let (X, µ) be a regular merotopic space and A a collection which is concentrated. Consider B := {B ⊂ X | A ∪ {B} is near in (X, µ)} and an arbitrary cover U ∈ µ. There exists a uniform cover V which is a regular reﬁnement of U. Since the collection A is micromeric there exist A ∈ A and V ∈ V such that A ⊂ V . We choose U ∈ U such that {U, X \ V } is a uniform cover. Because A is near and A is contained in V , we have that U ∩ A = ∅ for all A ∈ A. Moreover for any B ∈ B, there exists an element W of {U, X \ V } such that W ∩ B = ∅ and W ∩ A = ∅. So W = U . Finally U ∩ B = ∅ for every B ∈ B, which implies that B is near. 5.2.3. Proposition. Sep is concretely reﬂective in Mer and SepNear is concretely reﬂective in Near. Proof. We prove that SepNear is initially closed in Near. Let (fi : (X, µ) → (Xi , µi ))i∈I be an initial source in Near, where (Xi , µi ) are separated nearness spaces for i ∈ I. Let A be a concentrated collection in (X, µ), B := {B ⊂ X | A ∪ {B} is near in (X, µ)} and U a uniform cover of (X, µ). We can determine a ﬁnite subset J of I and Uj ∈ µj for j ∈ J such that j∈J fj−1 (Uj ) < U. For any j ∈ J, fj (A) is concentrated in (Xj , µj ) and hence the collection Bj = {B ⊂ Xj | fj (A) ∪ {B} is near in (Xj , µj )} is near in (Xj , µj ). So there exists an element Uj of Uj whose intersection with arbitrary elements of Bj is non-empty. In particular / Bj and then also fj−1 (Xj \ Uj ) ∈ / B. Because B is a grill, it follows that Xj \ Uj ∈ −1 {fj (Xj \ Uj ) | j ∈ J} = X \ {fj−1 (Uj ) | j ∈ J} is not a member of B. This means that any B ∈ B has a non-empty intersection with {fj−1 (Xj \ Uj ) | j ∈

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J} ⊂ U and hence B ∩ U = ∅, for any B ∈ B. We can conclude that B is near in (X, µ). 5.3. T0 -objects in RegNear and SepNear. In [94] the categorical deﬁnition of a T0 -object in a topological construct X was given. We recall the deﬁnition. 5.3.1. Definition. An X-object X is a T0 -object of a topological construct X if every morphism f : I2 → X is constant, where I2 is the indiscrete 2-point object of X. For later use in completion theory we need the explicit meaning of a T0 -object in RegNear. We obtain the following characterization. 5.3.2. Proposition. The following are equivalent for an object (X, µ) of RegNear. (1) (X, µ) is a T0 -object of RegNear. (2) The topological coreﬂection is a T0 -space. (3) The topological coreﬂection is a T2 -space. (4) The topological coreﬂection is a T3 -space. Proof. Note that on any set Z every non-empty A ⊂ P(Z) is micromeric in the indiscrete nearness structure. Since this space is regular and topological, a function f : Z → (X, µ) is uniformly continuous if and only if f : Z → T (X, µ) is uniformly continuous. The rest follows from the symmetry and regularity of the topological coreﬂection. We denote the full subcategory of RegNear whose objects are the T0 -objects by RegNear0 . Recall that a reﬂective subconstruct A in B is said to be quotient reﬂective if all the reﬂection morphisms are quotients. 5.3.3. Proposition. RegNear0 is quotient reﬂective in RegNear. Given a regular nearness space (X, µ), the relation ρ on X given by x ρ y ⇔ clµ {x} = clµ {y} deﬁnes an equivalence relation. When X|ρ is equipped with the quotient µρ of (X, µ) and fρ : X → X|ρ is the canonical map then fρ : (X, µ) → (X|ρ , µρ ) is the RegNear0 -reﬂection morphism for (X, µ). Proof. The result follows from the general theory on T0 -objects [94]. Here we give an explicit proof. First we give an explicit despription of µρ by showing that µρ = ν, where ν is the ﬁnal lift of fρ : ((X, µ) → X|ρ ) in the construct Mer, i.e., ν = {U cover of X|ρ | fρ−1 (U) ∈ µ}. Then by showing that (X|ρ , ν) satisﬁes the regularity condition, it will follow that it is a regular nearness space coinciding with (X|ρ , µρ ). Let U be a uniform cover of ν. Then fρ−1 (U) ∈ µ and by regularity of (X, µ) there exists a uniform cover V ∈ µ which is a regular reﬁnement of fρ−1 (U). Then intµ V is also a regular reﬁnement of fρ−1 (U). From the deﬁnition of fρ , it follows that fρ−1 (fρ (intµ A)) = intµ A for any subset A of X. Hence fρ (intµ V) ∈ ν. Moreover for any V ∈ V, there exists U ∈ U such that {X \ intµ V, fρ−1 (U )} ∈ µ. Then we also have {X|ρ \fρ (intµ V ), U } ∈ ν. We can conclude that fρ (intµ V) is a regular reﬁnement of U. So ν is a regular nearness structure and hence it equals µρ .

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In order to see that (X|ρ , µρ ) is a T0 -object, let a, b ∈ X|ρ such that a = b. Choose x, y ∈ X such that fρ (x) = a and fρ (y) = b. Since x ∈ / clµ {y}, we have that {X \{x}, X \{y}} ∈ µ. Because fρ−1 {X|ρ \{a}, X|ρ \{b}} = intµ {X \{x}, X \{y}} ∈ µ, {X|ρ \{a}, X|ρ \{b}} is a uniform cover of (X|ρ , µρ ) and hence clµρ {a} = clµρ {b}. It remains to show that fρ : (X, µ) → (X|ρ , µρ ) is the RegNear0 -reﬂection morphism. Let (Y, µ ) be a T0 regular nearness space and g : (X, µ) → (Y, µ ) a uniformly continuous map. Then from xρy we always have that g(x) = g(y). Indeed, when clµ {x} = clµ {y}, we have that {{x}, {y}} is near and hence {g{x}, g{y}} is near too. Since (Y, µ ) is T0 it follows that g(x) = g(y). Hence there exists a unique map h : X|ρ → Y satisfying h ◦ fρ = g. Because fρ is ﬁnal we can conclude that h : (X|ρ , µρ ) → (Y, µ ) is uniformly continuous. Recall that a topological construct is universal if it is the concretely reﬂective hull of the class of its T0 -objects. As is explained in the chapter on Categorical Topology in this book, this also means that the class of T0 -objects is initially dense. 5.3.4. Proposition. RegNear is universal. Proof. It suﬃces to verify that for every regular nearness space (X, µ) the RegNear0 -reﬂection morphism fρ : (X, µ) → (X|ρ , µρ ) is initial. Therefore we need to check that for any uniform cover U ∈ µ there exists a uniform cover U in µρ such that fρ−1 (U ) < U. If we take U = fρ (intµ U), this result follows since as noted before, we have that fρ−1 (fρ (intµ A)) = intµ A for any A ⊂ X. In the sequel we will also pay attention to the construct of T0 -objects of SepNear. Reasoning as in the proof of Proposition 5.3.2, we obtain the following characterization of a T0 -object of SepNear. 5.3.5. Proposition. The following are equivalent for an object (X, µ) of SepNear. (1) (X, µ) is a T0 -object of SepNear. (2) The topological coreﬂection is a T0 -space. (3) The topological coreﬂection is a T2 -space. We denote the full subcategory of SepNear whose objects are the T0 -objects by SepNear0 . From Proposition 5.3.2 it is clear that a regular nearness space is a T0 -object in RegNear iﬀ it is a T0 -object in SepNear. 5.4. cl-dense uniformly continuous maps in RegNear0 and SepNear0 . In this section we will show that the cl-dense uniformly continuous maps in RegNear0 (resp. SepNear0 ) are epimorphisms in RegNear0 (resp. SepNear0 ). It follows from the results in sections 2 and 3 that cl is an idempotent and hereditary closure operator on RegNear0 as well as on SepNear0 . 5.4.1. Proposition. A cl-dense uniformly continuous map in RegNear0 is an epimorphism in RegNear0 . Proof. Suppose f, g : X → A, A ∈ RegNear0 such that f |M = g|M for M some cl-dense subset of X. Apply the topological coreﬂection. Then T f, T g : T X → T A and T A ∈ RegTop0 , so in particular it has a Hausdorﬀ topology. Since f and g coincide on the dense subset M we now can conclude that they coincide on the whole of X. Analogous reasoning for the construct SepNear0 leads to the following result.

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5.4.2. Proposition. A cl-dense uniformly continuous map in SepNear0 is an epimorphism in SepNear0 . 6. Completion theory for nearness spaces We start with some guiding example for a completion theory. Given a T0 uniform space X, it is well known that the space X can be densely embedded into ˜ This extension which is called the completion of X, a complete T0 -uniform space X. has some interesting features. First of all it is a reﬂection which implies that every morphism f : X → Y with Y a complete T0 -uniform space, has a unique extension ˜ → Y . Secondly the completion is unique, meaning that any complete T0 f˜: X ˜ The uniform space which contains X as a dense subspace is isomporphic to X. behaviour of the completion in Unif 0 is exemplary for completion theories in many other categories. We brieﬂy recall the material on unique completions with respect to some given class of morphisms developed in [26]. It is called a “ﬁrm” completion theory since it resembles the completion theory for uniform spaces as we will explain below. Later on we prove that on the subconstruct RegNear0 such a “ﬁrm” completion theory exists with uniqueness of completion obtained with respect to the class of all cl-dense embeddings. 6.1. A ﬁrm completion theory. For the subconstruct X0 consisting of all T0 -objects in some topological construct X, we consider a reﬂective full subconstruct R with reﬂection functor R : X0 → R. Further we ﬁx some class U of morphisms and we assume that the morphisms in U are embeddings and that all isomorphisms belong to U. 6.1.1. Definition. R is said to be U-reﬂective if the reﬂection morphisms rX : X → R(X) belong to U. Note that the reﬂection R(X) of an object X then becomes an extension, i.e., the reﬂection morphisms rX are embeddings. Such a U-reﬂection is said to be a U-completion. Uniqueness of the completion with respect to U means that whenever f : X → Y is in U with Y in R, then the unique morphism f ∗ : R(X) → Y such that f ∗ ◦rX = f is an isomorphism. The proof of the next proposition is easy and can be found in [26]. 6.1.2. Proposition. If the class U moreover satisﬁes the condition that U is closed under composition and if R is a U-completion then the completion is unique with respect to U if and only if R(f ) is an isomorphism whenever f is in U. If this condition is fulﬁlled R is said to be subﬁrmly U-reﬂective. We denote by L(R) the class of all morphisms u : X → Y for which R(u) is an isomorphism. Using this notation, by the previous proposition, to say that R is subﬁrmly U-reﬂective is equivalent to U ⊂ L(R). 6.1.3. Definition. R is said to be ﬁrmly U-reﬂective if U = L(R). There is a simple criterion to see whether subﬁrm U-reﬂectivity implies ﬁrm U-reﬂectivity. It is based on the following deﬁnition.

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6.1.4. Definition. A morphism class U is said to be coessential if for every morphism u ∈ U and for every morphism f we have u ◦ f ∈ U ⇒ f ∈ U. The proof of the next proposition can again be found in [26]. 6.1.5. Proposition. If R is subﬁrmly U-reﬂective and U is coessential then R is ﬁrmly U-reﬂective. It is our purpose to show in the next section that by restricting Near to a suitable subconstruct, a ﬁrmly U-reﬂective completion exists, with respect to the class U of all cl-dense embeddings. 6.2. Complete nearness spaces. Classically completeness of nearness spaces is deﬁned by means of clusters. In order to be able to later compare with completeness for Cauchy spaces we will add equivalent descriptions in terms of micromeric collections. 6.2.1. Definition. A cluster in a merotopic space is a maximal non-empty near collection, where the set of all near collections is ordered by inclusion. There is a simple relation between clusters and certain micromeric collections. If (X, µ) is a merotopic space, then A is a cluster if and only if sec A is a minimal micromeric stack. We remark that every cluster and every minimal micromeric stack is a concentrated collection. 6.2.2. Proposition. For a merotopic space (X, µ) we have: (1) Every cluster is a near grill and every minimal micromeric stack is a Cauchy ﬁlter. (2) Every near grill and every Cauchy ﬁlter is a concentrated collection. Proof. In order to prove (1) let A be a cluster of (X, µ). Then clearly A = ∅, ∅∈ / A and from A ∈ A and A ⊂ B ⊂ X it follows that B ∈ A. It remains to show that from B1 ∈ / A and B2 ∈ / A it follows that B1 ∪ B2 ∈ / A. Let i ∈ {1, 2}. Since Bi ∈ / A, we have that A∪{Bi } is not near and hence Ui = {X \A | A ∈ A}∪{X \Bi } is a uniform cover of (X, µ). But then {X \ A|A ∈ A} ∪ {X \ (B1 ∪ B2 )} also is a uniform cover, because U1 ∧ U2 < {X \ A | A ∈ A} ∪ {X \ (B1 ∪ B2 )}. Hence A ∪ {B1 ∪ B2 } is not near in (X, µ) and thus B1 ∪ B2 ∈ / A. In order to prove (2) let A be a near grill of (X, µ). Then sec A is a micromeric ﬁlter contained in A. This implies sec A ⊂ sec(sec A) = A. We can conclude that A is micromeric and hence concentrated. Since the sec operator interchanges the roles between Cauchy ﬁlters and near grills it follows that Cauchy ﬁlters are concentrated too. We obtain the following characterization of separatedness. 6.2.3. Proposition. The following are equivalent for a merotopic space (X, µ): (1) (X, µ) is separated. (2) For every concentrated stack A there exists a unique cluster containing A. (3) For every concentrated stack A there exists a unique minimal Cauchy ﬁlter contained in A.

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Proof. (1) ⇒ (2). Suppose (X, µ) is separated and let A be a concentrated stack and let C = {C ⊂ X | {C} ∪ A near in (X, µ)}. Since every near collection containing A is contained in C and since C itself is near, it is the unique cluster containing A. (2) ⇔ (3). Let A be a concentrated stack and consider sec A. Since this is a concentrated stack too, we can apply (2). The unique cluster C containing sec A is a near grill. It follows that sec C is the unique minimal Cauchy ﬁlter contained in A. The converse follows in an analogous way. (2) ⇒ (1). Let A be concentrated and let G be the unique cluster containing A. Let B ⊂ X be such that A ∪ {B} is near. Then also A ∪ {B} is contained in a unique cluster which has to coincide with G. Hence B ∈ G. So we conclude that {C ⊂ X | {C} ∪ A near in (X, µ)} ⊂ G and so the collection {C ⊂ X | {C} ∪ A near in (X, µ)} is near. 6.2.4. Proposition. In a separated merotopic space the clusters are exactly the maximal near grills and the minimal micromeric stacks are exactly the minimal Cauchy ﬁlters. Proof. Let (X, µ) be a separated merotopic space. Since every cluster is a near grill and every near grill is contained in a cluster, it follows that the clusters are exactly the maximal near grills. The rest follows from the properties of the sec operator. 6.2.5. Definition. A nearness space is called complete if it satisﬁes one of the following equivalent conditions. (1) Every cluster has an adherence point in the topological coreﬂection. (2) Every minimal micromeric stack converges in the topological coreﬂection. The results in Propositions 6.2.3 and 6.2.4 now enable us to characterize completeness for separated nearness spaces in several diﬀerent ways. 6.2.6. Proposition. For any separated nearness space (X, µ) the following conditions are equivalent. (1) (X, µ) is complete. (2) Every maximal near grill has an adherence point. (3) Every near grill has an adherence point. (4) Every concentrated collection has an adherence point. (5) Every concentrated collection converges. (6) Every Cauchy ﬁlter converges. (7) Every minimal Cauchy ﬁlter converges. Proof. The equivalence between the conditions (1), (2) and (7) follows from Proposition 6.2.4. The only non-trivial remaining implication is from (4) to (5). This implication follows from the duality between near collections and micromeric collections determined by the sec operator and the result which states that a collection A of subsets of X converges to x ∈ X if and only if sec A adheres to x. From this result it now follows that the completeness notion deﬁned here extends the classical notion of completeness for uniform spaces, since a uniform nearness space is complete (as deﬁned in 6.2.5) if and only if it is complete as a uniform

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space (in the usual sense). Moreover a topological nearness space is clearly always complete. 6.3. A ﬁrm completion theory for RegNear0 . In order to develop a completion theory for RegNear0 we ﬁrst ﬁx a class of morphisms. Let U be the class of all cl-dense embeddings. This class is closed under composition. Further let cRegNear0 be the subconstruct of RegNear0 consisting of all complete spaces. Next we show that cRegNear0 is U-reﬂective in RegNear0 . Our proof will consist in constructing the actual Ureﬂection of a T0 -nearness space. 6.3.1. Definition. Given a separated T0 -nearness space (X, µ), the canonical completion iX : (X, µ) → (X ∗ , µ∗ ) is constructed as follows: (1) X ∗ is the set of all minimal Cauchy ﬁlters of (X, µ). (2) iX : X → X ∗ : x → N (x) is the inclusion map, which sends a point x of X to the neighbourhood ﬁlter of x with respect to the topological coreﬂection of (X, µ). (3) Given a cover U of X, we deﬁne o(U) = {o(U ) | U ∈ U} where o(U ) = {F ∈ X ∗ | U ∈ F} . We deﬁne U ∗ to be a uniform cover of (X ∗ , µ∗ ) iﬀ there exists a uniform cover U of (X, µ) such that o(U) < U ∗ . 6.3.2. Proposition. Given a separated T0 -nearness space (X, µ), (1) (X ∗ , µ∗ ) is a T0 -nearness space. (2) iX : (X, µ) → (X ∗ , µ∗ ) is an embedding. (3) clµ∗ (iX (X)) = X ∗ . (4) (X ∗ , µ∗ ) is complete. (5) If (X, µ) is regular so is (X ∗ , µ∗ ). Proof. (1). To show that for every U ∈ µ, the collection o(U) is a cover of X ∗ , let F be an element of X ∗ . Since F is a micromeric stack, there exists U ∈ U such that U ∈ F and hence F ∈ o(U ). It follows that o(U) is a cover of X ∗ . Clearly {X ∗ } ∈ µ, since o({X}) = {X ∗ }. To conclude that (X ∗ , µ∗ ) is a merotopic space, it remains to show that when U and V are uniform covers of (X, µ), then o(U) ∧ o(V) ∈ µ∗ . This follows from the observation that o(U) ∧ o(V) = o(U ∧ V) since o(A) ∩ o(B) = o(A ∩ B), for every A, B ⊂ X. In order to see that (X ∗ , µ∗ ) satisﬁes the nearness axiom it suﬃces to observe that for any set A we have intµ∗ o(A) = o(A). Indeed let F ∈ o(A). Then we have A ∈ F and since F is a minimal micromeric stack, the collection F \ {B | B ⊂ A} is not micromeric. So there exists a uniform cover U ∈ µ having no element in common with F \ {B | B ⊂ A}. For this cover we now have o(U) < {X ∗ \ {F}, o(A)}. This implies F ∈ intµ∗ o(A). It follows that for any V ∈ µ we have {intµ∗ o(V ) | V ∈ V} = o(V) which implies the nearness axiom. To show that (X ∗ , µ∗ ) is a T0 -space, let F = G in X ∗ . Let U = {U ⊂ X | U ∈ F or U ∈ G}. Then clearly by minimality of F and G the collection U is a uniform cover of (X, µ). It follows that {X ∗ \ {F}, X ∗ \ {G}} is a uniform cover of (X ∗ , µ∗ ). Hence {{F}, {G}} is not near.

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(2). The map iX : X → X ∗ is injective because (X, µ) satisﬁes the T0 property. That iX : (X, µ) → (X ∗ , µ∗ ) is an initial map follows easily since for any U ∈ µ we have that intµ U = i−1 X (o(U)). Indeed let A ⊂ X, then x ∈ intµ A iﬀ N (x) ∈ o(A) and thus intµ A = i−1 X (o(A)). (3). Let F ∈ X ∗ . We need to prove that {F, iX (X)} is near. It is suﬃcient to show that for any uniform cover U of (X, µ), there exists U ∈ U such that F ∈ o(U ) and iX (X) ∩ o(U ) = ∅. Let U ∈ µ. The family F is a micromeric stack in (X, µ) and hence intµ (U) ∩ F = ∅. So there exists U ∈ U such that F ∈ o(intµ U ) ⊂ o(U ) and iX (X) ∩ o(U ) = iX (intµ U ) = ∅. (4). Let A∗ be a cluster on (X ∗ , µ∗ ). We show that A = {A ⊂ X | iX (A) ∈ A∗ } is a cluster on (X, µ). To show that A is near, let U ∈ µ. Because A∗ is near there exists U ∈ U such that for any A ∈ A : o(U ) ∩ iX (A) = ∅, and hence also U ∩ A = ∅. Therefore A is near. Next we show that A is micromeric. First observe that iX (X) ∈ A∗ . Indeed using the density of iX (X) it follows that A∗ ∪ {{iX (X)}} is near, and applying maximality of A∗ we get the result. Clearly A is a non empty stack and also a grill. It follows that sec A ⊂ A. Hence sec A being micromeric, A is also micromeric. To show that A is a maximal near collection, let B ⊂ X such that A ∪ {B} is near. Then iX (A) ∪ {iX (B)} is also near. Since iX (A) and iX (A) ∪ {iX (B)} are concentrated collections and A∗ is the unique cluster which contains iX (A), we obtain that iX (B) ∈ A∗ and thus B ∈ A. So A is a cluster on (X, µ). In case the cluster A has an adherence point x ∈ X it follows that A ∪ {{x}} is near and therefore we have {x} ∈ A. But then {iX (x)} ∈ A∗ and this implies that A∗ adheres to iX (x). If A does not have an adherence point, then sec A ∈ X ∗ since it is a minimal Cauchy ﬁlter on X. The collection A∗ ∪ {{sec A}} still is a near collection. This follows from the fact that sec A ∈ o(U ) whenever o(U ) ∈ sec A∗ . This now implies that {sec A} ∈ A∗ and hence sec A is an adherence point of A∗ . (5). Next we show that o(V) is a regular reﬁnement of o(U) in (X ∗ , µ∗ ) whenever V is a regular reﬁnement of U in (X, µ). Let V be a regular reﬁnement of a uniform cover U ∈ µ. Then for any V ∈ V there exists a U ∈ U such that {X \ V, U } ∈ µ. So {o(X \ V ), o(U )} ∈ µ∗ and then also {X ∗ \ o(V ), o(U )} is a uniform cover of (X ∗ , µ∗ ). Therefore o(U ) is a uniform neighbourhood of o(V ) and thus o(V) is a regular reﬁnement of o(U). Finally we can conclude that (X ∗ , µ∗ ) is a regular nearness space whenever (X, µ) is. 6.3.3. Proposition. Given a T0 separated nearness space (X, µ), the canonical completion (X ∗ , µ∗ ) is uniform if and only if (X, µ) is uniform. Proof. It is suﬃcient to show that (X ∗ , µ∗ ) is a uniform space whenever (X, µ) is a uniform T0 -space. Let U ∗ ∈ µ and U ∈ µ such that o(U) < U ∗ . Since (X, µ) is uniform there exists a star reﬁnement V ∈ µ of U. This implies that for any V ∈ V, there exists U ∈ U such that St(V, V) ⊂ U . But then we also have St(o(V ), o(V)) ⊂ o(U ). Indeed let V ∈ V such that o(V ) ∩ o(V ) = ∅. Since o(V ) ∩ o(V ) = o(V ∩ V ), it follows that V ∩ V = ∅. So we get that V ⊂ U and hence o(V ) ⊂ o(U ). We can conclude that o(V) is a star reﬁnement of o(U) and hence also of U ∗ . The following properties are the basic ingredients needed to prove that the extension constructed so far is a reﬂection.

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6.3.4. Proposition. Let f : (X, µ) → (Y, ν) be a function between nearness spaces and let (A, µA ) be a cl-dense subspace of (X, µ). Suppose (Y, ν) is regular, the restriction f |A : (A, µA ) → (Y, ν) of f to A is uniformly continuous and the restriction f |A∪{x} : (A ∪ {x}, µA∪{x} ) → (Y, ν) of f to A ∪ {x} is continuous for every x ∈ X. Then f : (X, µ) → (Y, ν) is uniformly continuous. Proof. Let U ∈ ν. Then we choose a regular reﬁnement V of U. This implies that f |−1 A V ∈ µA . There exists an open uniform cover W for (X, µ) such that −1 {W ∩ A | W ∈ W} reﬁnes f |−1 U. A V. We now show that W is a reﬁnement of f For W ∈ W we choose V ∈ V such that W ∩ A ⊂ f −1 V and U ∈ U such that U is a uniform neighbourhood of V . For this choice we now show that W ⊂ f −1 U . If x ∈ W then using openness of W and density of A it follows that x ∈ clX (W ∩ A) and then also by continuity f (x) ∈ clY f (W ∩ A) ⊂ clY V . Since U is a uniform neighbourhood of V in Y it follows that clY V ⊂ U . We conclude that x ∈ f −1 U . Hence f −1 U is a uniform cover and our conclusion follows. 6.3.5. Proposition. If f : (X, µX ) → (Y, µY ) is a cl-dense embedding in Near, then any uniformly continuous map g : (X, µX ) → (Z, µZ ) to a complete and regular T0 -nearness space can be uniquely extended to a uniformly continuous map g : (Y, µY ) → (Z, µZ ), i.e., satisfying g ◦ f = g. Proof. Let g : (X, µX ) → (Z, µZ ) be a uniformly continuous map. For any y ∈ Y the collection B(y) = {B ⊂ X | y ∈ clµY B} is concentrated in (X, µX ). Uniform continuity of g implies that g(B(y)) is concentrated in (Z, µZ ). By completeness and the T0 property of (Z, µZ ), the collection g(B(y)) has a unique adherence point g(y) in (Z, µZ ). Moreover when x ∈ X, then g(f (x)) is an adherence point of g(B(f (x)) and hence one has g(f (x)) = g(x) for each x in X. So g ◦ f = g. Due to the construction the topological coreﬂection of the restriction map g|f (X)∪{y} : (f (X) ∪ {y}, µY |f (X)∪{y} ) → (Z, µZ ) is continuous for every y ∈ Y . Hence from Proposition 6.3.4 it now follows that g : (Y, µY ) → (Z, µZ ) is a uniformly continuous map. The uniqueness of g follows from the fact that the dense map f is an epimorphism. 6.3.6. Proposition. The construct cRegNear0 of all complete and regular T0 nearness spaces is a U-reﬂective subconstruct of RegNear0 . Proof. Since the canonical completion iX : (X, µ) → (X ∗ , µ∗ ) of a regular T0 -nearness space (X, µ) is a cl-dense embedding, it follows from Proposition 6.3.5 that iX : (X, µ) → (X ∗ , µ∗ ) is a reﬂection morphism which moreover belongs to U. 6.3.7. Proposition. Given a regular T0 -nearness space (X, µ), then two complete and regular T0 -spaces in which (X, µ) is cl-densely embedded are necessarily isomorphic. Proof. Let (X1 , µ1 ) and (X2 , µ2 ) be cRegNear0 -objects and suppose that j1 : (X, µ) → (X1 , µ1 ) and j2 : (X, µ) → (X2 , µ2 ) are two cl-dense embeddings. In view of Proposition 6.3.5, the uniformly continuous maps j1 and j2 are cRegNear0 reﬂection morphisms for (X, µ). By uniqueness of reﬂection it follows that (X1 , µ1 ) and (X2 , µ2 ) are isomorphic. 6.3.8. Proposition. The construct cRegNear0 is ﬁrmly U-reﬂective in the construct RegNear0 .

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Proof. In view of the previous results RegNear0 is subﬁrmly U-reﬂective in RegNear0 . By Proposition 6.1.4 to conclude that RegNear0 is ﬁrmly U-reﬂective in RegNear0 , it suﬃces to check that the class U is coessential. Let g and f be morphisms such that g and g ◦ f belong to U. By hereditariness of the closure operator cl, it follows that f belongs to U. We remark that regularity is an essential assumption in order to obtain a convenient completion theory in the setting of Near. When restricting to the category of separated T0 -nearness spaces the complete objects still deﬁne an epireﬂective subcategory of SepNear0 . An extensive study of the complete reﬂection, which is called the simple completion of a separated T0 -nearness space, can be found in [10]. 6.3.9. Definition. Given a separated T0 -nearness space (X, µ), the simple µ completion eX : (X, µ) → (X, ) is constructed as follows: (1) X is the set of all the clusters in X, is deﬁned by eX (x) = {A ⊂ X | x ∈ clµ A}, (2) eX : X → X is uniform iﬀ it satisﬁes the following two conditions: (3) a cover U of X −1 • eX U is a uniform cover of (X, µ). there exists U ∈ U such that A ∈ U and e−1 (U ) ∈ • for each A ∈ X X sec A. µ ) of a separated T0 -nearness space The simple completion eX : (X, µ) → (X, (X, µ) is a cl-dense embedding in SepNear0 and it can be shown to be a reﬂection. A disadvantage of this completion theory is that it behaves rather badly in so far as it destroys the properties uniform and regular, whereas the canonical completion preserves them. Moreover the simple completion is not unique. 6.3.10. Example. Let [0, 1] be the closed unit interval with its usual nearness structure (i.e., its usual topology), and let (X, µ) be the nearness subspace of [0, 1] determined by the set [0, 1] \ { n1 | n ∈ N0 }. Then (X, µ) is a uniform and hence regular T0 -nearness space. The canonical completion of (X, µ) is isomorphic to the unit interval. However in the simple completion of (X, µ) the sequence ( n1 )n∈N0 fails to converge to 0. So the canonical completion of (X, µ) is not isomorphic to the simple completion of (X, µ), from which it follows that the simple completion can be neither regular nor uniform. This example illustrates that the simple completion of regular T0 -nearness spaces does not have to be isomorphic with its canonical completion. Therefore completeness in the category of separated T0 -nearness spaces is not unique in the sense that the construct of complete separated T0 -nearness spaces does not determine a subﬁrmly U-reﬂective subcategory of the category SepNear0 with respect to the class U of all cl-dense embeddings. 7. Separation and regularity for Cauchy spaces 7.1. Closure operator on Lim and Chy. We start this subsection by ﬁxing some notation. For the material on closure operators we refer to 3.2.8 or to the book by Dikranjan and Tholen [50]. 7.1.1. Definition. For a subobject m : M → (X, q) in Lim, whenever x ∈ X deﬁne x ∈ cX M ⇔ ∃F ∈ F(M ) : stackX (F) →q x

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and let cX m : cX M → X be the corresponding subobject. 7.1.2. Proposition. (cX )X is a hereditary closure operator on Lim. Given a limit space (X, q) the closure in fact determines the pretopological reﬂection of (X, q). Proof. The operator cX is extensive and monotone. Moreover if f : (X, q) → (Y, p) is a morphism in Lim then applying the reﬂector to Pretop we get that f : (X, cX ) → (Y, cY ) is continuous. To see that the closure operator is hereditary let (X, q) be a subobject of (Y, p) and M a subset of X. For a ﬁlter F on M and x ∈ X we clearly have stackX (F) → x ⇔ stackY (F) → x. Hence cX M = X ∩ cY M . Note that (cX )X need not be idempotent. Next we deﬁne a closure operator on C and by further restriction on Chy. 7.1.3. Definition. For a subobject m : M → (X, γ) in C, whenever x ∈ X deﬁne x ∈ cX M ⇔ ∃F ∈ F(M ) : stackX (F) ∩ x˙ ∈ γ and let cX m : cX M → X be the corresponding subobject. The closure on a ﬁlter space in C or on a Cauchy space (X, γ) is obtained by ﬁrst going to the limit coreﬂection and then applying the closure derived from the limit structure. 7.1.4. Proposition. (cX )X is a hereditary closure operator on C and so is its restriction on Chy. Proof. Given a space (X, γ) in C the closure is obtained by ﬁrst going to the limit coreﬂection and then proceeding exactly as in Proposition 7.1.2. We again obtain a closure operator. The application of the coreﬂection preserves subobjects, so from Proposition 7.1.2 we can conclude that the operator is hereditary also on C. 7.1.5. Definition. If X = (X, γ) is a C-object and A ⊂ X then A is said to be dense in X if cX A = X. A is said to be strictly dense in X if for every Cauchy ﬁlter F on X there exists a Cauchy ﬁlter G on A for the subobject structure on A such that cX stack G ⊂ F. A morphism f : X → Y is said to be dense (strictly dense) if the image f X is dense (strictly dense) in Y . We remark that a strictly dense subset (morphism) is dense. This can be easily seen starting from x ∈ X and considering F = x. ˙ If G is a ﬁlter on A such that cX stack G ⊂ x˙ then in particular we have x ∈ cX A. 7.2. Separation for Cauchy spaces. 7.2.1. Proposition. The following are equivalent for an object (X, γ) of Chy. (1) (X, γ) is a T0 -object of Chy. (2) If x˙ ∩ y˙ is a Cauchy ﬁlter in (X, γ) then x = y. (3) The limit coreﬂection of (X, γ) is a T2 -limit space in the sense that ∃F ∈ F(X), F → x and F → y ⇒ x = y.

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Proof. (1) ⇒ (2). Note that for the Cauchy structure on I2 we have that {{0, 1}} is a Cauchy ﬁlter. Suppose x˙ ∩ y˙ is a Cauchy ﬁlter in (X, γ). Then (X, γ) has an indiscrete subspace {x, y}. So we can conclude that x = y. (2) ⇒ (3). Follows from the fact that the limit coreﬂection is reciprocal. (3) ⇒ (1). If the limit coreﬂection is T2 then in (X, γ) principal ﬁlters of points have unique limits, so clearly there cannot be a non-trivial indiscrete subspace. In case the limit coreﬂection of a Cauchy space X is T2 , then we say that X is a T2 -Cauchy space and in view of the previous proposition we denote the full subcategory of Chy whose objects are the T2 -Cauchy spaces by Chy0 . The next result follows from the general theory in [94]. 7.2.2. Proposition. Chy0 is quotient reﬂective in Chy. 7.2.3. Proposition. Dense maps are epimorphic in Chy0 . Proof. Let u : (X, γ) → (Y, γ ) be a dense map and suppose f : (Y, γ ) → (Z, δ) and g : (Y, γ ) → (Z, δ) are uniformly continuous with (Z, δ) a T2 -Cauchy space. Apply the coreﬂector to Limr as described in section 4. Then u : (X, qγ ) → (Y, qγ ) is a dense map and f : (Y, qγ ) → (Z, qδ ) and g : (Y, qγ ) → (Z, qδ ) are (uniformly) continuous with (Z, qδ ) a T2 -limit space. For y ∈ Y choose a ﬁlter F on X such that stack uF → y. Then stack(f ◦ u)F = stack(g ◦ u)F. So their unique limits f (x) and g(y) in (Z, qδ ) coincide.

7.3. Regularity for Cauchy spaces. Given F ∈ F(X) we use the following notation for its closure. cX F = stack{cX F | F ∈ B}. Remark that F = stack(B) for some base B implies cX F = stack{cX B | B ∈ B}. 7.3.1. Definition. A Cauchy space (X, γ) is Cauchy-regular if cX F is a Cauchy ﬁlter whenever F is a Cauchy ﬁlter. A Cauchy space (X, γ) is said to be a T3 -Cauchy space if it is Cauchy-regular and T2 . 7.3.2. Remark. We remark that the terminology we use here is diﬀerent from what is commonly used in the literature on Cauchy spaces. “Cauchy-regularity” is simply called “regularity” in papers dealing with Cauchy spaces. However we want to stress that this notion diﬀers from the regularity that is commonly used for merotopic spaces as introduced in section 5. So later in section 9 when we will be comparing the diﬀerent notions that are applicable, we will carefully use the appropriate terminology. However in this subsection and also in section 8 where no confusion can occur we will drop the preﬁx “Cauchy” in “Cauchy-regular”. The subconstruct consisting of the regular Cauchy spaces is denoted by RegChy and the construct of T3 -Cauchy spaces is denoted by T3 Chy. An important example of T3 -Cauchy spaces are T3 -limit spaces. For later use we recall the following deﬁnition. 7.3.3. Definition. A limit space (X, q) is said to be regular if F → x ⇒ cX F → x. A limit space is said to be T3 if it is regular and T2 .

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7.3.4. Proposition. The subconstruct RegChy is concretely reﬂective in Chy. Proof. Let (fi : (X, γ) → (Xi , γi ))i∈I be an initial source in Chy and assume that all spaces (Xi , γi ) are regular. Let F be Cauchy in (X, γ). Since for every F ∈ F we clearly have fi (cX F ) ⊂ cXi fi (F ) we can conclude that cXi fi (F) ⊂ stack(fi (cX F)). Hence also (X, γ) is regular. For a given Cauchy space X let kX : X → K(X) be the RegChy-reﬂection. Then K(X) is the ﬁnest regular Cauchy space on the same underlying set and coarser than the given X. We now give an explicit construction of the reﬂection based on the following notations. Let X be a Cauchy space and M ⊂ X. For a successor ordinal α the α-th iteration of the closure of M (in the associated limit space) is deﬁned recursively β as cX (cα−1 X (M )) and for a limit ordinal α as β<α cX (M ). For any ﬁlter F on X and any natural number n we deﬁne cnX F to be the ﬁlter generated by {cnX F | F ∈ F}. Let X be a Cauchy space. A transﬁnite sequence of spaces on X is deﬁned recursively as follows: Let k0 X = X and further let k1 X be the Chy-reﬂection of the ﬁlter space deﬁned by: a ﬁlter F is Cauchy iﬀ there exist n ∈ N and a Cauchy ﬁlter G in X such that cnX G ⊂ F. Let k2 X be the Chy-reﬂection of the ﬁlter space deﬁned by: a ﬁlter F is Cauchy iﬀ there exist n ∈ N and a Cauchy ﬁlter G in X such that cnk1 X G ⊂ F . . . . Let kα X be the Chy-reﬂection of the ﬁlter space deﬁned by: a ﬁlter F is Cauchy iﬀ there exist n ∈ N and a Cauchy ﬁlter G in X and β < α such that cnkβ X G ⊂ F. The smallest ordinal γ such that kγ X = kγ+1 X is the length of the regularity sequence and it is denoted by lK X. 7.3.5. Proposition. If lK X ≤ α then we have kα X = K(X). Proof. Let lK X ≤ α. We prove that kα X is regular. It clearly is suﬃcient to check the regularity condition on ﬁlters F that are Cauchy ﬁlter for the ﬁlter space constructed at level α. For such a ﬁlter there exists n ∈ N and a Cauchy ﬁlter G in X and β < α such that cnkβ X G ⊂ F. Now we have n cn+1 kα X G ⊂ ckα X ckβ X G ⊂ ckα X F

and so ckα X F is a Cauchy ﬁlter. It follows that kα X is regular. Since K(X) is the ﬁnest regular space coarser than X, it follows that kα X = K(X). 8. Completion theory for Cauchy spaces Completeness is a natural concept for a Cauchy space; it means that all the Cauchy ﬁlters converge to some point of the underlying limit space. As before, we ﬁx a class of morphisms. This time we let U be the class of all strictly dense embeddings. We prove that the full subconstruct cChy0 consisting of the complete T0 -objects, is U-reﬂective in Chy0 with respect to the class U and the actual reﬂector is known as the Wyler completion. However completeness in this setting is not unique. Nevertheless when restricting to a suitable subconstruct P of Chy uniqueness of completion can be obtained, and the actual completion is the regular reﬂection of the Wyler completion.

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8.1. Completeness on Chy0 . 8.1.1. Definition. A Cauchy space (X, γ) is said to be complete if every Cauchy ﬁlter F converges in the limit coreﬂection (X, qγ ) to some point x ∈ X. Let cChy0 be the full subconstruct of Chy0 consisting of all complete T2 -Cauchy spaces. The complete Cauchy spaces coincide with the reciprocal limit spaces. A completion theory will now be developed on Chy0 . First we prove that cChy0 is reﬂective in Chy0 . In order to do so we introduce the following construction which is known as the Wyler completion. 8.1.2. Definition. For any T2 -Cauchy space (X, γ) we consider the set of all equivalence classes for the following relation on Cauchy ﬁlters: F ∼ G ⇔ F ∩ G is a Cauchy ﬁlter. For a Cauchy ﬁlter F we denote [F] its equivalence class. Let ˜ = {[F] | F ∈ γ} X ˜ be the map deﬁned by jX (x) = [x]. and let jX : X → X ˙ Then clearly jX is injective. ˜ as follows. A ﬁlter h on X ˜ is said to A ﬁlter structure γ˜ is now deﬁned on X ˜ be Cauchy if there exist an α in X and a Cauchy ﬁlter F ∈ α such that stack(jX (F)) ∩ α˙ ⊂ h. The construction described here is known as the Wyler completion. Some authors rather use “simple completion”. ˜ γ˜ ) is a complete T2 -Cauchy space and jX : (X, γ) → 8.1.3. Proposition. (X, ˜ γ˜ ) is a strictly dense embedding. The only members of γ˜ containing X ˜ \ jX (X) (X, are principal ultraﬁlters. ˜ qγ˜ ) is T2 . Suppose h is Proof. First we check that the limit coreﬂection (X, a ﬁlter such that there exist α and β and ﬁlters F ∈ α and G ∈ β on X such that stack(jX (F)) ∩ α˙ ⊂ h and stack(jX (G)) ∩ β˙ ⊂ h. Then F # G. So F and G are equivalent and ﬁnally α = β. To see that jX is an embedding it suﬃces to observe that for a ﬁlter F on X we have (stack(jX (F)) →qγ˜ α) ⇒ F ∈ γ. ˜ choose F ∈ α Finally we check that jX is strict. For a Cauchy ﬁlter h on X such that stack(jX (F)) ∩ α˙ ⊂ h. Then cX˜ (jX (F)) ⊂ h. The proof of the last assertion follows immediately from the deﬁnition of γ˜ . 8.1.4. Proposition. If f : (X, γ) → (Z, δ) is uniformly continuous and (Z, δ) is a complete T2 -Cauchy space, then there exists a unique uniformly continuous ˜ γ˜ ) → (Z, δ) such that f˜ ◦ jX = f . map f˜: (X, ˜ \ jX (X) Proof. We deﬁne f˜ as follows. If x ∈ X let f˜([x]) ˙ = f (x). For α ∈ X ˜ we put f (α) = z by choosing a ﬁlter F ∈ α such that z is the unique limit of stack(f (F)) in (Z, δ). That for any choice of F the image has a (unique) limit follows from the uniform continuity of f and the fact that F is a Cauchy ﬁlter. The function f˜ is well deﬁned since it does not depend on the choice of F. Indeed, two ﬁlters F and F with the same properties are equivalent and we can make use of their intersection to prove that their images have the same limit. It is straightforward to see that f˜([x]) ˙ = f (x).

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Finally we have to prove that f˜ is a morphism. Since domain and codomain are (reciprocal) limit spaces it is suﬃcient to check continuity. Suppose h → α and choose stack(jX (F)) ∩ α˙ ⊂ h with F ∈ α. Clearly stack f (F) ⊂ stack f˜(jX (F)), hence stack f˜(jX (F)) → f˜(α). It follows that stack(f˜(h)) → f˜(α). Uniqueness of f˜ with the properties formulated in the theorem follows from the fact that by ˜ γ˜ ) is an epimorphism in Chy0 . Proposition 7.2.3 jX : (X, γ) → (X, We refer to section 6.1 for the terminology on U-completions. Consider the class U consisting of all strictly dense embeddings in Chy0 . From Propositions 8.1.3 and 8.1.4 we conclude: 8.1.5. Theorem. cChy0 is U-reﬂective in Chy0 , and the Wyler completion is a U-completion. 8.1.6. Remark. Uniqueness of completion with respect to U however is not fulﬁlled. For instance take for X the pair Q endowed with the Cauchy structure whose Cauchy ﬁlters are those in the usual uniformity induced by the real line. Let Y be the set R endowed with the usual topology considered as a complete T2 Cauchy space. The natural embedding X → Y belongs to U. However Y cannot be isomorphic to the Wyler completion of X since there are convergent ultraﬁlters containing R \ Q that are non principal. 8.2. Uniqueness of completion on a suitable subconstruct. In order to obtain uniqueness of completion, we will have to restrict the setting. Given a ˜ = (X, ˜ γ˜ ) be the Wyler completion T2 -Cauchy space X = (X, γ), as before let X ˜ ˜ = K(X, ˜ γ˜ ) be its regular with reﬂection morphism jX : X → X and let K(X) ˜ → K(X). ˜ reﬂection as constructed in section 7.3 with reﬂection morphisms kX˜ : X ˜ and K(X) ˜ lives yet another interesting ﬁlter space that will be useful In between X in the completion theory. It is based on the so called “Σ-operator”. 8.2.1. Definition. For a T2 -Cauchy space X = (X, γ) and A ⊂ X let ˜ | A ∈ G for some G ∈ [F]}. Σ(A) = {[F] ∈ X ˜ generated by {ΣF | F ∈ F}. If F is a ﬁlter let ΣF be the ﬁlter on X Further let ˜ | ∃G ∈ [F] : (ΣG) # (ΣA)}. Σ2 (A) = {[F] ∈ X ˜ generated by {Σ2 F | F ∈ F}. If F is a ﬁlter let Σ2 F be the ﬁlter on X The following result is straightforward. 8.2.2. Proposition. With the notations described above, with the closure cX˜ ˜ γ˜ ) and for A ⊂ X we have ΣA = c ˜ (jX (A)). taken in (X, X ˜ as follows. A ﬁlter h 8.2.3. Definition. A ﬁlter structure γ˜1 is deﬁned on X ˜ is Cauchy if there is a Cauchy ﬁlter G on X such that ΣG ⊂ h. on X ˜ γ˜1 ) constructed in 8.2.1 need not We remark that in general the ﬁlter space (X, be a Cauchy space. The next conditions are introduced to remedy this. 8.2.4. Definition. Let P be the full subconstruct of Chy consisting of all regular T2 -Cauchy spaces (X, γ) satisfying the following conditions: (1) If F and G are Cauchy ﬁlters and (ΣF) # (ΣG) then [F] = [G],

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(2) If F is a Cauchy ﬁlter there exists a Cauchy ﬁlter G such that ΣG ⊂ Σ2 F. ˜ γ˜1 ) 8.2.5. Proposition. Suppose (X, γ) belongs to P. Then the ﬁlter space (X, is a limit space, since it satisﬁes [F] = α ⇒ ΣF ⊂ α. ˙ It is regular and T2 and hence determines a complete and regular T2 -Cauchy space ˜1. ˜ as underlying set. We denote the Cauchy space (X, ˜ γ˜1 ) also simply by X with X Proof. The ﬁrst assertion follows from Proposition 8.2.2 and the fact that ˜ The ﬁrst condition in 8.2.4 clearly for [F] = α the ﬁlter stackX˜ jX (F) → α in X. ˜ γ˜1 ) implies that this limit structure is T2 . So it deﬁnes a complete Cauchy space (X, which in turn is T2 . By application of Proposition 8.2.2 we get that the Cauchy ˜ γ˜1 ) moreover is regular. space (X, ˜ γ˜1 ) coincides with 8.2.6. Proposition. Suppose (X, γ) belongs to P then (X, ˜ the concrete regular reﬂection K(X) constructed in Proposition 7.3.4. ˜ γ˜1 ) is regular, so we have Proof. By the previous result we have that (X, ˜ ≥ K(X) ˜ since the latter is the ﬁnest regular Cauchy space coarser than X. ˜ On X ˜ ˜ the other hand, since K(X) is regular and coarser than X the ﬁlters ΣF have to be included as Cauchy ﬁlters whenever F is Cauchy on X. The other inequality also holds. Let cP be the subconstruct of P consisting of all its complete objects. We remark that this class coincides with the construct of all T3 -limit spaces. This ˜ = X for complete X and ΣF = cX F for all Cauchy ﬁlters F. follows in view of X The T3 -property then clearly implies the deﬁning conditions (1) and (2) in 8.2.4. 8.2.7. Proposition. cP is reﬂective in P. Proof. Let f : (X, γ) → (Z, δ) be uniformly continuous with (X, γ) ∈ P and ˜ where rX = k ˜ ◦ jX . By Propositions 8.2.5 (Z, δ) ∈ cP. Let rX : X → K(X), X ˜ and 8.2.6 the space K(X) is a complete T2 -regular Cauchy space. Hence it belongs to cP. By Propostion 8.1.4 there exists a unique uniformly continuous function ˜ → Z such that f˜ ◦ jX = f . By application of Proposition 7.3.4 there exists a f˜: X ˜ → Z such that fˆ ◦ kX = f˜. Finally unique uniformly continuous function fˆ: K(X) ˆ we have f ◦ rX = f and since on the underlying sets kX is the identity, uniqueness of f˜ follows. As before let U be the class of all strictly dense embeddings between P-objects. ˜ deﬁned by rX = 8.2.8. Theorem. cP is U-reﬂective in P and rX : X → K(X) kX˜ ◦ jX is the U-reﬂection of a P-object X. Proof. In view of the previous proposition it suﬃces to prove that the reﬂection morphisms rX belong to U. We prove that for X belonging to P the morphism rX is an embedding. First ˜ 1 . For a Cauchy ﬁlter F on X clearly ΣF ⊂ stack mX (F), consider mX : X → X ˜ 1 . Conversely suppose stack mX (F) is in X ˜ 1 , then for so stack mX (F) belongs to X some Cauchy ﬁlter G on X we have ΣG ⊂ stack mX (F). Restricting to X we get that stack cX G ⊂ F and by regularity this implies that F is Cauchy on X. Since ˜ 1 = K(X) ˜ we are done. X

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In order to see that rX is strictly dense we proceed in the same way by ﬁrst considering mX . For a Cauchy ﬁlter F on X we clearly have cX˜ 1 (mX (F)) ⊂ cX˜ (mX (F)) = ΣF. ˜ 1 and K(X) ˜ strictness of So mX is strictly dense, and again using the equality of X rX follows. We remark that the class U contains all isomorphisms and that composition of a morphism in U with an isomorphism (on either side) still is in U. However in general this class is not closed under composition, so not all standing assumptions made in [26] are fulﬁlled. However uniqueness of completion, as deﬁned in section 6, can be proved. 8.2.9. Proposition. Whenever u : (X, γ) → (Y, γ ) belongs to U with (X, γ) a T2 -Cauchy space and (Y, γ ) a complete T2 -Cauchy space and f : (X, γ) → (Z, δ) is uniformly continuous with (Z, δ) in cP then there exists a unique uniformly continuous map f : (Y, γ ) → (Z, δ) such that f ◦ u = f . Proof. To construct the function f , let y ∈ Y and choose a Cauchy ﬁlter F on X such that stack u(F) converges to y in the limit coreﬂection of (Y, γ ). Such a ﬁlter exists by density of u. We deﬁne f (y) = lim stack f (F) where lim stands for the unique limit in the complete T2 -Cauchy space (Z, δ). This deﬁnition is independent of the choice of F. Indeed if G is another Cauchy ﬁlter for which stack u(G) converges to y then F and G are equivalent and so lim stack f (F) = lim stack f (F ∩ G) = lim stack f (G). For x ∈ X we have ˙ = f (x). f (u(x)) = lim stack f (x) ˙ = lim f (x) Finally to prove that f is uniformly continuous on (Y, γ ) let h be a Cauchy ﬁlter on Y . Using strict density of u we can choose a Cauchy ﬁlter F on X such that stack cY (u(F)) ⊂ h. Straightforward veriﬁcation shows that stack cZ (f (F)) ⊂ stack f (h). The regularity of (Z, δ) ﬁnally ensures that stack f (h) is a Cauchy ﬁlter. Uniqueness of f with the properties formulated in the theorem follows from the fact that u is an epimorphism in Chy0 by Proposition 7.2.3. 8.2.10. Theorem. Whenever f : (X, γ) → (Z, δ) is a morphism in P belonging ˜ the unique morphism to U with (Z, δ) in cP, then (Z, δ) is isomorphic to K(X), ˜ → Z such that fˆ ◦ rX = f is an isomorphism. fˆ: K(X) Proof. By the previous proposition f : (X, γ) → (Z, δ) is a reﬂection in cP. By ˜ are isomorphic. uniqueness of reﬂections we can conclude that (Z, δ) and K(X) 9. A comparison of the completion theories for subtopological spaces In this section we will make a comparison of the completion theories developed in sections 6 and 8. For this comparison we combine the notions regular nearness space and ﬁlter space. The class of spaces we end up with coincides with the class of regular subtopological spaces. In this common setting we prove that both completion theories developed in sections 6 and 8 coincide.

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9.1. Separated ﬁlter spaces. Separated merotopic spaces were characterized in Proposition 6.2.3 as those spaces in which every concentrated stack contains a unique minimal Cauchy ﬁlter. For ﬁlter spaces we can further improve this result. 9.1.1. Proposition. A ﬁlter space (X, γ) is separated if and only if every Cauchy ﬁlter contains a unique minimal Cauchy ﬁlter. Proof. For one implication it is suﬃcient to observe that Cauchy ﬁlters are concentrated stacks. For the other one observe that in a ﬁlter space for every concentrated stack there is a Cauchy ﬁlter contained in it. 9.1.2. Proposition. For a Cauchy space (X, γ) the following are equivalent: (1) (X, γ) is separated. (2) Every Cauchy ﬁlter contains a minimal Cauchy ﬁlter. (3) Every Cauchy equivalence class contains a minimum Cauchy ﬁlter. Proof. The only non-trivial implications follow from the fact that in a Cauchy space two equivalent minimal Cauchy ﬁlters are equal. 9.1.3. Proposition. Every separated ﬁlter space is a Cauchy space. Proof. Let F and G be Cauchy ﬁlters such that F # G. Let W be an ultraﬁlter ﬁner than F and than G. By Proposition 9.1.3 F, G and W each contain a smallest Cauchy ﬁlter. It follows that there is a minimal Cauchy ﬁlter H contained in F as well as in G. So F ∩ G is a Cauchy ﬁlter. 9.1.4. Proposition. Separated ﬁlter spaces can be viewed as Cauchy spaces in which every Cauchy equivalence class has a minimum element. In order to ﬁnd out how the nearness axiom can be expressed in this context we ﬁrst investigate the relation between the various closure operators we encountered in sections 3 and 7 respectively. Consider a ﬁlter space (X, γ) in C. The closure operator (clX )X on Mer was deﬁned in Proposition 3.2.9 and we now consider the induced closure operator on C. Another closure operator (cX )X on C was deﬁned in 7.1.3. 9.1.5. Proposition. Let (X, γ) be a ﬁlter space in C. Then the closures clX and cX coincide. Proof. Let A ⊂ X and suppose x ∈ cX (A). For a ﬁlter F on X converging to x in the limit coreﬂection and containing A we clearly have F ∩ x˙ ⊂ sec{A, {x}} and then sec{A, {x}} ∈ γ. Conversely suppose x ∈ clX A. Since γ is a ﬁlter structure we can ﬁnd a Cauchy ﬁlter F ∈ γ such that F ⊂ sec{A, {x}}. It follows that F # {A}. Let H be the ﬁlter generated. Finally x ∈ cX A because F = F ∩ x˙ ⊂ H ∩ x. ˙ In the sequel we will use the notation (clX )X to denote either of the two closure operators. If the C-space moreover satisﬁes the nearness axiom then we know that the closure operator (clX )X is idempotent and determines the underlying topology. In the sequel we refer to this topology when we use openness. Another way of looking at separated ﬁlter spaces satisfying the nearness axiom is based on the following notion.

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9.1.6. Definition. A T0 -nearness space is called a subtopological space provided it is a ﬁlter space. Using this deﬁnition we now have the following equivalence. 9.1.7. Proposition. The following are equivalent for a T0 -nearness space (X, γ): (1) (X, γ) is a separated and subtopological. (2) The canonical completion X ∗ is a T2 -topological space. (3) (X, γ) is a Cauchy space in which every Cauchy equivalence class has a minimum element with an open base. Proof. (1) ⇒ (2). Let (X, γ) be T0 , separated and subtopological. We ﬁrst prove that the canonical completion is topological. Let U ∗ be such that intµ∗ U ∗ is a cover. Consider the collection U = {U ⊂ X | ∃U ∗ ∈ U ∗ , o(U ) ⊂ U ∗ }. In order to prove that U is a uniform cover for X, let F be a minimal Cauchy ﬁlter. We choose U ∗ ∈ U ∗ such that F ∈ intµ∗ U ∗ . Since {U ∗ , X ∗ \ {F}} ∈ µ∗ it follows that A = {A ⊂ X | o(A) ⊂ U ∗ or o(A) ⊂ X ∗ \ {F}} is a uniform cover for X. We choose U ⊂ X such that U ∈ F ∩ A. Then clearly o(U ) ⊂ U ∗ . Finally we can conclude that U ∈ U ∩ F which proves that U ∗ is a uniform cover for X ∗ . Next we prove that the topology is T2 . Let F and G be diﬀerent minimal Cauchy ﬁlters and let U = {U | U ∈ F or U ∈ G}. Choose U ∈ G, U ∈ F and V ∈ F, V ∈ G such that U ∩ V = ∅. Then o(U ) and o(V ) are disjoint neighbourhoods of G and F respectively. (2) ⇒ (3). If X ∗ is a T2 -topological space then it is a separated Cauchy space and X is a subspace in Mer so that it is a separated Cauchy space too. By Propostion 9.1.4 every Cauchy equivalence class has a minimum element. Moreover the induced merotopic structure is a nearness structure and so every uniform cover is reﬁned by an open uniform cover. It follows that the minimal Cauchy ﬁlters have an open base. (3) ⇒ (1). Let (X, γ) be a Cauchy space in which every Cauchy equivalence class has a minimum element with an open base. By Proposition 9.1.4 (X, γ) is a separated ﬁlter space. To see that it also is a nearness space, let U be a uniform cover and let F be a minimal Cauchy ﬁlter. We choose F ∈ F ∩ U. Since F has an open base intX F ∈ F ∩ intX U. This fact implies that the two completeness notions we encountered coincide when we restrict them to subtopological spaces. 9.1.8. Proposition. A T0 separated subtopological space is complete (in the nearness sense 6.2.5) if and only if it is complete as a Cauchy space (in the sense of 8.1.1). Proof. Let (X, γ) be a T0 separated subtopological space. It is complete as a nearness space if and only if every minimal Cauchy ﬁlter converges. Since by Proposition 9.1.7 every Cauchy ﬁlter is equivalent to a minimal one, this is equivalent to completeness in the Cauchy sense. In the setting of nearness spaces the class of morphisms for which there is uniqueness of completion consists of all (clX )X -dense embeddings. In the setting of Cauchy spaces we considered the class of all strictly dense embeddings with respect to the closure operator (cX )X . Those classes of maps form the basis of the respective completion theories. Next we compare these classes when considered in the restricted setting of subtopological spaces.

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9.1.9. Proposition. Let u : (X, γ) → (Y, δ) be an embedding between T0 separated subtopological spaces. Then u is dense if and only if it is strictly dense. Proof. Strictly dense maps are always dense as we remarked in section 7.1. To prove the other implication, let F be a Cauchy ﬁlter for (Y, δ). Since (Y, δ) is a Cauchy space, by Proposition 9.1.3 we can apply 9.1.2 and choose a minimum element M in its Cauchy equivalence class. Since M has an open base by Proposition 9.1.7, it has a trace on uX. The ﬁlter H = M|uX is a Cauchy ﬁlter on uX. For any M ∈ M open we have M ⊂ clY (M ∩ uX), by the density of u(X). We can conclude that clY H ⊂ M, which proves strict density of u. 9.2. Regular subtopological spaces. In section 5 we introduced regularity for merotopic spaces. In section 7 we discussed Cauchy-regularity for Cauchy spaces. As we remarked in 7.3.2 these notions are diﬀerent, even in the restricted common setting of separated subtopological nearness spaces. In what follows we carefully make the distinction between the two notions Cauchy-regular and regular. As we know from Proposition 5.1.2, regular merotopic spaces are nearness spaces and they are always separated. The next result says that for ﬁlter spaces regularity is stronger than Cauchy-regularity. 9.2.1. Proposition. A regular ﬁlter space is a Cauchy-regular Cauchy space (in which every Cauchy equivalence class has a minimum element with an open base). Proof. Let (X, γ) be a regular ﬁlter space. Since regular spaces are separated, we know by Proposition 9.1.3 that (X, γ) is a Cauchy space. On the other hand (X, γ) also is a nearness space. In view of Proposition 9.1.7 we only have to prove that Cauchy-regularity holds. Let F be a Cauchy ﬁlter and let U be a uniform cover. W = {W ⊂ X | ∃U ∈ U : W < U } is also a uniform cover. Let F ∈ F ∩ W. Then there exists U ∈ U such that F < U . For x ∈ cl F we have that {X \ {x}, X \ F } is not a uniform cover and therefore it cannot be reﬁned by {X \ F, U }. It follows that x ∈ U . So we have proved that cl F ⊂ U and so (stack cl F) ∩ U = ∅. The converse is not valid. In Example 7.1 of [90] a T0 -Cauchy regular Cauchy space (D, C) is constructed in which Cauchy equivalence classes have a minimum element with an open base. However this Cauchy space is shown not to be a subspace of a regular topological space. So by Proposition 9.1.7 (D, C) is a separated subtopological nearness space, the canonical completion of which is not regular. It follows from Proposition 6.3.2 (5) that (D, C) is not regular. 9.2.2. Proposition. A T0 regular subtopological space (X, γ) belongs to P and the canonical completion (X ∗ , γ ∗ ) of X (as a nearness space) coincides with the ˜ of X (as a Cauchy space). completion K(X) Proof. Let (X, γ) be a T0 regular subtopological space. By Proposition 9.1.7 the canonical completion (X ∗ , γ ∗ ) is a T3 -topological space. By the previous results we know that it is a Cauchy-regular Cauchy space and every Cauchy equivalence class has a minimum element. So there is a bijective correspondence between X ∗ ˜ We remark that using the notation introduced in section 8, for a subset and X. A ⊂ X we have clX ∗ A = ΣA. Suppose F and G are Cauchy ﬁlters and ΣF # ΣG, then clX ∗ F and clX ∗ G have the same convergence point and so F = G. Moreover for any Cauchy ﬁlter F we

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have ΣF ⊂ Σ2 F since the closure in X ∗ is idempotent. Hence (X, γ) belongs to P. Since (X ∗ , γ ∗ ) is a T3 -topological space it belongs to cP and moreover i : (X, γ) → (X ∗ , γ ∗ ) is a strictly dense embedding. By Theorem 8.2.10 (X ∗ , γ ∗ ) is isomorphic ˜ to K(X). 10. Function spaces The lack of natural function spaces in a topological construct which is not cartesian closed, such as e.g. Top, has long been recognized as a disadvantage for various applications. In 1967 Steenrod published a paper [112], suggesting to replace Top by the subcategory of all compactly generated Hausdorﬀ spaces for use in homotopy theory and topological algebra. In their paper from 1971 [51] Dubuc and Porta show the importance of cartesian closedness in the setting of topological algebra, in particular Gelfand duality theory. In inﬁnite dimensional diﬀerential calculus the advantage of working in a cartesian closed setting has convincingly been demonstrated by several authors [56, 57, 98, 99]. In this section we will show that by going beyond Top to some of the constructs we encountered as subconstructs of Mer, a lot of cartesian closed candidates are available. 10.1. Cartesian closed topological constructs. Recall from the chapter Categorical Topology in this book, that a topological construct C is cartesian closed if for each C-object A the functor A × − has a right adjoint. Also recall that in the case of a well-ﬁbered topological construct, this condition can be expressed as follows: A well-ﬁbered topological construct C is cartesian closed if and only if for every object A ∈ C and for every ﬁnal episink (fi : Bi → B)i∈I the episink (1A × fi : A × Bi → A × B)i∈I is ﬁnal. It is often more informative to describe a topological construct as being cartesian closed if it has canonical function spaces in the following sense. A well-ﬁbered topological construct C is cartesian closed if and only if for every pair A, B of C-objects the set hom(A, B) can be equipped with the structure of a C-object, denoted [A, B] which fulﬁlls the following properties: (CC1) The evaluation map ev : A×[A, B] −→ B : (x, f ) → f (x) is a C-morphism, (CC2) For each C-object C and C-morphism f : A×C −→ B, the map f ∗ : C −→ [A, B] deﬁned by f ∗ (c)(a) := f (a, c) is a C-morphism. ev / A × [A, B] u: B O uu u uu 1A ×f ∗ uu f uu A×C 10.2. Function spaces for ﬁlter spaces and for Cauchy spaces. In this section we prove that the constructs Fil and Chy are cartesian closed and this will be done by describing function space structures that satisfy (CC1) and (CC2). 10.2.1. Proposition. The construct Fil of ﬁlter spaces is cartesian closed. Proof. Using the notations of Theorem 4.1.5 let A = (X, ϕ) and B = (Y, ϕ ) be ﬁlter spaces and let Hom(A, B) be the set of all uniformly continuous maps from A to B. We endow this set with a structure Φ deﬁned by: G ∈ Φ ⇔ ∀F ∈ ϕ : stack(ev(G ⊗ F)) ∈ ϕ ,

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where G is a ﬁlter on Hom(A, B). That this is indeed a ﬁlter structure follows from the following observations. For a uniformly continuous map f : A → B we have stack(ev(f˙ ⊗ H)) = stack(f (H)), so it is a Cauchy ﬁlter on B whenever H is a Cauchy ﬁlter on A. Consequently (F1) is fulﬁlled. If G ⊂ G then G ⊗ F ⊂ G ⊗ F for every Cauchy ﬁlter F on A. Since the evaluation preserves the order and is uniformly continuous we are done. For a uniformly continuous map h : A × C → B with C = (Z, ϕ ), we consider ∗ h : C → (Hom(A, B), Φ). For ﬁlters H ∈ ϕ and F ∈ ϕ we have stack(h(F ⊗ H)) ⊂ stack(ev(h∗ (H) ⊗ F)) and hence we can conclude that stack(h∗ (H)) ∈ Φ, which proves that h∗ is uniformly continuous. 10.2.2. Proposition. The construct Chy of Cauchy spaces is cartesian closed. Proof. Since products in Chy are formed as in Fil we can use essentially the same proof as for Fil. Assume that A = (X, ϕ) and B = (Y, ϕ ) are Cauchy spaces. It suﬃces to prove that (Hom(A, B), Φ) satisﬁes (CH). Let G and G belong to Φ such that G # G . Let F be a Cauchy ﬁlter on A, then stack(ev(G ⊗ F)) # stack(ev(G ⊗ F)). Therefore stack(ev(G ⊗ F)) ∩ stack(ev(G ⊗ F)) is a Cauchy ﬁlter on B coarser than stack(ev((G ∩ G ) ⊗ F)). Hence this ﬁlter is also a Cauchy ﬁlter on B. Therefore G ∩ G belongs to Φ. This completes the proof that (Hom(A, B), Φ) is a Cauchy space. The canonical function space structure on the set Hom(A, B) obtained in the proposition above is called the Cauchy continuous Cauchy structure. However the construct C which is situated in between Fil and Chy is not cartesian closed. 10.2.3. Proposition. C is not cartesian closed. Proof. We show that in C a ﬁnite product of quotient maps can fail to be a quotient map. For that purpose let X be [0, 1] with the symmetric topological structure deﬁned by selecting any two diﬀerent free ultraﬁlters U and W on [0, 1] and deﬁning W ∩ 0˙ to be a neighborhood ﬁlter of 0, U ∩ 1˙ to be the neighborhood ﬁlter of 1, and x˙ to be the neighborhood ﬁlter of x for x = 0 and x = 1. Then X is a ﬁlter space in C. Let f : X → [0, 1] be the identiﬁcation of 0 and 1 and let Y be the ﬁlter space such that f becomes a quotient in Fil. Let Z be the C-reﬂection of Y , so that f : X → Z is a quotient map in C. Let A be R+ with the merotopic structure deﬁned by: A is micromeric iﬀ either A contains a set of cardinality at most one or E ⊂ stack(A), where E is the ﬁlter of ﬁnite complements on R+ . Then A is a space in C. In fact A is even a Cauchy space. Let A × X and A × Y be the products in Fil and G be the C-reﬂection of A × Y . The ﬁlter stack(E ⊗ (U ∩ W)) is a Cauchy ﬁlter on A × Z but not on G. Therefore 1 × f : A × X → A × Z is not a quotient in C. 10.3. Function spaces for limit spaces. In this paragraph we consider the constructs Limr and Lims . Since Limr is (ﬁnitely) productive and concretely coreﬂective in Chy we can apply section 10.2.2 and we immediately obtain the next result.

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10.3.1. Proposition. Limr is cartesian closed. The canonical function space structure in Limr is the continuous convergence deﬁned as follows. Let A and B be reciprocal limit spaces and Hom(A, B) is the set of all (uniformly) continuous maps from A to B. Then a ﬁlter G on Hom(A, B) converges continuously to some g ∈ Hom(A, B) if and only if stack(ev(G ⊗ F)) → f (x) for every ﬁlter F on A and x ∈ A such that F → x. The larger construct Lims is also cartesian closed and the function spaces are formed in exactly the same way via continuous convergence. 11. Relations to other constructs At some stage in the development of generalized uniform spaces the paths of Uniformity and Topology crossed. In 1967 Cook and Fischer [40] introduced uniform limit structures as a generalization of uniform spaces, thus providing a setting for the study of uniform continuity. Instead of considering one ﬁlter of entourages as Weil did, they gave axioms for a collection of ﬁlters on X × X. Wyler [114] continued the line of thought of Cook and Fisher and slightly modiﬁed the axioms of uniform limit structures in 1971. The purpose for this change in the axioms is to make sure that canonical function spaces exist, and so it is motivated by categorical observations. The actual proof that the construct of uniform limit spaces is cartesian closed was given by Lee in [89]. In another chapter of this book the theory of uniform limit spaces is explained. In this section we will indicate how the category of uniform limit spaces is related to one of the constructs we studied in this chapter. We start with Wyler’s deﬁnition. 11.1. Uniform limit spaces. In the next deﬁnition we use the following notation. For two ﬁlters Φ and Ψ on X × X we use the notation Φ ◦ Ψ for the ﬁlter generated by all sets U ◦ V where U ∈ Φ and V ∈ Ψ. We say that Φ ◦ Ψ (as a ﬁlter) exists if it does not contain the empty set, i.e., U ◦ V = ∅ for all U ∈ Φ and V ∈ Ψ. We let Φ−1 = {U −1 | U ∈ Φ}. 11.1.1. Definition. A uniform limit structure on a set X is given by a collection L of ﬁlters on X × X such that the following properties are fulﬁlled for all ﬁlters Φ and Ψ on X × X: (UL1) (UL2) (UL3) (UL4) (UL5)

∀x ∈ X : stack(x˙ ⊗ x) ˙ ∈ L, If Φ ∈ L and Φ ⊂ Ψ then Ψ ∈ L, If Φ ∈ L and Ψ ∈ L then Φ ∩ Ψ ∈ L, If Φ ∈ L then Φ−1 ∈ L, If Φ ∈ L and Ψ ∈ L then Φ ◦ Ψ ∈ L whenever it exists.

The pair (X, L) is called a uniform limit space and a function f : X −→ Y between uniform limit spaces (X, LX ) and (Y, LY ) is called uniformly continuous if for each ﬁlter Φ ∈ LX one has that stack(f × f )(Φ) ∈ LY . The category of uniform limit spaces and uniformly continuous maps is denoted ULim.

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ULim contains the construct Unif of uniform spaces as a fully embedded subconstruct. Next we investigate the relation with the construct Chy. In a uniform limit space a Cauchy ﬁlter is deﬁned as a ﬁlter F for which stack(F ⊗ F) belongs to L. The class ϕL of Cauchy ﬁlters in a given uniform limit space L satisﬁes the axioms (F1) and (F2) of a ﬁlter space and moreover it fulﬁlls the axiom (CH) we encountered in section 4. In fact the next proposition states that this set of axioms is precisely what is needed for a collection of ﬁlters to be the collection of Cauchy ﬁlters in some uniform limit space. 11.1.2. Proposition. A ﬁlter space (X, ϕ) is a Cauchy space if and only if there exists a uniform limit space L such that the class of all L-Cauchy ﬁlters is ϕ. Proof. That given a uniform limit space (X, L) the class ϕL satisﬁes (F1) and (F2) is easy. Next suppose that F and G are ﬁlters in ϕL such that F#G. Observe that under these assumptions stack(F ⊗ G) = (F ⊗ F) ◦ (G ⊗ G). So by (UL5) we have stack(F ⊗ G) ∈ L. Next we use the identity (F ∩ G) ⊗ (F ∩ G) = (F ⊗ F) ∩ (F ⊗ G) ∩ (G ⊗ F) ∩ (G ⊗ G) to conclude that F ∩ G ∈ ϕL . In order to prove that the Cauchy axiom on ϕ is suﬃcient put n Lϕ = {Φ ∈ (X × X) | ∃F1 , . . . Fn ∈ ϕ : stack(Fi ⊗ Fi ) ⊂ Φ}. i=1

Clearly Lϕ satisﬁes (UL1)–(UL4). To see that (UL5) is fulﬁlled consider Φ and Ψ in Lϕ such that Φ ◦ Ψ exists. We can choose ﬁnite collections Fi with i = 1, . . . , n and Gj with j = 1, . . . , m of ﬁlters in ϕ such that n

stack(Fi ⊗ Fi ) ⊂ Φ,

i=1

Obviously m

n

stack(Gj ⊗ Gj ) ⊂ Ψ.

j=1

stack(Fi ⊗ Fi ) ◦

i=1

n j=1

stack(Gj ⊗ Gj ) =

(Fi ⊗ Fi ) ◦ (Gj ⊗ Gj )

(i,j)∈K

where K consists of those indices (i, j) for which (Fi ⊗ Fi ) ◦ (Gj ⊗ Gj ) exists. For (i, j) ∈ K the collection Fi # Gj and consequently Hij = Fi ∩ Gj belongs to ϕ. Since Hij ⊗ Hij ⊂ stack(Gj ⊗ Fi ) = (Fi ⊗ Fi ) ◦ (Gj ⊗ Gj ) we conclude that Φ ◦ Ψ belongs to Lϕ . Finally we have to prove that the class of Cauchy ﬁlters for the uniform limit space (X, Lϕ ) coincides with ϕ. Suppose F is a Cauchy ﬁlter for Lϕ . Then there exists a ﬁnite collection F1 , . . . , Fn ∈ ϕ such that n stack(Fi ⊗ Fi ) ⊂ stack(F ⊗ F). i=1

Without loss of generality we may assume that for i = k the ﬁlter Fi ∩ Fk ∈ ϕ. It follows that for i = k the ﬁlters Fi and Fk do not mesh, and hence we can choose Fi ∈ Fi for ni ∈ {1, . . . , n} such that Fi ∩ Fk = ∅ for i = k. Next choose F ∈ F with F × F ⊂ i=1 Fi × Fi . Then F ⊂ Fk for some k ∈ {1, . . . , n}. From this it can be deduced that for this index k we have Fk ⊂ F. So F ∈ ϕ.

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The transition from Chy to ULim is in fact functorial. With the notations of the previous proposition we have the following. 11.1.3. Proposition. Chy → ULim described on objects by (X, ϕ) → (X, Lϕ ) deﬁnes a concrete full embedding. Proof. Suppose that f : (X, ϕ) → (X, ϕ ) is a uniformly continuous map between Cauchy spaces and let Φ ∈ Lϕ . We can choose a ﬁnite collection Fi with i = 1, . . . , n of ﬁlters in ϕ such that n

stack(Fi ⊗ Fi ) ⊂ Φ.

i=1

Then we have n i=1

stack(f (Fi ) ⊗ f (Fi )) =

n

stack((f × f )(Fi ⊗ Fi ))

i=1 n ⊂ stack (f × f )( (Fi ⊗ Fi ) i=1

⊂ stack((f × f )(Φ)). Since stack f (Fi ) ∈ ϕ for each i = 1, . . . , n the ﬁlter stack(f × f )(Φ) ∈ Lϕ . So the transition is functorial. That the functor is injective on objects was shown in the previous proposition. Moreover the embedding is full. The relation among the categories Unif, Chy and ULim is given in the diagram in Figure 4. Chy qF Unif m F ww M FFc w r w FF w FF ww ww ULim Figure 4. The relationships among the categories Unif, Chy, and ULim

12. Where to ﬁnd more information What we presented in this chapter can be seen as an introduction to the theory of completions and function spaces in the setting of merotopic spaces. There are several books and surveys that treat completion theory more extensively, either in the setting of nearness spaces or in the setting of Cauchy spaces. Our section 9 also includes a comparison of these two approaches. This comparison is related to the work in [107, 19, 20, 109], to the work of Bentley on subtopological spaces [4] and also contains some new observations. We have chosen to make extensive use of the categorical theory on completions as developed by [26] and [27] focusing on uniqueness of completions with respect to a given ﬁxed class of morphisms.

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12.1. Books and surveys. A very nice German-language book dealing with nearness spaces is [72]. It includes most of the topics treated in our sections 2 up to 6, even much more detailed than what is presented here and it is written as a textbook for students. It has been an important source when compiling sections 2, 3, 5, and 6. Other books containing a lot of material on nearness spaces or merotopic spaces are [100] and [103]. The latter also deals with uniform limit spaces and with Cauchy spaces. Both of these references also discuss function spaces. In the book [91] Cauchy spaces play an important role in the theory of extensions of maps. The motivation for going beyond Top comes partly from important applications of the theory of convergence to functional analysis as described in the following interesting books [3, 23, 56, 57, 98, 99]. An important survey describing the main results in categorical topology obtained in the period 1971–1981 is [70]. Other surveys relating the theory of nearness spaces to constructions in Top are [17] and [75]. Some surveys describe the historical development of the theory. These papers moreover contain extensive lists of references on the topic of this chapter as well as a list of open problems. We cite here a few important ones [16, 92, 73, 74, 76, 59]. There are a few very recent papers discussing particular open problems in the setting of nearness spaces. For the problem of ﬁnding the epireﬂective hull of Tops in Near we refer to [7, 8, 15]. 12.2. Information on merotopic spaces and nearness spaces. Here we mention some original papers where the notions of merotopic and nearness space were introduced for the ﬁrst time. By weakening the axioms of the description of uniform spaces in terms of small members by Sandberg [108] a particularly elegant solution for describing both topological and uniform concepts was oﬀered by Katˇetov around 1963–1965 by his introduction of merotopic spaces and uniformly continuous maps [78, 79]. The basic idea was to present an axiomatization of collections of subsets that contain arbitrary small elements. These collections were called micromeric. Katˇetov used the concept of micromeric (also known as Cauchy) collections as a primitive in his axiomatization of merotopic spaces. Katˇetov has shown how the embedding of the construct of limit spaces in Mer is constructed. He used convergent nets for that purpose. In 1975 Robertson [107] translated Katˇetov’s results into the setting of ﬁlter convergence. Katˇetov paid a lot of attention to the objects deﬁned as ﬁlter spaces. A categorical study of the construct of all ﬁlter spaces was developed in [20] in terms of grills rather than ﬁlters. In 1974 Herrlich published his papers introducing the nearness concept. “A concept of Nearness” [65, 66] and “Topological Structures” [67] are the starting point of a vast literature on nearness spaces. Herrlich proved that one can also use near collections or uniform covers as a primitive concept to deﬁne merotopic spaces. We remark that Herrlich’s early work on nearness spaces in his introductory papers was already written using categorical terminology. He pointed out the advantage of working with products in the category Near [68, 69]. Several other papers contributed to the development of the theory [28, 77, 95, 96, 30, 31, 60, 62, 63, 64]. A lot of attention went to the study of completions and extensions and in this respect the development of the theory of regular nearness spaces became an important tool [97, 58, 9, 10, 4, 5, 6, 11, 12, 29, 32, 34, 35, 36, 49, 33, 21, 14]. Later work on extensions in which merotopic spaces are

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involved was done by Cs´asz´ar [41, 42, 43, 44] and by Cs´ asz´ar and De´ ak [45, 46, 47, 48]. 12.3. Information on limit spaces and Cauchy spaces. Motivated by the example of so called closed convergence of sets, Choquet introduced pseudotopological and pretopological limit spaces in [37]. Independently and rather inspired by continuous convergence Fischer and Cook and Fischer introduced “Limesr¨ aume” [39, 52]. Motivated by order convergence Kent introduced Convergence spaces [81] and studied basic constructions [82]. Motivated by the applications of convergence theory in analysis and in particular by the development of the theory of limit vector spaces, an appropriate notion of completeness was needed. Kowalsky in 1954, starting with a limit space, gave an axiomatic description for Cauchy ﬁlters compatible with a given “Limitierung” [88]. Soon after that, in 1968 in his paper [80], Keller formulated “Cauchyness” as a primitive concept, independent of any structure given in advance, thus introducing what we know as Cauchy spaces. He proved that his set of axioms is necessary and suﬃcient for a collection of ﬁlters to be the collection of Cauchy ﬁlters in some uniform convergence space. The relation to semi-uniform limit spaces was later investigated by Preuss [101, 102]. For our section 11 on the relations of Cauchy spaces to other structures, the original paper of Keller [80] was our main source. For our section 4 on ﬁlter spaces and limit spaces we followed the exposition given in [19]. The important contribution in that paper is that constructs such as Lims of limit spaces and Chy of Cauchy spaces are embedded in the superconstruct Mer. Apart from bringing uniﬁcation in the theory of convergence on one hand and of nearness spaces on the other hand, it makes it particularly easy to describe the functorial relation between the various constructs. Cauchy spaces proved to be extremely useful in the theory of completions. In 1970 applications of Cauchy spaces in completion and compactiﬁcation theory were described by Ramaley and Wyler in [105, 104]. This was the starting point for the development of the theory of Cauchy completions. Soon it became clear that in order to complete uniform limit spaces the basic tool is the completion of the associated Cauchy space [106]. Two basic constructions are fundamental in the development of completion theory: The Wyler completion [113] and the strict completion [85]. Some of the pioneers of this theory are Kent, Richardson, and Friˇc, [85, 86, 87, 83, 84, 53, 54, 55, 61]. A lot of eﬀort went to investigations on the existence of regular and strict regular completions. Later contributions are [90, 13]. For the compilation of our sections 7 and 8 we mainly used these pioneering papers on regularity and completions, in particular [85, 86] (for completions) and [84] (for the regularity sequence of a Cauchy space). 12.4. Function spaces. Katˇetov [79] proved that in the construct of ﬁlter spaces, which he introduced, canonical function spaces exist. Later Bentley, Herrlich and Robertson [20] proved an even stronger result that in Fil arbitrary products of quotients are quotients. Binz and Keller [24] were the ﬁrst to observe that the construct of limit spaces is cartesian closed. Cartesian closedness, exponential objects and cartesian closed topological hulls in the setting of convergence theory were investigated by the French school in [2, 25] and by Machado in [93] and later

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A CONVENIENT SETTING FOR COMPLETIONS AND FUNCTION SPACES

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Contemporary Mathematics Volume 486, 2009

Proximity: a powerful tool in extension theory, function spaces, hyperspaces, boolean algebras and point-free geometry Anna Di Concilio Abstract. Generally speaking, topology is a closeness between points and sets, proximity is a closeness between sets and uniformity is a measure of closeness between sets made by a class of objects. Topology, proximity and uniformity are substantially distinct structures. Proximity, located between topology and uniformity, is a more powerful tool than topology having an intensive interaction with uniformity. Proximity is also an exhaustive machinery for the construction of T2 -compactiﬁcations. Furthermore, proximity extends to complex frameworks such as function spaces, homeomorphism groups and hyperspaces. Then, proximity, in its dual formulation as strong inclusion, lends itself to a lattice theoretical approach. Tools, arguments, procedures and all techniques in the classic proximity theory are intensively employed in formalisations of point-free geometries and in constructive topology. Proximity spaces have enough structure to feel comfortable within them!

Contents 1. Introduction 2. Axiomatisations 3. Topology associated with a given proximity space 4. Interplay between proximity and uniformity 5. Compactiﬁcations and proximities: Smirnov theorem 6. Leader’s contribution 7. Proximity in complex frameworks 8. Lattice theoretical approach and point-free geometry References

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2000 Mathematics Subject Classiﬁcation. 54E05, 54B20, 54C35, 54D35, 54D80, 54H12, 57S05.

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1. Introduction The ﬁrst model of proximity was given by F. Riesz, who formulated it during an impressive, ahead of its time lecture, as far back as 1908, [55]. His idea was to design a good concept of a mathematical continuum, nowadays known as a topological space, that he meant as the natural framework into which the notion of an accumulation point, and consequently that of continuity, would be deﬁnable. In an attempt to capture the richness of the various forms of continuity, Riesz derived the concept of an accumulation point from the notion of enchained sets. To make a good concept of an accumulation point, he displayed an axiomatisation for a closeness between sets, called enchainment, which was a proximity ante litteram. Riesz felt that enchainment was the right background for an accumulation point. Choosing enchainment as vehicle for topology, he shifted attention from relations between points and sets to relations between sets, so extending well beyond topology. Riesz ﬁrmly believed that a space carrying two diﬀerent enchainments could have the same notion of accumulation point. Besides this, he posed the problem of constructing a suitable extension of a set carrying an enchainment in which two sets are enchained if and only if they share a common accumulation point. In spite of the strength and groundbreaking nature of his ideas, Riesz’s project was not furthermore developed for a very long time, because of the more inﬂuential work of F. Hausdorﬀ and K. Kuratowski. Indeed, in 1914, F. Hausdorﬀ planned to set a formal theory for topology. The common ground shared by metrics on a same set with the same convergence for sequences, followed by the same continuity for functions, was assumed as the base of his investigation. Within two years, Hausdorﬀ achieved his objective formulating the theory of topological spaces, as presently known. He introduced topology as the geometry which studies the properties of topological spaces invariant under homeomorphisms. Later on, K. Kuratowski characterised the point-set proximity relation point of adherence. His seminal work made topology to turn out as closeness between points and sets and, consequently, continuity as preservation of closeness between points and sets. It was just in 1952, [17], that V.A. Efremoviˇc launched proximity by elaborating a complete set-theoretical approach. He supplemented the axiomatisation sketched by F. Riesz with an additional very strong axiom, nowadays known as the Efremoviˇc property, which revealed a very powerful tool and an essential statement. In 1949, [15], focusing his attention on uniform properties of diﬀerentiable manifolds, Efremoviˇc achieved the following issue: even though homeomorphic, the Euclidean and Lobachevskii spaces are not equimorphic. In other words: between them there is no homeomorphism that carries inﬁnitely close sets, sets located at distance zero, to inﬁnitely close sets. By the use of his celebrated Lemma, Efremoviˇc established that every function between metric spaces which takes inﬁnitely close sets to inﬁnitely close sets is uniformly continuous. In this way, he discriminated the topology of diﬀerentiable manifolds from their uniform geometry. This was the premise that allowed to outline Efremoviˇc’s program on geometry of proximity. Objects of the new geometry are the inﬁnitesimal or proximity spaces, which are sets carrying a notion of inﬁnitely close sets. Morphisms are the proximity functions, which are maps taking inﬁnitely close sets to inﬁnitely close sets. Further, starting from the concept of δ-neighbourhood, Efremoviˇc recast his axiomatisation in terms of double containment or strong inclusion. By applying

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the Urysohn Lemma, he proved that two far sets can be separated by a real-valued proximity function. So, Efremoviˇc created the base for the functional method as an outside standpoint in the investigation of proximity spaces. Proximity arises very naturally in rich mathematical structures such as metric spaces. The metric proximity is quite simple and intuitive. In a metric space there is an extremely natural notion of inﬁnitely close sets, or dually of distant sets, when the usual distance between sets is assumed as the numerical measure of closeness between sets. Essentially, Efremoviˇc converted a very intuitive idea such as metric proximity into a rich abstract domain by drawing features and issues of uniform spaces into proximity. As a matter of a fact, uniform spaces carry a natural measure of closeness between sets, operated by the class of diagonal neighbourhoods, or surroundings, as follows: two sets A and B are close if and only if they are not distinguishable by a diagonal neighbourhood, i.e., U ∩ A × B = ∅ for every diagonal neighbourhood U . Uniform spaces originated from the full impetus of A. Weil in lieu of the dominant but unnecessary use of countability properties, [72]. Weil modeled his theory of uniformity by highlighting the natural meeting ground between metric spaces and topological groups, thanks to his great expertise on topological groups. The uniform world, designed by A. Weil, is the natural setting where metric uniform continuity, together with uniform convergence for functions and completeness, achieves its full explanation. Weil soon realized that the underlying topology of a uniform space had to be completely regular. That was because a unique natural uniformity emerges from a compact space and, as proved by A.N. Tychonoﬀ, every completely regular space is a subspace of a compact one. Consequently, by generalizing a method used by L.S. Pontrjagin, Weil proved that any uniform space is completely regular. In 1939 P. Samuel, [56], formulated a method of compactifying any uniform space. He exclusively used an inside standpoint and the notion of ultraﬁlter as main tool. For every uniform space (X, U), Samuel constructed a compact space s(X), known as the Samuel compactiﬁcation of (X, U), whose unique compatible uniform structure induces on X a totally bounded uniformity U ∗ , weaker than U, compatible with the topology induced by U. So, he produced a many-to-one correspondence, U → U ∗ , as a tool for studying the diﬀerent uniform structures compatible with the topology of a uniformisable space. At ﬁrst glance, Efremoviˇc’s proximity theory may look like an elegant generalisation of the uniform topology in metric spaces, or, also, like an instrument of simpliﬁcation in the realm of uniformities compatible with an assigned topology. According to the Efremoviˇc Lemma, metric spaces are rigid: uniform geometry and proximal geometry agree within the metric context; that is, any uniform character of a metric space is also a proximity invariant. Nevertheless, the proximal ﬁeld had been actively developed by Ju.M. Smirnov, who established an organic approach, which converted the proximity theory into an exhaustive machinery for the construction of T2 -compactiﬁcations, [59, 58]. His investigation was based on “the connection between proximity spaces and ordinary topological spaces considering them as a natural superstructure on contemporary general topology.” Smirnov’s research was motivated by the question posed by P.S. Alexandroﬀ: Which topological spaces admit an Efremoviˇc proximity relation

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compatible with the given topology? Smirnov answered Alexandroﬀ’s query by showing that a topological space admits a separated Efremoviˇc proximity relation if and only if it is a subspace of a T2 compact space. More precisely, he stated that every separated EF-proximity space is a dense subspace of a unique T2 compact space, in which two sets are close if and only if their closures share a common point. Smirnov essentially considered T2 compact spaces as a topological counterpart of proximities. He also established, up to equivalence, an order preserving isomorphism between the upper-semilattice of T2 -compactiﬁcations of a Tychonoﬀ space and the one of compatible separated EF-proximities. Furthermore, Smirnov proved that the class of EF-proximisable topological spaces agrees with that of completely regular ones. Can proximity be reduced into a suitable tool for the investigation of topological spaces? As a matter of fact, proximity is a useful tool to study compactiﬁcation but the reverse is not true. Indeed, some proximity properties cannot be expressed in topological terms. An essential feature of proximity lies in the fact that many of the interesting properties of uniform spaces turn out to be not only uniform but more proximity invariants. To sum up, though uniformity has been largely drawn into proximity, proximity theory encompasses the uniform results and generates new issues. The 1950s was the golden age for generalisations of topology such as proximity and uniformity. T. Shirota, by proving, in [57], that the collection of all open normal coverings of a completely regular space was really a covering uniformity, shed light on the richness of uniformity and provided the proximity investigation with a new tool. Later on, in a large and intensive production, [28, 34], by introducing the manageable concept of cluster, S. Leader gave new impulse to proximity with the reformulation of Smirnov’s compactiﬁcation theorem. Unifying proximity and boundedness in the concept of local proximity, he renewed interest in proximity. Leader also interpreted a local proximity as a localisation of an EF-proximity modulo a free cluster. On the basis of the Alexandroﬀ–Smirnov method for the construction of T2 -compactiﬁcations, he established, for each Tychonoﬀ space X, a correspondence, up to equivalence, between the class of all T2 -local compactiﬁcations of X and that of all separated local proximities compatible with X. Furthermore, by extending proximity to function spaces, Leader consolidated and greatly expanded proximity. Successively, by drawing hyperuniform issues into proximity, L.J. Nachman approached proximity in hyperspaces in [42, 41]. Then, in a largely close cooperation, S.A. Naimpally and collaborators planned to ﬁnd out and capture the proximity counterpart of set-open topologies and of hypertopologies which split in two halves. The basic idea was simple: replace the set-theoretical inclusion with a strong one, [7, 10]. The revival of the proximity lattice theoretical approach in constructive mathematics and in point-free spatial theory is really impressive, [11, 19, 69]. Generally speaking, while topology is a closeness between points and sets (K. Kuratowski) and proximity is a closeness between sets (V.A. Efremoviˇc), uniformity is a measure of closeness between sets made by a class of objects (A. Weil). Thus, proximity, located between topology and uniformity, is a more powerful abstract tool than topology. Topology, proximity and uniformity are substantially distinct structures. Proximity spaces have more structure than topological ones but carry less structure than metric ones.

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Which are the applied techniques? What is the apparatus? What are the results? What and where is the application? Which are the main tools or methods? What are the basic suggestive facts? Who are the main contributors and originators? Which are the future perspectives and ﬁelds where proximal techniques can be directly applied? This overview will try to answer these queries, by analysing diﬀerent levels of generalisation in several approaches. Given the vast amount of the literature related to relaxations, they are not touched herein. Who wishes to delve further into the truly interesting area of relaxations can refer to [45]. In conclusion, we draw the attention on the following fact: we mean the Urysohn Lemma as the procedure for constructing continuous functions separating non-empty disjoint closed sets, when a nice interposition property holds. 2. Axiomatisations 2.1. Efremoviˇ c’s axiomatisation. We are now ready to formalise Efremoviˇc’s set of proximity axioms. If X is a set, then Exp(X) stands for the powerset of X. If δ is a binary relation over the powerset Exp(X) of a set X, we denote the dual non-δ relation as δ. A binary relation δ over the powerset Exp(X) of a non-empty set X is called an Efremoviˇc proximity or, in short, an EF-proximity, if it satisﬁes the following axioms for A, B, C ⊂ X: (P1) If A δ B then A = ∅ and B = ∅. (P2) If A ∩ B = ∅ then A δ B. (P3) If A δ B then B δ A (symmetry). (P4) A δ (B ∪ C) if and only if A δ B or A δ C (additivity). (P5) If A δ B then A δ C and B δ X \ C for some C ⊂ X (normality). The axiom (P5) is called the Efremoviˇc property. The pair (X, δ) is called an Efremoviˇc space or, in short, an EF-proximity space. An EF-proximity is called separated if it satisﬁes: (P6) {x} δ {y} implies x = y. We generally say that A is close or near to B if A δ B and dually A is remote or far from B if A is not close to B. We have listed below some basic properties of proximity spaces: (i) If A is a subset of C, B a superset of D and A is close to B, then C is close to D. (ii) If there exists a point x which is close to both A and B, then A is close to B. (iii) In a separated EF-proximity space no two distinct singleton sets are close. The simplest examples of EF-proximities are the following two: Example 2.1.1 (The trivial or indiscrete proximity on a non-empty set X). Let A, B ⊂ X and set A δ B if and only if A = ∅ and B = ∅. In general, a trivial proximity is not separated. Example 2.1.2 (The discrete proximity on a non-empty set X). Let A, B ⊂ X and set A δ B if and only if A ∩ B = ∅. The discrete proximity is a separated EF-proximity. We now proceed by giving some relevant examples of EF-proximities on topological spaces with special properties.

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Example 2.1.3 (The standard or elementary proximity on a normal topological space X). Let X be a topological space that is normal, i.e., for any two non-empty closed disjoint subsets A, B of X there are two disjoint open subsets U , V of X containing A and B respectively. The standard or elementary proximity is deﬁned on X by saying that, for A, B ⊂ X, A δ B if and only if their closures share a common point. The standard or elementary proximity on a normal space is an EF-proximity. It is separated if and only if X is a T0 space. Every T2 compact space, equipped with the relative elementary proximity, is a separated EF-proximity space. Example 2.1.4 (The metric proximity). Let (X, d) be a metric space, A, B ⊂ X, and deﬁne the metric proximity by saying that A δ B if and only if d(A, B) = 0, where d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}. In other words, two sets A, B are close in the metric proximity associated with d if and only if they are inﬁnitely close in d: for each natural number n there is a point an in A and a point bn in B such that d(an , bn ) < n1 . The metric proximity is a pivotal focus in proximity theory. We will dedicate then due attention to this in the later subsection 4.4 of this paper. ˇ Example 2.1.5 (The functionally indistinguishable or Cech proximity on a completely regular space X). Let X be a completely regular space, i.e., for any point x in X and any closed non-empty subset C of X not containing x there is a continuous function f : X → [0, 1] so that f (x) = 0 and f (C) = 1. Two subsets A, B of X are functionally distinguishable if and only if there is a continuous function f : X → [0, 1] such that f (A) = 0 and f (B) = 1. Deﬁne δF in X by requiring that A δ F B if and only if A and B are functionally distinguishable. Then, we obtain ˇ the functionally indistinguishable or Cech proximity, δF , which is an EF-proximity. The proximity δF is separated if and only if the space X is T1 . Furthermore, by the Urysohn Lemma, δF is the elementary proximity if and only if the space X is normal, see Example 2.1.3. Example 2.1.6 (The Alexandroﬀ proximity on a T2 locally compact space X). Let X be a T2 locally compact space and A, B two non-empty subsets in X. We deﬁne the Alexandroﬀ proximity on X by setting, for A, B ⊂ X, A δ B if and only if their closures don’t intersect and at least one of their closures is compact. The Alexandroﬀ proximity is a separated EF-proximity. Example 2.1.7 (The Freudenthal proximity on a rim-compact T2 space X). Let X be a rim-compact T2 space. Recall that a topological space is rim-compact if and only if it admits arbitrarily small neighbourhoods with compact boundaries at any of its points. It can be also proven that a Tychonoﬀ space X is rim-compact iﬀ and only it admits a T2 -compactiﬁcation γ(X) whose remainder γ(X) \ X has arbitrarily small neighbourhoods with boundaries contained in X at any of its points. We deﬁne the Freudenthal proximity by saying, for A, B ⊂ X, that A δ B if and only if their closures are separated in X by some compact set, or equivalently A δ B if and only if for some compact set K in X, X \ K = G ∪ H where G and H are disjoint open sets in X with cl(A) ⊂ G and cl(B) ⊂ H. 2.2. Strong inclusions. We now illustrate a diﬀerent approach towards the complementary perspective of strong inclusions. Let (X, δ) stand for an EF-proximity space. We say that a subset B is a pneighbourhood or δ-neighbourhood of a subset A if and only if A is remote from the

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complement of B. In the case that B is a p-neighbourhood of A we say that A is strongly contained in B and use the notation A δ B. This deﬁnition naturally induces a binary relation over the powerset Exp(X) of X which we refer to as the p-neighbourhood relation or strong inclusion induced by the proximity δ. We take a new set-theoretic view point. A binary relation over the powerset Exp(X) of a non-empty set X satisfying the following axioms: (1) (2) (3) (4) (5)

∅ ∅; A B implies A ⊂ B; A B implies X \ B X \ A; A B and A C implies A B ∩ C; If A C then A B C for some B ⊂ X (density property);

has been named in the literature in several ways as: strong inclusion, double containment, non-tangential inclusion or well-inside relation. When the above mentioned set of axioms is supplemented by: (6) x = y implies {x} X \ {y} then is called separated. Now, it can be shown that the p-neighbourhood relation δ over Exp(X) relative to an EF-proximity δ satisﬁes axioms ( 1) through ( 5) and further the axiom ( 6) if and only if δ is a separated EF-proximity. Conversely, any strong inclusion , which satisﬁes the axioms (1) through (5), generates an EF-proximity δ, by putting: A δ B if and only if A X \ B. The starting strong inclusion is in turn exactly the p-neighbourhood relation determined by δ. Further, the proximity space (X, δ) is separated if and only if the dual strong inclusion δ satisﬁes the property (6). The relations δ and δ are interdeﬁnable. So, by using the properties (1) through (6), the deﬁnition of proximity space can be recast in terms of δ . If X is indiscrete, then any non-empty subset of X is strongly contained only in the entire X. If X is discrete, then the dual strong inclusion is just the usual inclusion. If X is a normal space and δ is the elementary proximity, then A δ B just means cl(A) ⊂ int(B). If X is T2 rim-compact space and δ is the Freudenthal proximity, then A δ B if and only if there exists a closed set C with compact boundary such that A ⊂ int(C) ⊂ C ⊂ B. Finally, if X is a T2 locally compact space and δ is the Alexandroﬀ proximity, then A δ B if and only if there exists a compact set K such that A ⊂ int(K) ⊂ K ⊂ B. Recall that in a metric space (X, d) for any non-empty subset A of X the collar around A is deﬁned as the subset S [A] := {x ∈ X : d(x, A) < }, being a positive real number. A subset B of X is a p-neighbourhood of A if and only if B contains an -collar around A. The metric example makes the following dual version of the Efremoviˇc property (P5) (of subsection 2.1) to be easily understandable: Proposition 2.2.1. Any two remote sets have disjoint p-neighbourhoods.

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2.3. Proximity functions. Let (X, δ) and (Y, δ ) be two EF-proximity spaces. A function f from X to Y is a proximity function, in short, either a p-function or a δ-function, if and only if wheneverA is δ-close to B in X, then f (A) is δ -close to f (B) in Y . It is easy to verify that f : X → Y is a proximity function if and only if whenever C is remote from D in Y , then f −1 (C) is remote from f −1 (D) in X. In the language of strong inclusions, a function f : X → Y is a proximity function if and only if whenever C D in Y , then f −1 (C) f −1 (D) in X. Trivially, if (X, δ) is discrete or (Y, δ ) is indiscrete, then any function from X to Y is a proximity function. A bijective map f : X → Y is called either a p-isomorphism or a δ-isomorphism if it preserves proximities in both directions: that is for any pair of subsets A, B of X, A δ B in X if and only if f (A) δ f (B) in Y or, alternatively, A B if and only if f (A) f (B). Since the usual composition of proximity functions is in turn a proximity function, the proximity spaces with proximity functions form a category. Proximity invariants are all properties of proximity spaces preserved by any p-isomorphism. They constitue the proximity geometry. ˇ If X and Y are completely regular spaces both carrying the Cech or functionally indinstinguishable proximity, see Example 2.1.5, then any continuous function f ˇ from X to Y is also a proximity function relative to Cech proximities. If two subsets C and D in Y are separated by a Urysohn function h : Y → [0, 1], then their preimages f −1 (C) and f −1 (D) are separated in X by the Urysohn function h ◦ f . If X and Y are both T2 locally compact spaces, both carrying the Alexandroﬀ proximity, see Example 2.1.6, then any homeomorphism between X and Y is a pisomorphism too. The essential reason lies in: any continuous image of a compact set is compact. If X and Y are both T2 rim-compact spaces, both carrying the Freudenthal proximity, see Example 2.1.7, then any homeomorphism between X and Y is a p-isomorphism too. The essential reason lies in: homeomorphic image of a closed set with compact boundary is again a closed set with compact boundary. Let R be the set of real numbers equipped with the metric d deﬁned as: x y d(x, y) := − , ∀x, y ∈ R. 1 + |x| 1 + |y| As known, the metric d induces the Euclidean topology on R. Any self-homeomorphism of R, being a monotone function, sends inﬁnitely close sets in d to inﬁnitely close sets in d. In other words, any self-homeomorphism of the real numbers space R with the Euclidean topology is a p-isomorphism from R, endowed with the metric proximity naturally associated with d, to itself, see Example 2.1.4. 2.4. Comparison. If δ1 and δ2 are EF-proximities on a same set X, we say δ1 is ﬁner than δ2 , or δ2 is coarser than δ1 , if and only if A δ1 B implies A δ2 B. Hence, two remote sets remain so in any ﬁner proximity. In the language of strong inclusions, 1 is ﬁner than 2 iﬀ A 2 B implies A 1 B. If X is a T2 locally compact space, then the Alexandroﬀ proximity is coarser ˇ than the Freudenthal proximity, which in turn is coarser than the Cech proximity.

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When X is the set of the real numbers equipped with the Euclidean topology, then the hierarchy is strict. The ﬁner-than relation is a partial order. Every family of EF-proximities on a non-empty set X admits inﬁmum and supremum w.r.t. the ﬁner-than relation. To verify the existence of inﬁma and consequently of suprema, we recall that a dyadic rational r is any rational number in the unit interval [0, 1] of the type 2mn for some integer n > 0 and m = 0, 1, . . . , 2n . Given a family {δλ } of EF-proximities on a same set X, say that A B if and only if for each dyadic rational r in the open unit interval ]0, 1[ there exists a subset Ur in X so that whenever r < s then: A δλ Ur δλ Us δλ B,

∀δλ .

In this way we produce a p-neighbourhood relation on X. Furthermore, the relation induces the ﬁnest EF-proximity coarser than each δλ in the given family. Of course, the supremum of {δλ } is obtained as the inﬁmum of all proximities ﬁner than each δλ . The trivial and discrete proximity are indeed, respectively, the minimum and the maximum of the ﬁner-than relation. 3. Topology associated with a given proximity space 3.1. Some basics. We explore topologies underlying EF-proximities and collect some of their basic properties. When restricted to points and sets, every EF-proximity δ on a set X determines a topology on X by the closure operator naturally associated to the proximity δ. We take the δ-closure of any subset A as the set of all points x in X such that the singleton set {x} is δ-close to A: clδ (A) = {x ∈ X : {x} δ A}. The topology τ (δ) induced by the closure operator clδ on X is known as the topology naturally associated with the proximity δ or as the natural topology underlying δ. We would like to point out that: (i) An EF-proximity δ is separated if and only if (X, τ (δ)) is a T1 space. (ii) The neighbourhoods of a point x in (X, τ (δ)) are the sets U in X for which {x} is δ-remote from X \ U . (iii) A closed set A and a compact set B are remote in any EF-proximity if and only if they don’t share a common point. (iv) A δ B if and only if clδ (A) δ clδ (B). (v) The intersection of all p-neighbourhoods of a subset A is the closure of A in the underlying topology. (vi) If A δ B, then B is a τ (δ)-neighbourhood of A, but the converse fails. When no confusion is possible we drop the δ. If (X, τ ) is a topological space, we say that X admits a compatible EF-proximity, or is EF-proximisable, if there is an EF-proximity δ on X such that τ = τ (δ). Trivially, the discrete proximity induces the discrete topology and the indiscrete proximity the indiscrete topology as well. All EF-proximities on topological spaces with special properties, introduced in Examples 2.1.3 through 2.1.7, induce as natural topologies the initial ones. In any of those cases a point x is close to a subset A if and only if x belongs to the topological closure of A.

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Diﬀerent proximities may give rise to the same topology. Thus, the correspondence between proximities on X and topologies on X is many-to-one, so that a proximity on X actually carries more structure on X than a topology. Taking A to be a point, we achieve the following: Proposition 3.1.1. Every proximity function between two EF-proximity spaces is also a continuous function between the underlying topological spaces. In general the converse is not true, but it is true in the case where the space X is compact, as we will see in the latter subsection. Proposition 3.1.2. Every p-isomorphism between two proximity spaces is also a homeomorphism between the underlying topological spaces. Of course, any topological invariant is a proximity invariant too. Recall that a closed set in a topological space is said to be regular closed if it is the closure of an open set. The use of the Urysohn Lemma in proximity is justiﬁed by the following betweenness property . The density axiom (δ 5) is strengthened by locating a regular closed set in between: Proposition 3.1.3. Let δ be an EF-proximity. If A δ B then there exists a regular τ (δ)-closed set C such that A δ int(C) ⊂ C δ B. By applying the Urysohn Lemma it results that: Theorem 3.1.4. Let (X, δ) be an EF-proximity space. If A is δ-remote from B, then there exists a continuous function f from X to the unit interval [0, 1], equipped with the Euclidean metric, such that f (A) = 0 and f (B) = 1. So taking A as a point, it follows that: Theorem 3.1.5. If (X, δ) is an EF-proximity space, then the underlying topological space (X, τ (δ)) turns out to be completely regular. Furthermore, (X, τ (δ)) is Tychonoﬀ if and only if δ is separated. Having seen that a completely regular space admits the functionally indistinˇ guishable or Cech proximity as a compatible EF-proximity, we now can deduce: Theorem 3.1.6. A topological space is EF-proximisable if and only if it is completely regular. 3.2. New proximities from old ones. Subspaces. Given an EF-proximity space (X, δ) and A ⊂ X, an EF-proximity δA is induced on A in a simple way by saying that B δA C if and only if B δ C. The proximity δA is called the relative proximity on A and we refer to A with this proximity as a subspace of (X, δ). There is a nice match: the topology induced by δA is the relative topology on A. Products. For each α ∈ A, let (Xα , δα ) stand for an EF-proximity space. The product proximity on X = Xα is deﬁned as the coarsest proximity for which each projection πα is a proximity function. The topology underlying the product proximity is the product topology. Quotients. Let ﬁnally (X, δ) stand for an EF-proximity space and Y for a quotient of X. The quotient proximity on Y is the ﬁnest that makes the canonical map of X onto Y into a proximity function. Unfortunately, in general, the quotient proximity does not induce the quotient topology.

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Inﬁma and suprema. Now, let X stand for a completely regular space. Every family of EF-proximities which are all compatible with X, i.e., having as natural underlying topology that one carried by the space X, admits as supremum an EFproximity once again compatible with X. Unfortunately, the same result doesn’t hold for the inﬁma, even in the ﬁnite case. Thus, the class of all EF-proximities compatible with a same completely regular space X, when it is carrying the ﬁner-than partial order, is a complete uppersemilattice, whose ﬁnest proximity is the functionally indistinguishable one. That class admits minimum, equivalent being a complete lattice, if and only if the space X is T2 locally compact. In that case the minimum results in the Alexandroﬀ proximity. 4. Interplay between proximity and uniformity Uniform spaces are a natural example of EF-proximity spaces and among them metric spaces play a pivotal role. We will brieﬂy illustrate the intensive interaction between uniformity and proximity. 4.1. Some background. We propose two possible approaches to uniformity: the diagonal one, largely accepted as the orthodox concept, [70], and the covering one, rejected as being heretical but extensively adopted in [25]. For a complete view see [27]. Diagonal deﬁnition. A diagonal uniformity on a non-empty set X is a collection U of subsets of X×X called diagonal neighbourhoods or surroundings, which satisﬁes the following axioms: (1) If U ∈ U, then U contains the diagonal ∆ = {(x, x) : x ∈ X}. (2) If U is in U and V is a subset of X × X containing U , then V is in U. (3) If U and V are in U, then U ∩ V is in U. (4) If U is in U, then there exists V in U such that V ◦ V is contained in U , i.e., whenever (x, y) and (y, z) are in V then (x, z) is in U . (5) If U is in U, then there exists V in U such that V −1 = {(x, y) : (y, x) ∈ V } is contained in U , When X carries such a structure, we call the pair (X, U) a uniform space. The uniform space (X, U) is called separated if and only if {U : U ∈ U} = ∆. A base of a uniformity U is a subcollection B of U such that any diagonal neighbourhood of U contains some member of B. For x ∈ X and U ∈ U, the U -enlargement of x is deﬁned by: U [x] := {y ∈ X : (x, y) ∈ U }.

And the U -enlargement of a subset A of X is deﬁned as: U [A] := x∈A U [x]. If (X, d) is a metric space, the metric uniformity naturally associated with d, usually denoted as U(d), admits as basic diagonal neighbourhoods the subsets V of X × X deﬁned by: V := {(x, y) ∈ X × X : d(x, y) < } being a positive real number. It is well-known that the family {U [x] : x ∈ X, U ∈ U} is neighbourhood-system for a topology τ (U) naturally associated with U. A uniform space (X, U) is said to be totally bounded if for each diagonal neighbourhood U in U there exists a ﬁnite number of points x1 , . . . , xn in X such that

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X = U [x1 ] ∪ · · · ∪ U [xn ]. Whenever the underlying topology τ (U) is compact, then U is totally bounded. A uniformity U is ﬁner than a uniformity V if U contains V. Covering deﬁnition. Before introducing the concept of covering uniformity on a set X, we recall that a cover C of a non-empty set X is a collection of subsets of X whose union is X. Further, we recall that, given two covers C, D of a same set X, we say that C is a reﬁnement of D, and write C < D, if any set in C is contained in some set in D. If C is a cover of X and A ⊂ X, then the star of A with respect to C is deﬁned as: St(A, C) := {C ∈ C : C ∩ A = ∅}. A cover of a uniform space (X, U) is a uniform cover if it is reﬁned by a cover of the form {U [x] : x ∈ X} for some U ∈ U, or consisting, as it is usual to say, by sets of the same size. Recall that a cover C is called a star reﬁnement of a cover D, written C ∗< D, if the cover St C = {St(A, C) : A ∈ C} reﬁnes D. A covering uniformity on a set X is a collection µ of covers of X with the following properties: (a) if C1 , C2 ∈ µ then there exists C3 ∈ µ such that C3 ∗< C1 and C3 ∗< C2 ; (b) if C < C and C ∈ µ, then C ∈ µ; where < stands for “reﬁnes” and ∗< for “star reﬁnes”. If µ is a covering uniformity, a subcollection ν of µ is a base for µ if any cover in µ is reﬁned from some cover in ν. Given any family µ of covers of a set X which satisﬁes (a) and (b), the collection of all supersets of {A × A : A ∈ C}, as C runs in µ, is a diagonal uniformity on X, whose uniform covers are precisely the elements of µ. Conversely, the collection of all covers reﬁned from a uniform cover of a diagonal uniformity generates a covering uniformity that, in turn, induces the starting one as diagonal counterpart. The dual diagonal uniformity U of a covering uniformity µ is totally bounded if and only if µ admits a base consisting of ﬁnite covers. 4.2. Proximity naturally associated with a uniformity. Let (X, U) stand for a uniform space. Say that A is U-remote from B if and only if there exists a diagonal neighbourhood U such that U [A] ∩ B = ∅. If we denote by δ(U) the relation U-remote, then δ(U) is an EF-proximity, usually known as either the natural EF-proximity associated with the uniformity U or the natural EF-proximity underlying the uniformity U. Also, the proximity δ(U) is separated if and only if the uniformity U is separated. In the covering language, A is U-remote from B if and only if there is a uniform cover C such that St(A, C) ∩ St(B, C) = ∅. By the Urysohn Lemma A is U-remote from B if and only if there is an uniformly continuous Urysohn function that separates A and B. A uniformity U and the associated proximity δ(U) both induce on the base space the same topology. But the proximity induced by a product uniformity does not need to be the product proximity, even though both produce the product topology. In other words, products of p-isomorphic uniform spaces don’t need to be p-isomorphic.

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Let (X, U) be a uniform space, denote by FU the family of all real-valued, uniformly continuous and bounded functions on X. Set: U (f, ) := {g ∈ FU : |f (x) − g(x)| < ,

∀x ∈ X}.

as f runs in FU and is a positive real number and S := {U (f, ) : f ∈ FU , > 0}. ∗

Then, the collection U of supersets of ﬁnite unions of elements of S is a diagonal totally bounded uniformity, coarser than U and compatible with X. But more so, if δ, δ ∗ are the proximities naturally induced by U, U ∗ respectively, then: A δ B if and only if A δ ∗ B. The uniformity U ∗ , introduced in [56], is known as the Samuel uniformity associated with U. Moreover, the uniform completion of the uniform space (X, U ∗ ) is known as the Samuel compactiﬁcation of (X, U). Thus, we now would like to point out that a diagonal uniformity yields more structure on a set X than a proximity, because every uniformity, as we have seen above, generates a proximity in a very natural way, while diﬀerent uniformities may give rise to the same proximity. As such, proximity appears to be located somewhere between uniformity and topology. Let now (X, U) and (Y, V) be uniform spaces. Then: Proposition 4.2.1. Every uniformly continuous function f : (X, U) → (Y, V) is also a proximity function between the natural underlying EF-proximity spaces. So, we have that the proximity invariants of uniform spaces are also uniform invariants. We would like to underline that an essential feature of proximity lies in the fact that many of the interesting properties of uniform spaces turn out not only to be uniform but more proximity invariants. Moreover: Proposition 4.2.2. If V is a totally bounded uniformity, then a function f : (X, U) → (Y, V) is uniformly continuous if and only if it is a proximity function relative to the natural underlying EF-proximities. Hence: Proposition 4.2.3. A totally bounded uniformity is coarser than any uniformity that induces the same proximity. A consequence of previous Proposition 4.2.3 and of the fact that any uniformity compatible with a compact space is totally bounded is the following central result: Theorem 4.2.4. Any T2 compact space admits as unique compatible EF-proximity the elementary one: two sets are close iﬀ their closures intersect. 4.3. Duality between proximities and totally bounded uniformities. Proximities are revealed here as counterparts of totally bounded uniformities. Let (X, δ) be an EF-proximity space. We say U = {Uα : α ∈ A} is a p-cover of the EF-proximity space (X, δ) if and only if there is a cover V = {Vα : α ∈ A} of X such that Vα δ Uα for each α ∈ A. By using the uniform covering technique, it can be proven that:

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Theorem 4.3.1. Let (X, δ) be an EF-proximity space. Then: (a) The collection of all ﬁnite p-covers of X is a base for a totally bounded uniformity on X, µδ . (b) The EF-proximity naturally associated with µδ is precisely δ. (c) µδ is the only totally bounded uniformity naturally giving the proximity δ. We can easily deduce that the assignment (X, δ) → (X, µδ ) is an equivalence between the category of proximity spaces with proximity functions and the category of totally bounded uniform spaces with uniformly continuous functions. 4.4. Metric proximity. Metric proximity is a pivotal focus in proximity theory. Any metric space (X, d) is carrier of the natural proximity: A, B ⊂ X, A δ B if and only if d(A, B) = 0, where d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}. Of course, metric proximity is a very special case and in the heart of metric proximity resides one of most impressive results: the Efremoviˇc Lemma. Lemma 4.4.1 (Efremoviˇc Lemma). Let > 0. If {xn }, {yn )} are two sequences for which d(xn , yn ) > , for each n ∈ N, then there exist subsequences {xnk }, {ynk } such that d(xnh , ynk ) ≥ 4 for each h, k in N. By the Efremoviˇc Lemma we deduce absolutely nontrivial corollaries. Corollary 4.4.2. Any function between metric spaces which is a proximity function relative their underlying metric proximities is also a uniformly continuous function relative to their underlying uniformities. An immediate consequence of Corollary 4.4.2 is: Uniform geometry and proximity geometry agree in the metric context! That precisely means: any uniform invariant of metric spaces is a proximity invariant too. By using the uniform version of the Efremoviˇc Lemma, see [45], we conclude with: Corollary 4.4.3. If a function from a metric space (X, d) to a uniform space (Y, U) is a proximity function relative to the metric proximity induced by d on X and the proximity δ(U) induced by U on Y , then it is uniformly continuous relative to the metric uniformity induced by d and U. 4.5. Proximity simpliﬁes uniformity. Let X be a completely regular space, therefore uniformisable and EF-proximisable. We say that two uniformities U, V, both compatible with X, are equivalent if and only if they share the same proximity on the space X. From (b) of Theorem 4.3.1 it follows that: Proposition 4.5.1. Each equivalence class of uniformities contains exactly one totally bounded uniformity which is also the smallest uniformity in the class. Unfortunately, the supremum of all uniformities of a same proximity class does not belong, in general, to that class. But, Corollary 4.4.2 yields: Proposition 4.5.2. If a proximity class of uniformities contains a metrisable uniformity, then this achieves the maximum in the class.

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We here approach a tricky situation. In [4], Alfsen and Njastad ˙ axiomatised the properties of the union of all uniformities of a same proximity class. We exhibit now another corollary of the Efremoviˇc Lemma: Corollary 4.5.3. Two metrics are uniformly equivalent if and only if they share the same proximity class. 5. Compactiﬁcations and proximities: Smirnov theorem 5.1. Ends. The Smirnov way of constructing compatiﬁcations for EF-proximity spaces is based on the notion of end. Let (X, δ) be a separated EF-proximity space. A ﬁlter F of X is called a δﬁlter or also a round ﬁlter of X if and only if any set F that belongs to F is a δ-neighbourhood of some set G also belonging to F, that is, to say for each F ∈ F there is G ∈ F such that G δ F . A round ﬁlter that is not contained in any other round ﬁlter is an end. Some basic properties hold: • Zorn’s Lemma implies that any round ﬁlter is contained in some end. • The system of neighbourhoods of any point is an end. • The separation axiom implies that any end can converge to a unique point. The following characterisation of compactness by ends plays a basic role in the Smirnov construction: Proposition 5.1.1. An EF-proximity space X is compact if and only if any end has a non-empty intersection. 5.2. Signiﬁcant steps in the construction of the Smirnov compactiﬁcation. Let (X, δ) be an EF-proximity space and s(X) stand for the set of all ends of X. A subset A of X absorbs a set of ends C if A belongs to any end in C. A suitable proximity can be induced in s(X) by saying: • C is close to D if and only if there exist two close subsets A, B in X absorbing C, D respectively. Equipped with this proximity: • s(X) turns out to be a separated EF-proximity space whose underlying topology is compact. s(X) is the Smirnov compactiﬁcation of (X, δ). Hence, from Theorem 4.2.4: • s(X) admits as unique compatible proximity the elementary one. Furthermore: • The function that identiﬁes any point of x in X with the end of all neighbourhoods of x is a proximity dense embedding of X in s(X). Thus: • Two sets in X are δ-close iﬀ their closures in s(X) don’t intersect. We summarise all previous issues in the most important theorem in proximity: Theorem 5.2.1 (Smirnov Compactiﬁcation Theorem). A topological space admits a compatible EF-proximity if and only if it is a subspace of a T2 compact space.

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Equivalently: Theorem 5.2.2. Any separated EF-proximity is the restriction of an elementary proximity: two subsets in X are δ-close if and only if their closures in the Smirnov compactiﬁcation of (X, δ), s(X), share a common point. We now list some lifting properties for proximity functions. • Any proximity function f : X → Y admits a unique continuous lifting from s(X) to s(Y ). • Any proximity function f : X → [0, 1] can be lifted continuously to s(X). The previously mentioned properties guarantee the essential uniqueness, up to homeomorphism, of the Smirnov compactiﬁcation s(X) of every EF-proximity space X, and also the characterisation of s(X) as the unique, up to homeomorphism, T2 -compactiﬁcation which any real-valued, bounded proximity function on X can be lifted continuously to. Moreover, the totally bounded uniformity naturally associated with the Smirnov compactiﬁcation s(X) happens to be the weakest one compatible with the starting EF-proximity space X. In conclusion this is a full and completely satisfactory answer to Alexandroﬀ’s query. We point out that the Smirnov compactiﬁcation associated with an EF-proximity δ and the Samuel compactiﬁcation, [56], associated with any uniformity generating the proximity δ agree. When they make sense, the Smirnov compactiﬁcations of a Tychonoﬀ space X associated with the Alexandroﬀ proximity, the Freudenthal proximity, the funcˇ tionally indistinguishable or Cech proximity are the one-point compactiﬁcation, the Freudenthal compactiﬁcation (the maximal, up to homeomorphism, T2 -compactificˇ ation γ(X) of X whose remainder γ(X)\X is zero-dimensional) and the Stone–Cech compactiﬁcation respectively. It can be shown that: Proposition 5.2.3. If δ1 and δ2 are compatible EF-proximities on a Tychonoﬀ space X, then δ1 is ﬁner than δ2 if and only if there is a continuous function f : s1 (X) → s2 (X) such that the restriction of f to X is the identity. The previous result yields the following issue: there is an order preserving isomorphim, up to equivalence, between the class of the T2 -compactiﬁcations of a Tychonoﬀ space X, carrying the usual partial order, and the uppersemilattice of all EF-proximities compatible with X, under the ﬁner-than relation. 6. Leader’s contribution 6.1. Clusters. A cluster c from an EF-proximity space X is a class of subsets of X satisfying the following three conditions: (a) If A and B belong to c, then A is close to B. (b) If A is close to every C in c, then A belongs to c. (c) If A ∪ B belongs to c, then either A or B belongs to c. It can be easily seen that the class cx of all sets close to a point x is a cluster. A cluster that contains a singleton set is a point-cluster, otherwise it is free.

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Leader established the following focal properties of clusters: • A is close to B if and only if there exists a cluster to which both A and B belong. • An EF-proximity space X is compact if and only if every cluster from X is a point-cluster. Leader’s construction develops as a simulation of the already examined Smirnov procedure in which the concept of end is replaced by the more manageable notion of cluster. In his investigations S. Leader concentrated his attention on the functional method of reconstructing proximities, an external standpoint which uses the channel of rings of real-valued functions. Namely: Theorem 6.1.1. A is remote from B in an EF-proximity space X if and only if there exists a proximity function of X into the unit interval [0, 1], equipped with the Euclidean metric proximity, which takes A onto {0} and B onto {1}. Theorem 6.1.2. The class of all bounded, real-valued, proximity functions on an EF-proximity space is a Banach algebra under the uniform norm. Furthermore, Leader gave the following proximity version of the Stone–Weierstrass theorem: Theorem 6.1.3. Let A be the algebra of all bounded, real-valued, proximity functions on an EF-proximity space X. Let B be a subalgebra of A such that: (1) B contains a function whose range is bounded away from 0, (2) Given A remote from B in X, there exists some f in B such that f (A) is remote from f (B). Then B is uniformly dense in A. 6.2. Local proximity spaces. Blending proximity with boundedness gives local proximity. Local proximity spaces play the same role in the construction of T2 -local compactiﬁcations of a Tychonoﬀ space as classical proximities in the construction of T2 -compactiﬁcations. Basically, Leader parallels the Smirnov compactiﬁcation procedure. Let X be a Tychonoﬀ space. Any given T2 -local compactiﬁcation l(X) of X takes up two features of X. The former one is the separated EF-proximity on X induced by the one-point compactiﬁcation of l(X). The latter one is the boundedness formed by all subsets of X whose closures in l(X) are compact. By joining proximity and boundedness in the unique concept of local proximity, Leader put this example in abstract. Surprisingly, in the sequel, see next subsection 7.3, looking for completely regular hypertopologies, we meet again a combination of proximity and boundedness even though under the diﬀerent name of uniformly Urysohn family. A non-empty collection B of subsets of a set X is called a boundedness in X if and only if: (i) A ∈ B and B ⊂ A implies B ∈ B, and (ii) A, B ∈ B implies A ∪ B ∈ B. The elements of B are called bounded sets. Of course, singletons, hence ﬁnite sets, are bounded. A local proximity space (X, β, B) consists of a set X together with an EFproximity β on X and a boundedness B in X which is subject to the following axioms:

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(a) If A ∈ B, C ⊂ X and A C then there exists some B ∈ B such that A B C, where is the strong inclusion of β. (b) If A β C, then there is some B ∈ B such that B ⊂ C and A β B. Whenever β is separated, then (X, β, B) is said to be a separated local proximity space. Every separated local proximity space can be seen as a localisation of a separated EF-proximity modulo a free cluster. Namely, the unbounded sets in a local proximity space over a non locally compact space ﬁll a free cluster. Conversely, every free cluster c from an EF-proximity δ determines a local proximity, when declaring all sets in c unbounded and declaring two subsets close if they both are unbounded or δ-close. An exhaustive way of producing local proximities compatible with a Tychonoﬀ space X consists by removing just one point from any Smirnov compactiﬁcation of X. We conclude with the following result: Proposition 6.2.1. Let (X, τ ) be a Tychonoﬀ space. Then there exists a bijection between the set of all up to equivalence T2 locally compact dense extensions of (X, τ ) and the set of all separated local proximities on (X, τ ). 7. Proximity in complex frameworks After acquiring basic results in the proximity theory it appears natural to extend proximity to more complex frameworks such as function spaces and hyperspaces. The ﬁrst attempts in these directions were thanks to S. Leader and L.J. Nachman, [42]. 7.1. Proximity in function spaces. The value of a functional convergence resides in preserving characteristic features in the limit passage. For instance, in the proximity context a convergence for functions, whose range is an EF-proximity space, is nice if it guarantees the limit of a net of proximity functions to be a proximity function in turn. Let X be a non-empty set and Y a Tychonoﬀ space. Denote the set of all functions from X to Y as Y X . Uniformities on Y yield a uniform control on Y X . We will see how. Every uniformity U compatible with the range space Y induces on Y X the uniformity of uniform convergence w.r.t. U, having as basic diagonal neighbourhoods the sets: ˆ := {(f, g) ∈ Y X × Y X : (f (x), g(x)) ∈ U, U

∀x ∈ X}

as U runs over all diagonal neighbourhoods in U. The uniformity of uniform convergence w.r.t. U in turn generates on Y X the uniform topology or the topology of uniform convergence w.r.t. U. Leader, as a pioneer, approached proximity in function spaces by generalising the Weierstrass uniform convergence, as previously described. Let X be a set and (Y, δ) an EF-proximity space. Leader introduced the convergence in proximity, that we also call Leader convergence, by saying that: a net {fλ } in Y X converges in proximity to a function f in Y X if and only if for A ⊂ X and B ⊂ Y then f (A) δ B implies fλ (A) δ B eventually. It is easy to show that, whenever X is a topological space, the collection C(X, Y ) of all continuous functions from X to Y is closed under any convergence in

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proximity. Also, whenever X and Y are both EF-proximity spaces, the collection P (X, Y ) of all proximity functions from X to Y is closed under the convergence in proximity. Furthermore, he proved that: Theorem 7.1.1. If X is a non-empty set and (Y, δ) is an EF-proximity space, then the convergence in proximity on Y X is weaker than the uniform convergence relative to any uniformity compatible with the assigned proximity δ on the range space Y . Leader conjectured that the converse was not true. But, later on, in [42], Nachmann veriﬁed Leader’s conjecture as being correct. Moreover, he proved the following very nice result indeed. Theorem 7.1.2. If X is a non-empty set and (Y, δ) is an EF-proximity space, then the convergence in proximity on Y X agrees with the uniform convergence relative to the unique smallest totally bounded uniformity compatible with the assigned proximity δ on Y . Recall that, whenever X and Y are topological spaces, a set-open topology on C(X, Y ) has as subbasic open sets the ones: [C, A] := {f ∈ C(X, Y ) : f (C) ⊂ A}, where C runs over a given family α, usually a network, of closed subsets in X and A is open in Y . A collection α of subsets of a topological space X is said to be a network on X provided that for any point x in X and any open subset A of X containing x there is a member C in α which contains x and is contained in A. A network α is a closed network if any element in α is closed and a hereditarily closed network if any closed subset of any element in α is again in α. When α is the family of all compact subsets of X, then we get the compact-open topology, which is the prototype in the class of set-open topologies. When restricted to C(X, Y ), the topology of convergence in proximity can be reformulated as a set-open type topology. Namely, it can be proven that it admits as subbasic open sets those ones of the form: (∗)

[C, A]δ := {f ∈ C(X, Y ) : f (C) δ A},

where C runs over all closed subsets in X and A over all open subsets in Y . This recasting takes up the opportunity of a closeness between functions, whose range carries a proximity, on members of a network. Going inside this example, after capturing the proximal nature of the compact-open topology (every compact set, if contained, is strongly contained in every open set in every EF-proximity), a new wide class of function space topologies was introduced in [10]. Let X be a topological space, α a network in X and (Y, δ) an EF-proximity space. The proximal set-open topology, in short PSOT, on C(X, Y ) induced by α and δ, is generated by the subbasic open sets already introduced in this subsection by (∗) but where now C is forced to run in α, while A once again is open in Y . Of course, when α is the family of all closed sets, then we get the topology of convergence in proximity relative to δ. While, when α is the family of all compact sets in X, we get the compact-open topology. The proximal set-open topologies have nice properties. We like to exhibit the following Arens-type result.

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Theorem 7.1.3. Let α a closed, hereditarily closed network in X and δ an EF-proximity on Y . Then the PSOT relative to α and δ is the topology of uniform convergence on members of α w.r.t. the unique totally bounded uniformity compatible with δ. 7.2. Proximity and homeomorphism groups. Let (X, δ) be an EF-proximity space, denote as H(X) the full group of self-homeomorphisms of (X, τ (δ)) and as PSOTδ the topology of convergence in proximity relative to δ. It easy to show that: Theorem 7.2.1. If X is a Tychonoﬀ space, then any PSOT, relative to a compatible EF-proximity, makes the evaluation function e : (f, x) ∈ H(X) × X → f (x) ∈ X into a continuous function. And furthermore: Theorem 7.2.2. If (X, δ) is an EF-proximity space, then PSOTδ makes the full group of p-isomorphisms of X into a topological group, i.e., provides continuity of the usual product and the inverse function. We summarise the previous two results in: Theorem 7.2.3. If (X, δ) is an EF-proximity space, then the full group of pisomorphisms of X equipped with PSOTδ , continuously acts on X by the evaluation function e : (f, x) ∈ H(X) × X → f (x) ∈ X. But more so: Theorem 7.2.4. Whenever X ia a T2 locally compact space, then the PSOT associated with the Alexandroﬀ proximity, known as the g-topology, is the weakest topological group topology on H(X) which makes the evaluation function into a continuous action of H(X) on X. And also: Theorem 7.2.5 ([9]). Whenever X ia a T2 , rim-compact and locally connected space, then the PSOT associated with the Freudenthal proximity is the weakest topological group topology on H(X) which makes the evaluation function into a continuous action of H(X) on X. And ﬁnally: Theorem 7.2.6 ([9]). Whenever X is the rational numbers space Q, equipped ˇ with the Euclidean topology, then the PSOT associated with the Cech proximity is the weakest topological group topology on H(Q) which makes the evaluation function into a continuous action of H(Q) on Q. 7.3. Proximity in hyperspaces. A hyperstructure is interesting if it shares nice features such as metrisability, uniformisability, completeness and the like with the basic structure. In [41], L.J. Nachman approached proximity in the hyperspace by drawing uniform issues into proximity. Let X stand for a topological space and CL(X) denote the set of all closed non-empty subsets of X, which we refer to as the hyperspace of X. It is well-known that if X is a Tychonoﬀ space, every diagonal uniformity U compatible with it naturally induces the Hausdorﬀ hyperuniformity, usually denoted as H(U), on the hyperspace CL(X). The Hausdorﬀ hyperuniformity H(U)

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admits as subbasic diagonal neighbourhoods all sets of the kind: H(U ) := {(A, B) ∈ CL(X) × CL(X) : A ⊂ U [B], B ⊂ U [A]}, where U runs over all diagonal neighbourhoods in U. Whenever the uniformity U is metrisable and d is a bounded metric compatible with it, then the Hausdorﬀ hyperuniformity H(U) is metrised by the Hausdorﬀ metric dH , deﬁned on CL(X) by the usual formula: dH (A, B) := max{e(A, B), e(B, A)}, where e(A, B) := sup{d(a, B) : a ∈ A}. If now (X, δ) is an EF-proximity space, every uniformity U compatible with it yields on the hyperspace CL(X) the hyperproximity, δ(H(U)), naturally associated with the Hausdorﬀ hyperuniformity H(U). But hyperproximities which are produced by the Hausdorﬀ lifting method run over a wide class. Thus, Nachman limited himself to dealing with two basic options appearing to be the most natural ones. Obviously, Nachman considered ﬁrst the hyperproximity deriving, via the Hausdorﬀ lifting method, from the weakest uniformity compatible with δ. He referred to it as the weak hyperproximity and denoted it as Hw (δ). But Hw (δ) presents a strong liability: Hw (δ) is not metrisable when δ is. And then he went on to consider the hyperproximity naturally associated with the supremum of all Hausdorﬀ hyperuniformities generated by uniformities compatible with the base space. Nachman called it the strong hyperproximity and denoted it as Hs (δ). As opposed to the weak hyperproximity, the strong hyperproximity is metrisable (completely metrisable) if and only if the proximity base space is metrisable (completely metrisable). Two facts guarantee this property. The former one: whenever a proximity space is metrisable then the metric uniformity is the maximum in the associated proximity class, see Proposition 4.5.2. The latter one: the Hausdorﬀ metric associated with a metric d is complete if and only if d is complete. Trivially, the weak and strong hyperproximities are usually distinct and they coincide if and only if the proximity space is completely bounded, that is, it admits only one compatible uniformity. This digression makes it clear that the depicted method provides no canonical way of equipping hyperspaces of proximity spaces with proximity. This subject was taken up again by S.A. Naimpally and collaborators, who, in an extensive and close cooperation, planned to capture the proximity counterpart of hypertopologies splitting in two halves. Let (X, τ ) be a topological space. The classical Vietoris topology on CL(X), prototype of hyperspace topologies, splits in two natural halves: the hit part and the miss part. The hit part of the Vietoris topology, V − , is generated by the collection of hit sets: F − := {E ∈ CL(X) : E ∩ A = ∅, ∀A ∈ F} where F is a ﬁnite family of open sets in X. The miss part of the Vietoris topology, V + , is generated by the collection of miss sets: A+ := {E ∈ CL(X) : E ⊂ A} where A is open in X. The classical Vietoris example has been naturally generalised in the notion of hit and miss hypertopology. Let ∆ be a family of closed of X. The hit and miss topology on CL(X) associated with ∆, τ (∆), is generated by the join of the hit sets

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F − , where F is a ﬁnite family of open sets in X, with the miss sets A+ , where now A is an open subset of X whose complement is in ∆. When ∆ is formed by the compact sets X, then τ (∆) is the Fell topology, see [7]. If the space X carries an EF-proximity δ then a proximity variation of τ (∆) can be displayed by replacing the miss sets with far-miss sets: A++ := {E ∈ CL(X) : E δ A}. Let once again ∆ be a family of closed sets in X. Then σ(∆), the hit and far-miss topology on CL(X) associated with ∆, is generated by the join of the hit sets F − , where F is a ﬁnite family of open sets in X, with the miss sets A++ , where A is once again an open subset of X whose complement is in ∆. But, in general, a hit and far-miss topology is not completely regular and thus not EF-proximisable. In looking for completely regular hypertopologies of hit and far-miss type, we hit the notion of uniformly Urysohn family, that is a proximity character despite the name. Let (X, δ) be an EF-proximity space and ∆ a family of non-empty closed sets of X containing all singletons, and for simplicity, closed under ﬁnite unions. Then ∆ is said to be a uniformly Urysohn family if whenever D is in ∆, A is in CL(X) and D is remote from A, then there exists D in ∆ such that D δ D ⊂ X \ A. Whenever the space X is normal, then CL(X) is uniformly Urysohn in the elementary EF-proximity. The collection of all non-empty compact sets of a T2 space X is uniformly Urysohn if and only if the space X is locally compact. A uniformly Urysohn family is very close to the boundedness in a local proximity space, see subsection 6.2. The nice interposition property characterising uniformly Urysohn families yields the following: Theorem 7.3.1. Let (X, δ) be an EF-proximity space and ∆ a family of nonempty closed sets in X, closed under ﬁnite unions, containing all singletons. Then the hit and far-miss topology on CL(X), σ(∆), associated with ∆ is proximisable if and only if ∆ is a uniformly Urysohn family. See [7]. Variations of the hit part are also possible, for example forcing F to run over all locally ﬁnite open families or over all locally ﬁnite open families of cardinality less than or equal to a ﬁxed cardinal number. The mixing of proximity variations of the miss part with variations of the hit part is not only a fascinating game to play but also an eﬃcient method of producing hyperproximities. 8. Lattice theoretical approach and point-free geometry 8.1. Proximity lattices. Proximity lends itself to a lattice theoretical approach. The ﬁrst attempt to built a point-free analogue to the notion of proximity ˇ space is that of proximity distributive lattice of A.S. Svarc, under the leadership of Efremoviˇc, [67]. Proximity lattice arises as an abstraction of the lattice of closed sets of an EFproximity space. EF-proximities, strong inclusions, proximity functions, clusters, basic construction procedures and many links among them took part in the lattice theoretic machinery. There is a link between proximity relations in their dual formulation and apartness spaces that are a framework for constructive topology, [67].

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The following sounds as a intriguing invitation, [19]: “Apartness seems particularly attractive in view of its potential applications in areas of semantics in which continuous phenomena play a role; there, it is distinctness between diﬀerent states which is observable, not equality.” 8.2. Point-free geometries. Tools, arguments, procedures and all techniques used in the classic proximity have been recast and intensively employed in formalisations of region-based theory of space or point-free geometries. In the Euclidean geometry, points are the elementary entities, while point-free geometry refers directly to sets, in the case regions, and relations between sets rather than referring to points and sets of points. Point-based spatial construction is dominant but the region-based spatial theory is quoted in a naive knowledge of space, as shown in recent and past literature. Proximity boolean algebra arises as an abstraction of the complete boolean algebra of all regular closed subsets of an EF-proximity space. The connection algebras, traditional frameworks for point-free geometries, and local connection algebras can be interpreted as boolean algebras of regular closed sets of an EFproximity space or of a local proximity space respectively, [11]. The reference list is far from exhaustive. Textbooks [7, 20, 24, 27, 45, 73] are strongly recomended for the necessary background in topology, uniformity, proximity, function spaces, hyperspaces and for an extensive relative bibliography. References [1] P. Aleksandrov, On two theorems of Yu. Smirnov in the theory of bicompact extensions, Fund. Math. 43 (1956), 394–398 (Russian). MR0084128 (18,813e) [2] E. M. Alfsen and J. E. Fenstad, On the equivalence between proximity structures and totally bounded uniform structures, Math. Scand. 7 (1959), 353–360, Correction: ibid. 9 (1961), 258. MR0115156 (22 #5958) , A note on completion and compactiﬁcation, Math. Scand. 8 (1960), 97–104. [3] MR0126247 (23 #A3543) [4] E. M. Alfsen and O. Njastad, ˙ Proximity and generalized uniformity, Fund. Math. 52 (1963), 235–252. MR0154257 (27 #4207a) , Totality of uniform structures with linearly ordered base, Fund. Math. 52 (1963), [5] 253–256. MR0154258 (27 #4207b) [6] B. Banaschewski and J.-M. Maranda, Proximity functions, Math. Nachr. 23 (1961), 1–37. MR0165486 (29 #2768) [7] G. Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers, Dordrecht, 1993. MR1269778 (95k:49001) ´ Cs´ [8] A. asz´ ar and S. Mr´ owka, Sur la compactiﬁcation des espaces de proximit´ e, Fund. Math. 46 (1959), 195–207. MR0100827 (20 #7255) [9] A. Di Concilio, Topologizing homeomorphism groups of rim-compact spaces, Topology Appl. 153 (2006), no. 11, 1867–1885. MR2227033 (2007b:54026) [10] A. Di Concilio and S. A. Naimpally, Proximal set-open topologies, Boll. Unione Mat. Ital. Sez. B. Artic. Ric. Mat. 3 (2000), no. 1, 173–191. MR1755708 (2001c:54013) [11] G. Dimov and D. Vakarelov, Contact algebras and region-based theory of space: proximity approach. II, Fund. Inform. 74 (2006), no. 2–3, 251–282. MR2284195 (2007j:68138b) [12] D. Do˘ıˇ cinov, A uniﬁed theory of topological spaces, proximity spaces and uniform spaces, Doklady Akad. Nauk SSSR 156 (1964), 21–24 (Russian), English translation: Soviet Math. Dokl. 5 (1964) 595–598. MR0167956 (29 #5221) [13] T. E. Dooher and W. J. Thron, Proximities compatible with a given topology, Dissertationes Math. Rozprawy Mat. 71 (1970), 41. MR0276920 (43 #2660)

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[41] L. J. Nachman, Hyperspaces of proximity spaces, Math. Scand. 23 (1969), 201–213. MR0251692 (40 #4919) , On a conjecture of Leader, Fund. Math. 65 (1969), 153–155. MR0251693 (40 #4920) [42] [43] S. A. Naimpally and B. D. Warrack, Clusters and ultraﬁlters, Publ. Inst. Math. (Beograd) (N.S.) 8 (22) (1968), 100–101. MR0229210 (37 #4784) , On completion of a uniform space by local clusters, Contributions to extension [44] theory of topological structures (Berlin, 1967) (J. Flacshmeyer, H. Poppe, and F. Terpe, eds.), Deutscher Verlag der Wissenschaften, Berlin, 1968. MR0248737 (40 #1988) , Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, [45] vol. 59, Cambridge University Press, London, 1970. MR0278261 (43 #3992) [46] O. Njastad, ˙ Some properties of proximity and generalized uniformity, Math. Scand. 12 (1963), 47–56. MR0159306 (28 #2523) , On real-valued proximity mappings, Math. Ann. 154 (1964), 413–419. MR0166757 [47] (29 #4030) , On uniform spaces where all uniformly continuous functions are bounded, Monatsh. [48] Math. 69 (1965), 167–176. MR0178453 (31 #2710) [49] , On Wallman-type compactiﬁcations, Math. Z. 91 (1966), 267–276. MR0188977 (32 #6404) [50] W. J. Pervin, Equinormal proximity spaces, Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math. 26 (1964), 152–154. MR0161306 (28 #4514) , On separation and proximity spaces, Amer. Math. Monthly 71 (1964), 158–161. [51] MR0165485 (29 #2767) [52] V. Z. Poljakov, Open mappings of proximity spaces, Doklady Akad. Nauk SSSR 155 (1964), 1014–1017 (Russian). MR0172240 (30 #2460) , Regularity, product and spectra of proximity spaces, Doklady Akad. Nauk SSSR 154 [53] (1964), 51–54 (Russian), English translation: Soviet Math. Dokl. 5 (1964) 45–49. MR0157347 (28 #582) , On some proximity properties determined only by the topology of the compactiﬁca[54] tion, Contributions to Extension Theory of Topological Structures (Berlin, 1967) (J. Flacshmeyer, H. Poppe, and F. Terpe, eds.), Deutscher Verlag der Wissenschaften, Berlin, 1969, pp. 173–178. MR0253279 (40 #6494) [55] F. Riesz, Stetigkeit und abstrakte Mengenlehre, Atti IV Congr. Intern. Mat. (Roma), vol. 2, 1908, pp. 18–24. [56] P. Samuel, Ultraﬁlters and compactiﬁcation of uniform spaces, Trans. Amer. Math. Soc. 64 (1948), 100–132. MR0025717 (10,54a) [57] T. Shirota, A class of topological spaces, Osaka Math. J. 4 (1952), 23–40. MR0050872 (14,395b) [58] Yu. M. Smirnov, On proximity spaces, Mat. Sbornik N.S. 31 (73) (1952), 543–574 (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 38 (1964) 5–35. MR0055661 (14,1107b) , On proximity spaces in the sense of V. A. Efremoviˇ c, Doklady Akad. Nauk SSSR [59] (N.S.) 84 (1952), 895–898 (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 38 (1964) 1–4. MR0055660 (14,1107a) [60] , On completeness of uniform spaces and proximity spaces, Doklady Akad. Nauk SSSR (N.S.) 91 (1953), 1281–1284 (Russian). MR0063014 (16,58e) , On the completeness of proximity spaces, Doklady Akad. Nauk SSSR (N.S.) 88 [61] (1953), 761–764 (Russian). MR0056906 (15,144c) , On the completeness of proximity spaces, Trudy Moskov. Mat. Obˇsˇ c. 3 (1954), [62] 271–306 (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 38 (1964) 37–73. MR0068199 (16,844c) [63] , On the dimension of proximity spaces, Doklady Akad. Nauk SSSR (N.S.) 95 (1954), 717–720 (Russian). MR0068200 (16,845a) , On completeness of proximity spaces. II, Trudy Moskov. Mat. Obˇsˇ c. 4 (1955), [64] 421–438 (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 38 (1964) 75–93. MR0072449 (17,286f) , On the dimension of proximity spaces, Mat. Sbornik N.S. 38 (80) (1956), 283–302 [65] (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 21 (1962) 1–20. MR0150728 (27 #715)

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[66] D. H. Smith, Hyperspaces of a uniformizable space, Proc. Cambridge Philos. Soc. 62 (1966), 25–28. MR0187200 (32 #4654) ˇ [67] A. S. Svarc, Proximity spaces and lattices, Ivanov. Gos. Ped. Inst. Uˇc. Zap. Fiz.-Mat. Nauki. 10 (1956), 55–60 (Russian). MR0087923 (19,436g) [68] J. W. Tukey, Convergence and uniformity in topology, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1940. MR0002515 (2,67a) [69] L. S. Vˆıt¸˘ a, Proximal and uniform convergence on apartness spaces, MLQ Math. Log. Q. 49 (2003), no. 3, 255–259. MR1979131 (2004c:54021) [70] A. D. Wallace, Separation spaces, Ann. of Math. (2) 42 (1941), 687–697. MR0004756 (3,57a) [71] A. J. Ward, On H-equivalence of uniformities: The Isbell–Smith problem, Paciﬁc J. Math. 22 (1967), 189–196. MR0215283 (35 #6125) [72] A. Weil, Sur les espaces a ` structure uniforme et sur la topologie g´ en´ erale, Hermann, Paris, 1937. [73] S. Willard, General topology, Addison-Wesley Publishing Co., Reading, MA, 1970, Reprinted Dover Publications Inc., Mineola, NY, 2004. MR0264581 (41 #9173) Department of Mathematics and Informatics, University of Salerno, 84100 Salerno, Italy

Contemporary Mathematics Volume 486, 2009

An Initiation into Convergence Theory Szymon Dolecki In memory of Gustave Choquet (1915–2006)

Abstract. Convergence theory oﬀers a versatile and eﬀective framework to topology and analysis. Yet, it remains rather unfamiliar to many topologists and analysts. The purpose of this initiation is to provide, in a hopefully comprehensive and easy way, some basic ideas of convergence theory, which would enable one to tackle convergence-theoretic methods without much eﬀort.

Contents 1. Introduction 2. Filters 2.1. Neighborhood ﬁlters 2.2. Principal ﬁlters 2.3. Coﬁnite ﬁlters 2.4. Sequential ﬁlters 2.5. Countably based ﬁlters 2.6. Grills 2.7. Order, ultraﬁlters 2.8. Decomposition 2.9. Stone transform 3. Basic classes of convergences 3.1. Topologies 3.2. Sequentially based convergences 3.3. Convergences of countable character 3.4. Pretopologies 4. Continuity 4.1. Initial convergences 4.2. Final convergences 4.3. Continuity in subclasses

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2000 Mathematics Subject Classiﬁcation. 54A20, 54A05, 54A25, 54B10, 54B15, 54B20, 54B30, 54C10, 54C35, 54D20, 54D30, 54D50, 54D55. I sent a preliminary version of this initiation to Professor Gustave Choquet who, with his habitual exquisite kindness, expressed his appreciation in a postcard of October 2005. c 2009 2008 American Mathematical Society

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4.4. Products 4.5. Powers 5. Adherences 6. Covers 7. Compactness 7.1. Compact families 7.2. Weaker versions of compactness 7.3. Countable compactness 7.4. Finite compactness 7.5. Sequential compactness 8. Adherence-determined convergences 8.1. Pseudotopologies 8.2. Narrower classes of adherence-determined convergences 9. Diagonality and regularity 9.1. Contours 9.2. Hausdorﬀ convergences 9.3. Diagonality 9.4. Regularity 9.5. Interactions between regularity and topologicity 10. Filter-determined convergences 11. Categories of convergence spaces 11.1. Abstract and concrete categories 11.2. Subcategories of convergence spaces 12. Functorial inequalities 13. Quotient maps 14. Power convergences 14.1. Topologicity and other properties of power convergences 15. Hyperconvergences 15.1. Sierpi´ nski topology 15.2. Upper Kuratowski convergence 15.3. Kuratowski convergence 16. Exponential hull of topologies 16.1. Bidual convergences 16.2. Epitopologies 17. Reﬂective properties of power convergences References

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1. Introduction Convergence theory oﬀers a versatile and eﬀective framework for topology and analysis. Yet, it remains rather unfamiliar to many topologists and analysts. The purpose of this initiation is to provide, in a hopefully comprehensive and easy way, some basic ideas of convergence theory, which would enable one to tackle convergence-theoretic methods without much eﬀort. Of course, the choice of what is essential, reﬂects the research experience of the author. A relation between the ﬁlters on a set X and the elements of X, denoted by x ∈ lim F,

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is called a convergence on X, provided that F ⊂ G implies lim F ⊂ lim G, and that the principal ultraﬁlter of every element is in this relation with the element. Each topological space deﬁnes a convergence space (on the same set) to the eﬀect that x ∈ lim F whenever F contains every open set containing x. A convergence deﬁned in this way is said to be topological. Non-topological convergences arise naturally in analysis, measure theory, optimization and other branches of mathematics: in topological vector spaces,1 there is in general no coarsest topology on the space of continuous linear forms, for which the coupling function is continuous; convergence in measure and convergence almost everywhere are, in general, not topological; stability of the minimizing set is, in general, non-topological. A non-topological convergence can be a natural formalism of a stability concept. On the other hand, often a non-topological convergence arises as a solution of a problem formulated in purely topological terms. A crucial example is the convergence structure resulting from the search for a power with respect to topologies τ and σ, that is, the coarsest topology θ on C(τ, σ) such that the natural coupling2 x, f = f (x) be (jointly) continuous from τ × θ to σ. It turned out that this problem has no solution unless τ is locally compact [4], but there is always a convergence, denoted by [τ, σ], which solves the problem. In other terms, the category of topological spaces is not exponential (in a predominant terminology, Cartesian closed ) but it can be extended to an exponential category, that of convergence spaces. Actually there exist strict subcategories of the category of convergence spaces that include all topologies and are exponential (for example, that of pseudotopologies).3 In a fundamental paper [8] Gustave Choquet studies natural convergences on hyperspaces 4, and concludes that some of them are not topological unless the underlying topology is locally compact, but are always pseudotopological. The notion of pseudotopology was born. From a perspective of posterior research, these nontopological convergences were power convergences with respect to a special coupling topology. I do not intend to give here a historical account of convergence theory. Let me only mention that non-topological convergences called pretopologies 5 were already ˇ studied by F. Hausdorﬀ [31], W. Sierpi´ nski [40] and E. Cech [7]. An actual turning point however was, in my opinion, the emergence of pseudotopologies in [8] of Gustave Choquet.

1See [5] of R. Beattie and H.-P. Butzmann. 2C(τ, σ) stands for the set of maps, which are continuous from a topology τ on X to a

topology σ on Z. 3G. Bourdaud showed in [6] that the least exponential reﬂective subcategory of convergences, which includes topologies, is that of epitopologies of P. Antoine [3]. 4A hyperspace is a set of closed subsets of a topological space (more generally, of a convergence space). 5by Choquet in [8]

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2. Filters A fundamental concept of convergence theory is that of ﬁlter. A non-empty family F of subsets of a set X is called a ﬁlter on X if (1)

∅∈ / F;

(2)

G ⊃ F ∈ F =⇒ G ∈ F;

(3)

F0 , F1 ∈ F =⇒ F0 ∩ F1 ∈ F.

Remark 1. The family 2X (of all subsets of X) is the only family that fulﬁlls (2) and (3) but not (1). We call 2X the degenerate ﬁlter on X. As 2X does not fulﬁll all the assumptions above, it is not considered, despite its name, as a (full right) ﬁlter. So by a ﬁlter I mean a non-degenerate ﬁlter, unless I explicitly admit a degenerate ﬁlter. A subfamily B of F such that for every F ∈ F there is B ∈ B with B ⊂ F is called a base of F. We say that B generates F; if B0 and B1 generate the same ﬁlter, then we write B0 ≈ B1 ; in particular if B is a base of F then B ≈ F. If you study topologies, you study ﬁlters, whether you like it or not. In fact, for every ﬁlter F on a given set X there is a unique topology on either X or on {∞} ∪ X determined by F. Let me explain this statement. 2.1. Neighborhood ﬁlters. If X is a topological space, then for every x ∈ X, the set N (x) (of neighborhoods of x) is a ﬁlter.6 Recall that a topology is prime if it has at most one non-isolated point.7 If H is a ﬁlter on Y and X ⊂ Y , then the trace H|X of H on X is deﬁned by H|X = {H ∩ X : H ∈ H}. This is a (non-degenerate) ﬁlter provided that H ∩ X = ∅ for every H ∈ H. A ﬁlter F is free whenever F ∈F F = ∅. If π is a Hausdorﬀ prime topology on Y and ∞ ∈ Y is not isolated, then the trace of Nπ (∞) on X is a free ﬁlter on X. Conversely, Proposition 2. For every free (possibly degenerate) ﬁlter F on X there exists a (unique) Haudorﬀ prime topology on {∞}∪X such that the trace of the neighborhood ﬁlter of ∞ is equal to F. Proof. If F is a free ﬁlter on X, then deﬁne a topology π on the disjoint union Y = {∞} ∪ X so that every x ∈ X is isolated, and Nπ (∞) = {{∞} ∪ F : F ∈ F}.8 Then the trace of Nπ (∞) on X is F. On the other hand,

F = ∅, then there is a unique Proposition 3. If F is a ﬁlter on X and F ∈F topology π on X such that F = Nπ (x) for every x ∈ F ∈F F , and x is isolated for each x ∈ / F ∈F F .9 6A set V is a neighborhood of x if there exists an open set O such that x ∈ O ⊂ V . 7

An element x of a topological space is isolated if {x} is a neighborhood of x.

8This topology is discrete if and only if F is degenerate. 9In fact, this topology π is deﬁned so that a set A is open if either A ∩ T F ∈F F = ∅ or

A ∈ F.

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2.2. Principal ﬁlters. If A ⊂ X then the family (A)• = {B ⊂ X : A ⊂ B} is a ﬁlter, called the principal ﬁlter of A. This ﬁlter is degenerate if and only if A = ∅, because A ∈ (A)• , and A is the least element of it. The family {A} is the smallest base of (A)• . 2.3. Coﬁnite ﬁlters. If A ⊂ X then the family (A)◦ = {B ⊂ X : card(A \ B) < ∞} is a ﬁlter, called the coﬁnite ﬁlter of A. This ﬁlter is degenerate if and only if A is ﬁnite, because A \ ∅ is ﬁnite whenever A is ﬁnite. The family {A \ F : F ⊂ A, card F < ∞} is a base of (A)◦ . 2.4. Sequential ﬁlters. We say that a sequence (xn )n of elements of X generates a ﬁlter S on X (in symbols S ≈ (xn )n ) if the family {{xn : n ≥ m} : m < ∞} is a base of S. A ﬁlter on X is called sequential if there exists a sequence (xn ) = (xn )n that generates it.10 2.5. Countably based ﬁlters. A ﬁlter is said to be countably based if it admits a countable base. Principal ﬁlters and sequential ﬁlters are special cases of countably based ﬁlters. Example 4. Let (An )n be a descending sequence of sets such that card(An \ An+1 ) = ∞ and n<∞ An = ∅. Then {An : n < ∞} is a base of a free countably based ﬁlter, which is not sequential. 2.6. Grills. Two families A, B of subsets of X mesh relation (in symbols, A # B) if A ∩ B = ∅ for every A ∈ A and B ∈ B; the grill of a family A of subsets of X is A# = {B ⊂ X : ∀A∈A A ∩ B = ∅}. A family A is isotone if B ⊃ A ∈ A implies B ∈ A. If A isotone then11 (4)

H∈ / A# ⇔ H c ∈ A. Let Ω ⊂ X × Y be a relation. Then the image by Ω of x is given by Ωx = {y ∈ Y : (x, y) ∈ Ω}.

Consequently the image ΩA of A ⊂ X by Ω is Ωx, ΩA = x∈A

and the preimage Ω− B of B ⊂ Y by Ω is the image of B by the inverse relation of Ω. Thus Ω− B = {x ∈ X : Ωx ∩ B = ∅}. We notice the following useful equivalence ΩA # B ⇔ A # Ω− B ⇔ (A × B) # Ω. In particular, if f : X → Y then f (A) and f − (B) are respectively the image and the preimage by the graph relation {(x, y) : y = f (x)}. In particular I denote by f − (y) the preimage of y by f . If F is a ﬁlter, then F ⊂ F # . Notice also that if F is a ﬁlter, then12 (5)

H0 ∪ H1 ∈ F # ⇒ (H0 ∈ F # or H1 ∈ F # ). 10A ﬁlter E is sequential if and only if E contains a countable set, and admits a countable

base B such that B1 \ B0 is ﬁnite for every B0 , B1 ∈ B. 11By deﬁnition, H ∈ / A# whenever there is A ∈ A such that H ∩A = ∅, equivalently H c ⊃ A, that is, H c ∈ A, because A is istone. 12In fact, if H ∈ # and H ∈ # c c c c 0 / F 1 / F , then H0 ∈ F and H1 ∈ F , hence H0 ∩ H1 ∈ F thus / F #. H0 ∪ H ∈

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2.7. Order, ultraﬁlters. We say that a ﬁlter F is coarser than a ﬁlter G (G is ﬁner than F) if F ⊂ G. If B is a base of F and B ⊂ G, then F ⊂ G. Remark 5. We can loosely say that the smaller the sets belonging to a ﬁlter, the ﬁner is the ﬁlter. Formally, if B0 is a base of a ﬁlter F0 , and B1 of F1 , and for every B ∈ B0 there is D ∈ B1 with D ⊂ B, then F0 ⊂ F1 . Of course, this partial order is induced on the set ϕX (of all ﬁlters on X) from that of all the families of subsets of X. We denote by F ∨ G the supremum and by F ∧ G the inﬁmum of two ﬁlters F and G. Remark 6. The supremum of ﬁlters F, G in the ordered set of ﬁlters exists if and only if F # G. If it does not exist, then the supremum of F and G in the complete lattice of all the families of subsets is, of course, equal to the degenerate ﬁlter. Remark 7. Occasionally I use the symbols ∨ and ∧ (respectively) for the supremum and the inﬁmum in the partially ordered set of ﬁlter bases. This notation is not ambiguous, because in the case when the considered ﬁlter bases are ﬁlters, their supremum and inﬁmum are ﬁlters too. The inﬁmum j∈J Fj exists for an arbitrary set {Fj : j ∈ J} of ﬁlters on X, and j∈J Fj = j∈J Fj (the coarsest ﬁlter is {X}, the principal ﬁlter of X), while j∈J Fj exists whenever ∅∈ / B = { j∈J0 Fj : J0 ⊂ J, card J0 < ∞}, and in this case B is a base of j∈J Fj . By the Zorn–Kuratowski lemma, for every ﬁlter F on X there exists a maximal ﬁlter U, which is ﬁner than F, called an ultraﬁlter. The set of all the ultraﬁlters ﬁner than F is denoted by βF. If f : X → Y and F is a ﬁlter on X, then f (F) = {f (F ) : F ∈ F} is a ﬁlter base on Y . We shall use the symbol f (F) also for the ﬁlter it generates.13 A ﬁlter F is an ultraﬁlter if and only if F # = F.14 It follows that If U is an ultraﬁlter, then f (U) is an ultraﬁlter. Moreover if W ∈ βf (F), then there exists U ∈ βF such that W = f (U).15 2.8. Decomposition. A ﬁlter F is called free if F ∈F F = ∅. The coﬁnite ﬁlter is free. In fact, if F is a free ﬁlter and A ∈ F then F ⊃ (A)◦ .16 Proposition 8 (Filter decomposition [15]). For every ﬁlter F on X, there exists a unique pair of (possibly degenerate) ﬁlters F ◦ , F • such that F ◦ is free, F • 13This abuse of notation should not lead to any confusion. 14If a ﬁlter F is not an ultraﬁlter, then there is a ﬁlter G F . Therefore F G ⊂ G # ⊂ F # .

Conversely, if there is H ∈ F # \ F , then F ∨ H is a ﬁlter that is obviously ﬁner than F and that contains H. Since H ∈ / F , the ﬁlter F is not maximal. 15Let W be an ultraﬁlter ﬁner than f (F ). Equivalently, f − (W) # F , hence there exists an ultraﬁlter U ﬁner than F ∨ f − (W), so that U ∈ βF and U # f − (W). The latter is equivalent to f (U) # W thus f (U) = W. 16Indeed, if x ∈ A and F is free, then there is F ∈ F such that x ∈ / F , that is, F ⊂ {x}c thus {x}c ∈ F . As each ﬁnite intersection of elements of a ﬁlter belongs to that ﬁlter, A \ D ∈ F for every ﬁnite subset D of A. Hence (A)◦ ∈ F .

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is principal, and17 F = F ◦ ∧ F • and F ◦ ∨ F • = 2X .

(6)

In particular, every sequential ﬁlter admits such a decomposition. Notice that Proposition 9. A ﬁlter is sequential and free if and only if it is the coﬁnite ﬁlter of a countably inﬁnite set. Proof. If (xn )n is a sequence, then we set An = {xk : k ≥ n}. Accordingly, (xn )n is free if and only if n<∞ An = ∅. The sequential ﬁlter generated by (xn )n is the coﬁnite ﬁlter of Am for each m, because Am \An is ﬁnite for every n < ∞ and (An )n is a base of it. Conversely, if S is the coﬁnite ﬁlter of a countably inﬁnite set A, then represent A = {xk : k < ∞}, where xm = xk if m = k. Then {An : n < ∞} is a base of S, so that S ≈ (xn )n . Proposition 10. A ﬁlter is sequential and principal if and only if it is the principal ﬁlter of a countable set. Proof. Indeed, if A is ﬁnite (of cardinality 0 < m < ∞) then we can represent A = {a1 , a2 , . . . , am } and (A)• is sequential, because the sequence a1 , a2 , . . . , am , a1 , a2 , . . . , am , a1 , a2 , . . . , am , . . . generates (A)• ; if A is countably inﬁnite, then we can represent A = {an : n ∈ N} where an = ak for n = k. Then the sequence a1 , a1 , a2 , a1 , a2 , a3 , . . . , a1 , a2 , . . . , an . . . generates (A)• .

2.9. Stone transform. Let me shortly mention the Stone space of a given set X, that is, the set βX of all ultraﬁlters on X endowed with the Stone topology. The Stone transform β associates to every ﬁlter F on X, the set βF of ultraﬁlters that are ﬁner than F. It is clear that F0 ⊂ F1 implies βF0 ⊃ βF1 .18 A base for the open sets of the Stone topology consists of the Stone transforms of principal ﬁlters, that is, {βA : A ⊂ X}. This topology is compact.19 It is well-known that Proposition 11. A subset of the Stone space is closed if and only if it is of the form βF, where F is a (possibly degenerate) ﬁlter.20 17Let F • be the principal ﬁlter of F = T ◦ c • F ∈F F . Then F = F ∨ F• is free, and obviously

(6) holds. If now (A)• is a principal ﬁlter ﬁner than F , then A ⊂ F• , hence F ∨ Ac is free if and only if A = F• , which shows the uniqueness of the decomposition. 18Moreover, β(F ∨ F ) = βF ∩ βF and β(F ∧ F ) = βF ∪ βF . 0 1 0 1 0 1 0 1 19If {βD : D ∈ D} is a cover of βX, then there is a ﬁnite subfamily A of D such that T T S βX ⊂ A∈A βA, for otherwise β( A∈A Ac ) = A∈A β(Ac ) = ∅ for each ﬁnite subfamily A of T D. In other words, { A∈A Ac : A ⊂ D, card A < ∞} is a ﬁlter base. If now U is an ultraﬁlter / U (equivalently U ∈ / βD) for that includes it, then a fortiori D c ∈ U for each D ∈ D, hence D ∈ each D ∈ D, which is a contradiction. 20If F is a ﬁlter and U ∈ / βF then there is H ∈ F \ U so that the Stone open (and closed) set T βH contains U and is disjoint from βF . Conversely, if A is a Stone closed set, then F = U∈A U is a ﬁlter such that βF = A. By construction each U ∈ A belongs to βF. If W ∈ / A there is an open, closed set of the form βW such that W ∈ βW and βW ∩ A = ∅, that is, β(W c ) ⊃ A, that / βF. is W c ∈ U for each U ∈ A, hence W c ∈ F and thus W ∈

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3. Basic classes of convergences If ξ is an arbitrary relation between the non-degenerate ﬁlters F on X and the elements x of X, then we write x ∈ limξ F

(7)

whenever (x, F) ∈ ξ and say that the ﬁlter F converges to x with respect to ξ (equivalently, x is a limit of F with respect to ξ). A relation ξ is a convergence if (8)

F ⊂ G =⇒ limξ F ⊂ limξ G,

(9)

∀x∈X x ∈ limξ (x)• ,

where (x)• is the principal ultraﬁlter determined by x.21 Remark 12. If ξ is a convergence on a set X, then the couple (X, ξ) is called a convergence space. The set, on which a convergence ξ is deﬁned, is called the underlying set of ξ. Notice that an underlying set is determined by a convergence thanks to (9); we denote the underlying set of a convergence ξ by |ξ|. Consequently a convergence determines the corresponding convergence space. Therefore I will use the terms convergence and convergence space interchangeably. Example 13. If τ is a topology on a set X then we deﬁne the associated convergence by x ∈ limτ F whenever Nτ (x) ⊂ F. This relation fulﬁlls (8) and (9),22 so that we can identify each topology with its associated (topological) convergence. Notice that a topological convergence fulﬁlls x ∈ lim N (x) for every x ∈ X, and the neighborhood ﬁlter N (x). Here is a basic example of a non-topological convergence. Example 14. If ν is the natural topology on R, then we deﬁne a convergence Seq ν on X by setting x ∈ limSeq ν F, whenever there exists a sequential ﬁlter E such that E ⊂ F and x ∈ limν E. This deﬁnes a convergence, which is not a topology. Indeed, Nν (0) = {E : 0 ∈ limSeq ν E} but there is no sequential ﬁlter which is / limSeq ν Nν (0).23 coarser than Nν (0), hence 0 ∈ We say that a convergence ξ is ﬁner than a convergence θ (in symbols, ξ ≥ θ) if limξ F ⊂ limθ F for every ﬁlter F. If Ξ is a set of convergences on X, then the supremum and the inﬁmum of Ξ are given by limW Ξ F = limξ F, limV Ξ F = limξ F. ξ∈Ξ

ξ∈Ξ

The greatest element of the set of all convergences on X is the discrete topology ι = ιX of X, which is deﬁned by x ∈ limι F whenever F is the principal ultraﬁlter 21There are several deﬁnitions of convergence. Many authors add to the set of our axioms, a third one like ´ lim F0 ∩ lim F1 ⊂ lim(F0 ∩ F1 ) (H. J. Kowalsky [36]), or x ∈ lim F ⇒ x ∈ ` lim (x)• ∩ F (e.g., D. C. Kent, G D. Richardson [34]). 22In fact, we have deﬁned a functor, which embeds the category of topologies in that of convergences. 23Indeed, N (0) ⊂ T{E : 0 ∈ lim / Nν (0) then ν Seq ν E} by the deﬁnition of Seq ν, and if A ∈ 1 c A ∈ Nν (0)# hence for every n < ∞ there is xn ∈ Ac ∩ {x : |x| < n }. In other words A belongs to a sequential ﬁlter (generated by) (xn ), which is ﬁner than Nν (0).

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of x.24 The least element is the chaotic topology (called also indiscrete topology) o = oX on X, deﬁned by limo F = X for every ﬁlter F on X. 3.1. Topologies. A subset O of a convergence space is open if lim F ∩ O = ∅ implies that O ∈ lim F. The family Nξ (x) of all the sets V such that there exists a ξ-open set O such that x ∈ O ⊂ V , is a ﬁlter, called the neighborhood ﬁlter of x for ξ and is denoted by Nξ (x). The family Oξ of all the open sets of a convergence ξ fulﬁlls all the axioms of open sets of a topology. Conversely, if T is the family of open sets of a topological space, then (10)

x ∈ limT F ⇔ (x ∈ T ∈ T ⇒ T ∈ F)

deﬁnes a convergence. Convergences constructed in this way are called topologies. For every convergence ξ there exists the ﬁnest topology that is coarser than ξ called the topologization of ξ and denoted by T ξ. This the convergence constructed with the aid of (10) with T = Oξ . It is straightforward that OT ξ = Oξ . Therefore T fulﬁlls ζ ≤ ξ ⇒ T ζ ≤ T ξ, T ξ ≤ ξ, T (T ξ) = T ξ, for every ζ and ξ. It follows that the set of all topologies (on a given set) is closed for arbitrary suprema. Moreover, the coarsest convergence on X is the chaotic topology on X. Example 15. Take the convergence Seq ν of Example 14. By deﬁnition of open set for a convergence, a set is open for Seq ν if and only if it is sequentially open, hence if it is open for ν, because ν is a sequential topology.25 Therefore T (Seq ν) = ν. The complement of an open set is said to be closed. The least ξ-closed set, which includes A is called the ξ-closure of A and is denoted by clξ A. We notice that (11)

x ∈ clξ A ⇔ A ∈ Nξ# (x).

Proof. We have that x ∈ / clξ A whenever there is an open set O such that x ∈ O and O ∩ A = ∅, that is, whenever A ∈ / Nξ# (x). 3.2. Sequentially based convergences. A convergence ξ is sequentially based if whenever x ∈ limξ F, then there exists a sequential ﬁlter E such that E ⊂ F and x ∈ limξ E. If (xn )n is a sequence that generates E then we write x ∈ limξ (xn )n whenever x ∈ limξ E.26 Denote by εX the set of sequential ﬁlters on X. If θ is an arbitrary convergence of X, x ∈ limSeq θ F ⇐⇒ ∃E∈εX (x ∈ limθ E and E ⊂ F) 24In fact, the convergence of the principal ultraﬁlter of x to x is postulated by the deﬁnition of convergence. 25A topology is sequential if every set closed for convergent sequences is closed. 26This is an example of the obvious and natural extension of convergences from ﬁlters to ﬁlter bases.

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deﬁnes a sequentially based convergence Seq θ. It is straightforward that for every ζ and ξ, ζ ≤ ξ ⇒ Seq ζ ≤ Seq ξ, ξ ≤ Seq ξ,

(12)

Seq(Seq ξ) = Seq ξ. A topology is a sequentially based convergence, if and only if each neighborhood ﬁlter is generated by a sequence. Therefore the sequential modiﬁer Seq can be used to produce numerous examples of non-topological convergences, as in Example 14. 3.3. Convergences of countable character. A convergence ξ is of countable character provided that if x ∈ limξ F then there exists a countably based ﬁlter E such that E ⊂ F and x ∈ limξ E. Each sequentially based convergence is of countable character; Example 14 is that of a topology of countable character, which is not sequentially based. The modiﬁer (of countable character) First θ of an arbitrary convergence θ is deﬁned similarly to Seq θ; the map First has the same properties as Seq in (3.2). 3.4. Pretopologies. A convergence ξ is called a pretopology if27 (13) limξ F ⊂ limξ F F ∈F

F ∈F

for every set F of ﬁlters. It follows that for every element x of a pretopological space, there exists a coarsest ﬁlter that converges to x. For every element x let F (14) Vξ (x) = x∈limξ F

be the vicinity ﬁlter of x for a convergence ξ; the elements of the vicinity ﬁlter of x are called vicinities of x. By (13) ξ is a pretopology if and only if x ∈ limξ Vξ (x) for every x. If A is a non-empty subset of a pretopological space (X, ξ) then Vξ (A) = x∈A Vξ (x) is the vicinity of A. A set is open if it is a vicinity of each of its elements.28 Every topology is a pretopology; if ξ is a topology, then Vξ (x) ⊂ Nξ (x) for every x, and since the inverse inclusion always holds, (15)

ξ = T ξ ⇒ Vξ (x) = Nξ (x)

Let us give an example of a non-topological pretopology. Example 16. Let X = {x∞ } ∪ {xn : n < ∞} ∪ {xn,k : n, k < ∞}, where all the elements are distinct. We deﬁne a convergence π by xn,k ∈ limπ F whenever F = (xn,k )• , xn ∈ limπ F whenever (xn )• ∧(xn,k )k ⊂ F, and x∞ ∈ limπ F provided that (x∞ )• ∧(xn )n ⊂ F. The convergence π is a pretopology and a sequentially based convergence. Indeed, for every element, there is a coarsest ﬁlter converging to that element, that is, a vicinity ﬁlter. The elements of the form xn,k are isolated, hence {xn,k } ∈ Vπ (xn,k ) for each n, k < ∞; Vπ (xn ) ≈ (xn )• ∧ (xn,k )k for every n < ∞; Vπ (x∞ ) ≈ (x∞ )• ∧ (xn )n . It is not a topology, because if O such that x ∞ ∈ O is an open set, then there is n0 such that xn ∈ O for every n ≥ n0 and there is κ(n) < ∞ 27This deﬁnition of pretopology is due to G. Choquet [8]. However an equivalent concept ˇ was considered by F. Hausdorﬀ [31], W. Sierpi´ nski [40] and E. Cech [7]. 28If A is open and x ∈ A, then A ∈ V(x), because every ﬁlter, which converges to x, contains A, and conversely.

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such that xn,k ∈ O for each k > κ(n). The neighborhood ﬁlter Nπ (x∞ ), which is generated by {x∞ } ∪ {xn : n > n0 } ∪ {xn,k : k > κ(n), n > n0 }, where n0 < ∞ and κ : N → N, does not converge to x∞ . Actually we have already described the topologization T π of π, namely NT π (x∞ ) = Nπ (x∞ ) was given above, and we have described NT π (x) = Vπ (x) if x = x∞ .29 It can be easily seen that the set of all pretopologies (on a given set) is closed for arbitrary suprema, and that the coarsest convergence on a given set is the chaotic topology on that set. This is equivalent to the existence of a map P associating with every convergence ξ the ﬁnest pretopology P ξ that is coarser than ξ. This map is called the pretopologizer and fulﬁlls ζ ≤ ξ ⇒ P ζ ≤ P ξ, P ξ ≤ ξ, P (P ξ) = P ξ, for every ζ and ξ. The pretopologizer can be easily written explicitly. Therefore x ∈ limP ζ F if and only if Vζ (x) ⊂ F. Remark 17. A network of a topological space (X, τ ) is afamily P of subsets of X such that for each x ∈ X and O ∈ Nτ (x) there is P ∈ P such that x ∈ P ⊂ O. A network is called a weak base whenever each subset B of X, with the property that for every x ∈ B there is P ∈ P such that x ∈ P ⊂ B, is open. For example, the family of all singletons is a network, which is not a weak base unless the topology is discrete. Let P be a family of subsets of X, which covers X. Then the family of ﬁnite intersections of {P ∈ P : x ∈ P } is a ﬁlter base; the ﬁlter VP (x) it generates is a vicinity ﬁlter of a pretopology, which we denote by πP . It follows immediately from the deﬁnitions that Proposition 18. A family P is a network of τ if and only if πP ≥ τ ; a family P is a weak base of τ if and only if T πP = τ . 4. Continuity Let ξ be a convergence on X and τ be a convergence on Y . A map f : X → Y is continuous (from ξ to τ ) if for every ﬁlter F on X, (16)

f (limξ F) ⊂ limτ f (F).

It follows that the composition of continuous maps is continuous. A bijective map f such that both f and f − are continuous is called a homeomorphism. 4.1. Initial convergences. For every map f : X → Y and each convergence τ on Y , there exists the coarsest among the convergences ξ on Xfor which f is continuous (from ξ to τ ). It is denoted by f − τ and called the initial convergence for (f, τ ).30 If V ⊂ X and θ is a convergence on X, then the initial convergence such that the embedding i : V → X is continuous is called a subconvergence of θ on V and is denoted by θ ∨ V . 29Notice that N (x) = N (x) for every convergence ξ and each x ∈ |ξ|. Tξ ξ 30Indeed, it follows from (16) that if f is continuous from ξ to τ , then lim F ξ

f − (limτ f (F)). Therefore limf − τ F = f − (limτ f (F )).

⊂

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Let τi be a convergence and fi : X → |τi | be a map for every i ∈ I. Then the coarsest convergence on X, for which fi is continuous for each i ∈ I, iscalled the initial convergence with respect to {fi : i ∈ I}. Of course, it is equal to i∈I fi− τi . It is straightforward that Proposition 19. If ξ is the initial convergence with respect to {fi : i ∈ I} then x ∈ limξ F if and only if fi (x) ∈ lim fi (F) for every i ∈ I. 4.2. Final convergences. For every map f : X → Y and each convergence ξ on X, there exists the ﬁnest among the convergences τ on Y for which f is continuous (from ξ to τ ). It is denoted by f ξ and called the ﬁnal convergence for (f, ξ) (or the quotient of ξ by f ).31 Let ξi be a convergence and fi : |ξi | → Y be a map for every i ∈ I. Then the is called the ﬁnal ﬁnest convergence on Y , for which fi is continuous for each i ∈ I, convergence with respect to {fi : i ∈ I}. Of course, it is equal to i∈I fi ξi . It is good to have in mind this immediate observation. Proposition 20. The following statements are equivalent: • f is continuous from ξ to τ ; • fξ ≥ τ; • ξ ≥ f −τ . 4.3. Continuity in subclasses. One easily sees that the preimage of an open set by a continuous map is open.32 Hence if τ is a topology, then f − τ is a topology.33 Similarly, if τ is a pretopology, then f − τ is a pretopology.34 Therefore if f is continuous from ξ to τ , then it is continuous also from P ξ to P τ , and from T ξ to T τ .35 It is also easy to notice that if ξ is a sequentially based convergence, then f ξ is also sequential.36 It follows (by Proposition 20 for instance) that if f is continuous from ξ to τ , then it is continuous also from Seq ξ to Seq τ . 4.4. Products. If ξ and υ are convergences on X and Y respectively, then the product convergence ξ × υ on X × Y is deﬁned by (x, y) ∈ limξ×υ F whenever there exist ﬁlters G on X and H on Y such that x ∈ limξ G, y ∈ limυ H and G × H ≤ F.37 In other words, a ﬁlter converges to (x, y) in the product convergence if and only if its projections converge to x and y respectively. 31It is straightforward that lim G = S fξ f (F )≤G f (limξ F ). Indeed y ∈ limf ξ G whenever

there exists a ﬁlter F such that limξ F ∩ f − (y) = ∅ and G ≥ f (F ). 32Let f be continuous from ξ to τ , let O ∈ O(τ ) and let x ∈ lim F ∩ f − (O). It follows that ξ f (x) ∈ limτ f (F) and f (x) ∈ O, hence O ∈ f (F ). Therefore f − (O) ∈ F . 33Let f − (O) ∈ F for every τ -open set O such that x ∈ f − (O). It follows that O ⊃ f f − (O) ∈ f (F) and f (x) ∈ O and thus f (x) ∈ limτ f (F ), hence x ∈ f − (f (x)) ⊂ f − (limτ f (F )) = limf − τ F . 34From the category theory point of view, topologies and pretopologies are concrete reﬂective subcategories of the category of convergences with continuous maps as morphisms. 35By Proposition 20, ξ ≥ f − τ , hence T ξ ≥ T (f − τ ). On the other hand f − τ ≥ f − (T τ ) and the latter convergence is a topology. Therefore T ξ ≥ f − (T τ ). 36From the category theory point of view, sequential convergences constitute a concrete coreﬂective subcategory of the category of convergences with continuous maps as morphisms. 37The product ﬁlter G × H is the ﬁlter generated by {G × H : G ∈ G, H ∈ H}.

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More generally, let Ξ be a set of convergences such that ξ is a convergence on Xξ for ξ ∈ Ξ. The product convergence Ξ = ξ∈ξ ξ is the coarsest convergence on ξ∈ξ Xξ , for which each projection pθ : ξ∈ξ Xξ → Xθ is continuous. In other words, Ξ = ξ∈ξ p− ξ ξ. In particular, each (convergence) product of topologies (respectively, of pretopologies) is a topology (respectively, a pretopology). 4.5. Powers. If X and Z are sets, hence Z X is the set of all maps from X to Z, then the map e = ·, · : X × Z X → Z deﬁned by e(x, f ) = x, f = f (x) is called the evaluation map. If ξ is a convergence on X and σ on Z, then C(ξ, σ) stands for the subset of Z X consisting of all the maps continuous from ξ to σ. The power (convergence) [ξ, σ] (of ξ with respect to σ) is the coarsest among the convergences τ on C(ξ, σ) for which the evaluation is continuous from ξ × τ to σ. The power [ξ, σ] exists for arbitrary convergences ξ and σ.38 Let us describe explicitly the power convergence. If G is a ﬁlter on X, ξ is a convergence onX, and F is a ﬁlter on C(ξ, σ), then G, F stands for the ﬁlter generated by { f ∈F f (G) : G ∈ G, F ∈ F}. Then f ∈ lim[ξ,σ] F if and only if f (x) ∈ limσ G, F for every x ∈ |ξ| and ﬁlter G on |ξ| such that x ∈ limξ G. The deﬁnition above was already given by H. Hahn [30] for sequential ﬁlters F. As mentioned in the introduction, power convergences constituted a decisive point in the development of convergence theory. And they remain a most important object of study till today. 5. Adherences An important notion in convergence theory is that of adherence. If ξ is a convergence on X and H is a family of subsets of X, then adhξ H = limξ F F #H

is the adherence of H. Therefore if U is an ultraﬁlter, then adhξ U = limξ U. Clearly, adhξ A ⊂ adhθ A if ξ ≥ θ. Recall that a family A is isotone if B ⊃ A ∈ A implies B ∈ A. If A, B are isotone families, then39 adh 2X = ∅; adh(A ∩ B) = adh A ∪ adh B. It follows that G ⊃ F implies adh G ⊂ adh F. 38Indeed, if ι is the discrete topology on C(ξ, σ), then e is continuous from ξ × ι to σ if and only if e(·, f ) is continuous from ξ to σ for every V f . Now, if T is a set of convergences on X such that ξ × τ ≥ e− σ for each τ ∈ T . then ξ × T τ ≥ e− σ, because (x, f ) ∈ limξ×Vτ ∈T τ H if and S only if there exist ﬁlters F and G such that x ∈ limξ F and f ∈ τ ∈T limτ G. 39Indeed, a ﬁlter F does not mesh neither A nor B if and only if there exist F , F ∈ F and 0 1 A ∈ A, B ∈ B such that F0 ∩ A = ∅ and F1 capB = ∅, equivalently F0 ∩ F1 ∩ (A ∪ B) = ∅. Because A, B are isotone, the elements of A ∩ B are of the form A ∪ B with A ∈ A, B ∈ B. Thus F does not mesh with A ∩ B, because F0 ∩ F1 ∈ F .

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If A is a subset of a convergence space (X, ξ), then adhξ A (an abbreviation for adhξ {A} = adhξ (A)• ) is the adherence of a set A. It follows that the operation of adherence of sets fulﬁlls adh ∅ = ∅; adh(A ∪ B) = adh A ∪ adh B; A ⊂ adh A. for every A and B. Therefore A ⊂ B implies adh A ⊂ adh B. Remark 21. The vicinity ﬁlter was deﬁned in (14) for an arbitrary convergence. Notice that (17)

x ∈ adhξ A ⇔ A ∈ Vξ# (x). If X is a ﬁxed set, then I denote Ac = X \ A for each A ⊂ X.

Proposition 22. If ξ is a topology, then the (set) adherence adhξ is idempotent and equal to the closure clξ . Proof. If x ∈ / adhξ A then there is V ∈ Vξ (x) such that V ∩ A = ∅, and if ξ is a topology, then by (15) there is an open set O such that x ∈ O and O ∩ A = ∅. Therefore x ∈ / O c ⊃ clξ A. Because clξ A is closed, and adhξ A ⊂ clξ A, also 2 adhξ A = adhξ (adhξ A) ⊂ clξ A. Remark 23. If ξ is a convergence on X and F is a ﬁlter on X, then we denote by adhξ F the ﬁlter generated by {adhξ F : F ∈ F}. Therefore we distinguish between the set adhξ F and the ﬁlter adhξ F. Similarly clξ F denotes the ﬁlter generated by {clξ F : F ∈ F}. Dual notions of adherence and of closure are those of respectively inherence and interior, namely inh A = (adh Ac )c ,

int A = (cl Ac )c .

Notice that x ∈ inh A if and only if A ∈ V(x), and x ∈ int A if and only if A ∈ N (x). 6. Covers Let (X, ξ) be a convergence space. A family P of subsets of X is a cover of B ⊂ X if limξ F ∩ B = ∅ implies that F ∩ P = ∅. As for every convergence each principal ultraﬁlter converges toits deﬁning point, each cover P of B is a set-theoretic cover of B, that is, B ⊂ P.40 Let us investigate the notion of cover in special cases. Example 24. If ξ is a pretopology, then the coarsest ﬁlter that converges to x is the vicinity ﬁlter Vξ (x). Therefore P is a cover of B in ξ if and only if for every x ∈ B there exists P ∈ P with P ∈ Vξ (x), equivalently x ∈ inhξ P . In other inh P . In particular, if words, P is a cover of B in ξ if and only if B ⊂ ξ P ∈P ξ is a topology, then this becomes B ⊂ P ∈P intξ P . In other words, P is a cover 40If ξ ≤ ζ then each ξ-cover of B is a ζ-cover of B. Then the last statement follows from the observation that P is a set-theoretic cover of B if and only if P is a cover of B for the discrete topology ι.

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of a subset B of a topological space if and only if {int P : P ∈ P} is an (open) set-theoretic cover of B.41 We denote Pc = {P c : P ∈ P}. Theorem 25 ([13]). A family P is a cover of B if and only if adh Pc ∩ B = ∅.

(18)

Proof. By deﬁnition, (18) means that if a ﬁlter F converges to an element of B then F does not mesh with Pc , that is, there exist F ∈ F and P ∈ P such that F ∩ P c = ∅, equivalently F ⊂ P , that is, F ∩ P = ∅, which means that P is a cover of A. Notice that in general Pc in (18) is not a ﬁlter,even not a ﬁlter base. A family R is an ideal if S ⊂ R ∈ R implies S ∈ R, and if T ∈ R for each ﬁnite T ⊂ R. Clearly, R is an ideal if and only if Rc is a (possibly degenerate) ﬁlter. Denote by P˜ ˜ c is the (possibly degenerate) ﬁlter generated the least ideal including P. Then (P) by the ﬁnite intersections of elements of Pc .42 Remark 26. In a topological space, if P is a family of open sets and P˜ is a cover of B, then P is also a cover of B,43 and on the other hand, for each cover P of B the family {int P : P ∈ P} is an open cover of B. 7. Compactness If A and B are subsets of a topological space X, then A is called (relatively) compact at B if for every open cover of B there exists a ﬁnite subfamily, which is a cover of A.44 It is known that A is compact at B if and only if for every ﬁlter H, A ∈ H# ⇒ adh H ∩ B = ∅.

(19)

If A and B are subsets of a convergence space X, then we take the characterization above for the deﬁnition.45 Proposition 27. A set A is compact at B if and only if A ∈ P for every ideal cover P of B. 41In each convergence space, a family of open sets is a cover if and only if it is a set-theoretic

cover. 42Notice that if B is base of a ﬁlter F then adh B = adh F . However adh H is (in general,

T strictly) bigger than adh{ G : G ⊂ H, card G < ∞}. For example, if H = {H0 , H1 } then adh H = adh H0 ∩ adh H1 while the adherence of the (ﬁlter generated by) ﬁnite intersections of elements of H is adh(H0 ∩ H1 ). 43Indeed, for every x ∈ B there is a ﬁnite subset T of P such that x ∈ S T , hence there is P ∈ T ⊂ P such that x ∈ P . 44Many authors say that a topological space X is compact if it is Hausdorﬀ and if is compact at X (in the sense of our deﬁnition). 45If A is compact at B in the topological sense, and P is an ideal cover of B, then by Remark 26 {int P : P ∈ P} is an open cover of B, hence S there is a ﬁnite subfamily R of P such that {int P : P ∈ R} is a cover of A, so that A ⊂ R ∈ P, thus by Propostion 27 A is compact at B in the convergence sense. Conversely, if A is compact at B in the convergence sense and P is ˜ is an ideal cover of B, thus by Propostion 27 there is a an open cover of B then by Remark 26 P S ﬁnite subfamily R of P such that A ⊂ R.

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Proof. Formula (19) means that adh H ∩ B = ∅ implies that A ∈ / H# , and c because H is isotone, A ∈ H by (4), hence by Theorem 25, if Hc is a cover of B then A ∈ Hc . As H is a ﬁlter, Hc is an ideal. In general convergence spaces there exists a notion of cover-compactness, which is (in general, stricly) stronger than that of compactness.46 If a subset A of a convergence space X is compact at X, then I call it relatively compact. A subset of a convergence space is compact if it is compact at itself. 7.1. Compact families. Our deﬁnitions have an obvious natural extension to families of sets [16]. Let A, B be families of subsets of X. Then A is compact at B if for every ﬁlter H, (20)

A # H ⇒ adh H ∈ B# .

A family A is relatively compact if it is compact at (the whole space) X, and compact if it is compact at itself.47,48 These notions generalize not only that of (relatively) compact sets, but also of convergent ﬁlters. In fact, Every convergent ﬁlter is relatively compact.49 More precise relationship between convergence and compactness will be given in Proposition 34. It is immediate that the image of a compact ﬁlter by a continuous map is compact. Theorem 28 (Tikhonov theorem). A ﬁlter (on a product of convergence spaces) is relatively compact if and only if its every projection is relatively compact. Proof. The necessity follows from the preceding remark. As for the suﬃciency, let F be a ﬁlter on Ξ. Let U be an ultraﬁlter with U # F. This implies pξ (U) # pξ (F) for each ξ ∈ Ξ, and since pξ (F) is ξ-relatively compact there is xξ ∈ Xξ such that xξ ∈ limξ pξ (U), which means that (xξ )ξ ∈ limQ Ξ U. No separation condition has been required in the deﬁnition of compactness. 46A is cover-compact at B if for each cover P of B there is a ﬁnite subfamily R of P which is a cover of A. If A is cover-compact at B then A is compact at B. Indeed, the condition holds in S particular for ideal covers, and a ﬁnite family R is a cover of A, then a fortiori A ⊂ R. It suﬃces to use Proposition 27 to conclude. The converse is not true in general. Take the pretopology from Example 16. Let A = {x∞ } ∪ {xn : n < ∞} and An = {xn } ∪ {xn,k : k < ∞}. The set A is compact at itself but not cover-compact at itself. In fact, every free ultraﬁlter on A converges to x∞ . On the other hand, the family P = {A} ∪ {An : n < ∞} is a cover of A but no ﬁnite subfamily is a cover of A. The subfamily {F } ∪ {Fn : n < m} is not a cover of F , because each ˜ is a fortiori a cover of vicinity of xm+1 includes all but ﬁnitely elements of Fm+1 . The ideal P A, for which no element is a cover of A. 47This is a terminological turnover with respect to the previous papers of mine and of my collaborators, where the term compactoid was used for all the sorts of relative compactness. The present choice is done for the sake of simplicity, and follows that of Professor Iwo Labuda of the University of Mississippi. The term compactoid space was introduced by Gustave Choquet [8] for compact space without any separation axiom. 48If B = {B} then we say compact at B instead of compact at B; if moreover, B = {x} then we say compactat x. 49 Actually if x ∈ lim F then F is compact at x. Indeed, if H#F then there is an ultraﬁlter U ﬁner than H ∨ F , hence x ∈ lim U = adh U ⊂ adh H.

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7.2. Weaker versions of compactness. I will now weaken the deﬁnition (20) of compactness by restricting the class of ﬁlters H. Let H be a class of ﬁlters. A family A (of subsets of a convergence space) is H-compact at B (another family of subsets of that space) if ∀H∈H H # A ⇒ adh H ∈ B# . If H is the class of all ﬁlters, then H-compactness is equivalent to compactness.50 7.3. Countable compactness. If H is the class of countably based ﬁlters, then H-compactness is equivalent to countable compactness. 7.4. Finite compactness. If H is the class of principal ﬁlters, then H-compactness is called ﬁnite compactness. This property is very broad (and useless) in the case of sets. Indeed, a subset A in a Hausdorﬀ topological space is ﬁnitely compact at a set B if and only if A ⊂ B. However the notion is far from being trivial and useless in the context of ﬁlters [11]. F.

Proposition 29. A ﬁlter F is ﬁnitely compact at a set B if and only if V(B) ⊂

Proof. By deﬁnition, F is ﬁnitely compact at B if adh H ∩ B = ∅ for every H ∈ F # . Equivalently, if adh H ∩ B = ∅, that is, if H c ∈ V(B) then H c ∈ F. 7.5. Sequential compactness. By deﬁnition, a convergence ξ is sequentially compact if for every sequential ﬁlter (equivalently, for every countably based ﬁlter) E there exists a sequential ﬁlter F ⊃ E such that limξ F = ∅. Notice that adhSeq ξ E = limξ F, F ∈εE

where εE stands for the set of sequential ﬁlters ﬁner than E. In other words,51 Proposition 30. A T1 convergence ξ is sequentially compact if and only if Seq ξ is countably compact. 8. Adherence-determined convergences 8.1. Pseudotopologies. A convergence ξ is a pseudotopology if x ∈ limξ F whenever x ∈ limξ U for every ultraﬁlter U ﬁner than F, that is, if (21) limξ F ⊃ limξ U. U∈βF

50A family A is relatively H-compact if it is H-compact at the whole space. A is Hautocompact if is H-compact at itself. So far I used the term H-compact for the property above, but Iwo Labuda convinced me that that terminology was not appropriate. In fact, if A is a family of subsets of X such that A = {A} with A X, then it is H-compact if A (with the convergence induced from X) is H-compact. This property is, in general, diﬀerent from that of H-autocompactness of A. Of course, the two notions coincide in case when H is the class of all ﬁlters. In other words, compactness of sets is absolute (that is, independent of environment). I prefer however the term H-autocompact to Labuda’s H-selfcompact, as the latter has a mixed (English-Latin) origin. 51This result is due to Ivan Gotchev [26, Theorem 3.6] for topologies T . 0

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This means that each pseudotopology is determined by the limits of all ultraﬁlters.52 Example 31 (non-pseudotopological convergence). In an inﬁnite set X distinguish an element ∞, and deﬁne a convergence on X as follows: the principal ultraﬁlter (x)• converges to x, and for each ﬁnite subset F of β◦ X, the set of all free ultraﬁlters on X, one has {∞} = lim U∈F U. This convergence is not a pseu/ lim F dotopology, because if F is a free ﬁlter such that βF is inﬁnite,53 then ∞ ∈ but ∞ ∈ lim U for each U ∈ βF.54 The set of pseudotopologies on a given set is stable for arbitrary suprema and contains the chaotic topology. As a result, for every convergence ζ there exists the ﬁnest among coarser pseudotopologies, the pseudotopologization Sζ of ζ. It is straightforward that limSζ F = limζ U. U∈βF

The pseudotopologizer is isotone, expansive and idempotent. As we have seen, this property holds also for the topologizer and the pretopologizer. The following property is particular for the pseudotopologizer: Proposition 32. If Θ is a set of convergences on X, then S( Θ) = Sθ. θ∈Θ

This proposition is very important for the sequel. Therefore, I shall provide its proof, even though it is straightforward and simple. Proof. By deﬁnition, limWθ∈Θ Sθ F = = = =

θ∈Θ

U∈βF

U∈βF

θ∈Θ

limθ U limθ U

limW Θ U

U∈βF S(limW Θ

F).

As for the topologizer and the pretopologizer, the pseudotopologizer fulﬁlls S(f − τ ) ≥ f − (Sτ ) for every convergence τ . The pseudotopologizer has another particular property (with important implications in topology). Namely, (22)

S(f − τ ) = f − (Sτ )

for every convergence τ and each map f . 52Each pseudotopology ξ on X can be characterized with the aid of the Stone transform. For

every x let Vξ (x) be the set of all ultraﬁlters which converge to x in ξ. Then by (21) x ∈ limξ F if and only if βF ⊂ Vξ (x). It follows that each map V : X → βX such that (x)• ∈ V(x) for each x deﬁnes a pseudotopology. 53It is known (e.g., [23, Theorems 3.6.11 and 3.6.14]) that if card(β F ) is inﬁnite, then it is ◦ ℵ at least 22 0 . 54This convergence is a prototopology, that is, fulﬁlling lim F ∩ lim F ⊂ lim(F ∩ F ). 0 1 0 1

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Proof. Indeed, if x ∈ limf − (Sτ ) F then equivalently f (x) ∈ limSτ f (F), that is, f (x) ∈ limτ U for every U ∈ βf (F). If now W ∈ βF, then f (W) ∈ βf (F) and thus f (x) ∈ limτ f (W), equivalently x ∈ limf − τ W, which means that x ∈ limS(f − τ ) F. Because the product is the supremum of initial convergences with respect to the projections on component spaces, we get an important Theorem 33 (prototheorem of Tikhonov). (23) S( Ξ) = Sξ ξ∈Ξ

for every set of convergences Ξ. The relationship between compactness and pseudotopological convergence is very close. In fact, Proposition 34. A ﬁlter F is ξ-compact at x if and only if x ∈ limSξ F. Proof. Indeed, x ∈ limSξ F if and only if x ∈ limξ U for every U ∈ βF, which is equivalent to the compactness of F at x. We notice that the generalization of the classical Tikhonov Theorem 28 can be easily deduced from the Tikhonov prototheorem (Theorem 33). 8.2. Narrower classes of adherence-determined convergences. If H is a class of ﬁlters, then (24) limAH ξ F = adhξ H HH#F

deﬁnes a convergence AH ξ obtained from the original convergence ξ. Of course, if H is the class of all ﬁlters, then AH is the pseudotopologizer. More generally, Theorem 35 ([10]). An AH -convergence is a (25) (26) (27)

pseudotopology ⇔ H is the class of all ﬁlters; paratopology ⇔ H is the class of countably based ﬁlters; pretopology ⇔ H is the class of principal ﬁlters;

Actually, paratopologies were deﬁned in [10] as the convergences fulﬁlling (26) of Theorem 35. Proof. (25). For each convergence ξ, if H # F then lim ξ F ⊂ adhξ H, hence limξ F ⊂ H#F adhξ H. If ξ is a pseudotopology, then H#F adhξ H ⊂ U∈βF limξ U ⊂ limξ F. Conversely, if H#F adhξ H ⊂ limξ F then U∈βF limξ U ⊂ limξ F, because for every ﬁlter H # F there is an ultraﬁlter U ≥ H ∨ F, that is, U ∈ βF and adhξ H ⊃ limξ U. (27). Suppose that ξ is a pretopology and let x ∈ adhξ H for every H ∈ F # . Since x ∈ adhξ H amounts to H ∈ Vξ# (x), we infer that F # ⊂ Vξ# (x), that is, F ⊃ Vξ (x), that is x ∈ limξ F. Conversely, suppose that H∈F # adhξ H ⊂ limξ F / limξ F. Hence there exists H ∈ F # such that x ∈ / adhξ H. and F ⊃ Vξ (x), but x ∈ # # # The latter means that H ∈ / Vξ (x), that is, F is not a subfamily of Vξ (x) = ∅, equivalently Vξ (x) is not a subfamily of F, which yields a contradiction.

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It turns out that (22) extends to the discussed modiﬁers. Indeed, if H is an F0 -composable class of ﬁlters (that is, if a H ∈ H is a ﬁlter on X and Ω ⊂ X × Y , then ΩH ∈ H) and AH is given by (24), then [21, Theorem 19] states that AH (f − τ ) = f − (AH τ ) for every convergence τ and each map f . In particular, the formula above holds for the pretopologizer and the paratopologizer. The topologizer can be also described by a formula of the type (24), but with a class H which depends on topologies. I prefer instead to give another, more direct, formula (28) limT ξ F = clξ H. # H∈F

Proof. One has x ∈ / limT ξ F if and only if there exists a ξ-open set O such that x ∈ O ∈ / F, equivalently x ∈ / O c ∈ F # , that is, there is H = clξ H ∈ F # such that x ∈ / H. As H ∈ F # implies clξ H ∈ F # , we infer (28). It turns out that each class of adherence-determined convergences corresponds to a version of compactness. Namely, Theorem 36 ([12]). Let H be a class of ﬁlters. A ﬁlter F is H-compact at x for ξ if and only if x ∈ limAH ξ F. Proof. A ﬁlter F is H-compact at x for ξ whenever x ∈ adhξ H for every ﬁlter H ∈ H such that H#F, that is, whenever x ∈ limAH ξ F. We conclude that compactness is of pseudotopological nature, countable compactness of paratopological and ﬁnite compactness of pretopological. Adherence-determined Compactness variant Pseudotopologies Compactness Paratopologies Countable compactness Pretopologies Finite compactness Figure 1. Adherence-determined nature of invariants It is easy to construct pseudotopologies, which are not pretopologies, using the following Remark 37. Recall that if ξ is a pseudotopology, and Vξ (x) stands for the set of ultraﬁlters that converge to x in ξ, then (29)

x ∈ limξ F ⇔ βF ⊂ Vξ (x).

A pseudotopology ξ is a pretopology if and only if Vξ (x) is closed with respect to the Stone topology for each x, [10, Proposition A.1].55 The following remark enables one to construct paratopologies which are not pretopologies. 55Indeed, β(V (x)) = cl V (x), where V (x) is the vicinity ﬁlter of x for ξ by virtue of (29). ξ β ξ ξ

Therefore ξ is a pretopology if and only if x ∈ limξ Vξ (x), that is, whenever Vξ (x) is β-closed. Actually, if VP ξ (x) is the set of all ultraﬁlters that converge to x in P ξ, then VP ξ (x) = clβ Vξ (x).

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Remark 38. Let Gδ β stand for the topology on βX such that a neighborhood base of U ∈ βX consists of Gδ subsets (with respect to the Stone topology of βX) which contain U. A pseudotopology ξ is a paratopology if and only if Vξ (x) is Gδ β-closed for each x, [10, Proposition A.2].56 Here is an example of a pseudotopology τ such that τ > Pω τ > P τ = T τ .57 Example 39. This is a pseudotopology τ on a countably inﬁnite set X, in which all elements but one are isolated, that is, if x is not equal to a distinguished element ∞, then (x)• is the only ﬁlter that converges to x. To deﬁne τ at ∞, let B be a subset of β◦ X (the set of all free ultraﬁlters on X), which is Gδ β-closed and is not Stone-closed.58 Let U ∈ B and set B0 = B \ {U}. Then B0 = clGδ β B0 = clβ B0 , where the latter stands for the Stone closure of B0 . If we set Vτ (∞) = B0 ∪ {∞}, then by virtue of the preceding remarks VPω τ (∞) = clGδ β B0 and VP τ (∞) = clβ B0 . Because all other points are isolated, P τ = T τ . 9. Diagonality and regularity

If N (x) is a neighborhood ﬁlter of a topology on X, then N (A) = x∈A N (x) is the neighborhood ﬁlter of a subset A of X. If A is a family of subsets, then let (30) N (A) = N (A). A∈A

In other words, B ∈ N (A) whenever there is A ∈ A such that B ∈ N (x) for each x ∈ A. Example 40. In particular, if A = N (x0 ) then N (N (x0 )) = N (x0 ). Indeed, N (N (x0 )) ⊂ N (x0 ) because if B ∈ N (N (x0 )) then there is A ∈ N (x0 ) such that B ∈ N (x) for each x ∈ A, in particular B ⊃ A hence B ∈ N (x0 ). Conversely, if B ∈ N (x0 ), that is, B is a neighborhood, then by a fundamental property of neighborhoods of a topological space, there is a neighborhood A of x0 such that B is a neighborhood of every x ∈ A, that is, B ∈ N (N (x0 )). 56If ξ is a paratopology and an ultraﬁlter U ∈ / Vξ (x), that is, x ∈ / limξ U by virtue of (29), then by (24) there is a countably based ﬁlter H, coarser than U and such that x ∈ / adhξ H. Let T (Hn )n be a decreasing sequence that generates H. Then βH = n<ω βHn . No W ∈ βH converges to x in ξ, that is, βH is a Gδ set disjoint from Vξ (x), which proves that Vξ (x) is Gδ β-closed. If ξ is a pseudotopology, but not a paratopology, then there exist x, an ultraﬁlter U such that / Vξ (x) x∈ / limξ U but x ∈ adhξ H for each countably based ﬁlter H coarser than U. Therefore U ∈ but βH ∩ Vξ (x) = ∅ for each βH from {βH : H ∈ Fω , H ≥ U}, which is a neighborhood base of U in Gδ β. This shows that Vξ (x) is not Gδ β -closed. Actually,

VPω ξ (x) = clGδ β Vξ (x). 57A slight modiﬁcation of this example [22, Example 5] provides a pseudotopology ξ such that ξ > Pω ξ > P ξ > T ξ. 58Such sets exist, because if (A ) is a descending sequence of inﬁnite subsets of the set N n n of natural numbers such that An \ An+1 is inﬁnite, then the supremum of coﬁnite ﬁlters (An )◦ is not coﬁnite but admits a ﬁner coﬁnite ﬁlter (of an inﬁnite set). In terms of the Stone transform, T the intersection of Stone open (and closed) sets A = n<∞ β◦ An is not open (is Gδ of course), but int A = ∅.

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Formula (30) deﬁnes a special contour. The example above shows that in topological spaces, the contour of the family of neighborhoods along a neighborhood of x0 , converges to x0 . We shall see that this means that topologies are diagonal convergences. Regularity can be also expressed in terms of contours, and is in some sense (that we will make precise in a moment) inverse to diagonality. It turns out that for Hausdorﬀ compact pseudotopologies, diagonality and regularity coincide.59 9.1. Contours. Consider a family F of subsets of a set X and for every x ∈ X, let G(x) be a family of subsets of Y . The contour of F over G(·) (or of G(·) along F) is the following family of subsets of Y : (31) G(F) = G(x). F ∈F x∈F When F = {F }, then we abridge G(F ) = x∈F G(x). Consequently, G(F) = F ∈F G(F ). If E is a ﬁlter generated by (xn )n and F(n) is a ﬁlter generated by (xn,k )k for every n, then the contour F(E) is denoted by (xn,k )k . (n)

If F is a ﬁlter on the underlying set |ξ| of a convergence ξ, then the symbol adhξ F denotes the ﬁlter generated by {adhξ F : F ∈ F}. In the particular case, where G(x) = Vθ (x) for every x,60 we have the following extension of Remark 21: A # Vθ (B) ⇐⇒ (adhθ A) # B.

(32)

Proof. Indeed, let A ∈ A and B ∈ B be such that A # x∈B Vθ (x), equiva lently A ∈ x∈B Vθ# (x), that is, A ∈ Vθ# (x) for some x ∈ B. This is tantamount to x ∈ adhθ A ∩ B and the proof is complete. 9.2. Hausdorﬀ convergences. A convergence is said to be Hausdorﬀ if every limit is at most a singleton. If a convergence ξ happens to be Hausdorﬀ, then we will often write x = limξ F as an abbreviation for {x} = limξ F. Compact Hausdorﬀ pseudotopologies are minimal among Hausdorﬀ pseudotopologies,61 namely Proposition 41. If ζ ≥ ξ are pseudotopologies, ζ is compact and ξ is Hausdorﬀ, then ζ = ξ. Proof. Because ξ and ζ are pseudotopologies, it is enough to show that they coincide for ultraﬁlters. Let U be an ultraﬁlter and x ∈ limξ U. By compactness, ∅ = adhζ U = limζ U ⊂ limξ U = {x} because ξ is Hausdorﬀ, thus x ∈ limζ U. A convergence ξ is topologically Hausdorﬀ if T ξ is Hausdorﬀ. Proposition 42. Each topologically Hausdorﬀ compact pseudotopology is a topology. Proof. If ξ is a compact pseudotopology and T ξ is Hausdorﬀ with ξ ≥ T ξ, then ξ = T ξ by Proposition 41. 59This is due to the minimality of compact pseudotopologies. 60V (x) is the vicinity ﬁlter of x with respect to θ. θ

61Like topologies.

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9.3. Diagonality. A convergence ξ on X is diagonal provided that if x0 ∈ limξ F and if x ∈ limξ G(x) for every x ∈ X, then x0 ∈ limξ G(F). A convergence ξ on X is pretopologically diagonal if x0 ∈ limξ F, then x0 ∈ limξ Vξ (F). Example 43. The pretopology π of Example 16 is not diagonal; its topologization T π is diagonal. Notice that (n) (xn,k )k is NT◦π (x∞ ) (the free part of the neighborhood ﬁlter of x∞ ). Proposition 44. A pretopology is a topology if and only if it is diagonal. Proof. By Example 40 each topology is a diagonal pretopology. To prove the converse implication it is enough to show that if π is a diagonal pretopology, then adh2π A ⊂ adhπ A because then clπ A ⊂ adhπ A (for every A). Indeed, if x0 ∈ adh2π A then there is a ﬁlter F on adhπ A with x0 ∈ limπ F. On the other hand, for each x ∈ adhπ A there is a ﬁlter G(x) on A with limπ G(x). Of course, A ∈ G(F), and by diagonality x0 ∈ limπ G(F), hence x0 ∈ adhπ A. 9.4. Regularity. A convergence ξ on X is regular with respect to a convergence θ on X (in short, θ-regular ) if for every ﬁlter F, (33)

limξ F ⊂ limξ adhθ F.

A convergence ξ is regular if it is ξ-regular (Fischer [24]), topologically regular if it is T ξ-regular. For each convergence ζ there exists the ﬁnest among the regular convergences that are coarser than ζ. It is called the regularization of ζ and is denoted by Rζ. An element x of a convergence space ξ is called irregular if there exists a ﬁlter F such that x ∈ limξ F \ limξ adhξ F. Example 45. Let X = {x∞ } ∪ {xn : n < ∞} ∪ {xn,k : n, k < ∞} be the set of Example 16. Deﬁne the convergence ζ by xn,k ∈ limζ F whenever F = (xn,k )• , xn ∈ limζ F whenever (xn )• ∧ (xn,k )k ⊂ F, and x∞ ∈ limζ F provided that (x∞ )• ∧ (n) (xn,k )k ⊂ F. This is a pretopology, which is not regular. Actually, the elements xn,k and xn are regular for each n, k < ∞ and x∞ is irregular. Notice that x∞ ∈ limRζ F whenever (x∞ )• ∧ (xn )n ∧ (n) (xn,k )k ⊂ F, that is, Rζ = T π, where π is the convergence of Example 16. The following proposition [14, Proposition 7.4] is the essence of classical examples of irregular topologies.62 Proposition 46. Let ξ be a pretopology of countable character. An element x is irregular with respect to ξ if and only if there exist a sequence (xn )n and for every n < ω a free sequence (xn,k )k such that xn ∈ limξ (xn,k )k x ∈ limξ (xn,k )k . (n)

but x ∈ / adhξ (xn )n . 62E.g., [23, Example 1.5.6]. Consider the unit interval [0, 1] in which a basic family of closed 1 : n < ω}. In this topology x = 0 sets consists of the closed sets for the natural topology and of { n 1 1 1 is irregular. Then xn = n and xn,k = n + k verify Proposition 46.

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Proof. An element x is irregular for ξ if and only if adhξ Vξ (x) does not converge to x, that is, whenever there is V ∈ Vξ (x) and a decreasing ﬁlter base (Vn ) / adhξ (xn )n , of Vξ (x) such that for every n < ω there is xn ∈ adhξ Vn \ V . Hence x ∈ and for each such n there exists a sequence (xn,k )k on Vn for which xn ∈ limξ (xn,k )k . Since (n) (xn,k )k is ﬁner than Vξ (x), it converges to x in ξ. If (xn,k )k were not free for inﬁnitely many n, then (n) (xn,k )k would be coarser than a subsequence of (xn )n , which does not converge to x in ξ. Therefore, (xn,k )k is free for almost all n, hence for all n, after having possibly dropped a ﬁnite number of them. Example 47. Consider a variant of σ of Example 45, where x∞ ∈ limσ F provided that F is ﬁner than the ﬁlter generated by the family {{x∞ } ∪ {xn,k : n ≥ m} : m < ∞}.

(34)

This is a pretopology of countable character with x∞ as a unique irregularity point. Notice that Rσ = Rζ, where ξ is deﬁned in Example 45. Proposition 46 implies [14] that if a point x of a Hausdorﬀ pretopological space ξ of countable character is irregular, then there exists a homeomorphic embedding i of the pretopology σ in ξ such that x = i(x∞ ).63 In some cases, the deﬁnition of Fischer coincides with that of Grimeisen [28, 29]: ξ is θ-regular if adhξ Vθ (H) ⊂ adhξ H

(35) for every ﬁlter H.

Proposition 48. A pseudotopology ξ is θ-regular if and only if (35) holds for every family H. Proof. Indeed, x ∈ adhξ Vθ (H) whenever there exists an ultraﬁlter F # Vθ (H) such that x ∈ limξ F. By (32) (adhθ F) # H and by (33) x ∈ limξ (adhθ F), hence x ∈ adhξ H. Conversely, if (35) holds, ξ is a pseudotopology and x ∈ / limξ adhθ F, then by (21) x ∈ / adhξ H for some ultraﬁlter H # adhθ F, and thus by (35) x ∈ / adhξ Vθ (H). By virtue of (32) F # Vθ (H) and thus x ∈ / limξ F. 9.5. Interactions between regularity and topologicity. As we have seen, regularity and diagonality are in some sense inverse properties. The minimality (under some uniqueness assumptions) of compact convergences in the complete lattice of convergences makes regularity and diagonality coincide. A convergence is normal indexnormal convergence if for any two disjoint closed sets A0 , A1 there exist disjoint open sets O0 , O1 such that A0 ⊂ O0 and A1 ⊂ O1 . Theorem 49. Each compact topologically regular convergence is normal. Proof. Suppose that a convergence on X is not normal: there exist disjoint closed sets A0 , A1 such that N (A0 ) # N (A1 ). Let U be an ultraﬁlter ﬁner than N (A0 ) ∨ N (A1 ). By compactness, there exists x ∈ lim U. If x ∈ / A0 , then Ac0 is an open set that contains x, hence by topological regularity there exists U ∈ U such 63Indeed, because ξ is of countable character, x ∈ lim R ξ (n) (xn,k )k implies that there exists

R a countably based ﬁlter E such that x ∈ limξ E and E ⊂ (n) (xn,k )k . Then, if needed, we can pick a subsequence of (xn )n and for each n present in this subsequence, a subsequence of (xn,k )k so that the ﬁlter of the type (34) restricted to these subsequences, converges to x.

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that cl U ∩ A0 = ∅, that is, U ∈ / N # (A0 ), which yields a contradiction. For the same reason, x ∈ A1 , and thus A0 ∩ A1 = ∅ contrary tothe assumption. A convergence ξ is a quasi-topology if P ξ = T ξ. By Proposition 22, this means that adhξ is idempotent. Theorem 50. Each Hausdorﬀ regular compact convergence is a quasi-topology. Proof. Let x ∈ adh2 A. By deﬁnition, there exists an ultraﬁlter U such that {x} = lim U (by Hausdorﬀness) and U # adh A, hence V(U) # A by virtue of (32). Therefore there is an ultraﬁlter W on A such that W # V(U), equivalently U # adh W. Therefore, by regularity and by compactness, {x} = lim U ⊃ lim adh W = lim W = ∅, which proves that x ∈ adh A.

Every regular quasi-topology is topologically regular. Indeed, the regularity of a convergence ξ is deﬁned with the aid of set adherence adhξ , which depends only on P ξ; if ξ is a quasi-topology, P ξ = T ξ, thus the regularity of ξ amounts to the topological regularity of ξ.Therefore by Theorems 50 and 49, each Hausdorﬀ regular compact convergence is normal, and in view of Proposition 42, we get [33, 35, 25]. Corollary 51. Each Hausdorﬀ regular compact pseudotopology is a topology. Unlike for topologies, a compact Hausdorﬀ convergence need not be regular, because there exist non topological compact Hausdorﬀ pseudotopologies (see e.g., the Kuratowski convergence in Subsection 15.3 and Proposition 93). Diagonality and regularity are antithetic properties. Therefore, in case of Hausdorﬀ compact pseudotopologies, each of them entails the other. This is due to the minimality of Hausdorﬀ compact pseudotopologies in the class of Hausdorﬀ pseudotopologies. Hence, the following extension of a classical result can be considered as a dual of Corollary 51. Theorem 52. Each Hausdorﬀ pretopologically diagonal compact pseudotopology is regular. Proof. If not, then there is x and a ﬁlter F such that x ∈ limξ F \limξ adhξ F. Because ξ is Hausdorﬀ, {x} = limξ W for every W ∈ βF, so that adhξ F = {x}. Let U be an ultraﬁlter such that U # adhξ F and x ∈ / limξ U. Consequently, Vξ (U) # F. As ξ is compact, there is y ∈ limξ U, thus y ∈ limξ Vξ (U), because ξ is pretopologically diagonal, hence y ∈ adhξ F. Therefore x = y, which is a contradiction. By virtue of Corollary 51, under the hypotheses of Theorem 52, the resulting convergence is also topological. 10. Filter-determined convergences Let H be a class of ﬁlters. A convergence ξ is said to be H-based if x ∈ limξ F implies the existence of a ﬁlter H ∈ H such that x ∈ limξ H and H ⊂ F. Example 53. If H is the class of sequential ﬁlters, then H-based convergences are precisely sequentially based convergences.

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Example 54. If H is the class of countably based ﬁlters, then H-based convergences are convergences of countable character. Example 55. A ﬁlter on a convergence space is said to be locally compact if it contains a compact set. The set H(X) is that of the ﬁlters on X, which are of the class H. If H(X) is the set of locally compact ﬁlters on X, then H-based convergences are called locally compact. Class H of ﬁlters Sequential Countably based Locally compact

H-based convergences sequentially based of countable character locally compact

Figure 2. Filter-determined convergences

11. Categories of convergence spaces We have seen that the sets of topologies, pretopologies and pseudotopologies on a given underlying set, is stable for arbitrary suprema, and contains the least convergence, that is, the chaotic topology. The initial convergence of a topology (respectively, a pretopology) is a topology (respectively, a pretopology). The set of sequentially based convergences on a given underlying set, is stable for arbitrary inﬁma, and contains the greatest convergence, that is, the discrete topology. The ﬁnal convergence of a sequentially based convergence is a sequentially based convergence. Such situations are well-known in category theory.64 At this point a use of category theory is not only illuminating, but essential for the eﬃciency of our investigation. I keep this use at a minimal level, because, on one hand, a comfortable employment of category theory requires itself a patient apprenticeship, and on the other, convergences form quite a simple category. The book [1] of J. Ad´ amek, H. Herrlich, and E. Strecker is a basic reference in category theory. 11.1. Abstract and concrete categories. Objects and morphisms are primitive notions. A category C consists of a class of objects and of a class of morphisms such that for every couple (ξ, τ ) of objects of C, there exists a set homC (ξ, τ ) of morphisms f : ξ → τ, so that for f ∈ homC (ξ, τ ) and g ∈ homC (τ, θ), the composition g ◦ f ∈ homC (ξ, θ), the composition is associative, and for each object ξ, the set homC (ξ, ξ) contains a neutral element 1ξ of the composition (called identity). A map F from the class of morphisms of a category C to the class of morphisms of a category D is called a functor (or, covariant functor ) if F (g ◦ f ) = F g ◦ F f,

F (1ξ ) = 1F ξ

for every ξ ∈ C. Therefore each functor F induces a map on objects, denoted also by F , to the eﬀect that F (ξ) = F (1ξ ). 64They correspond to concretely reﬂective and coreﬂective subcategories of a topological construct.

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Example 56. In the category of sets Set, the objects are sets, and the morphisms are maps. More precisely, for X, Y ∈ Set, one deﬁnes the set of morphisms homSet (X, Y ) = Y X (the set of all maps from X to Y ). A category C is called concrete (over Set) if there exists a functor |·| : homC (ξ, τ ) → homSet (|ξ|, |τ |), which is faithful (which, in the language of category theory, means injective). Therefore we can identify | homC (ξ, τ )| ⊂ homSet (|ξ|, |τ |). Consequently if C is concrete, then |ξ| is a set for every C-object ξ (the underlying set of ξ), and for every morphism ϕ ∈ homC (ξ, τ ), the image |ϕ| is a map from the set |ξ| to the set |τ |. Example 57. In the category of convergences C = Conv, the objects are convergences, and the morphisms are continuous maps. For every convergence ξ there is a unique set |ξ| on which the convergence is deﬁned, and every continuous map ϕ ∈ C(ξ, τ ) deﬁnes |ϕ| : |ξ| → |τ |. On the other hand, if f : X → Y , and ξ, τ are convergences respectively on X and Y , then there is at most one morphism ϕ ∈ homC (ξ, τ ) such that |ϕ| = f . In other words, ϕ ∈ homC (ξ, τ ) if and only if |ϕ| ∈ C(ξ, τ ). We have seen that there always exist initial and ﬁnal convergences associated with families of maps. In other words, the category of convergence spaces always admits initial and ﬁnal objects. Such categories are called topological constructs. The ﬁber of a set X of a concrete category C is {ξ ∈ C : |ξ| = X}. Each concrete category induces a partial order on its ﬁbers, that is, if |ξ| = |θ| then ξ ≥ θ whenever there is ϕ ∈ homC (ξ, θ) such that |ϕ| = i, the identity map on X. If C is a topological construct, then each ﬁber endowed with this partial order constitutes a complete lattice. In a topological construct C a map f ∈ homSet (|ξ|, |τ |) is a morphism of C whenever f ξ ≥ τ (equivalently, ξ ≥ f − τ ). Let C be a topological construct. A functor F : C → C is concrete if |F ξ| = |ξ| for every object ξ of C. The following is a special case of a result in [21]. Theorem 58. A map H on the class of convergences uniquely determines a concrete functor if and only if (36)

|Hξ| = |ξ|,

(37)

ζ ≥ ξ ⇒ Hζ ≥ Hξ,

(38)

f (Hξ) ≥ H(f ξ)

for every ζ, ξ and every map f from |ξ|. Proof. Let H be the restriction to objects of C of a concrete functor. If ζ ≥ ξ, that is, if the identity map i belongs to C(ξ, τ ) then i ∈ C(Hξ, Hτ ), equivalently Hξ ≥ Hτ . As f ∈ C(ξ, f ξ), also f ∈ C(Hξ, H(f ξ)), which means that f (Hξ) ≥ H(f ξ). Conversely, if f ∈ C(ξ, τ ) then f ξ ≥ τ , hence f (Hξ) ≥ H(f ξ) ≥ Hτ by (38) and (37), hence f ∈ C(Hξ, Hτ ). Therefore if ϕ ∈ homC (ξ, τ ) then Hϕ is the unique morphism in homC (Hξ, Hτ ) such that |ϕ| = |Hϕ|. If ϕ ∈ homC (ξ, τ ) and ψ ∈ homC (τ, θ) then ψ ◦ ϕ ∈ homC (ξ, θ). Then |H(ψ ◦ ϕ)| = |ψ ◦ ϕ| = |ψ| ◦ |ϕ| = |Hψ| ◦ |Hϕ|, thus by construction H(ψ ◦ ϕ) = Hψ ◦ Hϕ. Also |H(iξ )| = |iξ | = |iHξ |, which shows that H(iξ ) = iHξ .

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Notice that (36), (37) and (38) are equivalent to (36), (37) and (39)

H(g − τ ) ≥ g − (Hτ )

for every map f to |τ |65). 11.2. Subcategories of convergence spaces. A category D is a subcategory of a category C if the objects and morphisms of D are also, respectively, objects and morphisms of C. We have seen that the category of convergence spaces is concrete. A class of convergence spaces is a (concretely) reﬂective subcategory (of the category of convergence spaces) if (on every ﬁber of |·|) it (1) is stable for arbitrary suprema, (2) contains the least convergence, (3) is preserved by initial convergences. A class of convergence spaces (with continuous maps as morphisms) is a (concretely) coreﬂective subcategory (of the category of convergence spaces) if (on every ﬁber of |·|) it (1) is stable for arbitrary inﬁma, (2) contains the greatest convergence, (3) is preserved by ﬁnal convergences. If H stands for the objects of a (concretely) reﬂective subcategory, then there exists a map H associating with each convergence ξ (on X) the ﬁnest convergence Hξ (on X) from H, which is coarser than ξ. The map H is the corresponding reﬂector on ﬁx H = H, where ﬁx H = {ξ : Hξ = ξ}. The coreﬂector is deﬁned analogously. As in the sequel I will not consider non-concretely reﬂective and coreﬂective subcategories, the word concretely will be always omitted. Therefore topologies and pretopologies are reﬂective subcategories of the category of convergence spaces. The topologizer T and the pretopologizer P are the corresponding reﬂectors. Similarly, pseudotopologies and paratopologies are reﬂective subcategories of the category of convergence spaces. The pseudotopologizer S and the paratopologizer Pω are the corresponding reﬂectors. If W is a functor, then we say that a convergence ξ is W -regular if it is W ξ-regular. It can be easily checked that W -regular convergences form a reﬂective subcategory for every functor W . Sequentially based convergences form a coreﬂective subcategory of the category of convergence spaces. The sequential modiﬁcation Seq is the corresponding coreﬂector. Proposition 59 ([10]). If H is a class of ﬁlters (possibly depending on convergence) that includes all the principal ultraﬁlters, and such that H(ξ) ⊂ H(θ) if ξ ≥ θ and such that H ∈ H(ξ) implies that f (H) ∈ H(f ξ), then the class of H-based convergences is coreﬂective and limBH ξ F = limξ G H(ξ)G⊂F

is the coreﬂector. 65Suppose (38) and let τ = f ξ to get Hξ ≥ H(f − f ξ) ≥ f − (H(f ξ)). Apply f to both the sides of the inequality to the eﬀect that f (Hξ) ≥ f f − (H(f ξ)) ≥ H(f ξ). Conversely, if (39) holds, then set ξ = f − τ and apply f − to obtain H(f − τ ) ≥ f − f (H(f − τ )) ≥ f − H(f f − τ ) ≥ f − (Hτ ).

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Example 60. Because the class of countably based ﬁlters does not depend on convergence, and the image of a countably based ﬁlter is countably based, the convergences of countable character form a coreﬂective subcategory of convergences. The corresponding coreﬂector will be denoted by First. Example 61. If ξ ≥ θ, then each ξ-compact set is θ-compact, and thus each locally compact ﬁlter in ξ is locally compact in θ. As the continuous image of a compact set is compact, the continuous image of a locally compact ﬁlter is locally compact. Therefore, by Proposition 59, the locally compact convergences form a coreﬂective subcategory of convergences. The corresponding coreﬂector will be denoted by K. Reﬂectors and coreﬂectors are functors. Theorem 62 ([20]). For every functor H, the class of all convergences ξ such that ξ ≤ Hξ

(40) is reﬂective, and

Hξ ≤ ξ

(41) is coreﬂective.

As H is order-preserving, o ≤ Ho. If (40) holds for each ξ ∈ Ξ, then Proof. Ξ ≤ ξ∈ξ Hξ, and the latter is always less than H( ξ). Finally (40) implies f − ξ ≤ f − (Hξ), which is less than H(f − ξ) by (39), showing that (40) is preserved by initial convergences. This proves that the class (40) is reﬂective. The coreﬂectivity of (41) can be proved analogously. A category-theory concepts of initial and ﬁnal density are extremely useful in this quest. A class D of convergences is called initially dense in a subcategory M (of convergences) if for every (object) τ of M there exists a collection of maps {fι : ι ∈ I} such that fι− ζι τ= ι∈I

with ζι ∈ D for each ι ∈ I. A class D of convergences is called ﬁnally dense in a subcategory M if for every (object) ξ of M there exists a family of maps {fι : ι ∈ I} such that

ξ= fi τi . i∈I

with τι ∈ D for each ι ∈ I. 12. Functorial inequalities Some important classes of classical topologies can be characterized with the aid of convergence inequalities of the type (42)

ξ ≥ JEξ,

where J is a reﬂector and E is a coreﬂector [15, 10]. We shall call them JEconvergences. By Theorem 62, classes of JE-convergences form a coreﬂective subcategory of the category of convergence spaces. Restricted to topologies, they form a coreﬂective subcategory of the category of topological spaces provided that J ≥ T .

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Example 63. A topology is called sequential if each sequentially closed66 set is closed. A topology ξ is sequential if and only if (42) holds with J = T (the topologizer) and E = Seq (the sequential modiﬁer). Indeed (for an arbitrary convergence ξ) ξ ≥ T Seq ξ amounts to T ξ ≥ T Seq ξ ≥ T ξ (because T is order-preserving and Seq ξ ≥ ξ), which means that if a set is closed for Seq ξ then it is closed for ξ. Example 64. A topology is called Fr´echet if x ∈ clξ H implies the existence of a sequential ﬁlter E on H that converges to x in ξ. It turns out that a topology ξ is Fr´echet if and only if (42) holds with J = P (the pretopologizer) and E = Seq (the sequential modiﬁer). In fact, ξ ≥ P Seq ξ for a topology ξ whenever clξ H = adhξ H ⊂ adhSeq ξ H for every H, which means that x ∈ clξ H implies the existence of a sequential ﬁlter E on H that converges to x in ξ. Conversely, the latter statement implies that for every ﬁlter F, adhξ H ⊂ adhSeq ξ H, # # H∈F

H∈F

which, in view of Theorem 35 (27) means that limP ξ F ⊂ limP Seq ξ F, and thus ξ ≥ P ξ ≥ P Seq ξ. Example 65. A topology ξ is a k-topology if H ∩ C is closed in ξ ∨ C (the restriction of ξ to C) for every ξ-compact set C, then H is ξ-closed. It turns out that a topology ξ is a k-topology if and only (42) holds with J = T (the topologizer) and E = K (compact localization), that is, ξ ≥ T Kξ. JE-convergences for various special reﬂectors J and coreﬂectors E, are written in terms of standard classes of topologies, the deﬁnitions of which are furnished in the footnote below.67 J/E I S Pω P T

identity pseudotopologizer paratopologizer pretoplogizer topologizer

Seq

First

K

sequentially based sequentially based strongly Fr´echet Fr´echet sequential

countable character bisequential strongly Fr´echet Fr´echet sequential

locally compact locally compact strongly k

k k

Figure 3. JE-convergences. See [10] for a more complete table.

13. Quotient maps The objects of every reﬂective subcategory of convergence spaces can be represented as ﬁx J (the collection of ﬁxed points of a reﬂector J). If f : X → Y is a 66A set A is sequentially closed if the limit of every sequence with terms in A, is in A. 67A topology ξ is called strongly Fr´ echet if for every descending sequence (Hn )n of sets such

T that x ∈ n<∞ clξ Hn implies the existence of a sequence (xn )n such that x ∈ limξ (xn )n and xn ∈ Hn for each n < ∞. This is equivalent to the condition adhξ H ⊂ adhFirst ξ H for each countably based ﬁlter H. A topology ξ is called bisequential if adhξ H ⊂ adhFirst ξ H for every ﬁlter H. A topology ξ is called a k -topology if x ∈ clξ H implies that there is compact set C such that x ∈ clξ (H ∩ C). T A topology ξ is called a strongly k -topology if x ∈ n<∞ clξ Hn for a descending sequence (Hn )n of sets, then there exists a compact set C such that x ∈ clξ (Hn ∩ C) for each n < ∞.

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map, and ξ ∈ Ξ is a convergence on X, then there exists on Y the ﬁnest convergence from ﬁx J, for which f is continuous.68 This convergence is called the J-quotient of ξ by f . Actually, the J-quotient of ξ by f is equal to J(f ξ). Therefore, we say that a map f from a convergence space (X, ξ) to a convergence space (Y, τ ) is a J-quotient if τ ≥ J(f ξ).

(43)

It turns out that many types of classical maps in topology, like quotient maps, hereditarily quotient maps, countably biquotient maps, biquotient maps, almost open maps are J-quotient maps with respect to a reﬂective subcategory ﬁx J of convergence spaces. You will ﬁnd classical deﬁnitions of quotient, hereditarily quotient and biquotient maps in the examples below. Example 66. A map between topological spaces f : X → Y is quotient if a subset F of Y is closed whenever f − (F ) is closed. Example 67. A map between topological spaces f : X → Y is hereditarily quotient if y ∈ cl B implies that cl f − (B) ∩ f − (y) = ∅ for each subset B of Y . Example 68. A map between topological spaces f : X → Y is biquotient if y ∈ adh H implies that adh f − (H) ∩ f − (y) = ∅ for each ﬁlter H on Y . Theorem 69 ([10]). Let J be a class of ﬁlters and let J = AJ be the reﬂector on the subcategory of convergences, which are adherence determined by J. Then a map f : (X, ξ) → (Y, τ ) is a J-quotient if and only if for every ﬁlter J , (44)

J ∈ J, y ∈ adhτ J =⇒ f − (y) ∩ adhξ f − (J ) = ∅. Proof. Notice that (44) amounts to adhτ J ⊂ f (adhξ f − (J ))

for every ﬁlter J ∈ J, and as adhf ξ J = f (adhξ f − (J )), we conclude that limτ F ⊂ adhτ J ⊂ adhf ξ J = limAJ f ξ F, JJ #F

which is equivalent to τ ≥ J(f ξ).

JJ #F

The following observation is an initial step in the quest for preservation of various classes of spaces by variants of quotient maps. Proposition 70 ([10]). Let J be a reﬂector and E a coreﬂector. Then ξ is a JE-convergence if and only if the identity i : (X, Eξ) → (X, ξ) is J-quotient. Proof. By deﬁnition, i is J-quotient from Eξ to ξ whenever ξ ≥ J(i(Eξ)) = JEξ, that is, whenever ξ is a JE-convergence. Theorem 71 ([10]). The image of a JE-convergence by a continuous J-quotient is a JE-convergence. Proof. If ξ ≥ JEξ and τ ≥ J(f ξ) then f ξ ≥ f (JEξ) ≥ JE(f ξ) by (38). Therefore τ ≥ J(f ξ) ≥ JE(f ξ) ≥ JEτ because of the continuity. 68Indeed, the set Ξ of all the Ξ-convergences on Y for which f is continuous is non-empty, f because it contains the chaotic topology. As the supremum of Ξf belongs to Ξf , there exists the ﬁnest Ξ-convergence for which f is continuous.

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Theorem 71 is illustrated below (Figure 4) for various fundamental variants of quotient maps (and which are quotient with respect to reﬂective classes of convergence spaces) and for most known local properties of topologies.

almost open biquotient countably biquotient hereditarily quotient quotient

J I S Pω P T

Seq

First

K

sequentially based sequentially based Fr´echet strongly Fr´echet sequential

countable character bisequential strongly Fr´echet Fr´echet sequential

locally compact locally compact strongly k k k

Figure 4. Quotient variants The reﬂector with respect to which the quotient is considered is in the second column, the classical name of the type of quotient is in the ﬁrst column. The rows show which properties are preserved by which types of quotient maps. A more extensive table can be found in [10]. If f0 : X0 → Y0 and f1 : X1 → Y1 then the product map f0 × f1 : X0 × X1 → Y0 × Y1 is deﬁned by (f0 × f1 )(x0 , x1 ) = (f0 (x0 ), f1 (x1 )). The problem when the product of two quotient map is quotient has been studied by many authors. Special cases of it (when one of the maps is the identity) are answered by69 Theorem 72 (Whitehead–Michael). A regular (Hausdorﬀ ) topological space is locally compact if and only if the product of its identity map with every quotient map is quotient. Theorem 73 (Michael). A regular (Hausdorﬀ ) topological space is locally countably compact if and only if the product of its identity map with every quotient map from a sequential topological space is quotient. It turns out that there exists a simple convergence-theoretic scheme, which enables one to answer this question. The deﬁnition of J-quotient map (43) can be extended as follows. If M is a functor, then we say that a map f from a convergence space (X, ξ) to a convergence space (Y, τ ) is an M -quotient map if τ ≥ M (f ξ). Remark 74. If M = JE where J is a reﬂector and E a coreﬂector, then a map f , which is a JE-quotient from ξ to τ , is J-quotient from JEξ to τ .70 We denote by iX : X → X the identity map on X. Proposition 75 ([19]). Let M be a functor and L a reﬂector. For every convergence τ (45)

ξ × M τ ≥ L(ξ × τ )

if and only if i|ξ| × f is an L-quotient for every M -quotient f . 69[23, Theorem3.3.17], [37, Theorem 2.1 and 4.1] 70In fact, by (39) τ ≥ JJE(f ξ) ≥ J(f (JEξ)).

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Proof. Let f : τ0 → τ1 be an M -quotient, that is, τ1 ≥ M (f τ0 ). Then ξ × τ1 ≥ ξ × M (f τ0 ) ≥ L(ξ × f τ0 ) = L(i × f )(ξ × τ0 ). Conversely, if f = i|τ | , then M τ = M (i|τ | τ ) so that i|τ | is an M -quotient. By (45) i|ξ| × i|τ | is an L-quotient: ξ × M τ ≥ L(ξ × τ ). In Section 17 we will see that Theorems 72 and 73 are special cases of Proposition 75. 14. Power convergences One of the principal reasons for the occurrence of non-topological convergences was the fact that in general there exists no coarsest topology on the space of continuous maps (from one topological space to another) making the evaluation map continuous. As we shall see, closed subsets of a topological space can be identiﬁed with continuous maps (valued in the Sierpi´ nski topology). Therefore, in general, there is no coarsest topology on a hyperspace (space of closed subsets of a topological space) making the natural evaluation continuous. These facts were at the origin of the introduction of pseudotopologies by Gustave Choquet in [8] in 1947–1948. The space of continuous maps from a convergence ξ to a convergence σ is denoted by C(ξ, σ). If M is a functor, then (46)

C(ξ, σ) ⊂ C(M ξ, M σ)

by virtue of (39) (or equivalently of (38)). Lemma 76. If J is a reﬂector, and ζ and ξ are convergences, then ζ ≥ Jξ ⇔ ∀σ∈ﬁx J C(ξ, σ) ⊂ C(ζ, σ). Proof. Let ζ, ξ be convergences on X. The inequality ζ ≥ Jξ implies C(Jξ, σ) ⊂ C(ζ, σ) for every convergence σ, and if σ = Jσ then C(ξ, σ) ⊂ C(Jξ, σ) by (46). Conversely, suppose that C(ξ, σ) ⊂ C(ζ, σ) for each σ ∈ ﬁx J, in particular C(ξ, Jξ) ⊂ C(ζ, Jξ). As ξ ≥ Jξ, the identity i on X belongs to C(ξ, Jξ), hence i ∈ C(ζ, Jξ), that is, ζ ≥ Jξ. Proposition 77. Let D be initially dense in ﬁx J. If C(ξ, σ) ⊂ C(ζ, σ) for each σ ∈ D, then this holds for each σ ∈ ﬁx J. Proof. Let f ∈ C(ξ, σ) for some σ ∈ﬁx J. By initial density there exists a class of maps {gι : ι ∈ I} such that σ = ι∈I gι− ρι with ρι ∈ D for each ι ∈ I. − Therefore f ξ ≥ g ι ρι for each ι ∈ I, that is, gι ◦ f ∈ C(ξ, ρι ) ⊂ C(ζ, ρι ), or equivalently f ζ ≥ ι∈I gι− ρι = σ, which means that f ∈ C(ζ, σ). In Subsection 4.5 the power convergence (or the continuous convergence) [ξ, σ] (of ξ with respect to σ) was deﬁned as the coarsest among the convergences τ on C(ξ, σ), for which the evaluation e is continuous from ξ ×τ to σ, that is, the coarsest among the convergences θ for which (47)

ξ × θ ≥ e− σ,

where e− σ is the initial convergence of σ by e. The convergence σ in the deﬁnition above is called the coupling convergence. The exponential map t (48)

(t f )(y)(x) = f (x, y)

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is a bijection of the set Z X×Y (of all maps from X × Y to Z) onto the set (Z X )Y (of all the maps from Y to the set of maps Z X ). Its inverse associates by (48) with each g ∈ (Z X )Y an element gˆ of Z X×Y . If now ξ, τ and σ are convergences on X, Y and Z respectively, then an immediate consequence of the deﬁnition of power convergence is that (49)

C(ξ × τ, σ) ↔ C(τ, [ξ, σ]),

where ↔ denotes the restriction of that bijection. Moreover, Theorem 78. For convergences ξ, τ and σ, the exponential is a homeomorphism between the power convergences: (50)

[τ, [ξ, σ]] ≈ [ξ × τ, σ].

Proof. A map h ∈ Z X×Y belongs to lim[ξ×τ,σ] H if and only if for every x ∈ limξ F and for every y ∈ limτ G, we have h(x, y) ∈ limσ F × G, H. As F × G, H = F, G,t H, the preceding formula is equivalent to t h(y)(x) ∈ limσ F, G,t H, that is, to t h(y) ∈ lim[ξ,σ] G,t H, hence equivalent to t h ∈ lim[τ,[ξ,σ]] t H. If f : X → Y and Z is a ﬁxed set, then f ∗ : Z Y → Z X is deﬁned by f ∗ (h) = h◦f , that is, x, f ∗ (h) = f (x), h for every x ∈ X and h : Y → Z. If f ∈ C(ξ, τ ) then f ∗ (C(τ, σ)) ⊂ C(ξ, σ).71 Therefore for each f ∈ C(ξ, τ ) we will see f ∗ as restricted to C(τ, σ), that is, f ∗ : C(τ, σ) → C(ξ, σ). Theorem 79. If f ∈ C(ξ, τ ), then f ∗ is continuous from [τ, σ] to [ξ, σ]. Proof. Let h ∈ lim[τ,σ] H. In order to prove that h ∈ lim(f ∗ )− [ξ,σ] H, or equivalently, f ∗ (h) ∈ lim[ξ,σ] f ∗ (H), one must establish that x, f ∗ (h) ∈ limσ F, f ∗ (H), that is, (51)

f (x), h ∈ limσ f (F), H

for every x ∈ |ξ|, and each ﬁlter F such that x ∈ limσ F. Because f is continuous f (x) ∈ limτ f (F), and since by assumption, h ∈ lim[τ,σ] H (51) holds. In particular, if ξ and τ are convergences on a set X such that ξ ≥ τ , then the identity map i belongs to C(ξ, τ ), hence i∗ maps C(τ, σ) into C(ξ, σ) for every convergence σ. Therefore we consider now i∗ : C(τ, σ) → C(ξ, σ). Because i is the identity, i∗ is the injection of C(τ, σ) into C(ξ, σ). By Theorem 79, i∗ is continuous from [τ, σ] to [ξ, σ]. Hence by Theorem 79, i∗∗ = (i∗ )∗ maps C([ξ, σ], σ) into C([τ, σ], σ) and is continuous from [[ξ, σ], σ] to [[τ, σ], σ]. If A ⊂ B and iA,B : A → B is the injection and Z is a set, then i∗A,B : Z B → Z A is the restriction. Indeed, if h : B → Z, then by deﬁnition, i∗A,B (h) = h ◦ iA,B . In particular, i∗∗ is the restriction, wich associates with each map h : C(ξ, σ) → |σ| the map h ◦ i∗ : C(τ, σ) → |σ|. 71In fact, if h ∈ C(τ, σ) then f ∗ (h) = h ◦ f ∈ C(ξ, σ).

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Lemma 80. If ξ is a convergence on X and σ is a convergence on Z, then j(x)(h) = x, h deﬁnes an embedding j : X → C([ξ, σ], σ), which is continuous from ξ to [[ξ, σ], σ]. Proof. Indeed, if h ∈ lim[ξ,σ] H and x ∈ limξ F, then by deﬁnition, x, h ∈ limσ F, H = limσ j(F)(H). In the particular case F = (x)• this yields j(x)(h) = x, h ∈ limσ (x)• , H = limσ j(x)(H), which shows that j(x) ∈ C([ξ, σ], σ) for every x ∈ X. Corollary 81. For every ξ and σ, one has ξ ≥ j − [[ξ, σ], σ]. It is immediate that if ρ ≥ σ, then C(ξ, ρ) ⊂ C(ξ, σ) and the injection is continuous from [ξ, ρ to [ξ, σ], in symbols, [ξ, ρ [ξ, σ]. Indeed, h ∈ lim[ξ,ρ H whenever x ∈ limξ F implies h(x) ∈ limρ F, H, hence h(x) ∈ limσ F, H, that is, h ∈ lim[ξ,σ] H. In this way, we have deﬁned an order on C(ξ, o), where o stands for the indiscrete topology.72 Proposition 82. For each ξ and a set σ of convergences on a common underlying set, [ξ, σ] = [ξ, σ]. σ∈Σ

lim[ξ,W σ]

H. As we have seen h ∈ C(ξ, σ) and H can be exProof. Let h ∈ tended to a ﬁlter on C(ξ, σ) for each σ ∈ σ. By the deﬁnition of power convergence, x ∈ limξ F implies h(x) ∈ limW σ F, H = σ∈Σ limσ F, H, hence h ∈ lim[ξ,σ] H for each σ ∈ σ, that is, h ∈ limW H. σ∈σ [ξ,σ] If g : W → Z, then the map g∗ : W X → Z X is deﬁned by g∗ (h) = g ◦ h. If ρ is a convergence on W and σ is a convergence on Z, and g ∈ C(ρ, σ), then (the restriction of) g∗ maps C(ξ, ρ) to C(ξ, σ) and is continuous from [ξ, ρ → [ξ, σ]. In fact, Proposition 83. [ξ, g − σ] = g∗− [ξ, σ]. Proof. Let h ∈ lim[ξ,g− σ] H, that is, x ∈ limξ F implies h(x) ∈ limg− σ F, H, that is, by the deﬁnition of initial convergence, g(h(x)) ∈ limσ g(F, H). Because g(F, H) = F, g∗ (H), we conclude that this is equivalent to g∗ (h)(x) ∈ limσ F, g∗ (H), which means that g∗ (h) ∈ lim[ξ,σ] g∗ (H), that is, h ∈ limg∗− [ξ,σ] H. Corollary 84. If gi : Z → Zi is a surjection and σi is a convergence on Zi for each i ∈ I, then gi− σi ] = (gi )− [ξ, ∗ [ξ, σi ]. i∈I

i∈I

14.1. Topologicity and other properties of power convergences. As was said repeatedly, a power convergence of a topology with respect to another topology need not be a topology.73 Now we will see a suﬃcient and necessary condition for a power convergence to belong to a given reﬂective class. If J is a reﬂector, then J-convergences are those convergences τ for which Jτ ≥ τ . 72The open sets of the indiscrete topology on Z are ∅ and Z. 73A characterization of those underlying Hausdorﬀ regular topologies for which the power

convergence is topological will appear through Theorem 89 and Corollary 109.

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Proposition 85. Let J be a reﬂector. Then J[ξ, σ] ≥ [ξ, σ] for each convergence σ = Jσ if and only if ξ × Jτ ≥ J(ξ × τ ) for each convergence τ .74 Proof. Let J fulﬁll the condition and let σ ≤ Jσ. By deﬁnition, [ξ, σ] is the coarsest convergence on C(ξ, σ) for which ξ × [ξ, σ] ≥ e− σ. On applying J to the inequality above, we get by (39), ξ × J[ξ, σ] ≥ J(ξ × [ξ, σ]) ≥ J(e− σ) ≥ e− (Jσ) = e− σ, thus J[ξ, σ] ≥ [ξ, σ], and since J is contractive, [ξ, σ] is a J-convergence. Conversely, suppose J[ξ, σ] ≥ [ξ, σ]. By (76) it is enough to show that C(ξ × τ, σ) ⊂ C(ξ × Jτ, σ). If f ∈ C(ξ × τ, σ) then by Theorem 78 t f ∈ C(τ, [τ, σ]) ⊂ C(Jτ, J[τ, σ]) by (46). By Proposition 79, (t f )∗ ∈ C([J[ξ, σ], σ], [Jτ, σ]). On the other hand, ξ ≥ i− ([J[ξ, σ], σ]), which means that the injection i from |ξ| to |C(C(ξ, σ), σ)| belongs to C(ξ, [J[ξ, σ], σ]). Therefore the composition (t f )∗ ◦ i ∈ C(ξ, [Jτ, σ]), f )∗ ◦ i ∈ C(ξ × Jξ, σ). which means that f = (t We know already a reﬂector that commutes with (arbitrary) products, thus a fortiori fulﬁlls the assumption of Proposition 85. Corollary 86. If a coupling convergence σ is a pseudotopology (a fortiori, a topology), then the power convergence [ξ, σ] is a pseudotopology for each convergence ξ. A reﬂective class ﬁx J of convergence spaces is said to be exponential if J[ξ, σ] ≥ [ξ, σ] for every Jσ ≥ σ. In other words, a class is exponential if the power does not lead out of this class. We conclude that the class of pseudotopologies is exponential, but that of topologies is not. 15. Hyperconvergences 15.1. Sierpi´ nski topology. Sierpi´ nski topology I have mentioned that if the coupling convegence is the Sierpi´ nski topology, then the resulting power convergence is a hyperspace convergence. Let us investigate in detail this important special case of power convergences. In spite of its great simplicity, the Sierpi´ nski topology is a fundamental (nonHausdorﬀ) topology. It is often denoted by $ and is deﬁned on a two-element set, say, {0, 1} by its open sets {∅, {1}, {0, 1}}. This topology is not even T1 (the singleton {1} is not closed), but it is T0 . The Sierpi´ nski topology is compact; even more: every ﬁlter converges with respect to $. Observe that a subset A of a topological space (X, τ ) is closed if and only if the indicator function of A is continuous from τ to $.75 Therefore we can identify the set of all τ -closed subsets of X with C(τ, $). The Sierpi´ nski topology plays an exceptional role among other topologies: it is initially dense in the category of all topologies. This means that every topology τ is the initial convergence with respect to maps valued in the Sierpi´ nski topological space. Namely, (52) τ= f − $, f ∈C(τ,$)

74Notice that nothing was assumed about the underlying convergence ξ. 75The indicator function ·, A of A is deﬁned by x, A = 0 if x ∈ A and x, A = 1 if x ∈ / A.

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(where f − $ denotes the initial convergence of $ by f ). 15.2. Upper Kuratowski convergence. If $ is used as a coupling topology, then the coupling map goes from a subset of X × 2X to {0, 1} and is deﬁned by 0 if x ∈ A, x, A = 1 if x ∈ / A. A ﬁlter F on the set C(τ, $) (of closed subsets of a topological space (X, τ )) converges to A0 ∈ C(τ, $) in the upper Kuratowski convergence if for every x0 ∈ / A0 there exist a neighborhood V of x0 and F ∈ F such that V ∩A = ∅ for each A ∈ F , in other words, (53) clτ A ⊂ A0 . F ∈F

A∈F

The concept can be naturally extended to the case of an arbitrary underlying convergence τ . To this end, I will use the notion of reduced ﬁlter. If F is a ﬁlter on (a subset of) 2X then we denote by |F| the reduced ﬁlter of F, that is, the ﬁlter on X generated by { A∈F A : F ∈ F}. Let F be a set of ﬁlters on (a subset of) 2X . Then F = |F|. (54) F ∈F F ∈F Indeed, F ∈F F consists of the sets of the form F ∈F FF (where FF ∈ F for each F ﬁlter is generated by the unions of the elements of ∈ F), hence its reduced F . The ﬁlter |F| is generated by the sets of the form A. F ∈F F F ∈F F ∈F A∈FF Therefore the ﬁlters in (54) are equal. We say that F upper Kuratowski converges to A0 with respect to a convergence τ if (55)

adhτ |F| ⊂ A0 .

The formula above meansthat x0 ∈ / A0 and x0 ∈ limτ G imply that there exist F ∈ F and G ∈ G such that A∈F A ∩ G = ∅. In the particular case where τ is a topology the condition holds if and only if it holds for G = Nτ (x0 ). Therefore, if τ is a topology then (55) is equivalent to (53). Proposition 87. A ﬁlter F upper Kuratowski converges to A0 with respect to τ if and only if A0 ∈ lim[τ,$] F. Proof. Let τ be a convergence on X. By the deﬁnition of power convergence, A0 ∈ lim[τ,$] F if and only if x0 , A0 ∈ lim$ G, F for every x0 ∈ X and every ﬁlter G on X such that x0 ∈ limτ G. The ﬁlter G, F on {0, 1} is generated by {{x, A : x ∈ G, A ∈ F } : G ∈ G, F inF}. Because the only neighborhood of 0 in $ is the whole space {0, 1}, the convergence condition is restrictive only at 1. Therefore x0 , A0 ∈ lim$ G, F if and only if x0 , A0 = 1 (equivalently, x0 ∈ / A0 ) whenever there is G ∈ G and F ∈ F such that x, A = 1(equivalently, x ∈ / A) for every x ∈ G and A ∈ F , in other words, whenever G ∩ A∈F A = ∅, which amounts to adhτ |F| ⊂ A0 . We see that the upper Kuratowski convergence is a pseudotopology, because it is a power convergence with respect to a topological (hence, a fortiori pseudotopological) coupling convergence $. We know however no reason that it be a topology.

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The cocompact topology on C(τ, $) can be deﬁned by a base of open sets consisting of {A ∈ C(τ, $) : A ∩ K = ∅}, where K is a τ -compact set. Proposition 88. If τ is a Hausdorﬀ topology, then [τ, $] is ﬁner than the cocompact topology with respect to τ . Proof. If A0 ∈ lim[τ,$] F, and K is a compact set disjoint from A0 , then for each x ∈ K there exist Fx ∈ F and a neighborhood Vx of x such that Vx ∩ A∈Fx A = ∅. AsK is compact there are ﬁnitely many x in K, sayx1 , x2 , . . . , xm , such that K ⊂ 1≤j≤m Vxj . Then F = 1≤j≤m Fxj ∈ F and K ∩ A∈F A = ∅. In fact, by a theorem of Choquet [8], Theorem 89. Let τ be a (Hausdorﬀ ) regular topology. Then the upper Kuratowski convergence [τ, $] is a topology if and only if it coincides with the cocompact topology if and only if τ is locally compact. Proof. First let us show that if τ is a locally compact topology, then the cocompact topology with respect to τ coincides with [τ, $]. Let F converge to A0 in the cocompact topology and let x ∈ / A0 . As A0 is τ -closed and τ is locally compact there exists a compact neighborhood K of x such that K ∩ A0 = ∅, hence by assumption, there is F ∈ F such that K ∩ A∈F A = ∅, thus A0 ∈ lim[τ,$] F. If τ is not locally compact, then there is an element x0 , for every closed neighborhood W of which there exists W such that limτ UW = ∅. On an ultraﬁlter U the other hand, each element of W ∈Nτ (x) UW = clτ W =W ∈Nτ (x) UW is generated by the family consisting of W ∈Nτ (x) UW , where UW ∈ UW are all the possible selections. Therefore ( W ∈Nτ (x) UW ) # Nτ (x), hence x ∈ adhτ ( W ∈Nτ (x) UW ). Because all the singletons {x} are closed, the family {{{x} : x ∈ U } : U ∈ UW } generates a ﬁlter ZW on C(τ, $), the set of τ -closed sets. It is immediate that UW is the reduced ﬁlter of ZW , therefore ∅ ∈ lim[τ,$] ZW by virtue of (55) and Proposition 87. We will see that ∅ ∈ / lim[τ,$] ( W ∈Nτ (x) ZW ), and thus [τ, $] is not even a pretopology. Indeed, by (54) the reduced ﬁlter of W ∈Nτ (x) ZW is W ∈Nτ (x) UW , and since / lim[τ,$] ( W ∈Nτ (x) ZW ). adhτ ( W ∈Nτ (x) UW ) = ∅, we conclude that ∅ ∈ I shall characterize vicinities and open sets of [ξ, $] in terms of ξ.76 A family A of closed subsets of a topological space is stable if B ∈ A for every A ∈ A and each closed subset B of A. The vicinity ﬁlters of [ξ, $] admit bases of ξ-stable ﬁlters. Theorem 90. A family A is stable for ξ-closed subsets and a vicinity of A0 with respect to [ξ, $] if and only if Ac = Oξ (Ac ) and is ξ-compact at Ac0 . 76In order not to complicate the presentation, we shall present a special case of [20, Theorem 16.2, Corollary 16.3] where the underlying space is a topology.

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Proof. A family A is a vicinity of A0 for [ξ, $] if and only if A ∈ F for every ﬁlter on C(ξ, $) with A0 ∈ lim[ξ,$] F, equivalently if for each ﬁlter H on |ξ| such that adhξ H ⊂ A0 there exists clξ H = H ∈ H such that H ∈ A. Equivalently, H # Ac implies that adhξ H ∩ Ac0 = ∅, which means that Ac is ξ-compact at Ac0 . Corollary 91 ([17, Theorem 3.1]). A family A is open for [ξ, $] if and only if Ac = Oξ (Ac ) and is ξ-compact. Remark 92. The upper Kuratowski convergence is hypercompact, that is, every ﬁlter converges. 15.3. Kuratowski convergence. A ﬁlter F on C(ξ, $) lower Kuratowski converges to A0 if for every ﬁlter G such that limξ G ∩ A0 = ∅ and for each G ∈ G, there exists F ∈ F such that A ∩ G = ∅ for each A ∈ F [18, p. 306] (equivalently, for each G ∈ G each H ∈ F # there exists A ∈ H such that A ∩ G = ∅)77. Therefore F on C(ξ, $) lower Kuratowski converges to A0 whenever A0 ⊂ adhξ |F # |, where |F # | is the reduced grill. If ξ is a topology, then it is enough, in the deﬁnition above, to take for every x ∈ A0 the coarsest ﬁlter G = Nξ (x) that converges to x. Then the condition becomes: for every x ∈ A0 and each neighborhood V of x there exists F ∈ F such that V ∩ A = ∅ for each A ∈ F . Therefore if ξ is a topology, then the associated lower Kuratowski convergence is a topology. A ﬁlter F on C(ξ, $) Kuratowski converges to A0 if it lower and upper Kuratowski converges to A0 , that is whenever adhξ |F| ⊂ A0 ⊂ adhξ |F # |. If ξ is a topology, then the greatest element of C(ξ, $), to which a ﬁlter F converges in the lower Kuratowski topology, is78 Liξ F = cl A . ξ # H∈F

A∈H

As the least element to which F upper Kuratowski converges is clξ A , Lsξ F = F ∈F

A∈F

and Liξ F ⊂ Lsξ F for every ﬁlter F, a ﬁlter F Kuratowski converges to A0 if and only if Lsξ F ⊂ A0 ⊂ Liξ F. We infer that (without any assumption on the underlying topology): If the underlying convergence is a topology, then the Kuratowski convergence is a Hausdorﬀ pseudotopology. Proposition 93. If the underlying convergence is a topology, then the Kuratowski convergence is compact. Proof. For every ultraﬁlter U (on C(ξ, $)), U # = U, hence Lsξ U = Liξ U.79 If the underlying convergence is a topology, then the Kuratowski convergence is a topology if and only if the upper Kuratowski convergence is a topology.80 77This equivalence follows for [27]. 78(see [17]) 79If Ls U = Li U = ∅ then ∅ is the limit of the Kuratowski convergence with respect to ξ. ξ ξ 80

as the supremum (in a lattice of convergences) of two topologies.

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Consequently, if the underlying convergence ξ is a Hausdorﬀ regular topology, then by Theorem 89 and Corollary 51, the Kuratowski is regular if and only ξ is locally compact. 16. Exponential hull of topologies 16.1. Bidual convergences. Recall that if ξ, σ are convergences, then [ξ, σ] is a convergence on C(ξ, σ) and [[ξ, σ], σ] is a convergence on C([ξ, σ], σ) such that the injection j of |ξ| into C([ξ, σ], σ) is continuous from ξ to [[ξ, σ], σ]. Let Epiσ ξ = j − [[ξ, σ], σ]. It follows from Corollary 56 that81 (56)

ξ ≥ Epiσ ξ.

A convergence ξ is called bidual with respect to a convergence σ whenever ξ = Epiσ ξ. Proposition 94. Let f : X → Y and let ξ be a convergence on X. Then f (Epiσ ξ) ≥ Epiσ f ξ. Proof. Let y ∈ limf (Epiσ ξ) G, hence there exist x and F such that x ∈ limEpiσ ξ F, f (x) = y and f (F) = G. The ﬁrst equality means that x, k ∈ limσ F, K for every k ∈ lim[ξ,σ] K. To show that y ∈ limEpiσ f ξ G consider h ∈ C(f ξ, σ) and a ﬁlter H on C(f ξ, σ) such that h ∈ lim[f ξ,σ] H. It follows that f ∗ (h) ∈ lim[ξ,σ] f ∗ (H), hence y, h = f (x), h = x, f ∗ (h) ∈ limσ F, f ∗ (H) = limσ f (F), H = limσ G, H. Proposition 95. If ξ ≥ θ then Epiσ ξ ≥ Epiσ θ. Proof. Let x ∈ limEpiσ ξ F, that is, h(x) ∈ limσ F, H for every h ∈ C(ξ, σ) and each ﬁlter H on C(ξ, σ) such that h ∈ lim[ξ,σ] H. Because C(θ, σ) ⊂ C(ξ, σ) and the injection is continuous from [θ, σ] to [ξ, σ], if h ∈ lim[θ,σ] H then h ∈ lim[ξ,σ] H and thus h(x) ∈ limσ F, H, which proves that x ∈ limEpiσ θ F. It follows from Propositions 94 and 95 that Epiσ is a (concrete) functor. Proposition 96. [ξ, σ] = [j − [[ξ, σ], σ], σ]. Proof. Let h ∈ lim[ξ,σ] H. This means that x ∈ limξ F implies h(x) ∈ limσ F, H. In order to prove that h ∈ lim[j − [[ξ,σ],σ],σ] H, we need to show that h(x) ∈ limσ F, H for ﬁlter F on X and every x ∈ lim[[ξ,σ],σ] F. The latter means that g(x) ∈ limσ F, G for every g ∈ C(ξ, σ) and each ﬁlter G on C(ξ, σ) such that g ∈ lim[ξ,σ] G. In particular, this holds for g = h and G = H. In particular, with F = (x)• we conclude that C(ξ, σ) ⊂ C(j − [[ξ, σ], σ], σ). Conversely, by (56) the injection of C(Epiσ ξ, σ) into C(ξ, σ) is continuous from − [j [[ξ, σ], σ], σ] to [ξ, σ]. Therefore, Epiσ is idempotent. We have already noticed that Epiσ is a concrete functor. Therefore, on recalling (56), we conclude that 81Here is a direct proof. Let x ∈ lim F and let h ∈ lim ξ [ξ,σ] H. By deﬁnition of power

convergence, this means that h(x) ∈ limσ F , H. Hence x ∈ lim[[ξ,σ],σ] F .

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Theorem 97. For every convergence σ, the map Epiσ is a concrete reﬂector. For a given functor L, deﬁne EpiL ξ =

σ=Lσ

Epiσ ξ.

A convergence ξ is bidual with respect to a functor L if ξ = EpiL ξ. It follows that Corollary 98. For every functor L, the map EpiL is a concrete reﬂector. 16.2. Epitopologies. If L = T (the topologizer), then we abridge Epi = EpiT . A convergence ξ is called an epitopology if Epi ξ ≥ ξ. By Corollary 98 the class of epitopologies is a concretely reﬂective subcategory of the category of convergence spaces. Of course, Epi is the corresponding reﬂector, that we call the epitopologizer. Because the Sierpi´ nski topology $ is initially dense in ﬁx T (the category of topological spaces), by Corollary 84 implies that Epi ξ = Epi$ ξ. This fact is of great importance, because it reduces considerably the complexity of reasonings envolving the epitopologizer. Proposition 99. The epitopologizer commutes with ﬁnite products. Proof. By Proposition 96 [Epi ξ, $] = [ξ, $]. This and (50) imply that [ξ × Epi τ, $] ≈ [ξ, [Epi τ, $]] = [ξ, [τ, $]] ≈ [ξ × τ, $]. Hence Epi(ξ × Epi τ ) = j − [[ξ × Epi τ, $], $] = j − [[ξ × τ, $], $] = Epi(ξ × τ ), and thus ξ × Epi τ ≥ Epi(ξ × Epi τ ) = Epi(ξ × τ ). Therefore Epi ξ × Epi τ ≥ Epi(Epi ξ × τ ) ≥ Epi Epi(ξ × τ ) = Epi(ξ × τ ).

It follows from Proposition 85 that Corollary 100. The class of epitopologies is exponential. F. Mynard gave in [38] an explicit formula for the epitopologizer similar to that for adherence-determined convergences in (24), namely clξ (adhξ H), (57) limEpi ξ F = Fξ H#F

where Fξ stands for the class of the ξ-reduced ﬁlters, that is, the ﬁlters of the form |G| ≈ { B∈G B : G ∈ G} where G is a ﬁlter on C(ξ, $). Proof. By deﬁnition, x ∈ limEpi ξ F if and only if x, A ∈ lim$ F, G for every A ∈ C(ξ, $) and each ﬁlter G on C(ξ, $) such that A ∈ lim[ξ,$] G. Because the only $-neighborhood of 0 is the whole {0, 1}, the condition x, A ∈ lim$ F, G need be considered only in case x, A = 1. Therefore, and on recalling (55), x ∈ limEpi ξ F if and only if for every ﬁlter G on the set of ξ-closed sets such that adhξ |G| ⊂ A = clξ A if x ∈ / A then there is F ∈ F and G ∈ G such that F ∩ B∈G B = ∅, which means that F does not mesh with |G|. By taking A = clξ (adhξ |G|), our condition can be rephrased: x ∈ clξ (adhξ H) for every ξ-reduced ﬁlter H that meshes with F. In order to avoid introducing several other concepts, the following characterization of epitopologies is given here only for T1 -convergences, that is, those for which all the singletons are closed.82 82One of the characterizations of epitopologies (due to Bourdaud) is: A convergence is an epitopology if and only if it is a star-regular pseudotopology with closed limits, [20].

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Proposition 101. A T1 convergence is an epitopology if and only if it is a pseudotopology with closed limits. Proof. If a convergence ξ is T1 , then every ﬁlter on |ξ| is a ξ-reduced ﬁlter,83 hence in this case (57) becomes clξ (adhξ H). (58) limEpi ξ F = H#F

If ξ is an epitopology, it has closed limits, and is a pseudotopology, then by (58) because limSξ F = H#F adhξ H ⊂ H#F clξ (adhξ H). Conversely, if ξ is a pseudotopology with closed limits and x ∈ / limξ F then there is an ultraﬁlter U # F such that x ∈ / adhξ U = limξ U = clξ (adhξ U) because the limits are closed. Hence x∈ / limEpi ξ F showing that ξ is an epitopology. Proposition 102. Each topology is an epitopology. Proof. Obviously, each principal ﬁlter of a ξ-closed set is ξ-reduced. Hence by (57) and (28) each topology is an epitopology. It was mentioned in a footnote that some authors (e.g., H. J. Kowalsky [36]) deﬁne a convergence as a relation fulﬁlling (8), (9) (as I do) and an additional axiom (59)

lim F0 ∩ lim F1 ⊂ lim(F0 ∩ F1 ).

I call a convergence a prototopology if it fulﬁlls (59). It is straightforward that prototopologies constitute a concretely reﬂective subcategory of convergence spaces. An important fact is that Proposition 103. The class of topologies is ﬁnally dense in that of prototopologies. Proof. Let ξ be a prototopology on X. For every x ∈ X and each ﬁlter F such that x ∈ limξ F let τx,F be a topology on X such that F ∧ (x)•is the neighborhood ﬁlter of x and all other elements of X are isolated. Then ξ = (F ,x)∈ξ τx,F . Theorem 104 ([6]). The category of epitopologies is the least exponential reﬂective subcategory of prototopologies that includes all topologies. Proof. Let L be an exponential reﬂective subcategory of prototopologies that contains all topologies. By Proposition 103 for every prototopology ξ, there exist a family {τk : k ∈ K} of topologies such that ξ = k∈K τk . By virtue of Corollary 84, [ξ, $] = (j ∗ )− [τk , $]. k∈K

Because L is exponential and contains all topologies, [τk , $] ∈ L for every k ∈ K, and since L is reﬂective, [ξ, $] ∈ L as the initial object with respect to prototopologies in L. Therefore [[ξ, $], $] ∈ L because L is exponential, and thus the initial prototopology Epi ξ = j − [[ξ, $], $] belongs to L, because L is reﬂective. It follows that every epitopology belongs to L. 83Indeed, if H is a ﬁlter on |ξ| then take the ﬁlter G on C(ξ, $) generated by {{{x} : x ∈ H} : H ∈ H}. Then |G| = H.

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17. Reﬂective properties of power convergences We have seen that if M is an arbitrary functor, then the class of all the convergences τ such that M τ ≥ τ is reﬂective.84 An important problem consists in characterizing convergences ξ such that M [ξ, σ] ≥ [ξ, σ]

(60) for each σ ∈ D ⊂ ﬁx M .

85

Example 105. If M is equal to the topologizer T , then (60) is equivalent to the problem, the solution of which was given in Theorem 89 in the special case of Hausdorﬀ regular topologies ξ and the class D consisting of the Sierpi´ nski topology $. Let M be a functor, L a reﬂector. For a convergence σ, let (61)

EpiσM ξ = j − [M [ξ, σ], σ],

where j is the injection of |ξ| into C([ξ, σ], σ), and EpiσM ξ. (62) EpiL M ξ = σ=Lσ

Theorem 106 ([20, Theorem 15.2]). Let M be a functor, L a reﬂector, and ξ ≥ θ be convergences on the same underlying set. The following are equivalent: (63)

θ × M τ ≥ L(ξ × τ ) for each τ ;

(64)

M [ξ, σ] ≥ [θ, σ] for every σ = Lσ;

(65)

θ ≥ EpiL M ξ.

Proof. (63) ⇒ (64). If τ is a topology on a singleton86, then (63) implies θ ≥ Lξ, hence ξ ≥ θ ≥ Lξ. Therefore C(Lξ, σ) ⊂ C(θ, σ) ⊂ C(ξ, σ) for every convergence σ. On the other hand, C(ξ, σ) ⊂ C(Lξ, Lσ) for every functor L, so that if σ = Lσ then C(ξ, σ) ⊂ C(Lξ, σ). This implies that [ξ, σ] and [θ, σ] have the same underlying set. For τ = [ξ, σ] the inequality (63) becomes θ × M [ξ, σ] ≥ L(ξ × [ξ, σ]) ≥ L(e− σ) = e− σ, which means that M [ξ, σ] ≥ [θ, σ]. (64) ⇒ (65). On applying j − [·, σ] to (64), we get EpiσM ξ = j − [M [ξ, σ], σ] ≥ j − [[θ, σ], σ] ≤ θ for every σ = Lσ, thus (65). (65) ⇒ (63). By Lemma 76 it is enough to prove that C(ξ×τ, σ) ⊂ C(θ × M τ , σ) for each τ and every σ = Lσ. If f ∈ C(ξ × τ, σ) then by (49) t f ∈ C(τ, [ξ, σ]), hence by (46) t f ∈ C(M τ, M [ξ, σ]), and thus (t f )∗ ∈ C([M [ξ, σ], σ], [M τ, σ]) by Proposition 79. On the other hand, by (65), θ ≥ j − ([M [ξ, σ], σ]) for each σ ∈ ﬁx L, which means that the injection j belongs to C(θ, [M [ξ, σ], σ]). Therefore the composition (t f )∗ ◦ j ∈ C(θ, [M τ, σ]), which means that f = (t f )∗ ◦ j ∈ C(θ × M ξ, σ). In the particular case M = JE where J is a reﬂector, E is a coreﬂector and θ = ξ, Theorem 106 entails immediately87 the following special case of a theorem of F. Mynard [39, Theorem 3.1]. 84This does not mean in general that M is the reﬂector of the reﬂective class it deﬁnes. 85This is a very special case of problems thoroughly studied by F. Mynard (e.g., [39]). 86There is a unique topology on a singleton. 87on using JJ = J.

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Theorem 107. Let J, L be reﬂectors, E a coreﬂector, and ξ a convergence. The following are equivalent: (66)

ξ × Jτ ≥ L(ξ × τ ) for each τ ≥ JEτ (for each τ = Eτ );

(67)

JE[ξ, σ] ≥ [ξ, σ] for every σ = Lσ;

(68)

ξ ≥ EpiL JE ξ;

(69)

i × f is L-quotient for every J-quotient f with JE-domain. The last term of the equivalence above follows from Remark 74 and Theorem 71.

Remark 108. The list of equivalences in Theorem 106 can be extended. In particular, they hold if and only if M [ξ, ρ] ≥ [θ, ρ] for every ρ ∈ D,

(70)

where D is an initially dense subclass of ﬁx J. In fact, if σ ∈ ﬁx J, then there exists a class of maps {gι : ι ∈ I} such that σ = ι∈I gι− ρι with ρι ∈ D for each ι ∈ I. Therefore, by Corollary 84, (71) [ξ, σ] = [ξ, gι− ρι ] = (gι∗ )− [ξ, ρι ], ι∈I

ι∈I

hence, by (39), M [ξ, σ] ≥

ι∈I

M (gι∗ )− [ξ, ρι ] ≥

ι∈I

(gι∗ )− M [θ, ρι ],

so that M [ξ, σ] ≥ [θ, σ] by virtue of (70) and (71). On the other hand, (65) implies (72) θ≥ j − [M [ξ, ρ, ρ], ρ∈D

and (72) entails (63), because on one hand, our proof of the implication (63) by (65) hinges entirely on Lemma 76, and on the other, the conclusion of Lemma 76 depends on initially dense subclasses by virtue of Proposition 77. This fact has considerable importance in deciding whether a power convergence with respect to a topology is in the reﬂective class determined by M . Corollary 109. If M ≥ T is a functor, and ξ is a convergence, then M [ξ, σ] ≥ [ξ, σ] for every topology σ if and only if M [ξ, $] ≥ [ξ, $] if and only if ξ ≥ j − [M [ξ, $], $].88 By Theorem 89 the upper Kuratowski convergence with respect to a Hausdorﬀ regular topology is topological (equivalently, pretopological) if and only if the underlying topology is locally compact. Therefore Corollary 110. The topologizer T and the pretopologizer P do not commute with ﬁnite products. Example 111. Let us come back to the case where M is equal to the topologizer T and D consists of the Sierpi´ nski topology. Then the power convergences are upper Kuratowski convergences. Now Theorem 106 furnishes a suﬃcient and necessary condition for the topologicity of [ξ, $] for an arbitrary convergence (without any separation assumptions). We just need to interpret the condition ξ ≥ EpiT ξ, which amounts to j − [T [ξ, $], $] ≤ ξ. 88In fact, it can be shown [39] that EpiT ξ = j − [M [ξ, $], $]. M

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A topology ξ is called core-compact [9] if for every element x and each O ∈ Nξ (x) there exist V ∈ Nξ (x) such that V is ξ-compact at O. The following theorem was established in diﬀerent terms89 by Hofmann and Lawson in [32]; a more general result (for arbitrary convergences) was proved in [20]. Theorem 112. The upper Kuratowski convergence with respect to a topology ξ is topological if and only if ξ is core-compact. Proof. By Corollary 109 with M = T , we need show that ξ is core-compact if and only if ξ ≥ j − [T [ξ, $], $]. Because ξ is a topology, it is enough to show that for every element x the neighborhood ﬁlter Nξ (x) converges to x in j − [T [ξ, $], $]. In other words, we need prove that for every ξ-closed set A x, A ∈ limξ Nξ (x), N[ξ,$] (A).

(73)

In the Sierpi´ nski topology the only case of the formula above which is not always fulﬁlled is when x, A = 1 (equivalently, if x ∈ / A). In this case, there exist V ∈ Nξ (x) and A ∈ N[ξ,$] (A) such that V ∩D = ∅ for each D ∈ A. By Corollary 91 this is equivalent to the existence of a ξ-compact family B = Ac such that O ∈ B and B∈B B ⊃ V and thus B∈B B ∈ Nξ (x). Therefore if H is a ﬁlter such that V ∈ H# then H # B and by compactness adhξ H ∈ B# hence in particular, adhξ H ∩ O = ∅, that is, ξ is core-compact. Conversely if V is compact at O, then the family Oξ (V ) of all ξ-open sets which contain V , is ξ-compact at O. Similarly, if M = L = T Theorems 106 and 112, and Proposition 75 yield a generalization of Theorem 72: Theorem 113. A topological space is core-compact if and only if the product of its identity map with every quotient map is quotient. These results are instances of a general scheme, which enables one, for example, to characterize those ξ for which [ξ, $] is a T E-convergence. If in the deﬁnition of core-compactness we replace the topology ξ by T Seq ξ (equivalently, by T First ξ) then we get countable core-compactness: a topology ξ is called countably core-compact if for every element x and each O ∈ Nξ (x) and each countably based ﬁlter F which converges to x, there exists F ∈ F which is ξ-compact at O. Also Corollary 91 is a special case of a more abstract result, which in particular gives a characterization of sequentially open subsets of [ξ, $] in terms of countably compact families of open sets (see [2]). Because a topology τ is sequential whenever τ = T First τ , on replacing T by T First in Theorem 112 and in its proof, we conclude that Theorem 114. [19] The upper Kuratowski convergence with respect to a topology ξ is a sequential topology if and only if ξ is countably core-compact. On setting in Theorem 107 J = L = T and E = First, we recover this generalization of Theorem 73: Theorem 115. [19] A topological space is countably core-compact if and only if the product of its identity map with every quotient map from a sequential topological space is quotient. 89of the Scott topology

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Contemporary Mathematics Volume 486, 2009

Closure Marcel Ern´e Abstract. We start with a survey of the origins of the concept of closure since the late 19th century. Then, in Section 2, we present the essential facts about set-theoretical closure operators and closure systems in the classical setting, putting emphasis on a common treatment of topological and algebraic closure spaces. A uniform approach to diverse classes of closure spaces is given by socalled subset selections for ordered sets. Various kinds of standard completions are obtained as special instances. In the third section, we develop the modern theory of order-theoretical closure operations, providing a much wider ﬁeld of applications. Here, the correlations between closure operations, adjunctions and Galois connections form an important ingredient for the study of closure in the general context of ordered sets. In the fourth section, closure spaces and closure algebras are discussed from a categorical point of view. It turns out that many classical problems become more transparent and their solutions more elegant in the extended lattice-theoretical setting of complete closure algebras.

Figure 1. Closure operator, closure system and complete lattice 2000 Mathematics Subject Classiﬁcation. Primary: 06A15, 06A23, 54A05. Secondary: 18A20, 18A25, 18A32. Key words and phrases. adjunction, algebraic closure, closure algebra, closure operation, closure system, continuous, subset selection, standard completion, topological closure, topological category. c 2009 American Mathematical Society 163

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Contents 1. Origins and history of closure concepts 1.1. Numbers, lattices and closure operators 1.2. Closure in set theory, analysis and topology 1.3. Closure in logic and universal algebra 1.4. Closure in lattice theory, algebra and geometry 2. Closure spaces 2.1. Closure operators and closure systems 2.2. Union-closed closure systems 2.3. Standard completions of partially ordered sets 2.4. Lattice representation of closure spaces 3. Order-theoretical closure operations 3.1. Closure operations and closure ranges 3.2. Adjunctions and Galois connections 3.3. Join- and meet-closure in complete lattices 3.4. Z-distributivity and Z-continuity 4. Closure algebras 4.1. Monads, algebras and complete lattices 4.2. Topological constructs of closure spaces 4.3. Categories of closure algebras 4.4. Complete closure algebras over closure spaces References

165 165 166 167 169 172 172 177 183 187 190 190 201 208 211 215 216 220 221 229 234

The internal relation that orders a sequence is equivalent to the operation that generates one member by another. Ludwig Wittgenstein

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1. Origins and history of closure concepts In quite diverse domains of mathematics and their applications, certain limits or completion processes have to be performed. A fundamental tool for such constructions is that of closure or hull operators. The resulting collections of “closed sets” are stable under arbitrary intersections. If C is such a closure system on a set X, the pair (X, C) is referred to as a closure space. The associated closure operator maps any subset of X to the least member of C containing it. Readers not familiar with notions of closure are invited to read Section 2.1 before turning to this historical survey. The basic idea behind the concept of closure is that certain “closed” entities (like subalgebras, convex or topologically closed sets, deductively closed sets of sentences or formulas etc.) may be generated either “internally” by iterated application of suitable algorithms or processes (producing the linear, algebraic, convex, topological or deductive hull of suitable “bases”); or else by an “external” intersection process, cutting oﬀ all “superﬂuous material”, e.g., by intersecting subspaces or halfspaces. It is not easy to trace back the development of this fundamental mathematical idea to its historical roots. But certainly the following names have to be mentioned in connection with the origins of the concept of closure: Schr¨ oder, Dedekind, Canˇ tor, Riesz, Hausdorﬀ, Moore, Cech, Kuratowski, Sierpi´ nski, Tarski, Birkhoﬀ and Ore. Their legacy in the area of closure structures will be outlined in the following pages. 1.1. Numbers, lattices and closure operators. In connection with his pioneering development of Galois theory, Richard Dedekind related in his unpublished lectures on algebra [23], ﬁnished about 1860 and rediscovered more than hundred years later, the “greatest common divisor” (the meet) of Galois groups to the “least common multiple” (the join) of the corresponding ﬁelds. This might be the ﬁrst manifestation of a Galois connection between two closure systems (one of groups and one of ﬁelds), though the notion of closure system does not appear verbally in that context. Another early explicit creation of a mathematical object both by an internal and an external closure process is Dedekind’s famous “construction” of the natural numbers in his pioneering article from 1887 [20]: Was sind und was sollen die Zahlen? (What are and what is the purpose of numbers? ) At the beginning, he introduces the three typical properties of “containment”, today known as reﬂexivity, antisymmetry and transitivity — stimulating thereby the modern axiomatization of (partial) orderings. Having deﬁned union and intersection, he deduces some fundamental rules about these operations. Dedekind uses the symbol ≺ for the containment order. However, in most of his conclusions, ≺ could stand equally well for any partial order of a complete lattice (an ordered set in which all subsets have suprema and inﬁma — the order-theoretical generalization of a power set), though his axiomatic introduction of lattices (“Dualgruppen”) emerged only in his later work [21, 22]. Continuing with [20], Dedekind considers the lift ϕ to the power set PS of a self-map φ : S → S (not distingushing notationally between the two maps φ and ϕ). More generally, ϕ might denote any isotone self-map of a complete lattice C. Dedekind calls any K with ϕ(K) ≺ K a chain (“Kette”), while nowadays we would prefer to speak of ϕ-closed objects K. Of course, the term “chain” has not to be

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confused with the synonymous modern order-theoretical notion of “linearly ordered set”, though they are not unrelated: the collection Cϕ of all K with ϕ(K) ≺ K is closed under arbitrary meets, hence a closure system in the set-theoretical case; and the least member of Cϕ containing some ﬁxed element becomes then a model for the natural numbers (in particular, a chain), provided φ is injective (“deutlich”). Dedekind also observes that Cϕ is closed under unions as well in case ϕ is the lift of a map φ; in modern language, this observation may be expressed by saying that the subalgebra system of a unary algebra is closed under arbitrary, not only under directed unions. Given a subset A of the domain of a map φ with power set lift ϕ, Dedekind denotes the least member K of Cϕ that satisﬁes A ≺ K by ϕ0 (A). Thus, he introduces (without using that terminology) the closure of A and the closure operator ϕ0 associated with Cϕ . His crucial remark is that such closure operators enjoy the following properties: A ≺ ϕ0 (A) ∈ Cϕ and A ≺ K ⇔ ϕ0 (A) ≺ K for K ∈ Cϕ , which mean in modern order-theoretical terminology that Cϕ is a closure range and ϕ0 the corresponding closure operation (see Section 3.1). Dedekind also establishes two of the three characteristic properties of closure operators, viz. extensivity and isotonicity, A ≺ ϕ0 (A) and B ≺ A ⇒ ϕ0 (B) ≺ ϕ0 (A), but not explicitly the third one, idempotency: ϕ0 (ϕ0 (A)) = ϕ0 (A). Instead, he derives the formulas ϕ(A) ≺ ϕ0 (A) and B ≺ ϕ0 (A) ⇒ ϕ(B) ≺ ϕ0 (A), signifying that ϕ0 is the least closure operator “including” the lift of the map φ to the power set. And then Dedekind proves the most succinct description of closure operators: B ≺ ϕ0 (A) ⇔ ϕ0 (B) ≺ ϕ0 (A), which was reinvented a dozen of times during a whole century after him! However, since Dedekind mainly was an “algebraist”, he seems not to have realized the importance of closure in analysis. 1.2. Closure in set theory, analysis and topology. Nevertheless, the work of Dedekind was inﬂuenced by Georg Cantor’s new ideas opening the door to modern set theory [13], and vice versa. Many of Cantor’s constructions (like limits, adherence points, perfect sets etc.) required certain closure processes. But probably the explicit and precise concept of closure was introduced into analysis not before the twentieth century, when Friedrich Riesz wrote his pioneering articles Die Genesis des Raumbegriﬀs [76] (1906) and Stetigkeitsbegriﬀ und abstrakte Mengenlehre [77] (1909), E.H. Moore his Introduction to a Modern Form of General Analysis [67] (1910), and Felix Hausdorﬀ his monograph Grundz¨ uge der Mengenlehre [51] (1914). To Hausdorﬀ we owe a systematical treatment of topological closure, its relationship to boundary, neighborhoods etc. Moore pointed out the one-to-one correspondence between closure operators and their ranges, the closure systems, even now often referred to as Moore families. Birkhoﬀ [9] speaks of a closure property if the intersection of any collection of

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sets with the property in question again shares that property; he attributes the term closure system to Cohn [17] and mentions that Moore called such properties extensionally attainable, a terminology quite in the spirit of Ernst Schr¨ oder (see the next section), who had emphasized the correspondence between extent and intent of a concept — the basic ingredients for the modern formal concept analysis initiated by Wille [46], where the extents and the intents of a mathematical context form two dually isomorphic closure systems (cf. Birkhoﬀ’s polarities and Ore’s Galois connexions, to be discussed brieﬂy in Sections 1.4 and 3.2). Signiﬁcant progress in the theory of closure was made in the twenties and thirties of the last century by the Polish school, in particular by Kuratowski [57], ˇ Sierpi´ nski [84] and Tarski [87], and by Cech [15, 16]. The famous notion of Kuratowski closure includes, besides the usual axioms for topological closure, the T1 -axiom, requiring all singletons to be closed. In fact, Kuratowski proposed the following succinct system of axioms: (i) A = A (idempotency) (ii) A + B = A + B (additivity) (iii) {x} = {x} (T1 -axiom) with + being the symbol for union. Clearly, (ii) implies monotonicity — but where is the extensivity axiom A ⊆ A? It follows from (ii) and (iii): x ∈ A ⇒ x ∈ {x} + A = {x} + A = A. ˇ Cech also pursued a diﬀerent generalization of topological closure operators, keeping additivity but dropping idempotency. This path was continued later on by many other authors and led to the important theory of pretopological spaces and of (additive) categorical closure operators. Since our slogan is “closure makes things closed”, we shall not follow that trace here further. As the scope of the present book is to show up perspectives beyond topology, we are not going either to discuss here the widely known applications of closure operators in classical topology, but we aim at an understanding of more general structures involving (idempotent) closure operators. 1.3. Closure in logic and universal algebra. Probably the ﬁrst to realize the structure of closure systems in logic and its links to algebraic structures was Ernst Schr¨ oder, who employed them (using diﬀerent terminology) in his theory of Algorithms and Calculi, a forerunner of modern universal algebra, algebraic logic and formal concept analysis (see [39] for more details). About 1870, he began to study the formalism governing the interplay between logic and algebra, leaning on Boole’s ideas about the Laws of Thought [11], but mathematically much more precise than those, and culminating in his voluminous Algebra of Logic [83] (1890– 1895). For that purpose, he created the ﬁrst semi-formal axiom system of lattice theory (perhaps even prior to Dedekind). His “logical calculus” may be regarded as an early systematic lattice- and closure-theoretical treatment of classes, statements and conclusions. Schr¨ oder’s theory of “algorithms” was an investigation of equational theories and their models, including Galois connections between certain closure systems. For Schr¨oder, an algorithm or group was “. . . ein solches System von Formeln des Gebietes, welches keine ihm nicht bereits angeh¨ orende Formel des Gebiets kraft der ’Prinzipien’ nach sich zieht”

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(. . . a system of formulas in the ﬁeld not implying any formula by virtue of certain ’principles’ unless that formula already belongs to the system). In other words, by an algorithm he meant a deductively closed system. Schr¨ oder carefully distinguished between the (closure) system of all algorithms as a model for his “logical calculus” and what he called the “identical calculus”, the boolean system of all subsets of the given “universe”. Not only the deductively closed sets of sentences or equations, but also their models form certain closure systems and consequently (complete) lattices. Though the notion of “closure” seems not to occur explicitly in that early development, Schr¨ oder implicitly had a closure operation in mind when he spoke of a process deriving new elements from the given ones, so that after iteration of that process the given elements together with the derived ones form the generated “groups” — which are, of course, not groups in the modern sense but certain closure systems of subalgebras or formulas. Schr¨oder emphasized that, in contrast to the boolean case of power sets, the “sum” (= join) in such a closure system need not be the union but is — in modern terminology — the closure of the union. The closure systems occurring implicitly in the work of Schr¨ oder and Dedekind are not topological but algebraic; that is, the corresponding closure operators are ﬁnitary in the sense that any element in the closure of a set belongs to the closure of some ﬁnite subset. Otherwise stated, the closure systems are inductive — a typical phenomenon when inductive processes are used to obtain certain closures. In contrast to the corresponding closure operators, those occurring in analysis and topology rarely happen to be ﬁnitary. Schr¨ oder’s ﬁrst attempts to make logical thinking accessible to algebraic methods were perfected a few decades later by Alfred Tarski, whose closure-oriented work had enormous inﬂuence on modern mathematical logic. The main papers by Tarski from 1923 to 1938 dealing with closure systems in logics are collected in the volume Logics, Semantics, Metamathematics, translated by J.H. Woodger [87]. One of the major articles in that collection is entitled Fundamental concepts of the methodology of the deductive sciences. Here Tarski focusses on the notion of deductive(ly closed) system, by which he means the system of all sentences derivable by certain prescribed logical rules from given axioms — as proposed already by Schr¨ oder. What Tarski establishes ﬁrst is essentially the deﬁnition of a ﬁnitary closure operator (on a countable set of “sentences”): he states explicitly the laws of extensivity and idempotency, and postulates that the closure Cn(A) (“the set of consequences via inference rules”) of a set A is the union (written as a sum) of the closures of all ﬁnite subsets of A — a property that, he remarks, implies monotonicity. Tarski also points out that the closure of a union coincides with the closure of the union of the closures, a property that remains valid for arbitrary closure operators and is crucial when closure systems are regared from the lattice-theoretical point of view. Moreover, Tarski shows that ﬁnitary closure operators preserve directed unions. As a consequence, the system of “deductively” closed sets is not only stable under arbitrary intersections but also under directed unions — the modern deﬁnition of an inductive closure system. The further development is then of more logical nature and would bring us too far away from the mainstream of closure operators.

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However, we would like to mention a less known related topic, namely Tarski’s early discovery of strong connections between logical and topological structures. An epoch-making article in that area was Tarski’s 1937 paper Sentential calculus and topology, where he revealed a striking similarity between mathematical logic and topology, bijectively correlating with every sentence A of the logical language a sentence A1 of topology, in such a manner that A is provable in the two-valued calculus if and only if A1 holds in every topological (T1 ) space. These and many other connections between logic and topology have been elaborated thoroughly by later generations; see, for example, the monographs Closure Spaces and Logic by Martin and Pollard [62] and Topology via Logic by Vickers [90]. 1.4. Closure in lattice theory, algebra and geometry. In the thirties, Garrett Birkhoﬀ propagated the power of a lattice-theoretical point of view in the development of universal algebra and geometry. His Lattice Theory [9] became an inspiring source for generations of mathematicians. One of the guiding ideas is that every closure system is a complete lattice, in which not only arbitrary meets (the intersections) but also arbitrary joins (the closures of the unions) exist. It was the merit of Birkhoﬀ and Ore [69, 70] to have pointed out that the structure of groups, rings, vector spaces, lattices and other algebraic structures can be understood much better by a thorough study of two algebraic closure systems associated with any (ﬁnitary) general algebra: namely, that of subalgebras and that of congruences, providing a concise theory of homomorphisms. Concerning Birkhoﬀ’s immense work on geometric structures under a latticeor closure-theoretical perspective, it may suﬃce here to mention two typical ﬁelds that have been inﬂuenced primarily by his research in that area. For many subspace systems occurring in aﬃne, projective and other geometries, the associated closure operator − satisﬁes the Birkhoﬀ–Steinitz–Mac Lane Exchange Axiom x = y ∈ A ∪ {x} \ A ⇒ x ∈ A ∪ {y}. Algebraic closure spaces of that kind with the additional property that all singletons are closed (T1 -axiom) are called combinatorial geometries (see e.g. [1]). Abstractly, the Exchange Axiom is expressed by semimodularity (together with the postulate of enough compact atoms) of the corresponding complete lattices, usually referred to as geometric lattices [1, 9, 50]. On the other hand, closure systems of convex sets, subsemilattices, trees, closure ranges (see Section 3.1!) and other interesting objects in combinatorics, algebra and geometry satisfy the so-called Anti-Exchange Axiom x = y ∈ A ∪ {x} \ A ⇒ x ∈ A ∪ {y}. Closure spaces with that property are termed convex geometries and may be described by purely lattice-theoretical axioms as well (see e.g. Edelman and JamisonWaldner [28, 55]). Another basic manifestation of closure structures comes from Birkhoﬀ’s polarities associated with any relation between two sets: assigning to each subset A of one of these sets the whole of all elements in the other set that are related to each element in A, one obtains two opposite mappings between the respective power sets whose ranges are dually isomorphic closure systems — a phenomenon well-known from classical Galois theory. Nowadays, this interplay between “objects” and “attributes”, and the structural data exploration by means of the associated closure

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systems, regarded as concept lattices, is the main theme of formal concept analysis (see Ganter and Wille [46]). Some important examples of polarities, stemming from algebra, geometry and topology, will be discussed in Section 3.2. Parallel to Birkhoﬀ, Oystein Ore discovered a broad spectrum of relationships between lattice theory, topology and closure — in fact, either of these two pioneers gave priority to the closure-theoretical view in topology. Let us have a look at Ore’s fundamental paper Some studies on closure relations [72], published in 1943. As a ﬁrst crucial connection between closure systems and complete lattices (structures in his terminology), Ore remarks (referring to Birkhoﬀ) that not only every complete lattice C is isomorphic to a closure system, namely that of all principal ideals a = {c ∈ C : c ≤ a} (a ∈ C), but moreover, an arbitrary closure system is of that form exactly when every closed set is the closure of a unique point. (Ore generally assumes the empty set to be closed, which is not an essential restriction for the pure structure theory but excludes interesting applications; in the present situation, he seems to have overlooked the conﬂict between closedness of ∅ and the postulate that all closed sets be point closures.) We shall adopt Ore’s notation Γ(A) or A for the closure of a subset A in a closure space. Ore devotes a major part of his investigations to the notion of subspace (without using that term), obtained by relativization: for any subset B of a closure spaces (X, Γ), the induced subspace is given by (B, ΓB ), where ΓB (A) = Γ(A) ∩ B. He speaks of a distinct relativization when any two closed sets A1 = A2 of the whole space have distinct traces A1 ∩ B and A2 ∩ B in the subspace, and basically proves the equivalence of the following statements: (a) (b) (c) (d) (e)

(B, ΓB ) is a distinct relativization. A → A ∩ B is an isomorphism between the ranges of Γ and ΓB . A = Γ(A ∩ B) for all closed A ⊆ X. If A1 , A2 are closed in (X, Γ) with A1 ⊂ A2 then B ∩ (A2 \ A1 ) = ∅. B meets every nonempty set of the form Γ(A) \ A.

In connection with (e), Ore refers to an early paper by Kuratowski and Sierpi´ nski from 1921 [58]. Note that condition (d) may be restated by saying that B is dense in that space whose closed sets are generated by the closed and the open sets of the original space (X, Γ) (by forming intersections). In modern terminology, such subspaces B are called strictly or strongly dense. Ore pays particular attention to the characterization of all closure spaces whose lattice of closed sets is isomorphic to a given complete lattice C. First, he reduces the problem to the T0 case by observing that any closure space and its T0 -reﬂection (obtained by identifying points with equal closure) have isomorphic closure systems. Then, he shows that the closure system C of a T0 -closure space (X, Γ) is isomorphic to C iﬀ (X, Γ) is homeomorphic to a distinct relativization of the closure space associated with C (having the principal ideals as closed sets). Those subsets of C which are obtained by distinct relativization are just the bases or join-dense subsets of C, i.e., subsets B such that each element of C is a join of elements from B — or, equivalently, for a, c ∈ C with c ≤ a, there is a b ∈ B with b ≤ c but b ≤ a.

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The middle part of Ore’s paper deals with “additive spaces”, the topological spaces in our sense, and “completely additive spaces”, alias Alexandroﬀ spaces, in which arbitrary unions of closed sets are closed (cf. [4]). In the last part, Ore discusses some questions of compactness, a quite fresh theme at that time, inspired by the work of Tychonoﬀ and Wallman [91]. In a related paper entitled Combinations of closure relations [71], Ore thoroughly studies abstract properties of the lattice of all closure operators on a set and the dual lattice of all closure systems. While parts of Ore’s terminology did not survive (“complete intersection ring” for “closure system”, “topology” for “closure space”, “structure” for “lattice” etc.), his fundamental ideas remained vital and inspiring until today. For example, in [72], he created the nice notion of structure invariant spaces, i.e., closure spaces having isomorphic closure systems (recall that he used the word structure for lattice) and observed that structure invariant TD -closure spaces (in which all sets {x} \ {x} are closed) must be homeomorphic (cf. Thron [88]). Moreover, Ore established lattice representations for the closure systems of several important classes of closure spaces: T1 -closure spaces topological T1 -spaces TD -closure spaces topological TD -spaces Alexandroﬀ spaces

←→ ←→ ←→ ←→ ←→

atomistic complete lattices, atomistic coframes, complete lattices with a least basis, coframes with a least basis, frames with a least basis.

The word frame (or locale) is the modern term for a complete lattice satisfying the inﬁnite distributive law (where denotes joins and ∧ binary meets) a ∧ B = {a ∧ b : b ∈ B} for all elements a and subsets B, and a coframe enjoys the dual law. Atomistic lattices have a basis of atoms. For a more recent uniform discussion of the interplay between various kinds of closure spaces and complete lattice structures satisfying certain abstract laws, see [31, 36, 40] and Section 2.4. Replacing power sets with arbitrary sets, Ore also developped a general theory of closure operations and the related theory of Galois connections (Galois connexions in [73]), opening the door to the modern purely order-theoretical treatment of closure structures and (dually) adjoint maps. Readers interested in a comprehensive discussion of these ideas may consult the survey article Adjunctions and Galois connections: origins, history and development [39]. A few aspects of that useful theory will be dealt with in Sections 3.1 and 3.2. From the work of Birkhoﬀ and Ore we learn that the closure system of all meet-closed subsets of a complete lattice is dually isomorphic to the complete lattice of all closure operations. An internal characterization of the ranges of closure maps on arbitrary posets is due to R. Baer [5], who baptized them “partial ordinals”. We prefer the more suggestive name closure ranges (see Section 3.1). It is not the place here to report on the enormous later expansion of the theory of closure spaces and their order-theoretical generalizations, nor its applications to mathematically inﬂuenced ﬁelds like computer sciences (see, e.g., [85]), physics (see, e.g., [3, 65, 66]) and other sciences. But some of the basic mathematical mainstreams will be pursued in the sequel.

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2. Closure spaces As indicated in Section 1, the notion of closure spaces is one of the most frequently used concepts occurring simultaneously in algebra, analysis, geometry, topology and logic. Such spaces have two equivalent manifestations: an operational description by “closures” or “hulls” and one by suitable collections of subsets, stable under arbitrary intersections. Thus, the main ingredients are closure operators (or hull operators) and the associated closure systems (or hull systems). Surprisingly, only a minority of mathematicians seems to favorize the closed set approach in topology, while the majority prefers to think in terms of open sets. Of course, both methods are equivalent, but the framework of closure systems and closure operators makes the overlap and the diﬀerences between topology and algebra more transparent. We discuss the common features but also the diﬀerences between topological and algebraic closure, pointing out the importance of ﬁniteness and induction properties of the latter. Next, we introduce the adequate general structures including algebraic, topological and order-deﬁned (Alexandroﬀ) closure spaces, namely so-called Z-closure spaces. These have closure systems that are closed under Z-unions, where Z is a uniform subset selection for posets (e.g. ﬁnite, directed or arbitrary subsets). The general theory is then applied to standard completions of ordered sets, i.e., closure systems consisting of downsets and containing all principal ideals. Prominent examples are the Alexandroﬀ completion by all downsets, the Dedekind–MacNeille completion by cuts, the Banaschewski–Frink completion by ideals, and topological completions like the Scott completion. In the fourth section, we set up the fundamental duality between closure spaces and (join-)based complete lattices, the starting point for many Stone type dualities linking algebra with topology via lattice-theoretical structures.

2.1. Closure operators and closure systems. The following introductory remarks should help to avoid notational ambiguities. The use of certain capital letters in algebra and topology sometimes causes conﬂicts: while many topologists prefer to denote the entire space by X and its subsets by A etc., many algebraists do just the converse and denote the whole algebra by A and its subsets by X etc. Since our main purpose is a common treatment of both algebraic and topological structures, we shall make a compromise and adopt the following notational conventions. X, Y etc. stand for arbitrary sets, A, B etc. for subsets. (Partially) ordered sets are denoted by P or Q, complete lattices by C or D. We write PX for power sets (ordered by set inclusion) and use calligraphic letters like C, D, X , Y for systems on a set X, i.e., subsets of the power set PX. (From the purely axiomatic point of view, there is no distinction between sets and systems, but we wish to make clear by our notation whether we are dealing with single subsets of a certain “ground set” or with collections of subsets.) For elements of ordered structures, we prefer to use letters like a, b etc., while x, y etc. will denote elements of closure spaces or topological spaces. Of course, these distinctions cannot be maintained with absolute consequence, because certain sets will carry topological, algebraic and/or order-theoretical structures simultaneously.

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Also, it is common use to denote an ordered set or a space and its underlying set by the same symbol. For a function (= map) f : X → Y , the image of a subset A of X under f is denoted by f (A); in order to exclude confusion with images of elements, we sometimes write f (A) instead of f (A). Similarly, we prefer to write f (B) for the preimage of a subset B of Y ; (the notation f −1 (B) is a bit misleading, because it might suggest erroneously that f or f has an inverse). Iterated liftings like f and f etc. may now be distinguished exactly from each other. Greek letters denote maps between ordered structures. A closure space is a pair (X, C) consisting of a set X and a closure system C on X, that is, a subset of the power set PX with X ∈ C for all X ⊆ C (including the empty intersection ∅ = X). Alternatively, one may introduce closure spaces as pairs (X, Γ) where Γ is a closure operator on X, i.e., a map from PX to PX that is • extensive: A ⊆ Γ(A) • isotone: A ⊆ B ⇒ Γ(A) ⊆ Γ(B) • idempotent: Γ(Γ(A)) = Γ(A). These three conditions together are equivalent to the single condition A ⊆ Γ(B) ⇔ Γ(A) ⊆ Γ(B). It is convenient to write A instead of Γ(A) if the closure operator is understood. Isotonicity of Γ (meaning that Γ preserves inclusion) may be characterized by the condition Γ(A) ⊆ Γ(A∪B), and all three closure properties together by the inclusion A ∪ Γ(Γ(B)) ⊆ Γ(A ∪ B) or by the identity A ∪ Γ(A ∪ Γ(B)) = Γ(A ∪ B). This and the following fundamental connection between closure operators and closure systems, including proofs, will be resumed in Section 3.1, where we shall enter a more general order-theoretical area. Proposition 2.1. Passing from closure operators Γ on X to their ranges CΓ = Γ (PX) = {A ⊆ X : Γ(A) ⊆ A}, one obtains a bijection (in fact, a dual isomorphism) between the complete lattice of all closure operators (ordered by argumentwise inclusion) and the closure system (on PX!) of all closure systems on X. Indeed, the equivalence A ⊆ Γ(B) ⇔ Γ(A) ⊆ Γ(B) yields Γ(A) = {C ∈ CΓ : A ⊆ C} and for an arbitrary system X ⊆ PX, the map ΓX : PX → PX with ΓX (A) =

{B ∈ X : A ⊆ B}

is a closure operator with corresponding closure system CΓX = ΩX = { Y : Y ⊆ X }. Hence, any closure system C coincides with CΓC , and any closure operator Γ with ΓCΓ . Thus, both deﬁnitions of closure spaces are equally eﬀective, and one may change from one to the other whenever convenient. The members of a closure system are called closed and their complements open. Hence, the open sets form a kernel system, that is, a system closed under arbitrary unions. In the same manner

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as closure systems correspond to closure operators, kernel systems correspond to kernel operators or interior operators K, which are contractive, i.e., K(A) ⊆ A, instead of being extensive. The direct one-to-one correspondence between closure operators and kernel operators is obtained by passing from Γ to the operator K deﬁned by K(A) = X \ Γ(X \ A). An Alexandroﬀ topology is both a kernel system and a closure system, and the corresponding space is an Alexandroﬀ (discrete) space (see [4]). Interior, neighborhoods, boundary, subspaces etc. may be deﬁned just as in topology, so we omit here a detailed discussion of these subjects. However, always keep in mind that in general closure spaces, the union of two closed sets need not be closed, the intersection of two neighborhoods need not be a neighborhood, and so on. In particular, products of topological (closure) spaces diﬀer from those of closure spaces: the closure system of the product of closure spaces consists of all products formed by closed sets in the factors, while in the case of topological products, one has to take ﬁnite unions of all such products before forming arbitrary intersections. Anyway, the closure of a set is closed, and we feel this is a typical property of closure. Therefore, we shall not focus on closure operators in the sense ˇ of Cech [16], which preserve ﬁnite unions but are not necessarily idempotent. A map f between (the underlying sets of) closure spaces (X, Γ) and (Y, ∆) is continuous if preimages of closed sets are closed, or equivalently, preimages of open sets are open; and, as in the topological case, continuity of such a map f may also be characterized by the condition f (Γ(A)) ⊆ ∆(f (A)) for all A ⊆ X. In what follows, we assume familiarity with a few basic notions of category theory. For general background, we refer to the books Categories for the Working Mathematician by Mac Lane [60] and Abstract and Concrete Categories by Ad´ amek, Herrlich and Strecker [2]. Closure spaces together with continuous maps as morphisms form a category CSp, and there is an obvious “forgetful” functor from CSp to Set (the category of sets and functions), omitting the closure structure. The three most important full subcategories are • TCSp, the category of topological closure spaces • ACSp, the category of algebraic closure spaces • ATSp, the category of Alexandroﬀ (discrete) spaces The objects of these subcategories are determined by certain union closure properties of their closure systems: CSp ??? TCSp ACSp ?? ? ATSp

closure space topological algebraic Alexandroﬀ

closure system closed under . . . closure operator preserves . . . ﬁnite unions directed unions arbitrary unions

Figure 2. Categories of closure spaces Of course, the category TCSp is isomorphic to the usual category Top of topological spaces (via passage between closed and open sets). An ordered set is directed if every ﬁnite subset has an upper bound; for a system X of sets, directedness means

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that for every ﬁnite subsystem Y of X there is a B ∈ X with Y ⊆ B. The concepts of “ﬁnite” and “directed” are in some sense complementary: a system is closed under arbitrary unions iﬀ it is closed both under ﬁnite and under directed unions. On the other hand, a ﬁnite ordered set is directed iﬀ it has a greatest element. Though from the closure-theoretical point of view, ﬁniteness is typical for topology and directedness for algebra, the converse holds regarding operations: while in classical algebraic structures, one usually works with ﬁnitary operations, topological considerations of convergence etc. often require directed domains (of nets). A fundamental connection between closure spaces and ordered sets is obtained by associating with each system X on a set X the specialization order ≤X deﬁned by x ≤X y ⇔ for all A ∈ X , y ∈ A implies x ∈ A and interpreted as “x is more special than y, sharing with y all properties determined by X ” (caution: Ore [72] and others use the dual relation). Of course, specialization orders are always quasi-orders (alias preorders), i.e., reﬂexive and transitive; and they are partial orders (i.e., antisymmetric) if and only if X is a T0 system, meaning that for any two distinct points there is a member of X containing exactly one of these points. Note that the dual specialization order belongs to the system X c = {X \ A : A ∈ X } of all complements (the open sets if X is a closure system): x ≤X y ⇔ y ≤X c x. Continuous maps f between closure spaces preserve specialization: x ∈ {y} ⇒ f (x) ∈ {f (y)}. Hence, there is a forgetful functor from CSp, the category of closure spaces, to Qos, the category of quasi-ordered sets and isotone (= order-preserving) maps, i.e. maps ϕ satisfying a ≤ b ⇒ ϕ(a) ≤ ϕ(b). Conversely, for any quasi-ordered set Q = (X, ≤), there is a largest closure system A∧ Q having ≤ as specialization order, which is obtained as follows. The downset or lower set generated by A ⊆ X is ↓A = {b ∈ X : ∃a ∈ A (b ≤ a)}, and the upset or upper set ↑A is deﬁned dually. For the principal ideal ↓{a} one writes ↓a, and dually ↑a for the principal ﬁlter ↑{a}. It is straighforward to check that both ↓ and ↑ are union-preserving closure operators. Consequently, the Alexandroﬀ completion A∧ Q = A∧ (X, ≤) = {A ⊆ X : A = ↓A} = {↓A : A ⊆ X} is closed under arbitrary unions and intersections, hence an Alexandroﬀ topology. Moreover, it turns out that there is a one-to-one correspondence between Alexandroﬀ spaces and quasi-ordered sets, inducing one between T0 -Alexandroﬀ spaces (in which distinct points have diﬀerent closures) and partially ordered sets (due to Alexandroﬀ [4]): Proposition 2.2. Associating with any Alexandroﬀ topology T on a set X the specialization order ≤T , one obtains a dual isomorphism between the closure system of all Alexandroﬀ topologies on X and the closure system of all quasi-orders on X. The inverse isomorphism sends any quasi-order ≤ on X to A∧ (X, ≤). Moreover, the continuous maps between Alexandroﬀ spaces are precisely the isotone maps between the corresponding quasi-odered sets.

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Proof. Clearly, ≤T is a quasi-order, and its downsets are exactly the members of T , being unions of principal ideals (which belong to T , because T is closed under intersections). Conversely, the specialization order of any downset system A∧ (X, ≤) is the original quasi-order: x ≤ y ⇔ x ∈ ↓ y ⇔ ∀A ∈ A∧ (X, ≤) (y ∈ A ⇒ x ∈ A). Sending T to ≤T and, in the opposite direction, ≤ to A∧ (X, ≤) obviously inverts inclusion. Finally, if a map f : (X, ≤) → (Y, ≤ ) is isotone, then it is continuous as a map between the corresponding Alexandroﬀ spaces, since f (↓ A) ⊆ ↓ f (A) for all A ⊆ X. Categorically speaking, Proposition 2.2 assures that the forgetful functor from CSp to Qos restricts to a concrete isomorphism between ATSp and Qos, whose inverse is the functor sending Q = (X, ≤) to (X, A∧ Q). As remarked earlier, the Alexandroﬀ topologies are exactly those closure systems which are both topological and algebraic. The relation ACSp ∩ TCSp = ATSp # Qos may be restated informally as “Algebra ∩ Topology = Order Theory”. But alternate points of view are possible as well, justifying the informal equation (see [53]) “Algebra ∩ Topology = Compactness”. The name algebraic closure system results from the fact, probably pointed out ﬁrst by Birkhoﬀ [9], that the algebraic closure systems are precisely the systems of subalgebras of ﬁnitary algebras, i.e., sets X equipped with a (possibly inﬁnite) family of ﬁnitary operations fi : X ni → X. Denote, for any subset B of a closure space (X, Γ), the system of all ﬁnite subsets of B by FB and deﬁne the inductive or ﬁnitary closure by ΓI (B) = {Γ(E) : E ∈ FB}. Theorem 2.3. For a closure operator Γ and the corresponding closure system C on a set X, the following statements are equivalent: (a) Γ is ﬁnitary, i.e., Γ = ΓI . (b) Γ preserves directed unions. (c) C is inductive, i.e., closed under directed unions. (d) C consists of all subalgebras of a (ﬁnitary) algebra. (e) There is an algebra such that for each of its subsets B, the (underlying set of the) subalgebra generated by B is Γ(B). Proof. These facts are well-known and have been proved by various authors (see, e.g., Birkhoﬀ [9], Schmidt [81]). We add here a succinct implication circle establishing all claimed equivalences. (a) ⇒ (b). For any directed system Y ⊆ PX, compute Γ( Y) = {Γ(E) : E ∈ F( Y)} = {Γ(E) : ∃B ∈ Y (E ∈ FB)} = Γ(Y). (b) ⇒ (c). If Y ⊆ C is directed then Γ( Y) = Γ(Y) = Y ∈ C. (c) ⇒ (a). Since ΓI (A) = {Γ(E) : E ∈ FA} is a directed union of closed sets with A ⊆ ΓI (A) ⊆ Γ(A), it follows that Γ(A) = ΓI (A). (a) ⇒ (d). For each a = (a1 , . . . , an ) ∈ X n , put Ea = {a1 , . . . , an }. Given any b ∈ Γ(Ea ), deﬁne an n-ary operation fab : X n → X by fab (x) = b if x = a and

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fab (x) = x1 otherwise. Then, for B ∈ C and x ∈ B n , we get fab (x) ∈ B; hence, B is a subalgebra of the algebra (X, (fab : a ∈ X n , b ∈ Γ(Ea ))). Conversely, if A is (the underlying set of) a subalgebra then b ∈ Γ(A) implies b ∈ Γ(Ea ) for some a ∈ An , and it follows that b = fab (a) ∈ A. Thus, Γ(A) ⊆ A ∈ C. Note that the case n = 0 (constant operations) works as well: X 0 = {∅} and fab (∅) = b for b ∈ Γ(∅). (d) ⇒ (c) is easy, using the fact that all operations are ﬁnitary. (d) ⇔ (e) is clear by Proposition 2.1. 2.2. Union-closed closure systems. In order to bring various speciﬁc kinds of closure spaces under a common umbrella, we introduce now so-called Z-closure spaces, where Z is a subset selection for partially ordered sets (posets), that is, a (class) function associating with any poset (or quasi-ordered set) P a certain collection ZP of subsets. The theory of subset selections has been developed quite extensively (see, for example, [30, 33, 36, 37, 64, 92]) and proved very useful for a uniform treatment of diverse problems arising in order theory and its applications. Six of the most frequently occurring selections are listed below: Z A B C D E F

members of ZP arbitrary subsets binary subsets (with at most two elements) chains (nonempty linearly ordered subsets) directed subsets (whose ﬁnite subsets have upper bounds) 1-element subsets (singletons) finite subsets

If not otherwise stated, a collection X of sets is always thought of being ordered by inclusion, so that we may write ZX for Z(X , ⊆). With any subset selection Z and any poset P , we associate the system Z ∧ P = {↓ Z : Z ∈ ZP ∪ EP } of Z-downsets (or Z-ideals). Principal ideals are always Z-downsets. Two properties of certain subset selections are needed frequently: we say Z is • functorial if Z ∈ Z ∧ P implies ↓ ϕ(Z) ∈ Z ∧ Q for all isotone, i.e., orderpreserving maps ϕ : P → Q, • union complete if Y ∈ ZZ ∧ P implies Y ∈ Z ∧ P . The ﬁrst impulse to study functorial subset selections was given by Wright, Wagner and Thatcher [92], who considered a slightly more restricted notion of subset systems and pointed out their relevance to theoretical computer science. We note in passing that the name “functorial subset selection” is justiﬁed by the fact that any such Z gives rise to an endofunctor Z ∧ of the category Pos of partially ordered sets and isotone maps, sending any isotone map ϕ : P → Q to the Z-lift Z ∧ ϕ : Z ∧ P → Z ∧ Q, Z → ↓ ϕ(Z). In the case of a union complete functorial subset selection Z, the functor Z ∧ is part of an order completion monad (see [30] and Meseguer [64]). All six speciﬁc subset selections introduced before are functorial, but only A, D, E and F are union complete. (Although the union of any nonempty system of chains that is linearly ordered by inclusion must be a chain, too, the selection C

178

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of chains fails to be union complete, because the nested union of chain-generated downsets need not be generated by any chain.) More generally, we say a collection X of sets is Z-union closed if Y ∈ ZX implies Y ∈ X . When a closure system CΓ on X is Z-union closed, we call it a Z-closure system and the closure space (X, Γ) or (X, CΓ ) a Z-closure space. Thus, a collection X of sets is a Z-closure system iﬀ Y ∈ AX implies Y ∈ X , and Y ∈ ZX implies Y ∈ X . By deﬁnition, the E-closure spaces are simply the closure spaces in the original sense, the D-closure spaces are the algebraic ones, the F-closure spaces are the topological ones, and the A-closure spaces are the Alexandroﬀ spaces. While it is easy to see that F-union closedness is equivalent to B-union closedness, it requires sophisticated arguments (and choice principles) to show that the D-union closed systems coincide with the C-union closed ones and even with the W-union closed systems, where W denotes the selection of all nonempty well-ordered subsets (see [36] for an elementary proof). The case of closure systems is a bit easier. We shall use a certain set-theoretical induction principle that often helps to avoid ordinal numbers and transﬁnite induction; it is a typical application of closure methods. Proposition 2.4. If a system of sets is W-union closed and contains all ﬁnite subsets of a set X then it must include the whole power set of X. Proof. The trick is here to observe that the subsets of PX with the stated two properties constitute a closure system on PX, so that there is a least system X with these properties. Consider the system X = {A ⊆ X : A ∪ {x} ∈ X for all x ∈ X}. A straightforward inspection shows that X again has the two properties in question (namely W-union closedness and FX ⊆ X ), so that by minimality, X coincides with X . Now, Zorn’s Lemma yields a maximal member of X = X , which of course must be the whole set X. Since the same arguments hold for any subset of X, we even obtain PX ⊆ X . Corollary 2.5. For closure systems, W-, C- and D-union closedness are equivalent. Hence, a closure system that is closed under unions of nonempty wellordered subsystems is already inductive (= algebraic). Proof. Given a W-closure system CΓ on a set X, consider the system X = {A ⊆ X : Γ(A) = ΓI (A)}. Clearly, X contains all ﬁnite subsets of X, and if Y chain in C Γ , whence in X then Γ(Y) is a well-ordered is a well-ordered chain I I and so Γ( Y) = Γ(Y) = {Γ (A) : A ∈ Y} = Γ ( Y), i.e., Γ(Y) ∈ C Γ Y ∈ X . Now, Proposition 2.4 shows that X must be the whole power set PX, i.e., Γ is ﬁnitary and CΓ is an algebraic, hence D-union closed closure system (see Theorem 2.3). Henceforth, we assume that Z is a functorial subset selection. Often, that hypothesis is not needed in full strength (see [30, 33, 36, 37, 40]), but it will be convenient and suﬃcient forour present purposes. A map Γ : P → Q preserves Z-unions if Y ∈ ZP implies Γ( Y) = Γ(Y). For closure operators, this property reﬂects Z-union closedness of their ranges:

CLOSURE

179

Proposition 2.6. (1) If Γ is an inclusion and Z-union preserving map on a power set PX then CΓ = {A ⊆ X : Γ(A) ⊆ A} is a Z-closure system. (2) Conversely, if C is a Z-closure system then the associated closure operator ΓC with ΓC (A) = {B ∈ C : A ⊆ B} preserves Z-unions. (3) Sending C to ΓC and Γ to CΓ yields mutually inverse dual isomorphisms between the complete lattice of all Z-closure systems on X and the complete lattice of all Z-union preserving closure operators on X. The proof consists of a series of easy veriﬁcations. More general order-theoretical results will be established in Theorem 3.45. For any collection X of sets, there is a least Z-closure system C Z X containing X , just because the Z-closure systems on a set X constitute a new closure system (of course, not on X but on PX). In most cases occurring in practice, the closure C Z X is obtained in a one- or two-step process: one for adding the obligatory intersections, and one for the required unions. Surprisingly, we shall see that the order in which these two steps have to be performed varies from case to case, depending on the given subset selection. The following notations will be convenient: 0Z X = X ∪ { Y : Y ∈ Z(X , ⊆)}, ΩZ X = X ∪ { Y : Y ∈ Z(X , ⊇)}. By deﬁnition, X is a Z-closure system iﬀ 0Z Ω X = X . Instead of 0A and ΩA , we simply write 0 and Ω, respectively. Thus, for example, 0Ω X denotes the collection of all unions of sets that are intersections of members of X , while Ω0 X has the dual meaning (but coincides with 0Ω X ). On any power set PX, one has the following special closure operators: Proposition 2.7. Among all systems on X containing a ﬁxed system X : (1) ΩA X = Ω X is the least closure system. (2) 0A X = 0 X is the least kernel system. (3) 0F X is the least ∪-system (closed under ﬁnite unions). (4) ΩF X is the least ∩-system (closed under ﬁnite intersections). (5) ΩF 0F X = 0F ΩF X is the least ∪- and ∩-system. (6) 0ΩF X is the least topology. (7) Ω0F X is the least topological closure system. (8) 0D Ω X is the least algebraic closure system. (9) Ω0 X = 0Ω X is the least Alexandroﬀ topology. The ﬁrst ﬁve of these statements are easily veriﬁed. The remaining four claims or their duals are special instances of Theorem 2.8. Note that for topological generation (7), one has to form ﬁrst ﬁnite unions and then arbitrary intersections, while for algebraic generation (8), one has to form ﬁrst arbitrary intersections and then directed unions! For any cardinal number m, the subset selection Am picking all subsets of cardinality less than m is functorial; and m is a regular cardinal iﬀ Am is union complete (this may be taken as the deﬁnition of regular cardinals). Sometimes, it is convenient to admit the symbol ∞ for m, in which case A∞ stands for A. Another transﬁnite sequence of functorial union complete subset selections is obtained by deﬁning Dm P to be the system of all m-directed subsets, where D is m-directed if each A ∈ Am D has an upper bound in D. That Dm is a functorial

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180

selection is evident from the fact that isotone maps transport upper bounds, and forming images cannot increase cardinalities. The reader is encouraged to verify that union completeness of the subset selections Dm does not require regularity of m but the Axiom of Choice if m > ω. Theorem 2.8. (1) Put Z = Am for a regular cardinal m or m = ∞. Then, for each system X on a set X, the least Z-closure system on X containing X is C Z X = Ω0Z X (but not 0Z Ω X in general). (2) Put Z = Dm for an arbitrary cardinal m. Then, for each system X on a set X, the least Z-closure system on X containing X is C Z X = 0Z Ω X (but not Ω0Z X in general). Again, we shall prove more general statements about so-called m-frames and Z-continuous lattices in Sections 3.3 and 3.4. Note that not only the case m = ω (the ﬁrst inﬁnite cardinal) but also the case m = ω1 (the ﬁrst uncountable cardinal) is of particular interest for topology, analysis and measure theory: while Aω = F selects all ﬁnite subsets, Aω1 selects all countable subsets. Thus, Aω1 -union completeness occurs, for example, in connection with countably compact spaces, Lindel¨ of spaces, zero-sets of continuous functions, and measure algebras. In the following table, the cases in which ΩY 0Z is always a closure operator are marked by the sign +, the other cases by the sign −. By reasons of duality, the same table is obtained for Ω exchanged with 0. Y A B C D E F

Z

A

B

C

D

E

F

+ − − + + −

− − − − − −

− − − − − −

− − − − − −

+ − − − + +

+ − − − + +

A few comments and (counter-)examples are in order. Whereas the equation T = 0ΩF X means that X is a subbasis for the topology T , the system ΩF 0 X will not be a topology in general, and dually, the system 0F Ω X need not be a (topological) closure system. Example 2.9. The system X of all closed real intervals (bounded or unbounded) is both a closure system and a subbasis for the closed sets in the usual topology on the reals, while the system 0F Ω X = 0F X of all ﬁnite unions of closed intervals is certainly not a closure system. Notice that Ω0B X coincides here with the closure system Ω0F X , because a ﬁnite union of closed intervals may be represented as a ﬁnite intersection of binary unions of intervals. With the inductive (or algebraic) closure, the situation is just opposite: while 0D Ω X is the inductive closure system generated by X , the closure system Ω0D X need not be inductive.

181

CLOSURE

Example 2.10. Let Z denote the set of integers, N the set of positive integers, so that ω = N ∪ {0} is the set of nonnegative integers. The system X = {{k ∈ N : k ≤ n} ∪ {−n} : n ∈ N} ∪ {ω} ⊆ PZ has no comparable members and is therefore trivially inductive, whereas Ω X = ΩF X = Ω0D X contains the ascending chain {{1, . . . , n} : n ∈ N} but not its union N. Hence, Ω0D X = ΩF 0D X is not even W-union closed. d Z d d d p p p d ω X p @@@ p @@@ p @@@ d @@ d @d Figure 3. A non-algebraic closure system That ΩB 0Z is not a closure operator in general, no matter what subset selection Z is taken, is demonstrated by Example 2.11. Let X be any set with at least four elements. Then X = {∅, X} ∪ {X \ {x} : x ∈ X} is a kernel system on X. Hence, X = 0 X = 0Z X and ΩB 0Z X = ΩB X ⊂ ΩB ΩB X = ΩB 0Z ΩB 0Z X

(⊂ means proper inclusion),

because X \ {x, y} and X \ {x, z} belong to ΩB X but X \ {x, y, z} ∈ ΩB X , provided x = y = z = x. On the other hand, K = ΩB X is a kernel system, whence 0Z K = K; and as K is not closed under binary intersections, none of the operators 0Z ΩB is idempotent. Dually, the operators ΩZ 0B need not be idempotent either. Next, to the operators ΩC and 0C . By duality, it suﬃces to verify that neither 0 ΩZ nor ΩZ 0C is always a closure operator, for any selection Z. C

Example 2.12. The ﬁrst uncountable ordinal ω1 is the set of all smaller ordinals, and ω1+ = ω1 ∪ {ω1 } is its ordinal successor. Obviously, X = Fω1 ∪ {ω1+ } is a topological closure system, whence ΩZ X = X for any Z. The members of the system 0C X are ω1+ and all countable subsets of ω1 , because any chain of ﬁnite sets must be countable (the cardinality function being an injection into ω). But ω1 is the union of all smaller ordinals, which form a (well-ordered) chain under inclusion, and all members of that chain are countable, hence elements of 0C X . Thus, 0C X is not C-union closed, and 0 C ΩZ X = ΩZ 0C X = 0C X ⊂ 0C 0C X ⊆ 0C ΩZ 0C ΩZ X . The reader might wonder why, in contrast to 0A and 0F , the operator 0D is not idempotent, although directed unions of directed sets are directed and D is union complete. A counterexample is constructed as follows:

´ M. ERNE

182

Example 2.13. For m ∈ N and n ∈ ω, put Amn = {a ∈ ω ω : 1 ≤ a0 ≤ m and am ≤ n}, Am = {a ∈ ω ω : 1 ≤ a0 ≤ m}. Then Amn ⊆ Am n holds if and only if m = m and n ≤ n . Consequently, each of the systems Xm = {Amn : n ∈ ω} is an ω-chain with union Am , and {Am : m ∈ N} is an ω-chain with union A = {a ∈ ω ω : 1 ≤ a0 } ⊂ X = ω ω . For the countable system X = {Xm : m ∈ N} = {Amn : m ∈ N, n ∈ ω} it follows that A ∈ 0W 0W X = 0C 0C X = 0D 0D X but not A ∈ 0D X , because each directed subset of X is contained in some Xm .

A3 d A2d `` A1 `` `` ``d `` `` d d d A12 A22 A32 d d d A11 A21 A31 d d d A10 A20 A30

A` d ` ` ` ``

dX

Figure 4. 0D is not idempotent This example also demonstrates that the operator 0D ΩD (and dually ΩD 0D ) may fail to be a closure operator, because here we have X = ΩD X . With slightly more eﬀort one proves that the operator 0D ΩF is not idempotent either in this example: each member B = X of the ∩-system ΩF X is of the form B = {a ∈ ω ω : ami ≤ ni , i ≤ k, 1 ≤ a0 ≤ mB } for suitable k, mi ∈ N, ni ∈ ω, and mB = min{mi : i ≤ k}. Put mB = max{mi : i ≤ k}. If B ⊆ C ∈ ΩF X then mC ≤ mB (because the coordinate set {cmC : c ∈ C} and therefore also {bmC : b ∈ B} is bounded, i.e., ﬁnite), and as mC ≤ mC , it follows that C is contained in AmB . Thus, any directed subset of ΩF X \ {X} has an upper bound Am , whence A cannot be its union, i.e., A ∈ 0D ΩF X . Contrast this with the fact that 0D ΩX is always the inductive closure system generated by X (cf. Proposition 2.7 (8) and Theorem 2.8 (2)). The latter observation may be sharpened as follows (see Corollary 3.55 for an order-theoretical extension):

CLOSURE

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Proposition 2.14. (1) For any inclusion preserving Γ : PX → PX, the ﬁnitary modiﬁcation ΓI : PX → PX, A → {Γ(E) : E ∈ FA} is inductive and preserves directed unions. (2) For any closure operator Γ, the map ΓI is the greatest inductive closure operator contained in Γ argumentwise. (3) For any system X on X, the range of the associated closure operator Γ = ΓX is Ω X , the closure system generated by X , while the range of ΓI is C D X = 0D Ω X , the inductive closure system generated by X . Proof. (1) is easy. For (2), use A ⊆ {Γ({a}) : a ∈ A} ⊆ ΓI (A) and note that A ⊆ ΓI (B) entails Γ(E) ⊆ ΓI (B) for all E ∈ FA, i.e., ΓI (A) ⊆ ΓI (B). Thus ΓI is a closure operator; it is inductive by (1), and by deﬁnition, it contains any inductive closure operator that is contained in Γ argumentwise. The ﬁrst part of (3) is straightforward. For the second, note that the members of CΓI are directed unions of sets Γ(E) ∈ Ω X = CΓ and are therefore elements of 0D Ω X ⊆ C D X . For the reverse inclusion C D X ⊆ CΓI , simply use the fact that CΓI is an inductive closure system containing X . 2.3. Standard completions of partially ordered sets. Every poset P carries a few important “intrinsic” closure operators and closure systems, which provide certain natural completions, i.e., minimal complete lattices containing (an isomorphic copy of) P and having certain desirable additional properties. As a ﬁrst example, we already got to know the union preserving downset operator ↓P = ↓, sending each subset to the downset generated by it, and the associated Alexandroﬀ completion A∧ P = {↓A : A ⊆ P }. Recall that every union-preserving closure operator is the downset operator of a unique quasi-ordered set, which is a poset iﬀ distinct points have diﬀerent closures (cf. Proposition 2.2). By a standard extension of a poset P we mean a collection of downsets containing at least all principal ideals, and by a standard completion [43] a standard extension that is a closure system. Such completions have been investigated already in the ﬁfties by Banaschewski [6] and slightly later by Schmidt [82]. For an investigation of their distributivity properties, see [43]. The specialization order of any standard extension of a given poset P is just the order relation of P . While the largest of all standard completions is the Alexandroﬀ completion A∧ P , the smallest one is the Dedekind–MacNeille completion [19, 61] or normal completion N P , whose members are the cuts, the intersections of principal ideals. Note that A↓ = {↓ a : a ∈ A} and A↑ = {↑a : a ∈ A} are the sets of all lower and upper bounds of A, respectively (not to be confused with the downset ↓ A and the upset ↑A), whereas ∆P (A) = A↑ ↓ = {↓ b : b ∈ A↑ } = {↓ b : A ⊆ ↓ b} is the cut generated by A (cf. [9, 29, 35]). Hence, ∆P is a closure operator, the so-called cut operator of P = (X, ≤). The pair (X, ∆P ) is referred to as a cut space, and as a complete cut space in case P is a complete lattice. The latter holds iﬀ every

184

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cut is a principal ideal or, equivalently, P is isomorphic to its normal completion N P (via the principal ideal map sending a to ↓ a). An easy veriﬁcation conﬁrms that a map ϕ : C → D between complete lattices is continuousas a map between the associated cut spaces iﬀ it preserves arbitrary joins, i.e., ϕ( C A) = D ϕ(A) for all A ⊆ C (cf. Proposition 3.26). At ﬁrst glance, complete cut spaces look rather special, but they generate all T0 -closure spaces (in which distinct points have diﬀerent closures) by relativization (cf. Section 1.4 and [72]): Proposition 2.15. Every T0 -closure space (X, C) is a subspace of the cut space associated with a complete lattice isomorphic to C. C Proof. Let X be the range of the map ηX : X → C, x → {x} = ΓC ({x}), C and put N C|X = {Y ∩ X : Y ∈ N C}. Then ηX induces an isomorphism between C (X, C) and the subspace (X , N C|X ) of (C, N C). Indeed, ηX is injective by the T0 C axiom, and it induces a bijection between C and N C|X , since ηX (A) = ↓N C A ∩ X = ∆C ({A}) ∩ X for A ⊆ X.

A map f between closure spaces (X, C) and (Y, D) is said to be strictly dense if each closed set B ∈ D is the closure of its trace B ∩ f (X) — or, equivalently, if the contravariant lift f : D → C is a lattice isomorphism. Now, we are in a position to give a short proof for Ore’s characterization of closure spaces with a prescribed abstract lattice of closed sets [72]: Proposition 2.16. The closure system of a T0 -closure space (X, C) is isomorphic to a given complete lattice C iﬀ (X, C) admits a strictly dense embedding in the cut space (C, N C) or, equivalently, (X, C) is homeomorphic to a subspace of (C, N C) obtained by relativization to a basis. Proof. Let ϕ : C → D = N C # C be a lattice isomorphism and deﬁne f : X → C by {f (x)} = ϕ({x}). By the T0 -axiom, this determines f uniquely, and as ϕ is an isomorphism, f is one-to-one. For B ∈ D and x ∈ X, the equivalences x ∈ f (D) ⇔ f (x) ∈ D ⇔ ϕ({x}) ⊆ D ⇔ {x} ⊆ ϕ−1 (D) ⇔ x ∈ ϕ−1 (D) show that f is the isomorphism ϕ−1 : D → C. Furthermore, a subset B of C is a basis (i.e., join-dense) iﬀ the inclusion map from B into C is a strictly dense embedding in the associated cut space; indeed, B is join-dense iﬀ for ↓ c ⊆ ↓ a there is a b ∈ B ∩ ↓ c with b ∈ B ∩ ↓ a, which means that the relativization to B induces a one-to-one map from N C onto N C|B . Closely related to the representation of closure spaces by complete lattices is a universal property of the Dedekind–MacNeille completion: Theorem 2.17. For any closure space (X, C), the pair (C, N C) is the universal complete cut space over (X, C) in the following sense: C (1) The map ηX : (X, C) → (C, N C), x → {x} is continuous. (2) For any continuous map f from (X, C) into a complete cut space (D, N D), C there is a continuous map f ∨ : C → D with f =f ∨ ◦ ηX . ∨ ∨ (3) The map f is uniquely determined: f (A) = f (A) for A ∈ C. C C is continuous: ηX (↓C A) = {x ∈ X : {x} ⊆ A} = A for A ∈ C. Proof. (1) ηX

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(2) f ∨ : C → D with f ∨ (A) = f (A) is continuous, since for b ∈ D, f ∨ (↓ b) = {A ∈ C : f (A) ≤ b} = {A ∈ C : f (A) ⊆ ↓ b} = ↓C f (↓ b) belongs to N C (note that f (↓ b) is closed by continuity of f ). C = f again follows from the continuity of f : The identity f ∨ ◦ ηX C f ∨ ◦ ηX (x) = f ({x}) = f (x) since ↓D f ({x}) = f ({x}) = {f (x)}. C satisﬁes (3) Any continuous, i.e., join-preserving ϕ : C → D with f = ϕ ◦ ηX C ϕ(A) = ϕ( {{x} : x ∈ A}) = ϕ ◦ ηX (A) = f (A) = f ∨ (A).

Categorically speaking, the essence of Theorem 2.17 is that the functor assigning to any continuous map f between closure spaces (X, C) and (Y, D) the map f − : C → D, A → f (A) between the cut spaces (C, N C) and (D, N D) is a reﬂector from the category CSp0 of T0 -closure spaces to the full subcategory of complete cut spaces, which is isomorphic to the category CL∨ of complete lattices and joinpreserving maps (cf. Section 4). Under the perspective that the standard completions of a poset P are those closure systems whose specialization order is the original order of P , Theorem 2.17 may be reformulated as follows (see Schmidt [82]): Corollary 2.18. Every standard completion C of a poset P is universal: the map ηPC : P → C, a → ↓ a is C-continuous (preimages of principal ideals belong to C), and every C-continuous map from P into a complete lattice uniquely factorizes through ηPC and a join-preserving map. An order-embedding of a poset P in a complete lattice C is a function ϕ : P → C with a ≤ b ⇔ ϕ(a) ≤ ϕ(b), and it is a join-completion if its range ϕ(P ) is join-dense in C. Two join-completions ϕ : P → C and ψ : P → D are said to be equivalent if there is an isomorphism ι : C → D with ψ = ι ◦ ϕ. Theorem 2.19. The standard completion maps ηPC (where C runs through the standard completions of P ) form a set of representatives for the join-completions of P . In other words, every join-completion of P is equivalent to exactly one ηPC . Proof. Given any join-completion ϕ : P → C, put C = {ϕ (↓ c) : c ∈ C}. Then C is a closure system, since {ϕ (↓ c) : c ∈ B} = ϕ (↓ B) for B ⊆ C, and it is a standard completion, having as specialization order the order of P :

a ≤C b ⇔ ∀c ∈ C(b ∈ ϕ (↓ c) ⇒ a ∈ ϕ (↓ c)) ⇔ ϕ(a) ≤ ϕ(b) ⇔ a ≤ b. By Theorem 2.17, the join-preserving map ϕ∨ satisﬁes ϕ∨ ◦ ηPC = ϕ. Moreover, in the present situation, it is surjective since ϕ(P ) is join-dense in C, and injective, because for A = ϕ (↓ c) and B = ϕ (↓ d) in C, the inequality ϕ∨ (A) = ϕ(A) ≤ ϕ∨ (B) = ϕ(B) entails c=

(ϕ(P ) ∩ ↓ c) = ϕ(ϕ (↓ c)) = ϕ(A) ≤ · · · ≤ (ϕ(P ) ∩ ↓ d) = d,

hence A ⊆ B. Thus, ϕ∨ is an isomorphism satisfying ϕ = ϕ∨ ◦ ηPC .

´ M. ERNE

186

Finally, if two standard completion maps ηPC and ηPD are equivalent, they must already be equal, because an isomorphism ι : C → D with ι ◦ ηPC = ηPD has to satisfy, for each a ∈ P and A ∈ C, the equivalences a ∈ A ⇔ ηPC (a) ⊆ A ⇔ ηPD (a) = ι(ηPC (a)) ⊆ ι(A) ⇔ a ∈ ι(A),

and therefore ι must be the identity map.

By a global standard extension or global standard completion, respectively, we mean a subset selection that associates with each poset a speciﬁc standard extension or completion. An arbitrary subset selection Z gives rise to the global standard extension Z ∧ of all Z-downsets, but also to several global standard completions [33, 36], e.g., the Z-cut completion Z deﬁned by

Z P = {A ∈ A∧ P : ∀Z ∈ ZP (Z ⊆ A ⇒ ∆P (Z) ⊆ A)} and the Z-join ideal completion Z ∨ deﬁned by Z ∨ P = {A ∈ A∧ P : ∀Z ∈ ZP (Z ⊆ A, a =

Z ⇒ a ∈ A)}.

Its members are the Z-join ideals. Clearly, for Z-complete posets P (in which all subsets Z ∈ ZP have joins), Z P coincides with Z ∨ P . Of the various interesting global standard completionslying between the extremal ones, the Dedekind–MacNeille completion N = A and the Alexandroﬀ completion A∧ = E , we mention here the following: Example 2.20 (The Banaschewski–Frink ideal completion; see [6, 45]). IP = F P is the smallest algebraic closure system containing all principal ideals. By Proposition 2.14, it consists of all directed unions of cuts, and the corresponding ﬁnitary closure operator is ∆IP , where ∆IP (A) = {∆P (F ) : F ∈ FA} is the ideal generated by A. Observe that many authors mean by an ideal a directed downset (as, for example, in [47, 48]), but the system D∧ P of all such directed ideals is in general not a closure system and properly contained in IP . On the other hand, the system F ∨ P of all ∨-ideals is an algebraic closure system that contains IP , and for arbitrary posets, the containment may be proper. But for ∨-semilattices S with least elements, all three types of ideals agree: IS = D ∧ S = F ∨ S.

Example 2.21 (The upper topology). F ∧ P , the system of ﬁnitely generated downsets, is closed under ﬁnite unions but rarely a closure system unless P is ﬁnite. The least closure system UP containing F ∧ P is the topological closure system generated by the set of all principal ideals: see Proposition 2.7 (7). The corresponding topology is referred to as the upper topology [47, 48] or weak topology, because it consists of upper sets and is the weakest topology making all principal ideals closed. Example 2.22 (The Scott topology). Another topological closure system is D P , the cut completion associated with the selection D of all directed subsets. Indeed, if A ∈ D P and B ∈ D P then also A ∪ B ∈ D P , since any directed D ⊆ A ∪ B is contained either in the downset A or inthe downset B. The corresponding topology is the weak Scott topology [30]. D P contains the least topological standard completion UP (properly in general). On the other hand, D P is contained in the so-called Scott completion D ∨ P which is also a topological closure system. The corresponding open sets form the Scott topology [47, 48]. For arbi trary posets, the inclusion D P ⊂ D∨ P may be proper, but if all directed subsets have a join, both kinds of completion coincide.

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A∧ P? ?? D∨ P? F ∨ P? ?? ?? A∨ P D P F P IP ? ??? D∧ P? ??? N

UP F ∧P

Figure 5. Global standard extensions Much more material about global extensions and completions, in particular, their universal properties, may be found in [30, 32, 33] and [37]. 2.4. Lattice representation of closure spaces. The idea to represent a closure space abstractly by the lattice of its closed sets plus some additional information about certain distinguished (join) bases dates back to the pioneering work by Birkhoﬀ [9] and Ore [72] (see Section 1.4 and Theorem 2.16). A systematic investigation of that topic, its categorical aspects and the involved methods are to be found in [31], [36] and [40]. Here, we sketch only some basic ideas. In fact, let us consider so-called (join-)based lattices: these are pairs (B, C) consisting of a complete lattice C and a basis (a join-dense subset) B of C. Appropriate morphisms between such objects (B, C) and (E, D) are maps ϕ : C → D that preserve not only arbitrary joins but also the distinguished bases, i.e., ϕ(B) ⊆ E. This gives a category BCL∨ . Now, the passage from closure systems to complete lattices and vice versa is made “functorial” as follows. For any closure space (X, C), the point closures {x} (x ∈ X) form a basis BC of the complete lattice C (because any closed set is the union, a fortiori the join of the point closures contained in it). In other words, the pair G(X, C) = (BC , C) is a based lattice. Any continuous map f between closure spaces (X, C) and (Y, D) gives rise to a join- and basis-preserving “lifted” map Gf : G(X, C) → G(Y, D), A → f (A). Indeed, denoting joins in C by C , the continuity of f yields Gf ( C Y) = Gf ( Y) = f ( Y) = D Gf (Y) and Gf ({x}) = {f (x)}. In that way, we have constructed a functor G from the category CSp of closure spaces to the category BCL∨ of based lattices. In the oppositite direction, a functor H from BCL∨ to the category CSp0 of T0 -closure spaces is obtained by sending any based lattice (B, C) to the closure space (the hull-kernel space or spectrum) H(B, C) = (B, {B ∩ ↓ c : c ∈ C}) (which is T0 because {a} = B ∩ ↓ a = B ∩ ↓ b = {b} entails a = b), and any joinand basis-preserving map ϕ : (B, C) → (E, D) to its restriction Hϕ = ϕ|E B : B → E.

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This is a continuous map between H(B, C) and H(E, D), on account of the equation ϕ (E ∩ ↓ d) = {b ∈ B : ϕ(b) ≤ d} = {b ∈ B : b ≤ ϕ(d)} = B ∩ ↓ ϕ(d) where ϕ : D → C with ϕ(d) = {b ∈ B : ϕ(b) ≤ d} is the upper adjoint of ϕ (see Section 3.2). Furthermore, the map ι(B,C) sending c to B ∩ ↓ c is an isomorphism between the based lattices (B, C) and G ◦ H(B, C) = ({B ∩ ↓ b : b ∈ B}, {B ∩ ↓ c : c ∈ C}) (by join-density of B in C). On the other hand, for any T0 -closure space (X, C), the map η(X,C) , sending x to the point closure {x}, is an isomorphism between (X, C) and the hull-kernel space H ◦ G(X, C) = (BC , {{{x} : x ∈ A} : A ∈ C}). Using the fundamental notion of categorical equivalence (see [2, 60]), the previous considerations amount to Theorem 2.23. The lattice representation functor G and the spectrum functor H induce an equivalence between the category CSp0 of T0 -closure spaces and the category BCL∨ of based lattices. In order to cover various important classes of spaces on the one hand and of complete lattices on the other hand, we consider invariant point selections for complete lattices. These are (class) functions assigning to each complete lattice C a certain subset ΣC such that every isomorphism ϕ : C → D maps ΣC onto ΣD. Typical examples are the subsets of all atoms p a ≤ p ⇒ a = 0 = C or a = p ∨-irreducible elements p p = a ∨ b ⇒ p = a or p = b, and p = 0 ∨-prime elements p p≤a ∨ b ⇒ p ≤ a or p ≤ b, and p = 0 -irreducible elements p p = A ⇒ p ∈ A -prime elements p p ≤ A ⇒ p ∈ ↓A compact elements p p ≤ A for a directed A ⇒ p ∈ ↓ A By a Σ-complete closure space we mean a T0 -closure space (X, C) such that the members of ΣC are precisely the point closures {x} with x ∈ X. On the lattice side, we mean by a Σ-based lattice a complete lattice C for which ΣC is a basis (i.e., join-dense in C). Its Σ-spectrum is the closure space (ΣC, {ΣC ∩ ↓ c : c ∈ C}). We list various frequently used instances of these notions in the table below. elements of ΣC all elements atoms ∨-prime elements ∨-prime atoms -irreducible elements -irreducible ∨-primes -prime elements -prime atoms compact elements compact exchange atoms

Σ-based lattices complete lattice atomistic lattice spatial coframe atomistic coframe minimally based lattice minimally based coframe superalgebraic lattice ABC lattice algebraic lattice geometric lattice

Σ-complete space complete cut space T1 -closure space sober topological space topological T1 -space TD -closure space topological TD -space T0 -Alexandroﬀ space discrete space ideal space combinatorial geometry

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For sober spaces and spatial frames, see e.g. [56] or [48]; a coframe is a dual frame. An ideal space is equipped with the closure system of all ideals of a joinsemilattice with 0. An atom is minimal above the least element, and an exchange atom (or semimodular atom) is an atom p such that a ≤ b < a ∨ p implies a = b. Concerning atomistic, algebraic and geometric lattices, see e.g. Birkhoﬀ [9] and Gr¨ atzer [50]. By an ABC lattice, there is meant an atomistic boolean complete lattice. Any Σ-based lattice C may be regarded as a based lattice (ΣC, C). Under that identiﬁcation, Theorem 2.23 amounts to Theorem 2.24. Let Σ be any invariant point selection for complete lattices. Then the lattice representation functor G and the spectrum functor H induce an equivalence between the category of Σ-complete closure spaces and the category of Σ-based lattices. In particular, the Σ-based lattices are exactly the isomorphic copies of closure systems of Σ-complete closure spaces, and the Σ-complete closure spaces are exactly the Σ-spectra of Σ-based lattices. Furthermore, two Σ-complete closure spaces are homoeomorphic iﬀ they have isomorphic closure systems, and two Σ-based lattices are isomorphic iﬀ they have homeomorphic Σ-spectra. This very general theorem was established ﬁrst in [31] (see also [36] and [40] for supplements and extensions). It provides a broad variety of representation theorems that were discovered and rediscovered by diverse authors during the second half of the last century. Speciﬁcally, given a functorial subset selection Z and the associated invariant point selection ΣZ of all Z-prime elements p, satisfying p ∈ ↓ A for all A ∈ ZP with p ≤ A, we see that the category of ΣZ -complete closure spaces is equivalent to the category of ΣZ -based lattices. Note that every ΣZ -complete space is a Zclosure space. Indeed, the Z-closure spaces are those in which all point closures are Z-prime closed sets, while the ΣZ -complete spaces are those T0 -spaces whose point closures are exactly the Z-prime closed sets.

Figure 6. Hull operation, generating a complete hull system

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3. Order-theoretical closure operations The most natural framework for closure structures are ordered sets (posets). Besides functions, the only notion needed for an axiomatic introduction of closure is that of order, manifested by certain transitive relations (reﬂexivity and antisymmetry are sometimes dispensable but will be generally assumed for convenience). Note that every relation is contained in a smallest quasi-order, its reﬂexive-transitive hull (closure strikes again!) The one-to-one correspondence between set-theoretical closure operators and closure systems extends to one between closure operations and closure ranges in posets. Particularly convenient is the theory of closure for complete lattices, where arbitrary joins and meets are available, and the closure ranges are nothing but the meet-closed subsets. Among many other results, we shall derive various useful ﬁxpoint theorems with the order-theoretical machinery of closure operations. A vital theme to be discussed here is the fundamental connection between adjoint mappings and closure operations: either concept induces and determines the other completely, at least up to isomorphism. A deeper study will be devoted to subsets of complete lattices that are closed under the formation of certain speciﬁed joins and meets, giving rise to a broad spectrum of closure operators. The general notions of Z-distributive and Z-continuous lattices are crucial for an eﬀective theory extending the results about set-theoretical closure operators and closure systems to a general order-theoretical setting. The homomorphism theory of Z-continuous lattices, their Z-subalgebras and the associated closure operators is more advanced and requires some routine with the handling of joins and meets in posets and lattices. 3.1. Closure operations and closure ranges. The classical concept of (settheoretical) closure operators naturally extends to arbitrary posets instead of power set lattices through the following deﬁnition: a closure map, closure operation or hull operation on a poset P = (X, ≤) is a map γ : P → P (or, more accurately, γ : X → X) enjoying the following three properties for all a, b in P : (c1)

a ≤ γ(a)

(γ is extensive or expansive)

(c2)

a ≤ b ⇒ γ(a) ≤ γ(b)

(γ is isotone or order preserving)

(c3)

γ(γ(a)) = γ(a)

(γ is idempotent).

Each of these three basic properties is of particular importance in its own right: extensivity means that the operation “enlarges” the objects on which it acts; isotone maps “respect” or “transport” the order relations under consideration; and idempotency requires that the process of enlarging stops after the ﬁrst application. (Observe that elsewhere in the literature the terms monotone or increasing are used for isotone maps, whereas in other contexts, by an increasing map an extensive one is meant.) As remarked earlier, the convention that closure operations have to be idempotent is not generally adopted. In the modern theory of categorical closure operators (see, for example, [14, 26, 27]), one takes into account certain families of operations that satisfy categorical variants of (c1) and (c2) but not necessarily (c3). However, our primary intuition is that a closure operation produces “closed” objects in a single step; therefore, we generally assume idempotency of closure operations γ and

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call the elements γ(a) closed. Thus, an element c is closed iﬀ it is a ﬁxpoint of γ, i.e., γ(c) = c. On the other hand, in many mathematical theories, one is working with maps that satisfy (c2) and (c3) but not necessarily (c1). With the obvious examples from algebra, functional analysis and physics in mind, one speaks of projection operators or projections if isotone and idempotent maps are meant (see [47, 48] for more mathematical background). Of course, there is a notion dual to that of closure maps, namely that of kernel maps or kernel operations (sometimes also baptized coclosure maps); these are projections κ that are contractive (or regressive) instead of being extensive, i.e., κ(a) ≤ a. Our ﬁrst result concerns the characterization of closure operations (alias hull operations) by one single condition, interpreted by the phrase: a hull contains an object if and only if it contains the hull of that object. Proposition 3.1. A self-map γ on a poset P is a closure operation iﬀ (c)

a ≤ γ(b) ⇔ γ(a) ≤ γ(b).

Proof. For (c) ⇒ (c1) take a = b; for (c) ⇒ (c2) use the proven inequality b ≤ γ(b) to obtain a ≤ b ⇒ a ≤ γ(b) ⇒ γ(a) ≤ γ(b), and for (c) ⇒ (c3), apply (c1) and the implication a ≤ γ(b) ⇒ γ(a) ≤ γ(b) to the case a = γ(b). Conversely, a projection γ satisﬁes a ≤ γ(b) ⇒ γ(a) ≤ γ(γ(b)) = γ(b), and extensivity yields the implication γ(a) ≤ γ(b) ⇒ a ≤ γ(b). If a subset A of aposet P has aleast upper bound (supremum, join) b then we indicate this by b = A or by b = P A. Dually, we write c = A or c = P A if c is the greatest lower bound (inﬁmum, meet) of A. Proposition 3.2. For a self-map γ on a poset P to be a closure operation, the following condition is necessary and suﬃcient: If A ⊆ P has a join then A ≤ γ( A) = γ(P ) γ(A), (cW ) and if γ(A) has a join, too, then γ( A) = γ( γ(A)). Proof. Suppose γ : P → P is a closure operation. (c1) entails A ≤ γ( A), and (c2) assures that γ( A) is an upper bound of γ(A) in γ(P ); if b ∈ γ(P ) is any upper bound of γ(A) then A ≤ b by (c1) and b = γ(b) by (c3), hence γ( A) ≤ b by (c). Thus γ( A) = γ(P ) γ(A). Furthermore, if γ(A) exists then (c1) and (c2) yield A ≤ γ(A) ≤ γ( A), and then (c2) together with (c3) gives γ( γ(A)) ≤ γ(γ( A)) = γ( A) ≤ γ( γ(A)). Concerning suﬃciency of the condition (cW ), note ﬁrst that the special case A = {a} immediately gives (c1) and (c3). And if a ≤ b then A = {a, b} has the join b, whence γ(a) ≤ γ(P ) γ(A) = γ( A) = γ(b), proving (c2). Recall that a join-semilattice is a poset in which any two-element subset {a, b} has a join a ∨ b, and a complete join-semilattice is a poset in which every nonempty subset has a join. Thus, the complete lattices are exactly the complete join semilattices possessing a least element 0 = ∅. Corollary 3.3. On join-semilattices, closure operations may be characterized by one inequality, for example: (c≤ )

a ∨ γ(γ(b)) ≤ γ(a ∨ b)

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192

or by one identity, for example: (c= )

a ∨ γ(a ∨ γ(b)) = γ(a ∨ b).

Proof. By Proposition 3.2, every closure operation γ on a join-semilattice satisﬁes γ(a ∨ γ(b)) = γ(γ(a) ∨ γ(γ(b))) = γ(γ(a) ∨ γ(b)) = γ(a ∨ b), hence (c= ); this in turn entails (c≤ ), because a = b in (c= ) yields (c1), and substituting γ(b) for b leads to a ∨ γ(γ(b)) ≤ a ∨ γ(a ∨ γ(γ(b))) = a ∨ γ(a ∨ γ(b)) = γ(a ∨ b). Finally, (c≤ ) returns the closure properties: for a = b, (c≤ ) gives (c1) and (c3), and then a ≤ b implies γ(a) = γ(γ(a)) ≤ γ(a ∨ b) = γ(b). This corollary has the interesting consequence that the closure join-semilattices, i.e., the join-semilattices equipped with a closure operation, form a variety, being equationally deﬁned. Hence, substructures, products and homomorphic images of closure join-semilattices are again such objects. Another immediate consequence of Proposition 3.2 is: 3.4. Corollary A self-map γ of a complete lattice C is a closure map iﬀ A ≤ γ( A) = γ( γ(A)) = γ(C) γ(A) for all subsets A of C. Let us return to the general case of arbitrary partially ordered sets. With any self-map γ of a poset P , we may associate four intrinsic subsets: the range γ(P ) = {γ(a) : a ∈ P }, the upper pre-ﬁxpoint set P γ = {a ∈ P : γ(a) ≥ a}, the lower pre-ﬁxpoint set Pγ = {a ∈ P : γ(a) ≤ a}, and the ﬁxpoint set Pγ= = {a ∈ P : γ(a) = a}. By deﬁnition, γ is idempotent iﬀ γ(P ) = Pγ= , and a closure operation iﬀ γ is isotone, P = P γ and γ(P ) = Pγ . Now, by a closure range, we mean a subset C of a poset P such that for each a ∈ P , there is a least c ∈ C with a ≤ c. This terminology is justiﬁed by the following crucial correspondence: Proposition 3.5. Assigning to each closure operation on a poset P its range, one obtains a dual isomorphism between the pointwise ordered set Clo(P ) of all closure operations and the set Cl(P ) of all closure ranges, ordered by inclusion. Proof. The range Pγ of any closure operation γ has the property that for each a ∈ P , there is a least c ∈ Pγ with a ≤ c, namely c = γ(a). If δ is a closure operation with γ(a) ≤ δ(a) for all a ∈ P then Pδ is contained in Pγ , as δ(a) ≤ a implies γ(a) ≤ a. On the other hand, if C is any closure range in P then the map γC sending a ∈ P to the least c ∈ C above a is a closure map, and C is its range. Being obviously order-reversing, the assignments γ → Pγ and C → γC are mutually inverse dual isomorphisms between the poset of closure operations and that of closure ranges. Let us mention three natural examples of closure operations that are deﬁned on non-complete posets whose order relation is not set inclusion. Example 3.6. Fix an inner point 0 of some star-like solid and send each other point x to the nearest (often the unique) common point of the surface and the ray from 0 through x. This gives a closure operation on the space minus 0, strictly ordered by x < y iﬀ x sits closer to 0 than y on a ray through 0. The corresponding closure range is the surface of the solid.

CLOSURE

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Example 3.7. The function ceil on the reals, assigning to each real number the least integer above or equal to it, is clearly a closure operation whose range are the integers. Dually, the function ﬂoor sending a real number to the greatest integer below or equal to it is a kernel operation whose range are again the integers. Hence, the integers form both a closure range and a kernel range that has neither arbitrary meets nor arbitrary joins. Example 3.8. The previous example may be generalized as follows. Let P be any linearly ordered set (in which any two elements a, b are comparable). Here, the closure maps may be vizualized as “staircase functions”:

Figure 7. A staircase function In order that the closure operations on P form a closure range in the pointwise ordered set I(P ) of all isotone self-maps of P , it is suﬃcient that P is a complete join-semilattice, i.e., every nonempty subset has a join, or equivalently, every lower bounded subset has a meet. In that case, the least closure operation ϕ above an isotone map ϕ : P → P is given by ϕ(a) = {b ∈ P : a ∨ ϕ(b) ≤ b} (see Theorem 3.16 (4)). For example, on the half-open real unit interval P = ]0, 1], the closure ranges in P and in I(P ), respectively, form a closure system, although P and I(P ) are not complete (missing a least element) and have inﬁnite ascending chains (cf. Proposition 3.14). In contrast to that example, on the chain ω of natural numbers or on the chain of integers, there is no closure operation above the isotone map ϕ with ϕ(x) = x + 1. There is a simple characterization of closure operations in terms of order relations alone. Calling a quasi-order R on a set X a principal quasi-order for a poset P = (X, ≤) if each of the principal ideals Rb = {a ∈ X : a R b} is also a principal ideal of P (i.e., Rb = {a ∈ X : a ≤ c} for some c ∈ X), we have: Proposition 3.9. Associating with any closure operation γ on a poset P the quasi-order ≤γ deﬁned by (q)

a ≤γ b ⇔ a ≤ γ(b) ⇔ γ(a) ≤ γ(b)

one obtains an isomorphism between Clo(P ) and Pq(P ), the set of principal quasiorders for P , hence a dual isomorphism between Cl(P ) and Pq(P ). Proof. For any γ ∈ Clo(P ), the equivalence (q) shows that ≤γ is a principal quasi-order with γ(b) = max{a : a ≤γ b}. On the other hand, if R is any principal quasi-order for P then γ(b) := max Rb deﬁnes a closure operation: indeed, b R b

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194

yields b ≤ γ(b), and a ≤ γ(b) entails γ(a) ≤ γ(b) on account of the relations γ(a) R a R b; further, by deﬁnition, a ≤γ b ⇔ a ≤ γ(b) ⇔ a R b. Finally, the equivalence γ ≤ δ ⇔ ≤γ ⊆ ≤δ ensures that the map γ →≤γ is in fact an isomorphism between Clo(P ) and Pq(P ). More results about closure operations and closure ranges are available in the presence of enough joins and meets. A subset C of P is called • meet-closed if for each A ⊆ C, b = P A implies b ∈ C, • meet-subcomplete if each A ⊆ C has a meet in P and P A ∈ C. Join-closed and join-subcomplete subsets are deﬁned dually. Proposition 3.10. The following implications hold between the previously deﬁned notions: meet-subcomplete ⇒ complete and meet-closed ⇓ ⇓ closure range ⇒ meet-closed For subsets of a complete lattice, all four properties are equivalent. Proof. A meet-subcomplete subset C is a closure range because for a ∈ P , the element γ(a) = {c ∈ C : a ≤ c} is the least element of C above a.A closure range C in turn is meet-closed: if a subset A of C has a meet b = P A in P then the least element c of C above b is a lower bound of A and must therefore coincide with b. On the other hand, it is clear that a meet-subcomplete subset is both complete and meet-closed, and that a meet-closed subset of a complete lattice is meet-subcomplete. Let us consider an example showing that no other implications than those stated in Proposition 3.10 generally hold between these four properties. Example 3.11. Consider the set {2, 3, 12, 18, 36}, ordered by divisibility. While the subposet C1 = {2, 12, 18, 36} is complete and meet-closed but not a closure range, the subposet C2 = {2, 3, 12, 36} is a closure range but not complete, and the subposet C3 = {12, 18, 36} is meet-closed but neither complete nor a closure range. 36t Z 12 t C Z t18 1 ZZ Z Z d3 2 t

36 t Z 12 t C Z d18 2 ZZ Z Z t3 2 t

36 t Z 12 t C Z t18 3 ZZ Z Z d3 2 d

Figure 8. Meet-closed subsets On account of the last observation in Proposition 3.10, the theory of closure operations becomes considerably easier and more convenient in the case of complete lattices: Corollary 3.12. Assigning to each closure operation on a complete lattice C its range yields a dual isomorphism between the complete lattice Clo(C) of closure operations on C (with pointwise meets) and the closure system Cl(C) of all meetclosed subsets of C.

195

CLOSURE

An obvious question arises immediately: is the system Cl(P ) of all closure ranges (or the dually isomorphic pointwise ordered collection Clo(P ) of all closure operations) always a complete lattice, for arbitrary posets P or at least for semilattices? That the answer is in the negative is shown by Example 3.13. In each of the following four diagrams, a closure range of the same ∨-semilattice S (the “kite”) is marked by bold black dots. These closure ranges form a four-element subposet of Cl(S) whose minimal elements A, B have no join in Cl(S) and whose maximal elements C, D have no meet in Cl(S). Hence, Cl(S) is neither a join- nor a meet-semilattice. ' $ ' s s @ @s @s s s @ @ @ @p @p pp pp C s c c s s c c s & % & @ @ @ @ ' $@' s s @ @c @s s c @ @ @ @p @p pp pp A c c c c c c c c & % &

$

D

%

$

B

%

Figure 9. A kite and closure ranges ' $ r @r r @ @ @p ppb ' $ @ ' $ r r @ 6 b @ @ @r @r r r 6 b @ @ @p @p 6 b pp & % ppr γC γD b 6 b r r 6 b 6 b r & % & % Figure 10. A join of two closure maps

196

´ M. ERNE

Note also that the closure operations on S do not form a closure range in the ordered set I(S) of all isotone maps on S: there is no least closure operation above the pointwise join in I(S) of the closure operations corresponding to the two closure ranges C and D. In a sense, this example is typical for the failure of completeness when dealing with posets of the form Cl(P ), as we shall see in the sequel (thanks to C. Ronse for having drawn the author’s attention to the paper by Ranzato [75] and related literature). For any subset A of a poset P , let M(A) denote the set of all maximal lower bounds, and put ΛB = {M(A) : A ⊆ B}. Subsets B with ΛB = B are called Λ-closed; clearly, they always form a closure system, and for subsets of complete lattices, “Λ-closed” means “meet-closed”. The poset P is said to be M-complete if every lower bound of any subset is dominated by a maximal one, i.e., A ⊆ ↑ b ⊆ P entails M(A) ∩ ↑ b = ∅. For example, every ﬁnite poset and every complete joinsemilattice is M-complete. Now, one easily proves a slight extension of the main result in [75]: Proposition 3.14. Every closure range in a poset P is Λ-closed, and the converse holds in M-complete posets P . For such posets, Cl(P ) is a closure system, and Clo(P ) is not only a complete lattice but also a closure range in EI(P ), the poset of all extensive and isotone self-maps of P . Proof. Let C be a closure range, A ⊆ C and m ∈ M(A). Then we have γC (m) ≤ γC (a) = a for all a ∈ A, hence γC (m) ∈ M(A) and so γC (m) = m by maximality of m. Thus, m ∈ C, showing that C is Λ-closed. Conversely, assume that C is a Λ-closed subset of an M-complete poset P . For a ∈ P , we have M(↑ a ∩ C) ⊆ C and ↑ a ∩ M(↑ a ∩ C) = ∅. Hence, ↑ a ∩ C meets M(↑ a ∩ C) and the only element in the meet is the minimum of ↑ a ∩ C. For ϕ ∈ E(P ), the ﬁxpoint set Pϕ is Λ-closed and therefore a closure range: given A ⊆ Pϕ and b ∈ M(A), one has b ≤ ϕ(b) ≤ ϕ(a) = a for all a ∈ A, hence b = ϕ(b) ∈ Pϕ . It follows that the map ϕ = γPϕ is the least closure operation above ϕ: indeed, ϕ(a) ∈ Pϕ entails ϕ(a) ≤ ϕ(ϕ(a)) ≤ ϕ(a), and if ϕ ≤ γ for some γ ∈ Clo(P ) then Pγ ⊆ Pϕ = Pϕ and so ϕ ≤ γ. Zorn’s Lemma is equivalent to the statement that every D-complete poset is M-complete, which implies by Proposition 3.14 the following Corollary 3.15. If P is a D-complete poset then Cl(P ) is a closure system, and Clo(P ) is a complete closure range in EI(P ). The latter result can even be established without invoking any choice or maximal principle, referring instead to the Fixpoint Theorem 3.25 (cf. [75]). Particularly convenient is the situation for complete join-semilattices: Theorem 3.16. Let C be a complete join-semilattice. (1) For any subset B of C, the following map is a closure operation: γB : C → C, a → {b ∈ B : a ≤ b}. (2) For any isotone self-map ϕ of C, the following set is a closure range: Cϕ = {a ∈ C : ϕ(a) ≤ a}.

CLOSURE

197

(3) For each B ⊆ C, the least closure range containing B is B = CγB = {a ∈ C : a = {b ∈ B : a ≤ b}}. Hence the system Cl(C) of all closure ranges in C is a closure range in AC. (4) For each isotone self-map ϕ of C, the least closure operation above ϕ is ϕ = γCϕ . Hence the pointwise ordered set Clo(C) of all closure operations on C is a closure range in I(C). (5) The maps B → γB and ϕ → Cϕ induce mutually inverse dual isomorphisms between the complete meet-semilattice Cl(C) and the complete join-semilattice Clo(C). If C is a complete lattice then so are Cl(C) and Clo(C). Proof. (1). γB is well-deﬁned since the sets {b ∈ B : a ≤ b} are lower bounded. γB is clearly extensive, and the implication a ≤ γB (c) ⇒ γB (a) ≤ γB (c) shows that γB is a closure operation. (2). For a ∈ C, the set {c ∈ C : a ∨ ϕ(c) ≤ c} has a meet a, being lower bounded bya. By deﬁnition, a is the least element of Cϕ above a (note that a ∨ ϕ(a) ≤ {c ∈ C : a ∨ ϕ(c) ≤ c} = a because ϕ is isotone). (3). By (1) and (2), CγB is a closure range containing B, and if D is any other closure range containing B then CγB ⊆ D, since D is meet-closed. (4). For any isotone ϕ : C → C, the map ϕ = γCϕ is a closure operation by (1) and (2), and for any other closure operation γ on C with ϕ ≤ γ, it follows that Cγ ⊆ Cϕ and then γCϕ ≤ γ. (5). If (ϕi : i ∈ I) isa nonempty family of isotone self-maps of C then the pointwise supremum ϕ = {ϕi : i ∈ I} is isotone, hence the join in I(C). Although ϕ need not be a closure map if all the ϕi s are closure maps, the equation Cϕ = {Cϕi : i ∈ I} shows that Cl(C) is closed under nonempty intersections, hence a complete meet-semilattice. The isomorphism statement is clear by (3) and (4). The last claim in (5) has been established in Corollary 3.12. There is no reason why the closure system of all closure ranges in a complete lattice should be topological or algebraic. For example, the complete chain {1/n : n ∈ N} ∪ {0} has no ascending subchains, but there is an obvious ascending chain of ﬁnite closure ranges {1/n : n ≤ m} (m ∈ N) whose union {1/n : n ∈ N} fails to be a closure range. However, closure systems of the form Cl(P ) have a strong property that is equivalent to the Anti-Exchange Axiom for convex geometries in the ﬁnite case (see Section 1.4) but stronger in the inﬁnite case: a closure space or its closure system is called (extremally) detachable (cf. Jamison-Waldner [55]) if for any closed set A and any point x outside A, the truncated closure A ∪ {x} \ {x} is again closed (in particular, such spaces are TD : the sets {x} \ {x} are closed). Recall that a lattice is lower semimodular if a ≺ a ∨ b implies a ∧ b ≺ b (where a ≺ c means that a < c but no b satisﬁes a < b < c; cf. [9] or [50]). Any detachable closure system is lower semimodular, because in such a system, A ≺ B can happen only if A diﬀers from B by one element. The most suggestive example of a detachable closure system is that of all convex subsets of a linear space. But we also have:

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198

Proposition 3.17. If Cl(P ) is a closure system (in particular, if P is a complete join-semilattice or satisﬁes the ACC) then Cl(P ) is detachable, hence lower semimodular. Proof. For a closure range A ∈ Cl(P ) and x ∈ P \ A, let B denote the least closure range containing A ∪ {x}. Observe that A ∪ ↓ x ∈ Cl(P ) (indeed, if a ∈ P but a ∈ ↓ x then γA (a) is the least element of A ∪ ↓ x above a); hence B ⊆ A ∪ ↓ x. Consider an arbitrary a ∈ P . If γB (a) is distinct from x then it is the least element of B \ {x} above a; otherwise, x = γB (a) < b for all b ∈ ↑a ∩ B \ {x}, whence ↑a ∩ B \ {x} ⊆ A ⊆ B \ {x}, and therefore γA (a) = min(↑a ∩ B \ {x}). Thus, B \ {x} is a closure range. For more results on the structure of Cl(P ) for ﬁnite P , see Hawrylycz and Reiner [52]. In the special case of power sets, the following property of the lattice of all closure ranges has been observed already by Ore [71]: Proposition 3.18. If C is a complete lattice in which the ∧-irreducible elements are meet-dense then Cl(C) is dually pseudocomplemented; that is, for each A ∈ Cl(C), there is a least B ∈ Cl(C) with A ∨ B = C in Cl(C). Proof. Let M denote the set of all ∧-irreducible elements m (satisfying m∈ F whenever F ⊆ C is a ﬁnite set with m = F ). For A ∈ Cl(C), put B = { D: D ⊆ M \ A}. Then B is meet-closed, and each c ∈ C is the meet of a = {m ∈ M ∩ A : c ≤ m} ∈ A and b = {m ∈ M \ A : c ≤ m} ∈ B. Thus, C is the only meet-closed subset of C containing A ∪ B. And if some B1 ∈ Cl(C) satisﬁes A ∨ B1 = C, i.e., each c ∈ C is of the form a ∧ b with a ∈ A and b ∈ B1 , then each m ∈ M \ A must belong to B1 , whence B ⊆ B1 . Since closure operations on complete lattices are so much better tractable than those on arbitrary posets, it is certainly desirable to reduce, as far as possible, the considerations to the setting of complete lattices. This can be done quite satisfactorily by a suitable completion process. Theorem 3.19. For any subposet P of a complete lattice C and any closure operation γ on P , the map γ C : C → C, a → C (Pγ ∩ ↑a) is the greatest closure operation on C that induces γ on P . If P is meet-dense in C (i.e., each element of C is a meet of elements in P ) then γ C is even the unique extension of γ to a closure operation on C, and in that case, the assignment γ → γ C yields an isomorphism between Clo(P ) and CloP (C) = {δ ∈ Clo(C) : δ(P ) ⊆ P }, the poset of closure maps on C that leave P invariant. Proof. We already know that γ C is a closure operation (see Theorem 3.16), and clearly γ C (a) = γ(a) ∈ P for a ∈ P . If δ : C → C is a closure operation with δ|P = γ then we get δ(P ) ⊆ P and Pγ = {a ∈ P : γ(a) = a} = {a ∈ P : δ(a) = a} = Cδ ∩ P, γ (a) = C (Pγ ∩ ↑a) = C (Cδ ∩ P ∩ ↑a) ≥ C (Cδ ∩ ↑a) = δ(a), and equality holds if P is meet-dense in C, i.e., b = (P ∩ ↑b) for b ∈ C. C

CLOSURE

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Conversely, if δ is an arbitrary closure operation on C such that δ(P ) ⊆ P , then γ = δ|P is a closure operation on P , since for a, b ∈ P , a ≤ γ(b) = δ(b) ⇔ γ(a) = δ(a) ≤ γ(b) and if P is meet-dense in C then we obtain, as before, γ C = δ. The monotonicity conditions γ1 ≤ γ2 ⇒ γ1C ≤ γ2C and δ1 ≤ δ2 ⇒ δ1 |P ≤ δ2 |P are obviously fulﬁlled. Thus, γ → γ C and δ → δ|P are mutually inverse isomorphisms between Clo(P ) and CloP (C). For each poset P , the principal ideal mapping a → ↓ a is a join-dense embedding in any standard completion of P , and a meet-dense embedding in the normal completion N P . Thus, using Theorem 3.19 and its dual, we arrive at Corollary 3.20. Every kernel operation on a poset uniquely extends to a kernel operation on each of its standard completions, and every closure operation uniquely extends to a closure operation on the normal completion. Observe that not only the range of any closure operation, but even the lower pre-ﬁxpoint set Pϕ = {a ∈ P : ϕ(a) ≤ a} of any isotone self-map ϕ of a poset P is meet-closed in P (cf. Propositions 3.5 and 3.10): if a is the meet of some B ⊆ Pϕ in P then for all b ∈ B, wehave a ≤ b and therefore ϕ(a) ≤ ϕ(b) ≤ b, whence ϕ(a) ≤ B = a, i.e., a = B ∈ Pϕ . This together with Proposition 3.10 gives Tarski’s Fixpoint Theorem [86]: Theorem 3.21. For any isotone self-map ϕ of a poset P , the lower pre-ﬁxpoint set Pϕ is meet-closed; dually, the upper pre-ﬁxpoint set P ϕ is join-closed in P , and both sets are invariant under ϕ. Hence, the ﬁxpoint set Pϕ= = Pϕ ϕ = P ϕ ϕ is join-closed in Pϕ and meet-closed in P ϕ . If P is complete then Pϕ , P ϕ and Pϕ= are complete lattices, and in particular, ϕ has a least and a greatest ﬁxpoint. For the existence of least ﬁxpoints for isotone self-maps, one needs not the full hypothesis of completeness, but W⊥ -completeness (requiring joins for well-ordered subsets) is enough, as we shall see soon. Recall that Part (4) of Theorem 3.16 provides the “external” description of a closure operation on I(C), the poset of isotone self-maps of C, sending ϕ to ϕ. The “internal” construction of the closure involves a transﬁnite recursion process: replacing ϕ by ϕ ∨ idC if necessary, one may assume that ϕ is not only isotone but also extensive. For any ordinal m, deﬁne inductively the powers ϕm by (+) ϕ0 = idC , ϕm+1 (a) = ϕ(ϕm (a)) and ϕm (a) = {ϕk (a) : k < m} if m is a limit ordinal. The process stops at an ordinal m embeddable in C, so that ϕm+1 = ϕm , and then transﬁnite induction yields ϕm ◦ ϕm = ϕm , i.e., ϕm is a closure operation — in fact, the least one above ϕ. Though the “internal” method has some intuitive appeal, it leans on a rather “external” tool, namely the huge tower of ordinal numbers. But there is a nice substitute for such transﬁnite constructions, which moreover shows that many algebraic properties of ϕ may be transferred to ϕ. The fundamental ingredient is a construction of well-ordered sets inside a given poset P by means of a closure process, extending Dedekind’s original construction of the natural numbers to a transﬁnite setting — however, without invoking any ordinal numbers. The subsequent quite general theorem is due to Bourbaki [12] (for even stronger results,

200

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see [32]). It provides an order-theoretical abstraction of Zermelo’s famous deduction of the Well-Ordering Principle from the Axiom of Choice [94, 95] and its reﬁnement due to Hessenberg [54]. Let ϕ be any self-map of a poset P , and call a subset T of P a ϕ-tower if it is join-closed and invariant under ϕ, i.e., ϕ(T ) ⊆ T . Theorem 3.22. For any extensive self-map ϕ of a poset P , the ϕ-towers form a closure system, and for each a ∈ P , the least ϕ-tower Tϕ (a) containing a is well-ordered by the induced order relation. The ordinal-free proof that Tϕ (a) is well-ordered is rather tricky (cf. [12, 29]). If P is W-complete, i.e., every well-orderd chain in P has a join, then max Tϕ (a) exists and is a ﬁxpoint of the map ϕ provided ϕ is extensive. Thus, we arrive at Lemma 3.23 (Bourbaki’s Fixpoint Lemma). Every extensive map on a nonempty W-complete poset has a ﬁxpoint. The Fixpoint Lemma is trivial in the light of Zorn’s Lemma, but the crucial point is that it has been derived in a choice-free manner, and that it provides in turn an immediate deduction of Zorn’s Lemma from the Axiom of Choice. Corollary 3.24. Let S be a W-union closed set of extensive and isotone maps on a poset with ϕ ◦ ϕ ∈ S whenever ϕ ∈ S. Then ϕ ∈ S implies ϕ ∈ S. Proof. Given ϕ ∈ S, put Sϕ = {ψ ∈ S : ϕ ≤ ψ ≤ ϕ}. The extensive squaring map ψ → ψ ◦ ψ on the W-union closed system Sϕ has a ﬁxpoint, and this is a closure operation above ϕ, hence equal to ϕ. If ϕ is an extensive and isotone homomorphism with respect to some ordered algebraic structure then ϕ is both a closure operation and a homomorphism, because ϕ ◦ ϕ is one, and directed unions of homomorphisms are again such. This applies, for instance, to the theory of nuclei in locales and quantales (see e.g. Johnstone [56] and Rosenthal [79]), a vital branch of the theory of closure operations, with many applications. Let us explain brieﬂy the involved basic notions. A quantale is a complete lattice C endowed with an extra semigroup operation (a, b) → ab that distributes over arbitrary joins, and it is a locale if the multiplication is the binary meet. A closure operation γ on a quantale C is called a nucleus if γ(a)γ(b) ≤ γ(ab), or equivalently, if γ induces a semigroup homomorphism between C and the range Cγ , which becomes a quantale when endowed with the multiplication aγ b = γ(ab). Now, if ϕ is a prenucleus, i.e., an extensive and isotone map satisfying ϕ(a)ϕ(b) ≤ ϕ(ab), then Corollary 3.24 shows that ϕ is a nucleus. Call a poset W⊥ -complete if it is W-complete and has a least element — in other words, if every well-ordered subset has a join. Now, we can prove: Theorem 3.25. (1) A poset P is W⊥ -complete iﬀ every isotone self-map of P has a least ﬁxpoint. (2) A meet-semilattice S is W⊥ -complete iﬀ every isotone self-map of S has a ﬁxpoint. (3) For a lattice L, the following conditions are equivalent: (a) L is complete. (b) L is W⊥ -complete. (c) Every isotone self-map of L has a ﬁxpoint. (d) Every isotone self-map of L has a least ﬁxpoint.

CLOSURE

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(e) The ﬁxpoints of any isotone self-map of L form a complete lattice. Proof. (1). If P is W⊥ -complete and ϕ : P → P is isotone then the set P ϕ = {a ∈ P : a ≤ ϕ(a)} is again W⊥ -complete, and ϕ restricts to an extensive self-map of P ϕ . Hence, the greatest element fϕ of the least ϕ-tower over 0 = ∅ is the least ﬁxpoint of ϕ (indeed, if b is any ﬁxpoint of ϕ then {a ∈ P ϕ : a ≤ b} is a ϕ-tower over 0 in P ϕ , whence fϕ ≤ b). On the other hand, if P has a well-ordered subset W with no least upper bound, then deﬁne ϕ : P → P by ϕ(a) = min{w ∈ W : w ≤ a} if a is not an upper bound of W , and ϕ(a) = a otherwise. Then a distinction of cases shows that ϕ is isotone; but a least ﬁxpoint of ϕ would be the join of W . (2). By (1), it suﬃces to construct, for any well-ordered subset W of a meetsemilattice S such that W has no join, a ﬁxpoint-free isotone map ϕ : S → S. Since the upper bounds of W form a meet-subsemilattice W ≤ , Zorn’s Lemma gives a dually well-ordered subset V in W ≤ with ↑V = W ≤ . Again, put ϕ(a) = min{w ∈ W : w ≤ a} if a ∈ W ≤ , yet ϕ(b) = max{v ∈ V : b ≤ v} if b ∈ W ≤ \ V≤ (where V≤ is the set of lower bounds for V ). These alternatives are exclusive, since any c ∈ W ≤ ∩ V≤ would be the join of W ; and ϕ is easily seen to be isotone and ﬁxpoint-free. (3). The implication (a) ⇒ (e) is Tarski’s Fixpoint Theorem 3.21. For the implication (c) ⇒ (b) apply (2), and for (b) ⇒ (a), note that by the Set Induction Principle 2.4, a W⊥ -complete (join-semi)lattice is already complete. Indeed, the W-union closed collection of all subsets possessing a join contains all ﬁnite subsets. (c) ⇒ (a) (with a more complicated proof) is due to Davis [18]. 3.2. Adjunctions and Galois connections. All closure structures arise from so-called adjoint pairs of maps in opposite directions between ordered sets. Such pairs of maps make it possible to shift certain entities from one side of an inequality to the other — a quite eﬀective tool in many situations [39]. Explicitly, an adjoint pair of maps ϕ /Q P o ψ

between (partially) ordered sets is characterized by the condition a ≤ ψ(b) ⇔ ϕ(a) ≤ b. Equivalently, ϕ and ψ are isotone maps satisfying the inequalities a ≤ ψ(ϕ(a)) and ϕ(ψ(b)) ≤ b. The left or lower adjoint ϕ and the right or upper adjoint ψ determine each other uniquely, so we write ϕ for ψ. Note that the class of (left resp. right) adjoint maps is closed under composition, with ϕ 2 ◦ ϕ 1 and ϕ1 ≤ ϕ2 ⇔ ϕ 2 ≤ ϕ 1 . 1 ◦ ϕ2 = ϕ In what follows, we shall collect together the main deﬁnitions and facts from the theory of adjunctions, with particular emphasis on their relevance to closure structures. No proofs will be given, since they are easy and there exists a comprehensive literature on the subject; e.g., the reader may refer to the book Residuation Theory by Blyth and Janowitz [10], the survey Adjunctions and Galois connections: origins, history and development [39], the Primer on Galois connections [42], and the Compendium of Continuous Lattices [47] (see also [48]).

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Maps having an upper or a lower adjoint are characterized as follows: Proposition 3.26. A map ϕ : P → Q has an upper adjoint iﬀ it is residuated, that is, the preimage under ϕ of any principal ideal in Q is a principal ideal in P . The upper adjoint ψ = ϕ is then given by ψ(b) = max{a ∈ P : ϕ(a) ≤ b}. Dually, a map ψ : Q → P has a lower adjoint iﬀ it is residual, that is, preimages under ψ of principal ﬁlters in P are principal ﬁlters in Q. The lower adjoint ϕ is then given by ϕ(a) = min{b ∈ Q : a ≤ ψ(b)}. Any residuated map between posets preserves all existing joins, and any residual map all existing meets. Conversely, any join- (resp. meet-) preserving map between complete lattices is residuated (resp. residual). It is also easy to see that a lower adjoint is injective iﬀ its upper adjoint is surjective, and dually; moreover, that the order isomorphisms are exactly those bijective maps which have both an upper and a lower adjoint, which then coincide with the inverse of the original map. Any map ϕ : P → Q naturally factorizes into a surjection ϕo : P → ϕ(P ) (obtained by restriction of the codomain to the range) and the inclusion map ϕo from ϕ(P ) into Q. If γ is a closure operation on P then γ o is not only the restriction of γ to Pγ = γ(P ) but also the upper adjoint of γo , on account of the equivalence a ≤ γ o (b) = b ⇔ γ(a) = γo (a) ≤ b for a ∈ P and b ∈ Pγ . Hence, closure operations are in one-to-one correspondence with (lower adjoints of) residual inclusion maps between ordered sets. On the other hand, any adjunction (ϕ, ψ) gives rise to a closure operation ψ ◦ ϕ and to a kernel operation ϕ ◦ ψ, by virtue of the inequalities and equalities idP ≤ ψ ◦ ϕ,

ϕ ◦ ψ ≤ idQ ,

ϕ ◦ ψ ◦ ϕ = ϕ,

ψ ◦ ϕ ◦ ψ = ψ.

Corollary 3.27. For a map γ : P → P , the following are equivalent: (a) (b) (c) (d)

γ is a closure operation. (γo , γ o ) is an adjoint pair with γ o ◦ γo = γ and γo ◦ γ o = idPγ . There is an adjoint pair (ϕ, ψ) with ψ ◦ ϕ = γ and ϕ ◦ ψ = idPγ . There is an adjoint pair (ϕ, ψ) with ψ ◦ ϕ = γ.

Moreover, the pairs (ϕ, ψ) in (c) are unique up to an automorphism of Pγ ; indeed, (c) holds if and only if γ is a closure operation with ϕ = α ◦ γo and ψ = γ o ◦ α−1 for some automorphism α of Pγ . We summarize the main facts about decompositions of adjunctions: Theorem 3.28. For any adjunction (ϕ, ψ) between posets P and Q, the range ϕ(P ) is also the range of the kernel operation κ = ϕ ◦ ψ, and ψ(Q) is also the range of the closure operation γ = ψ ◦ ϕ. The maps ϕ and ψ induce mutually inverse isomorphisms ϕoo and ψoo between ψ(Q) and ϕ(P ). The adjunction is determined by these maps through the equations ϕ = ϕoo ◦ γ and ψ = ψoo ◦ κ. Hence, there is a one-to-one correspondence between adjoint pairs and isomorphisms between kernel ranges and closure ranges.

203

CLOSURE

P ? γ ?? ?? ψ ϕ ?? −1 ?? α /P P? γ ? ?? ?? ? γo ?? γo Pγ

P ? O γ ?? ?? ψ ϕ ?? ?? /P ?? γ ? ?? α ?? γo ?? γo Pγ

Figure 11. Decomposition of closure maps ϕ /Q P oE E y BBB ψ | E y | EE y BB | E yy BB κ=ϕ◦ψ || γ=ψ◦ϕ || ϕo EE yy ψo BB yEE γo κo | y BB | y E | y E BB | y EE | y BB | y E | o y | " o ~| ϕo γ κo / o /Q ϕ(P ) ψ(Q) o P o ⊇ ⊆ ψo

Figure 12. Adjunction, closure and kernel map An elementary but nevertheless surprising consequence is that, up to composition with isomorphisms, all surjective residuated (i.e., lower adjoint) maps may be regarded as corestricted closure operations. More precisely, one has the following “Homomorphism Theorem for Residuated Maps”: Theorem 3.29. For an arbitrary residuated map ϕ : P → Q, the map γ = ϕ◦ϕ is a closure operation, κ = ϕ ◦ ϕ is a kernel operation, and ϕ factorizes through a residuated surjection γo : P → ϕ(Q) = γ(P ), a → γ(a), followed by an order isomorphism ϕoo : ϕ(Q) → ϕ(P ), a → ϕ(a), followed by a residuated inclusion map κo : ϕ(P ) → Q, a → a. In particular, the surjective residuated maps are in fact just the corestricted closure operations followed by isomorphisms, and the injective residuated maps are the isomorphisms followed by inclusions of kernel ranges. The following equivalences are also easily veriﬁed: Proposition 3.30. For any adjoint pair (ϕ, ψ) of self-maps of a poset, ϕ is a projection ⇔ ψ is a projection ⇔ ϕ is a kernel map ⇔ ψ is a closure map ⇔ ϕ is a closure map ⇔ ψ is a kernel map ⇔

ϕ◦ϕ=ϕ ϕ◦ψ =ϕ ϕ◦ψ =ψ

⇔ ψ ◦ ψ = ψ, ⇔ ψ ◦ ϕ = ψ, ⇔ ψ ◦ ϕ = ϕ.

Turning upside down the righthand domain in an adjunction, one obtains a Galois connexion (Ore [73]) or Galois connection (caution: some authors use that name as a synonym for “adjunction”). Thus, for us, a Galois connection is a pair of maps ϕ : P → Q and ψ : Q → P so that a ≤ ψ(b) ⇔ b ≤ ϕ(a).

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Equivalently, one may postulate that ϕ and ψ are antitone (that is, order reversing) maps and both composites, ϕ ◦ ψ and ψ ◦ ϕ, are closure operations. Corollary 3.31. The closure ranges of a Galois connection are dually isomorphic, and on the other hand, every dual isomorphism between closure ranges is induced by exactly one Galois connection between the domains of the corresponding closure operations. This fundamental result was established, for the complete case, by Ore [73] and, for the general case, by Schmidt [80]. The main problem when dealing with concretely given Galois connections is to determine, in a convenient fashion, the socalled Galois-closed objects, i.e., the elements in the ranges of the two involved maps — or, stated otherwise, the ﬁxpoints of the resulting two closure operations. An important example, hidden behind Theorem 3.16, is the Galois connection between subsets and isotone self-maps of a complete (join-semi)lattice C: there, we have B ⊆ Cϕ ⇔ ϕ ≤ γB , the Galois-closed subsets are precisely the closure ranges (the meet-closed subsets), and the Galois-closed self-maps are the closure operations. Practical applications proﬁt from the fact that all Galois connections between power sets, alias polarities (Birkhoﬀ [9]), arise in a unique way from relations between the underlying sets (Everett [44]). Theorem 3.32. Every relation R between two sets X and Y gives rise to a Galois connection between the power sets of X and Y , by mapping A ⊆ X to the set AR = {y ∈ Y : x R y for all x ∈ A}, and B ⊆ Y to the set BR = {x ∈ X : x R y for all y ∈ B}. The ranges of these two maps are dually isomorphic closure systems. Conversely, any Galois connection between power sets and any dual isomorphism ϕ between two closure systems originates in that way from a unique relation R between the underlying sets, where x R y means that y belongs to the image of the closure of {x} under ϕ. Indeed, if ϕ is a dual isomorphism between closure systems C on X and D on Y then the relation R = {(x, y) : y ∈ ϕ({x})} satisﬁes AR = {ϕ({x}) : x ∈ A} = ϕ( C {{x} : x ∈ A}) = ϕ(A) for A ∈ C, and dually BR = ϕ−1 (B) for B ∈ D. These equations show that C and D are the ranges of the polarity induced by R. Concerning uniqueness, note that if ϕ is induced by R then x R y ⇔ y ∈ {x}R = {x}R R R = ϕ({x}). In case X = Y is the same set, the semigroup of all relations on X is isomorphic to the semigroup of all polarities of PX. Let us consider a few interesting examples of Galois connections occurring in topology, algebra, geometry and order theory. Looking “beyond topology”, we shall see that many fundamental dual isomorphisms between certain closure systems arise from rather elementary relations. Example 3.33 (Dualities between standard completions). If X is quasi-ordered by a relation ≤, the Galois-closed sets are the cuts B≤ , which form the Dedekind– MacNeille completion. This closure system is dually isomorphic to that of all dual

CLOSURE

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cuts A≤ . On the other hand, taking for R the complement ≤ of the order relation, one obtains a polarity that induces a dual isomorphism (via complementation) between the upper and the lower Alexandroﬀ topology of (X, ≤). More generally, if ϕ is a dual isomorphism between a standard completion C of (X, ≤) and a standard completion D of the dual (X, ≥), then the unique relation R having C and D as closure ranges of the induced polarity is given by x R y ⇔ y ∈ ϕ(↓ x). Example 3.34 (Zero sets and varieties). If Y is a set of functions from the real euclidean space X = Rn (or another space) to R, then the relation x R f ⇔ f (x) = 0 leads to a Galois connection such that the Galois-closed subsets of X are the intersections of zero sets f ({0}), f ∈ Y . If Y is closed under pointwise multiplication then the closure system of all Galois-closed subsets is topological, on account of the identity AR ∪ B R = (AB)R for AB = {f g : f ∈ A, g ∈ B}. Important topological examples are obtained if Y is the ring of all continuous functions; in that case, the topologically closed sets are exactly the Galois-closed sets if and only if the space is completely regular. Prominent algebraic examples are obtained if Y is the ring of all polynomials in one or several variables. This leads to the polarity between varieties (algebraic zero sets) and ideals in polynomial rings, a basic tool in algebraic geometry. Here, the complements of the zero sets form the so-called Zariski topologies. Example 3.35 (Points and copoints in geometry). In synthetic geometry, based on the idea of incidence relations, Galois connections and closure systems of “ﬂats”, obtained by intersecting “hyperplanes”, play a crucial role. More generally, consider an algebraic closure system C on a set X and call its completely meet-irreducible members copoints (cf. [55]). Then, for the polarity associated with the incidence relation between points and copoints, one closure range is C itself, whereas the other, dual closure system consists of the principal ﬁlters of C. For this conclusion, one needs the full strength of the Axiom of Choice in form of Zorn’s Lemma, guaranteeing enough copoints that generate all closed sets via intersection. This is the famous Birkhoﬀ–Frink Theorem [9, 50] with many applications, not only in geometry but also in algebra and topology. Example 3.36 (Quasi-orders and Alexandroﬀ topologies). For a ﬁxed set X, deﬁne a relation R between PX and X ×X by A R (y, x) ⇔ (x ∈ A or y ∈ A). The Galois-closed subsets of PX with respect to the associated polarity are precisely the Alexandroﬀ topologies on X, while the Galois-closed subsets of X × X are precisely the quasi-orders on X. Thus, that polarity induces a dual isomorphism between the closure system of all quasi-orders and that of all Alexandroﬀ topologies on X, as described in Proposition 2.2. Example 3.37 (Convergence relations and topologies). Let FX denote the set of all ﬁlters on X, and consider the relation R between PX and FX × X with A R (F, x) ⇔ (x ∈ A or A ∈ F). The Galois-closed subsets of PX are now all topologies, while the Galois-closed subsets of FX × X are the topological convergence relations. Thus, we arrive, by a very elementary appoach, at the equivalent description of topological spaces by open sets and by convergence structures. The previous example of quasi-orders and Alexandroﬀ topologies may be regarded as the special case obtained by restriction to ﬁxed ultraﬁlters F = y˙ = {A ⊆ X : y ∈ A}.

206

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Example 3.38 (Stacks, ﬁlters and grills). For the symmetric relation # on power sets, given by A # B ⇔ A ∩ B = ∅, the associated polarity induces a selfinverse dual automorphism of the Alexandroﬀ topology SX of all stacks, i.e., upsets of the power set lattice PX. That dual automorphism sends any stack S to S # = {B ⊆ X : ∀A ∈ S (A # B)} = PX \ {X \ A : A ∈ S}. Moreover, this polarity induces a dual isomorphism between the closure system of ﬁlters and the kernel system of grills, i.e., complements of set ideals; the ﬁxpoints of that dual isomorphism are exactly the ultraﬁlters. Example 3.39 (Regular open sets). Now, take the complementary relation // (with A // B ⇔ A ∩ B = ∅) on PX, or more generally, on any topology T ⊆ PX. Here, the Galois-closed subsets of T turn out to be the principal ideals {U ∈ T : U ⊆ V } generated by regular open sets V (i.e., sets that coincide with the interior of their closure). Thus, the lattice of Galois-closed sets is isomorphic to the complete boolean lattice of all regular open sets. This construction extends from topologies to arbitrary pseudocomplemented semilattices S, in which for each a there is a greatest b with a ∧ b = 0, called the pseudocomplement of a and denoted by a∗ (cf. Birkhoﬀ [9], Frink [45], Glivenko [49]). In fact, the pair (∗ ,∗ ) is a Galois connection whose range S ∗ = S ∗∗ is a Boolean lattice with a ∨ b = (a∗ ∧ b∗ )∗ : it is meet-closed in S and self-dual under pseudocomplementation, has the least element 0 = 0∗∗ and the greatest element 1 = 0∗ , is complemented since a ∧ a∗ = 0 and a∨a∗ = (a∗ ∧a∗∗ )∗ = 0∗ = 1, and is distributive because the unary meet operations x → a ∧ x preserve arbitrary joins, being residuated: a ∧ b ≤ c∗∗ ⇔ c∗ ∧ a ∧ b = 0 ⇔ b ≤ (a ∧ c∗ )∗ . Example 3.40 (Closed subspaces of Hilbert spaces). The orthogonality relation ⊥ on a Hilbert space, deﬁned by x ⊥ y if the scalar product x, y = 0 vanishes, gives rise to the notion of (self-)adjoint linear operators, but also to an ordertheoretical dually self-adjoint map, sending each subset A to its orthogonal space A⊥ and inducing a dual automorphism on the lattice of all closed subspaces. Note that in ﬁnite-dimensional spaces, all linear subspaces are closed. The last three examples indicate the relevance of polarities generated by symmetric relations. Let us summarize the main properties of such “symmetric polarities” (all claims are straightforward): Proposition 3.41. Passing from relations to polarities, one obtains an isomorphism between the power set lattice R(X) = P(X × X) of all relations on X and the atomic boolean complete lattice of all dual isomorphisms between closure systems on X. Under that bijection, (1) the symmetric relations correspond to the involutions (self-inverse dual automorphisms) of closure systems; (2) the irreﬂexive symmetric relations (graphs) correspond to the strict orthocomplementations of closure systems C (i.e., involutions ⊥ satisfying A ∩ A⊥ = ∅ for all A ∈ C); (3) the point-separating symmetric relations S (i.e., those satisfying the implication {x}S ⊆ {y}S ⇒ x = y) correspond to the involutions of T1 -closure systems (in which singletons and ∅ are closed).

CLOSURE

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Generally, an involution of a poset P is a self-inverse order-reversing bijection on P , and an orthocomplementation of a bounded lattice or poset P is an involution ⊥ such that a ∧ a⊥ = 0 and consequently also a ∨ a⊥ = 1 (= 0⊥ ) for all a ∈ P . It is a challenging exercise to verify that an involution with a⊥ ≤ a for a = 1 is already an orthocomplementation (and conversely). In the theoretical foundations of quantum physics (see, e.g., Aerts et al. [3], Moore [65, 66]) so-called state systems play an important role. These are sets equipped with an irreﬂexive, symmetric and point-separating relation. Using the previous remarks in combination with Theorem 2.24, one concludes (cf. [65], where the morphisms are slightly diﬀerent): Corollary 3.42. The state systems form a category that is concretely isomorphic to the category of T1 -closure spaces with strict orthocomplementation (both categories with continuous maps as morphisms). On the other hand, the latter category is equivalent, under the functorial equivalence between T1 -closure spaces and atomistic complete lattices, to the category of atomistic complete lattices with orthocomplementation and join-preserving maps sending atoms to atoms. Composing polarities with involutions of power sets via complementation leads to so-called axialities, i.e., adjunctions between power sets; though equally natural as polarities, axialities seem to be less familiar (for some instructive examples, see [39, 42]). Given a relation R ⊆ X × Y , put AR = {y ∈ Y : ∃x ∈ X(x R y and x ∈ A)} (A ⊆ X), RB = {x ∈ X : ∃y ∈ Y (x R y and y ∈ B)} (B ⊆ Y ), ⇒

R B = {x ∈ X : ∀y ∈ Y (x R y ⇒ y ∈ B)} (B ⊆ Y ), and note the identity X \ R⇒ B = R(Y \ B). Corollary 3.43. The axiality associated with a relation R ⊆ X × Y is constituted by the following two adjoint maps: ΘR : PX → PY, A → AR,

ΘR : PY → PX, B → R⇒ B.

It induces a kernel operator ΘR ◦ΘR on X, a closure operator ΘR ◦ΘR on Y , and two mutually inverse isomorphisms between the closure system C(R) = {R⇒ B : B ⊆ Y } and the kernel system O(R) = {AR : A ⊆ X}. The open sets with respect to C(R) are exactly the members of the kernel system O(Rd ) of the dual relation Rd . Every axiality between PX and PY , and consequently every isomorphism ϕ between a closure system C on X and a kernel system K on Y , comes from a unique relation R, which is given by x R y ⇔ y ∈ ϕ({x}). For any relation R on one and the same set X, the system TR = {A ⊆ X : AR ⊆ A} is an Alexandroﬀ topology (closed under arbitrary unions and intersections). Indeed, TR is just the collection of all upsets with respect to the least quasi-order = {Rn : n ∈ ω}. containing R, the reﬂexive-transitive closure R A complete lattice is supercontinuous iﬀ it is the image of an Alexandroﬀ topology under a join- and meet-preserving map (see the next section). Recall that a relation R is idempotent iﬀ it is transitive and interpolative: x R z ⇔ ∃z(x R y R z).

208

´ M. ERNE

Now, a series of straightforward veriﬁcations (using 2.2, 3.30 and 3.43) yields Theorem 3.44. For any relation R on a set X, the maps ΘR and ΘR induce an adjoint pair of self-maps of the supercontinuous lattice TR . The following conditions are equivalent: (a) R is idempotent. (b) ΘR is idempotent, hence a projection. (c) ΘR is idempotent, hence a projection. (d) ΘR induces a kernel operation on TR . (e) ΘR induces a closure operation on TR . (f ) O(R) is the range of ΘR . (g) C(R) is the range of ΘR . (h) O(R) = {A ⊆ X : AR = A} and C(R) = {B ⊆ X : R⇒ B = B} are isomorphic supercontinuous lattices. 3.3. Join- and meet-closure in complete lattices. We are now going to collect together a series of order-theoretical results that include, when applied to the “classical” case of power set lattices, almost all of the facts about set-theoretical closure operators mentioned earlier. A part of these results may be found, for the special subset selection Z = D, in the Compendium of Continuous Lattices [47] or in its updated version, Continuous Lattices and Domains [48]. Other results are contained in the literature on subset selections (see, e.g., [7, 34, 37, 89]), but we present here some simpliﬁed or uniﬁed proofs and include several new results. Henceforth, let Z be a functorial subset selection (see Section 2.2) and C a complete lattice with order relation ≤ (generalizations to posets and to more general subset selections are possible but would require more technicalities; cf. the above references). Let A be a subset of C. Generalizing the operators 0Z and ΩZ from Section 2.2, we put VZ A = A ∪ { Z : Z ∈ Z(A, ≤|A )} ΛZ A = A ∪ { Z : Z ∈ Z(A, ≥|A )} and say A is Z- -closed or Z- -closed, respectively, in C, according towhether VZ A =A or ΛZ A = A. In particular, A-closed means join-closed ( -closed ) and A- -closed means meet-closed ( -closed ). We shall write V for VA and Λ for ΛA (where AC = {A : A ⊆ C}). Thus, by Theorem 3.16, Λ A is the range of the closure operator γA . Sometimes, a slightly stronger kind of closedness is needed: A is said to be strongly Z- -closed in C provided that Z ∈ ZC ∩ PA implies Z ∈ A. Of course, for absolute subset selections, satisfying Z(A, ≤|A ) = ZC ∩ PA for all A ⊆ C, both notions of Z- -closedness agree. Each of the subset selections introduced earlier (A, B, . . . , F, Am , Dm ) is absolute, but there exist union complete functorial subset selections that fail to be absolute, for example, the selection of all upper bounded subsets. Whereas the strongly Z- -closed sets form a closure system for arbitrary subsetselections Z, functoriality is needed to ensure that ar bitrary intersections of Z- -closed sets are Z- -closed. As in the set-theoretical case, we shall try to determine, by simple internal constructions, the Z-subalgebra AZ generated by a subset A, that is, the smallest and Z-closed subset of C containing A. Since arbitrary intersections of - and Z- -closed subsets are again such, we have in fact a closure operator Z on the ground set of C.

CLOSURE

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A few cases are of particular interest: the E-subalgebras are just the -closed subsets, while the A-subalgebras are closed under arbitrary joins and meets, the Dsubalgebras under directed joins and arbitrary meets, and the F-subalgebras under ﬁnite joins and arbitrary meets. Mixed cases, where one postulates closedness under Z-joins and under Z -meets for some Z , are more complicated and will not be discussed here. A map ϕ : C → D between complete lattices preserves Z-joins if ϕ( Z) = ϕ(Z) for all Z ∈ ZC. Recall that the least closure map above an isotone self-map ϕ is given by ϕ(a) = {c ∈ C : a ∨ ϕ(c) ≤ c}. Now, the order-theoretical version of Proposition 2.6 reads as follows: Theorem 3.45. Let C be a complete lattice. (1) If ϕ is an isotone and Z-join preserving self-map of C then the set Cϕ = {c ∈ C : ϕ(c) ≤ c} is a strongly Z- -closed and -closed subset of C, hence a Z-subalgebra whose associated closure operation ϕ is the least one above ϕ. (2) If B is Z- -closed in C then the corresponding closure operation γB : C → C, a → {b ∈ B : a ≤ b} preserves Z-joins. (3) Assigning to each closure operation its range, one obtains a dual isomorphism between the pointwise ordered set of all Z-join preserving closure operations and the closure system of all Z-subalgebras of C. Proof. (1). Cϕ is -closed since A ⊆ Cϕ entails ϕ( A) ≤ ϕ(A) ≤ A. Cϕ is stronglyZ- -closed since for Z ∈ ZC ∩ PCϕ , one obtains ϕ(Z) ⊆ ↓ Z and so ϕ( Z) = ϕ(Z) ≤ Z, i.e., Z ∈ Cϕ . That ϕ = γCϕ is the least closure operation above ϕ has been stated in Theorem 3.16 (4). (2). Again by Theorem 3.16, γB is a closure operation and B is its range. For ∧ Z ∈ γB isisotone and Z is functorial), hence ∈ Z B (because ZC, we get ↓ γB (Z) γB (Z) ∈ B and γB ( Z) = γB ( γB (Z)) = γB (Z). (3) is now an immediate consequence of (1), (2) and Proposition 3.5. As the proofshows, fortunately we need not distinguish between Z- -closed and strongly Z- -closed closure ranges. Given a closure operation γ on C, let us denote by γ Z the closure operation associated with the Z-subalgebra Cγ Z , regarded as a closure range. For A ⊆ C, the closure operation γA Z maps a ∈ C to the least element of the Z-subalgebra AZ above a (since AZ = Λ AZ and Λ A = CγA ). Corollary 3.46. The Z-subalgebras of any complete lattice C form a closure system, hence a closure range ClZ (C) in Cl(C), and the associated closure operator sends A to AZ , the least Z-subalgebra containing A. Dually, the Z-join preserving closure operations on C form a kernel range CloZ (C) in Clo(C), and the associated kernel operation sends γ to γ Z . Hence, γ Z is the greatest Z-join preserving closure operation below γ.

210

´ M. ERNE

Obviously each of the operators VZ and ΛZ , and consequently each composite of such operators like VZ Λ etc. is extensive and preserves inclusion. It is now desirable to ﬁnd conditions under which these operators are also idempotent, hence closure operators — in other words, conditions so that the Z-subalgebra generated by a subset of C is obtained already in a one- or two-step procedure by application of VZ and Λ. The ﬁrst case to be settled concerns the functorial absolute subset selection Z = Am picking all subsets of cardinality less than m; recall that Am is union complete if and only if m is a regular cardinal or the symbol ∞, in which case A∞ C stands for AC, the collection of all subsets. Proposition 3.47. If Z = Am for a regular cardinal m or m = Z∞, then for Z each subset A of C the set V A is the least Z-closed subset and Λ A the least Z- -closed subset of C containing A. Thus, VZ and ΛZ are closure operators. Proof. Let B ∈ Z VZ A. Each b ∈ B ⊆ VZ A satisﬁes b = Zb for some Zb ∈ ZA. Then, by regularity of m, the union Z = {Zb : b ∈ B} also belongs to ZA, and it follows that B = Z ∈ VZ A. Clearly, any Z- -closed subset containing A must also contain VZ A. By duality, analogous results hold for ΛZ . Note that Proposition 3.47 becomes false for singular cardinals, but also for quite natural subset selections like Z = D (see Examples 2.11 and 2.13). The two-step characterization of the closure operators that generate topologies, topological closure systems and Alexandroﬀ topologies, respectively, given in Proposition 2.7, may be extended to the lattice-theoretical setting as follows: call C an m-frame if it enjoys, for all Y ∈ Am AC (i.e., collections Y of less than m subsets of C), the distributive law (D) { A : A ∈ Y} = { χ (Y) : χ ∈ A∈Y A}. The ω-frames are merely the frames or locales [56], which may be characterized alternatively (using an easy induction) by the simpler identity a ∧ B = {a ∧ b : b ∈ B}. (D∧ ) On the other extreme, the ∞-frames are the completely distributive lattices (deﬁned via choice functions; see, e.g., [47] and [74]). Proposition 3.48. Let m be a regular cardinal or ∞, and put Z = Am . For any subset A of an m-frame C, the least -closed and Z- -closed subset containing A is VΛZ A, and this is an m-frame with the induced order. Hence, VΛZ is a closure operator on (the underlying set of ) any m-frame. Analogously, on any dual m-frame, ΛVZ is a closure operator generating the Z-subalgebras AZ , and these are again dual m-frames. Proof. We prove the second part; the ﬁrst part is then obtained by duality. Let C bea dual m-frame, A some subset of C, and B=V Z A. By Proposition 3.47, B is Z- -closed. If A ⊆ D ⊆ C and D is Z- - and -closed then B ⊆ D and Λ B ⊆ D. The inclusion A ⊆ B⊆ ΛB is also obvious, and ΛB is -closed; so it suﬃces to show that Λ B is Z- -closed. LetZ ∈ ZΛ B and a = Z. For each c ∈ Z ⊆ Λ B and Bc = B ∩ ↑c, we have c = Bc . As C is a dual m-frame and Z has less than m elements, we compute a = { Bc : c ∈ Z} = { χ(Z) : χ ∈ c∈Z Bc } ∈ Λ B.

CLOSURE

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Since arbitrary joins and Z-meets in VΛZ A coincide with those formed in C, it is easy to see that VΛZ A is a dual m-frame, too. The special case m = ∞ amounts to

Corollary 3.49. For any completely distributive lattice C, the least - and closed subset containing a subset A is ΛV A = VΛ A, and this is again a completely distributive lattice. By induction, a lattice L is distributive (i.e., any three elements a, b, c in L enjoy the identity a∧(b∨c) = (a∧b)∨(a∧c)) iﬀ L satisﬁes (D∧ ) for all ﬁnite subsets B; and similar thoughts as before show that for any distributive lattice L with top 1 and bottom 0 (but not necessarily complete) the least sublattice containing 0, 1 and a subset A is ΛF VF A = VF ΛF A. 3.4. Z-distributivity and Z-continuity. A ﬂexible distribution property is the following (cf. [8, 37]): a complete lattice C is said to be Z-distributive (in [35]: completely Z-distributive) if { A : A ∈ Y} = Y for all Y ⊆ Z ∧ C, where Z ∧ C is the system of all Z-downsets (i.e., downsets generated by members of ZC or by singletons; see Section 2.3). Using the cross operator # from Example 3.38 with Y # = {B ⊆ X : A ∩ B = ∅ for all A ∈ Y}, the Z-distributive law may be restated in a more symmetric form: { A : A ∈ Y} = { B : B ∈ Y # } for all Y ⊆ ZC ∪ EC. In the presence of enough choice functions, Z-distributivity is equivalent to the identity (D) for all subsets Y of ZC ∪ EC. But more comfortable for practical use is the following choice-free description of Z-distributivity in terms of the so-called Z-below sets ⇓Z a = ⇓Z {B ∈ Z ∧ C : a ≤ B} : C a= Lemma 3.50. A complete lattice C is Z-distributive iﬀ a = ⇓Z a (a ∈ C). See [8, 34, 35, 37] for this and many other results around the notion of Zdistributivity, and for a broad spectrum of examples. Recall that Dm denotes the subset selection of all m-directed sets, i.e., subsets D such that each A ∈ Am D has an upper bound in D. In the area between order, topology and closure, (m-)directed sets are a basic tool. While for m = ω the Dm -distributive lattices are precisely the continuous lattices in the sense of Scott [47, 48], for m ≤ 2 one obtains the supercontinuous lattices (alias constructively completely distributive lattices, cf. Rosebrugh and Wood [78]). Whereas complete distributivity of supercontinuous lattices and even of power sets is equivalent to the Axiom of Choice, no choice principles are needed in order to show that any power set and, more generally, any Alexandroﬀ topology is supercontinuous, hence Z-distributive for arbitrary subset selections Z. By deﬁnition, C is supercontinuous iﬀ the join map : A∧ C → C is a complete homomorphism (preserving arbitrary joins and meets); and on the other hand, it is rather obvious that the image of a supercontinuous lattice under a complete homomorphism is again supercontinuous. Similar results hold for Z-distributive lattices (see [37]).

212

´ M. ERNE

The notion of continuous and supercontinuous lattices is generalized to that of Z-continuous lattices and posets by replacing the selection D = Dω of directed subsets with an arbitrary subset selection Z: a poset P is Z-continuous if it is Z-complete (each Z ∈ ZP has a join) and each of the Z-below sets ⇓Z a = {B ∈ Z ∧ P : a ≤ B} is a Z-downset with join a [7, 37, 68, 89]. Equivalently, P is Z-continuous iﬀ the principal ideal map η Z : P → Z ∧ P , a → ↓ a has a lower adjoint (the join map : Z ∧ P → P ) which in turn has a lower adjoint (the Z-below map λ : P → Z ∧ P , a → ⇓Z a). By deﬁnition, every Z-continuous complete lattice is Z-distributive, while the converse often fails, even for seemingly harmless selections like F, the selection of all ﬁnite subsets: every power set is trivially F-distributive (being supercontinuous), whereas only the ﬁnite power sets are F-continuous (if ⇓F A = EA is ﬁnitely generated the A must be ﬁnite). However, for the selections Z = Dm of m-directed subsets, Z-continuity and Z-distributivity are equivalent for complete lattices: indeed, if Z is any subset selection and the Z-downsets of a Z-distributive lattice C form a closure system then C is already Z-continuous. This is always fulﬁlled for Z = Dm , because in that case, a downset is m-directed iﬀ it is Am - -closed. The Z-join ideals of a poset P , introduced in Section 2.3, are just the strongly Z- -closed downsets; they form a standard completion Z ∨ P . Let us recall some typical classes of Z-join ideals: • A-join ideals = principal ideals; • B-join ideals = F-join ideals = ∨-ideals (Example 2.20); • C-join ideals = D-join ideals = Scott-closed sets (Example 2.22); • E-join ideals = downsets. In order to ﬁnd an internal construction of Z-join ideals, deﬁne a modiﬁed cut Z operator ∆Z P = ∆ : AP → AP by ∆Z (A) = {∆P (Z) : Z ∈ ZP ∩ P(↓ A)} which for Z-complete posets (and so for complete lattices) amounts to ∆Z (A) = ↓{ Z : Z ∈ ZP ∩ P(↓ A)}. By deﬁnition, a Z-join ideal is just a ﬁxpoint of the operator ∆Z . But, for arbitrary subsets A, the “one-step closure” ∆Z (A) need not yet give a Z-join ideal; hence, ∆Z might fail to be a closure operator in our strict sense. For absolute subset selections Z, the earlier introduced operator VZ is related to ∆Z by the identity ∆Z = ↓ VZ ↓. The notion of Z-below sets for elements naturally extends to arbitrary subsets: for each subset A, its Z-below set is Z Z ∇Z {⇓ a : a ∈ A}. P (A) = ∇ (A) = Both ∆Z (A) and ∇Z (A) are always downsets. Henceforth, we assume that Z is a union complete functorial subset selection. This assumption has the useful consequence that any Z-continuous poset P is already strongly Z-continuous, i.e., its Z-below relation Z P deﬁned by Z c Z P a ⇔ c ∈ ⇓P a

CLOSURE

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is idempotent (interpolation property; see, e.g., [7] or [89]). All of the subsequent results remain valid for arbitrary functorial (not necessarily union complete) subset selections if “Z-continuous” is replaced by “strongly Z-continuous”. Under the above assumption, we may invoke the axiality (ΘR , ΘR ) resulting from the idempotent Z-below relation R = Z P (see Corollary 3.43 and Theorem 3.44) and obtain (cf. [8] and [37]): Theorem 3.51. For any Z-continuous poset P , the restrictions of the operators ∇Z and ∆Z to the downset lattice A∧ P form an adjoint pair. ∇Z induces a kernel operation on A∧ P , while ∆Z is a closure operator. Its range, the closure system of Z-join ideals, is isomorphic to the kernel system of Z-below sets. Both systems are supercontinuous lattices, being images of the Alexandroﬀ topology A∧ P under complete homomorphisms. Proof. We have already remarked that ∆Z is extensive, and as ⇓Z a is contained in ↓ a (use E ∧ P ⊆ Z ∧ P ), the map ∇Z is contractive on A∧ P . In view of Corollary 3.43 and Theorem 3.44, the only detail to be veriﬁed is that for the idem∧ R ⇒ potent relation R = Z P and each downset B ∈ A P , the set Θ (B) = R B = Z Z Z {a ∈ P : Ra = ⇓ a ⊆ B} coincides with ∆ (B). But a ∈ ∆ (B) means a ≤ Z for some Z ∈ ZP ∩ PB, which is equivalent to ⇓Z a ⊆ B, since ⇓Z a belongs to Z ∧ P and has join a. Example 3.52 (The upper topology on the reals). Its closed sets are the Djoin ideals, hence the Scott-closed sets. They form a complete chain (hence a supercontinuous lattice) that is isomorphic to the extended real line R ∪ {−∞, ∞} but also to the lower topology (the dual Scott topology) on the reals — which consists of all D-below sets! By the previous reasoning, the interpolation property ensures that the Z-join ideal closure is obtained already in the ﬁrst step: Corollary 3.53. For any subset A of a Z-continuous poset, the Z-join ideal generated by A is ∆Z (A), and it is equal to ↓ V Z ↓ A if Z is absolute. For the special case Z = D, the last corollary says that in a continuous lattice or poset, the Scott closure of a subset A is obtained simply by taking the downset generated by all directed joins in ↓ A — in other words, continuous lattices have “one-step Scott closure”. It is a nontrivial fact that for spatial frames with countable bases (join-dense subsets), the converse holds as well. As a topological consequence of that lattice-theoretical result, a second countable Hausdorﬀ (or at least sober) space is locally compact iﬀ its lattice of open sets has one-step Scott closure [41]. Notice that in contrast to the operator ↓ VD ↓, the operators VD and ↓ VD need not be idempotent on (D-)continuous lattices (and not even on power set lattices; see Example 2.13). For Z = A and generally for Z = Am with regular m, the situation is much easier: here, the operator ∆Z is always idempotent, on account of Proposition 3.47. In Corollary 3.46, we have considered the greatest Z-join preserving closure operation γ Z dominated by a given closure operation γ. In Z-continuous lattices, the external description of γ Z via intersection of closure ranges may be replaced

214

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by an internal construction “from below”, generalizing the construction of the inductive modiﬁcation of closure operations on power sets, where “D-below” means “ﬁnite subset” (cf. Theorem 2.3). Theorem 3.54. Let C be a Z-continuous complete lattice. Then, for any closure operation γ on C, the greatest Z-join preserving closure operation below γ is given by γ Z (a) = γ(⇓Z a) and its range is VZ Cγ . Hence, VZ Λ A is the Z-algebra generated by A ⊆ C, and VZ Λ is a closure operator. Proof. Put γˇ (a) = γ(⇓Z a). The equation γˇ = γ Z will follow at the end. (1) γˇ (a) ≤ γ(a) since ⇓Z a ⊆ ↓ a and γ is isotone, so that γ(a) = γ(↓ a). (2) Extensivity of γ entails extensivity of γˇ : b ≤ γ(b) for all b Z a implies a = ⇓Z a ≤ γ(⇓Z a) = γˇ (a). (3) γˇ is a closure operation because γ is one: It remains to be shown that γˇ (ˇ γ (a)) ≤ γˇ (a). As γ is isotone and ⇓Z a ∈ Z ∧ C, Z ∧ we have ↓ γ(⇓ a) ∈ Z C, and γˇ (a) = γ(⇓Z a) yields ⇓Z γˇ (a) ⊆ ↓ γ(⇓Z a). Since γ is a closure operation, now conclude that γ(⇓Z γˇ (a)) ⊆ ↓ γ(⇓Z a) and we Z Z γˇ (ˇ γ (a)) = γ(⇓ γˇ (a)) ≤ γ(⇓ a) = γˇ (a). (4) γˇ preserves Z-joins: By interpolation, Z ∈ ZC implies ⇓Z Z = {⇓Z a : a ∈ Z}, hence γˇ ( Z) = γ( {⇓Z a : a ∈ Z}) = { γ(⇓Z a) : a ∈ Z} = γˇ (Z). (5) If δ : C → C preserves Z-joins and satisﬁes δ ≤ γ then δ ≤ γˇ : δ(a) = δ( ⇓Z a) = δ(⇓Z a) ≤ γ(⇓Z a) = γˇ (a). Z In all, this establishes the claimed identity γˇ = γ . Now, it follows from Theorem 3.45 that Cγˇ is the least -closed and Z- -closed subset containing the closure range B = Cγ , whence VZ B ⊆ Cγˇ . Conversely, if a lies in Cγˇ then a = γˇ (a) = γ(⇓Z a) ∈ VZ B, since ⇓Z a ∈ Z ∧ C implies ↓B γ(⇓Z a) ∈ Z ∧ B. Combining both inclusions, we arrive at the equality VZ B = Cγˇ . And if A is an arbitrary subset of C then the previous consideration applies to the -closed set B = Λ A, whence AZ = BZ = VZ B.

The case Z = D gives the order-theoretical extension of Proposition 2.14: Corollary 3.55. For any closure operation γ on a continuous lattice, the greatest inductive (i.e., D-join preserving) closure operation below γ is given by γ D (a) = γ(⇓D a). The D-subalgebra generated by A is VD Λ A, and it is again a continuous lattice. A related useful property of projections has been established by Venugopalan [89] along the lines of the theory of continuous posets [47, 48]: Proposition 3.56. If γ is a Z-join preserving projection on a Z-continuous poset P then its range R = γ(P ) is a Z-continuous poset with Z ⇓Z R a = ↓R γ(⇓P a) for a ∈ R. = ∧ ) = Pγis Z-complete: Z ∈ ZR implies ↓P Z ∈ Z P ; thus, Proof. R = γ(P P Z exists and γ( P Z) = R Z. Given a = γ(a) ∈ R, one concludes: Z Z γ(⇓ a) ⊆ ⇓ a, since for Z ∈ ZR, a ≤ (1) ↓ R P R R Z implies a ≤ R γ(Z) = a) ⊆ ↓ γ(Z) ⊆ ↓ Z. γ( P Z) = R γ(Z) = R Z and then γ(⇓Z P P P Z ∧ Z ∧ ⇓Z (2) ⇓Z R P a ∈ Z P implies ↓R γ(⇓P a) ∈ Z a ⊆Z ↓R γ(⇓P a), as R Zand a = Z Z Z γ(a) = γ( P ⇓P a) = R γ(⇓P a). Thus ⇓R a = ↓R γ(⇓P a) and a = R ⇓R a.

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We are now prepared for another central theorem of this section: Theorem 3.57. Let P be a Z-continuous poset, Q a Z-complete poset, and ϕ : P → Q lower adjoint to ψ : Q → P . Then the following are equivalent: Z (a) ϕ preserves the Z-below relation: a Z P b ⇒ ϕ(a) Q ϕ(b). (b) ψ preserves Z-joins. (c) γ = ψ ◦ ϕ and κ = ϕ ◦ ψ preserve Z-joins. If one of these conditions is fulﬁlled then (d) ψ(Q) = Pγ and ϕ(P ) = Qκ are isomorphic Z-continuous posets. Proof. For (a) ⇔ (b), see [7] (where the roles of ϕ and ψ are interchanged). (b) ⇒ (c). Being lower adjoint, ϕ preserves Z-joins, and so do ψ ◦ ϕ and ϕ ◦ ψ. (c) ⇒ (b). By Theorem 3.28, ψ = γ o ◦ ψoo ◦ κo , where κo : Q → Qκ is the surjective corestriction of κ, ψoo : Qκ → Pγ is an isomorphism, and γ o is the restriction of γ to Pγ (the inclusion map into P ). First, Z ∈ ZQ implies Z1 = ↓ κo (Z) ∈ Z ∧ Qκ and κo ( Q Z) = κ( Q Z) = Q Z1 = Qκ Z1 . Second, the isomorphism ψoo sends Z1 to a Z-downset Z2 ∈ Z ∧ Pγ with ψoo ( Qκ Z1 ) = ∧ ∧ Pγ Z2 . Third, Z2 ∈ Z Pγ entails Z3 = ↓P Z2 ∈ Z P and γ o ( Pγ Z2 ) = γ(γ( P Z2 )) = γ( P Z3 ) = P γ(Z3 ) = P γ o (Z2 ) since γ is a closure map. Thus, ψ( Q Z) = γ o (ψoo (κo ( Q Z))) = γ o (ψoo ( Qκ Z1 )) = γ o ( Pγ Z2 ) = P γ o (Z2 ) = P γ o (ψoo (Z1 )) = P γ o (ψoo (κo (Z))) = P ψ(Z). (c) ⇒ (d). Pγ is the image of P under the closure map γ, hence Z-continuous by Proposition 3.56; and Pγ is isomorphic to Qκ by Theorem 3.28. Corollary 3.58. Let C be a Z-continuous complete lattice. (1) The image of C under a join- and Z-below-preserving map, or under a meet- and Z-join-preserving map, is again Z-continuous. (2) If D is complete and embedded in C under the preservation of Z-joins and meets, or of joins and the Z-below relation, then D is Z-continuous. (3) For A ⊆ C, the Z-subalgebra VZ Λ A generated by A is Z-continuous. Proof. (1). If ϕ : C → D preserves joins and the Z-below relation then Theorem 3.57 applies. If ψ : C → D preserves meets and Z-joins then it has a lower adjoint ϕ, and γ = ψ ◦ ϕ is a Z-join preserving closure operation. Hence, by Proposition 3.56, D = ψ(C) = Cγ is Z-continuous (cf. [7]). (2) is deduced from (1) by passing to adjoints (ϕ injective ⇔ ψ surjective). (3) follows from Theorem 3.54 and Proposition 3.56. 4. Closure algebras It is now time to provide the reader with a powerful categorical machinery that also helps to understand closure structures. We start with a brief account of the extremely useful notions of adjoint functors, the categorical extension of adjunctions between posets, and of monads, a categorical generalization of closure operations on posets. These concepts are applied to and illustrated by the power set monad and its algebras, the complete lattices. Then we show that not only the

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category of topological (closure) spaces but also many other categories of closure spaces occurring in practice (for example, that of algebraic closure spaces!) are topological constructs. Our main purpose is to elaborate the topological and algebraic nature of a very convenient category CCA, namely that of complete closure algebras. These are the abstract counterparts of closure spaces, and the morphisms between them are modelled exactly along the lines of continuous maps in the classical case. The most important and surprising fact is that the category of complete closure algebras is both topological over the self-dual algebraic construct CL∨ of complete lattices (with join-preserving maps) and algebraic over the topological construct of closure spaces. Therefore, all basic constructions like products and coproducts, initial sinks and ﬁnal sources, discrete and indiscrete objects, epi-mono-factorizations etc. are easily available in the category CCA. Concerning the role of closure spaces in the categorical treatment of spaces “beyond topology”, like convergence spaces and approach spaces, the reader is referred to the extensive literature on the subject (see e.g. [3, 24, 25, 59]. Also, another modern version of closure structures, namely the categorical deﬁnition of closure operators (as families of closure operations on objects of a category) is not detailed here, as it is not within the scope of this work. Interested readers may consult the two comprehensive monographs on that subject by Dikranjan and Tholen [27] and by Castellini [14]. 4.1. Monads, algebras and complete lattices. Though we assume the reader to be acquainted with basic categorical notions such as functors, natural transformations, various kinds of morphisms, (full) reﬂective subcategories, (co)products etc., we shall recall some of the main concepts needed in the sequel. As we have seen, closure and kernel maps are intimately related to adjunctions and Galois connections. Instead of the classical base category Set (of sets and functions), we propose for the investigations of closure structures a more general, order-theoretically oriented category CL∨ that has some substantial advantages compared with Set — the most important one being its self-duality. Its objects are the complete lattices, i.e., ordered sets in which all subsets have joins (and consequently also meets, obtained as joins of lower bounds); its morphisms are the maps preserving arbitrary joins — or equivalently, the residuated maps between complete lattices. This category CL∨ is concretely isomorphic to the category CL∧ of complete lattices with meet-preserving maps, just by turning the objects upside down (reversing the order); but CL∧ is also a categorical dual of CL∨ , namely under the functor that assigns to each join-preserving map its upper adjoint (which preserves all meets). Combining both processes, i.e., passing not only to the orderdual objects but also to the upper adjoints at the morphism level, one arrives at a self-duality of the category CL∨ that allows to dualize all categorical statements about it. There is an obvious functor from CL∨ to Set, “forgetting” the lattice structure and keeping the underlying sets and maps. In other words, CL∨ is a concrete category over Set or, for short, a construct. To capture constructs like CL∨ under the concept of algebraicity, one clearly must allow inﬁnitary operations — which in other contexts may create severe set-theoretical obstacles and other problems that do not occur in the ﬁnitary case. For instance, the category of complete lattices and maps that preserve not only arbitrary joins but also (at least ﬁnite) meets does

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not have free algebras over sets with more than two elements (see [38] for a short argument). But, fortunately, the category CL∨ does have free objects over arbitrary sets X, namely the power set lattices PX, as we shall see soon. During decades, there has been a vital and fruitful discussion, sometimes rather controversial, with respect to what the typical ingredients of an algebraic structure or category should be. Category theorists have developed a precise formal machinery for such situations, covering much more general concepts than that of ﬁnitary algebras in the classical sense of universal algebra. However, the exact meaning of the word “algebraic” varies from case to case. In order to make categorical tools accessible to categories like CL∨ , we shall enter the convenient area of (regularly) monadic functors and categories as the appropriate framework for algebraic theories, as established, for example, in [2]. In that context, the varieties are just the monadic constructs. Perhaps the most powerful tool in category theory are adjoint functors — the categorical generalization of adjoint maps. Any poset P = (X, ≤) may be regarded as a category, by taking the elements of X as objects and the pairs (a, b) with a ≤ b as morphisms; the composite (b, c) ◦ (a, b) is (a, c). The functors between posets, viewed as categories, are then merely the isotone maps. The categorical generalization of an adjoint pair (ϕ, ψ) of maps between posets P and Q is an adjoint situation or adjunction, constituted by two functors F : X → A, G : A → X and natural transformations η : idX → G ◦ F, called the unit, and ε : F ◦ G → idA , called the counit, (replacing the inequalities idP ≤ ψ ◦ ϕ and ϕ ◦ ψ ≤ idQ ), such that Fη ◦ εF = idF and Gε ◦ ηG = idG . (The corresponding poset identities ϕηa ◦ εϕ(a) = idϕ(a) and ψεb ◦ ηψ(b) = idψ(b) are automatic.) Alternately, the characteristic equivalence a ≤ ψ(b) ⇔ ϕ(a) ≤ b for adjoint pairs of maps may be replaced in the categorical setting by natural isomorphisms f → g = εA ◦ Ff and g → f = Gg ◦ ηX between the hom-sets X(X, GA) (of all X-morphisms f : X → GA) and A(FX, A) (of all A-morphisms g : FX → A), whence F is called the left adjoint of G, and G the right adjoint of F. If η and ε are isomorphisms, the adjunction (or either of the functors F and G) is a (categorical) equivalence. εA FGA A _?o ? ?? ?? ?? g ?? F f ?? ? F

GA ? _?? ?? ?? Gg f ?? ?? ?? / GFX X ηX

Figure 13. An adjoint situation In the same vein, closure operations are generalized categorically to so-called monads. Here, one considers triples T = (T , η, µ) consisting of an endofunctor T : X → X and two natural transformations η : idX → T , called the unit, and µ : T 2 = T ◦ T → T , called the multiplication. A T -algebra is then a pair (X, f ) constituted by an X-object X and an X-morphism f : T X → X (the structure

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morphism) such that f ◦ ηX = idX and f ◦ µX = f ◦ T f. In the poset case, the functor T is an isotone self-map γ : P → P , the existence of a unit means a ≤ γ(a), and the existence of a multiplication means γ(γ(a)) ≤ γ(a). The “γ-algebras” are just the γ-closed elements a = γ(a) (together with the pair (γ(a), a) as “structure morphism”). The triple T = (T , η, µ) is a monad (or standard construction) if for each object X, the pair (T X, µX ) is a T -algebra, called the free T -algebra over X, such that µX ◦ T ηX = idT X . (In the poset case, the “free γ-algebras” are the closed elements γ(a), so that every “γ-algebra” is free.) The Eilenberg–Moore category of T -algebras, denoted by XT , has as morphisms h : (X, f ) → (Y, g) the X-morphisms h : X → Y satisfying h ◦ f = g ◦ T h. By deﬁnition, monads and their algebras are determined by the following two commuting diagrams: T 2T ??? ?? T µ µT ?? ?? ?? µ µ / o 2 2 TO T _? ?T ?? ?? ?? id T ηT ?? ?? T η T

T 2 X? ?? ?? T f µX ?? ?? ?? f f / o XO T? X T X_? ?? ?? ?? idX ηX ?? ηX ?? X

Figure 14. Monads and algebras At ﬁrst glance, it may be hard to memorize these diagrams. But looking at the most important monad for us, the power set monad P = (P, η, µ) on Set, the diagrammatic relations become quite vivid: here, PX is thepower set of X, the values ηX (x) are the singletons {x}, and µX (Y) is the union Y for any Y ⊆ PX, i.e., Y ∈ P 2 X. What are the P-algebras in that case? By deﬁnition, a P-algebra is a pair (X, f ) where X is a set and f is a map from PX to X such that f ({x}) = x for all x ∈ X, and f ( Y) = f (f (Y)) for all Y ⊆ PX. It is now an elementary exercise to verify that x ≤ y ⇔ f ({x, y}) = y deﬁnes a partial order on X such that C = (X, ≤) becomes a complete lattice in which the supremum of an arbitrary subset A is given by f (A) (see, e.g., [29, 5.6]). Conversely, for any complete lattice C, the join map A C : PX → X, A →

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gives rise to a P-algebra (X, C ). Furthermore, the morphisms between two Palgebras are just the join-preserving maps between the corresponding complete lattices. In all, this shows (cf. [2, 20.5]): Proposition 4.1. The category of complete lattices and join-preserving maps is isomorphic to the category of algebras for the power set monad. In the same fashion as adjoint pairs (ϕ, ψ) of maps between posets induce closure operations γ = ψ◦ϕ, every adjoint situation (F, G, η, ε) gives rise to a monad (G ◦ F, η, GεF ). Conversely, in analogy to the epi-mono-factorization of closure operations γ into adjoint pairs (γo , γ o ), every monad T = (T , η, µ) determines an adjoint situation (F T , G T , η, ε) whose associated monad is the original one: G T is the “forgetful” functor from the category XT of T -algebras to the category h h X with G T ((X, f ) → (X , f )) = X → X , and F T is the “free” functor with f

Tf

F T (X −→ X ) = (T X,µX ) −→ (T X , µX ). For the power set monad P, the power set algebra (PX, ) is the free (join-)complete lattice over X. Hence, the modiﬁed power set functor F P from Set to CL∨ is left adjoint to the forgetful functor G P from CL∨ to Set. For any adjoint situation (F, G, η, ε) between categories A and X with associated monad T = (G ◦ F, η, GεF ), there is a unique functor K : A → XT , the comparison f

functor, such that F T = K ◦ F and G = G T ◦ K. It sends any A-morphism A → B Gf to the homomorphism (GA, εGA ) −→ (GB, εGB ). Monadic functors G : A → X have a left adjoint F such that the comparison functor of the induced monad T is an isomorphism, and consequently, the category A is isomorphic to the category of T -algebras. A monadic construct consists of a category A and a monadic functor G : A → Set. There is much evidence that such monadic constructs (or even more general kinds of categories, see [2, Ch. 23]) deserve the name “algebraic”. By the previous remarks, the forgetful functor from CL∨ to Set is a monadic functor. Let us note a few immediate consequences concerning the power set monad: Proposition 4.2. The category CL∨ of complete lattices is self-dual and a monadic construct. Hence, the injective resp. surjective join-preserving maps are precisely the (regular, extremal) mono- resp. epimorphisms in CL∨ . Furthermore, CL∨ is complete and cocomplete. Speciﬁcally, CL∨ has arbitrary products and coproducts, both with underlying cartesian products. Moreover, the category CL∨ has regular epi-mono-factorizations. These conclusions follow from general categorical principles (see e.g. [2, 20.23]). The last statement in Proposition 4.2 is, of course, also a direct consequence of the Homomorphism Theorem for Residuated Maps 3.29. The following notion is familiar from categories of topological spaces: A source, that is, a family of morphisms fi : A → Ai in a concrete category A with forgetful (i.e., faithful) functor U : A → X is initial if for each X-morphism g : X → UA satisfying Ufi ◦g = Uhi for certain A-morphisms hi , there is a (unique) A-morphism g A with Ug A = g. A single morphism f is said to be initial iﬀ so is the source (f ), and an embedding is an initial morphism with underlying monomorphism in the base category (cf. [2, 8.6]). Corollary 4.3. In the category CL∨ , the monomorphisms are exactly the (regular) embeddings, and they are also embeddings in the order-theoretical sense, characterized by the equivalence a ≤ b ⇔ ϕ(a) ≤ ϕ(b).

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The dual notions of initial sources, morphisms and embeddings are ﬁnal sinks, ﬁnal morphisms and quotient morphisms. Now, from Corollary 4.3 and the selfduality of the category CL∨ , we immediately infer the following Corollary 4.4. The epimorphisms in the category CL∨ are precisely the regular quotient morphisms (with underlying surjections). Notice that surjectivity of epimorphisms in monadic constructs is not obligatory: in the variety of bounded distributive lattices, the embedding of a 3-element chain in a 4-element boolean lattice is an epimorphism. The previous statements about the power set monad may be generalized to any union complete functorial subset selection Z instead of P. It turns out that for the resulting order completion monads, the algebras are, up to isomorphism, nothing but the Z-complete posets (see [30, 33, 64]). 4.2. Topological constructs of closure spaces. Recall that a concrete category with forgetful functor U : A → X is said to be topological if every “Ustructured” source (fi : X → UAi )i∈I in the base category X has a unique initial lift (fiA : A → Ai )i∈I in A such that UfiA = fi (see e.g. [2]), as familiar from the situation of initial structures for topological spaces. The same arguments go through for other categories of closure spaces, and we may note (cf. Section 2.1 and Theorem 4.6): Proposition 4.5. Each of the closure space categories CSp, TCSp, ACSp, ATSp (with the obvious forgetful functor) is a topological construct. But none of these constructs can be algebraic, because the forgetful functor to Set does not reﬂect isomorphisms (an obligatory property of algebraic constructs): a continuous bijection between (algebraic, topological or Alexandroﬀ-discrete) closure spaces need not be an isomorphism — the inverse map may fail to be continuous. Hence, the name “algebraic closure spaces” for objects of the category ACSp might be a bit misleading, whence many authors prefer to speak of ﬁnitary or inductive closure spaces. As demonstrated in [2], topological categories A and their forgetful functors U : A → X have the following useful properties, which are therefore fulﬁlled for each of the above closure space categories over X = Set: • U has a left adjoint (the discrete functor ); • U has a right adjoint (the indiscrete functor ); • U preserves and reﬂects mono-sources and epi-sinks; • the extremal A-monos are the inital extremal X-monos; • the extremal A-epis are the ﬁnal extremal X-epis; • A is complete and cocomplete; • A has epi-mono-factorizations iﬀ X has them and A is ﬁber-small. Note that the base category X = Set has all the required properties, and consequently, the above properties are fulﬁlled for each topological construct of closure spaces (which is ﬁber-small, because the closure systems on a ﬁxed set form a set, not a proper class). The subsequent considerations, extending Proposition 4.5, will show that there is a broad spectrum of such closure space categories. As earlier, Z denotes a functorial subset selection for posets. Recall that by a Z-closure space we mean one whose closure system C is Z-union closed, i.e., Y ∈ ZC implies Y ∈ C. Theorem 4.6. The construct ZCSp of Z-closure spaces is topological.

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Proof. Let ((Xi , Ci ) : i ∈ I) be a family of Z-closure spaces and suppose there is given a family of maps fi : X → Xi with common domain X. Put X = {fi (Ai ) : Ai ∈ Ci , i ∈ I} and let C Z X denote the least Z-closure system containing X (cf. Section 2.2). Then each fi is clearly continuous as a map from (X, C Z X ) to (X, Ci ). Consider now any Z-closure space (Y, D) and a map g : Y → X such that all the composed maps fi ◦ g are continuous. We claim that then the map g : (Y, D) → (X, C Z X ) must be continuous, too. The ﬁnal system B = {A ⊆ X : g (A) ∈ D} is closed under arbitrary intersections and under Z-unions, because g preserves them: Y ⊆ B ⇒ g ( Y) = g (Y) ∈ ΩD = D ⇒ Y ∈ B, Y ∈ ZB ⇒ ↓D g (Y) ∈ Z ∧ D ⇒ g ( Y) = g (Y) ∈ D ⇒ Y ∈ B. Moreover, the system X is contained in B, on account of the equations g (fi (Ai )) = (fi ◦ g) (Ai ) ∈ D for all Ai ∈ Ci , valid by continuity of fi ◦ g. Hence, the Z-closure system C Z X must also be a subset of B, which means that g : (Y, D) → (X, C Z X ) is continuous, as claimed. It remains to verify the uniqueness statement. Suppose C is a Z-closure system on X such that each fi : (X, C) → (Xi , Ci ) is continuous, and so that the map g : (Y, D) → (X, C) is continuous whenever each fi ◦g is continuous. Then X ⊆ C and so C Z X ⊆ C. Consider the identity map idX from (X, C Z X ) to (X, C). Each fi = fi ◦ idX : (X, C Z X ) → (Xi , Ci ) is continuous (see above). Hence, idX : (X, C Z X ) → (X, C) is continuous, which means C ⊆ C Z X . As soon as CSp is seen to be topological, the same statement for the subcategories ZCSp may be deduced from the general categorical fact that concretely coreﬂective subcategories of a topological category are again topological (see [2, 5.22 and 21.35]). Indeed, a few similar arguments lead to Proposition 4.7. The category ZCSp of Z-closure spaces is concretely coreﬂective in CSp, the category of closure spaces. The concrete coreﬂector sends each closure space (X, C) to the Z-closure space (X, C Z C). Being topological, the category ZCSp has also arbitrary ﬁnal sinks, which are easily constructed explicitly, as in the case of topological spaces. 4.3. Categories of closure algebras. As Ad´ amek, Herrlich and Strecker point out in Chapter 23 of their book [2], there is a duality principle for topological categories: if (A, U) is a topological category over some category X then the opposite category (Aop , U op ) is topological over Xop . However, as they emphasize, there is no duality principle available for topological constructs, i.e., concrete categories over Set, like CSp or TCSp # Top. The crucial lack is that Set fails to be a self-dual category.

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This undesired failure will now be remedied by passing from Set to CL∨ and from CSp to the more general category CCA of complete closure algebras, which has the following striking advantages: • CCA is a topological category over the self-dual and monadic (hence algebraic but not topological) construct CL∨ of complete lattices and joinpreserving maps. • CL∨ is therefore isomorphic to a reﬂective subcategory (of indiscrete closure algebras) but also to a coreﬂective subcategory (of discrete closure algebras) of CCA. • CCA is a regularly monadic, hence algebraic category over the topological (but not algebraic) construct CSp of closure spaces, which is isomorphic to a coreﬂective subcategory of CCA. • CCA is therefore a topologically algebraic construct with several pleasant additional properties (see [2, Ch. 25]). • The adjoint functors from the constructs CL∨ and CSp to Set lift to adjoint functors from CCA to the mutual other construct. • Composition of the respective adjunctions leads to the fundamental adjunction between CSp and CL∨ , which provides the framework for many Stone type dualities (see [36, 40]). CCA < F xxxx O cFFFFFF x x FFFF x x FFFF xxx F# x|xxxx / CL CSp Fo bFFFF xx; ∨ x FFFF xx FFFF xxxx FFF" xxxxxx {x Set Figure 15. Adjunctions between closure categories Before turning to the category CCA, it will be instructive to deﬁne ﬁrst a larger category ProA of projection algebras by taking as objects the projections (isotone and idempotent self-maps) γ on arbitrary posets P , or, if more suitable, the pairs (P, γ) (not to be confused with algebras of projections in linear algebra and functional analysis). As morphisms between two projection algebras (P, γ) and (Q, δ) we take so-called promorphisms; these are residuated maps ϕ : P → Q between the underlying posets (with upper adjoint ϕ : Q → P ) satisfying the equivalent conditions ϕ◦γ ≤δ◦ϕ ϕ◦γ◦ϕ ≤δ

γ◦ϕ ≤ϕ ◦δ γ≤ϕ ◦δ ◦ϕ

For the equivalence of these inequalities, verify the implication circle ϕ◦γ ≤δ◦ϕ⇒ϕ◦γ◦ϕ ≤δ◦ϕ◦ϕ ≤δ ⇒γ◦ϕ ≤ϕ ◦δ ⇒γ ≤γ◦ϕ ◦ϕ ≤ ϕ ◦δ ◦ϕ ⇒ϕ◦γ ≤ϕ◦ϕ ◦ δ ◦ ϕ ≤ δ ◦ ϕ.

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There are a few obvious full subcategories of ProA: category CA CCA KA CKA

objects closure algebras complete closure algebras kernel algebras complete kernel algebras

description closure maps = extensive projections closure maps on complete lattices kernel maps = contractive projections kernel maps on complete lattices

In each case, one may take again the pairs (P, γ) instead of the maps γ as objects. Observe that the underlying posets of closure algebras in the pioneering paper by McKinsey and Tarski [63] are boolean lattices. The category Pos∨ of posets and lower adjoint (= residuated) maps is selfdual under the functor sending each poset P to its order dual P and any lower adjoint map ϕ : P → Q to its upper adjoint ϕ (see Section 3.2), regarded as a to P. As remarked earlier, this self-duality induces one of (residuated!) map from Q the full subcategory CL∨ of complete lattices and join-preserving maps. Moreover, on account of the above equivalence ϕ ◦ γ ≤ δ ◦ ϕ ⇔ γ ◦ ϕ ≤ϕ ◦ δ, the self-duality of Pos∨ also lifts to ProA: Proposition 4.8. The dualization functor induces a self-duality of the category ProA of projection algebras and a duality between the category CA of closure algebras and the category KA of kernel algebras, which in turn induces a duality between the categories CCA and CKA of complete closure algebras and complete kernel algebras, respectively. We have obvious forgetful functors from the categories ProA, CA and KA to the category Pos∨ , sending a projection, closure or kernel algebra to the underlying poset. Thus, ProA, CA and KA are concrete categories over Pos∨ , and CCA and CKA are concrete categories over CL∨ . The main reason why we have introduced the categories CA and CCA is obvious: the categories CSp, Top # TCSp etc. are embedded (faithfully, but not fully) in CCA by assigning to any continuous map f : (X, C) → (Y, D) between closure spaces the union preserving image map f : (PX, C) → (PY, D) which is in fact a CCA-morphism, lower adjoint to the preimage map f : (PY, D) → (PX, C). In order to make the embedding full (so that the morphisms between two objects in the embedded category are exactly the same as those in the larger category), one has to restrict the morphism class suitably. The hint how to proceed here comes from the observation that for any continuous map f : (X, C) → (Y, D), the preimage map f : PY → PX has not only a lower adjoint (viz. the image map f ) but also an upper adjoint, because it preserves arbitrary joins (= unions). Calling a map between posets doubly residuated if it a has a residuated upper adjoint, we see that a map f between closure spaces (X, C) and (Y, D) is continuous if and only if there is a (unique) doubly residuated promorphism ϕ : PX → PY with ϕ({x}) = {f (x)}. (Indeed, an application of Theorem 3.57 to Z = A shows that a doubly residuated map between ABC lattices must send atoms to atoms, because these are the elements p with p A p.) This suggests to consider, for any

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subcategory C of ProA, that category C∨ whose objects are those of C, but whose morphisms are the doubly residuated promorphisms. Then, we have: Proposition 4.9. The category CSp of closure spaces is densely embedded in the full subcategory ABCCA of CCA whose objects are closure operations on ABC lattices, and it is equivalent to the full subcategory ABCCA∨ of CCA∨ . The category of topological spaces is equivalent to that full subcategory of ABCCA∨ whose closure operations preserve ﬁnite joins. The next theorem shows how adjunctions between closure algebras give rise to adjunctions between the associated closure ranges. Recall that for any closure operation γ : P → P , the restriction γ o to Pγ (the range of γ) is just the inclusion map from Pγ into P , and that γo denotes the surjective corestriction of γ to Pγ , whence γ o ◦ γo = γ and γo ◦ γ o = idPγ . For any map ϕ between closure algebras (P, γ) and (Q, δ), we put ϕc = δo ◦ ϕ ◦ γ o . Theorem 4.10. Given closure algebras (P, γ), (Q, δ) and an adjoint pair of maps ϕ : P → Q, ψ : Q → P , each of the following conditions is necessary and δ) → (P , γ) to be promorphisms (hence suﬃcient for ϕ : (P, γ) → (Q, δ) and ψ : (Q, ϕ a CA-morphism and ψ a KA-morphism): (a) ϕ ◦ γ ≤ δ ◦ ϕ. (a’) γ ◦ ψ ≤ ψ ◦ δ. (b) ϕ ◦ γ ◦ ψ ≤ δ. (b’) γ ≤ ψ ◦ δ ◦ ϕ. (c) δ ◦ ϕ ◦ γ = δ ◦ ϕ. (c’) γ ◦ ψ ◦ δ = ψ ◦ δ. (d) ∃ϕ : Pγ → Qδ : ϕ ◦γo = δo ◦ϕ. (d’) ∃ψ : Qδ → Pγ : ψ ◦δo = γo ◦ψ. (e) ϕc : Pγ → Qδ is lower adjoint to ψ c : Qδ → Pγ . (e’) ϕc and ψ c are the unique CL∨ -morphisms making the following diagram commute: ψ P o >Q I } || γ }} | δ | } || }} γo ~}} || /Q δo P `A ϕ AA o AAγ o AA A δo δ γo Pγ o Q ψc }}> δ ~ ~ } ~ }}}id ~~idPγ ~ Qδ } ~~ } / Qδ Pγ c ϕ

Proof. That (a), (a’), (b), (b’) are equivalent characterizations of CA-morphisms ϕ and KA-morphisms ψ was veriﬁed earlier (cf. Proposition 4.8). (a) ⇔ (c) and (a’) ⇔ (c’) are clear since γ and δ are closure maps. (c) ⇒ (e). ϕc and ψ c are isotone and satisfy the inequalities ϕc ◦ ψ c = δo ◦ ϕ ◦ γ ◦ ψ ◦ δ o = δo ◦ ϕ ◦ ψ ◦ δ o ≤ δo ◦ δ o = idQδ , ψ c ◦ ϕc = γo ◦ ψ ◦ δ ◦ ϕ ◦ γ o ≥ γo ◦ ψ ◦ ϕ ◦ γ o ≥ γo ◦ γ o = idPγ . (e) ⇒ (d). As idQ ≤ ψ ◦ ϕ ≤ ψ ◦ δ ◦ ϕ, we have γo ≤ γo ◦ ψ ◦ δ ◦ ϕ = ψ c ◦ δo ◦ ϕ; by the adjunction, it follows that ϕc ◦ γo ≤ δo ◦ ϕ ≤ δo ◦ ϕ ◦ γ = ϕc ◦ γo . Hence, ϕ = ϕc satisﬁes ϕ ◦ γo = δo ◦ ϕ.

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(d) ⇒ (c). First, ϕ ◦ γo = δo ◦ ϕ entails ϕ = ϕ ◦ γo ◦ γ o = δo ◦ ϕ ◦ γ o = ϕc , and then δ ◦ ϕ ◦ γ = δ o ◦ δo ◦ ϕ ◦ γ o ◦ γo = δ o ◦ ϕ ◦ γo = δ o ◦ δo ◦ ϕ = δ ◦ ϕ. (e) ⇒ (d’) ⇒ (c’) and (d’) ⇒ ψ = ψ c are obtained analogously. The equivalence (e’) ⇔ (c), (c’) is now clear from the previous implications. Characterization (e) is due to A. Pigors. By the characterization (e’), CA may be regarded as an arrow category (cf. [60, Ch. II.4]) over Pos∨ , but we shall not pursue that aspect further. The equivalence (a) ⇔ (c’) (together with γ(P ) = Pγ = Pγ= ) yields: Corollary 4.11. A residuated map ϕ : P → Q between closure algebras (P, γ) and (Q, δ) is a promorphism (hence a CA-morphism) iﬀ ϕ(Q δ ) ⊆ Pγ . For a continuous map f between closure spaces (X, C) and (Y, D), the residuated map f c : C → D is given by f c (A) = f (A), and its upper adjoint by the restricted preimage map f c : C → D. Each of the properties listed in Theorem 4.10 is a more or less obvious abstraction of continuity for maps between closure spaces (and Corollary 4.11 is the abstract version of the phrase “preimages of closed sets are closed”). In view of Proposition 4.9 one might call the morphisms of the category CA and its subcategories continuous (and sometimes that term is used even for arbitrary maps ϕ between closure algebras satisfying δ ◦ ϕ ◦ γ = δ ◦ ϕ or the slightly weaker inequality ϕ ◦ γ ≤ δ ◦ ϕ). On the other hand, a KA-morphism, that is, a residuated map ψ between kernel algebras (P, κ) and (Q, λ) satisfying the equivalent conditions λ ◦ ψ ≥ ψ ◦ κ, λ ◦ ψ ◦ κ = ψ ◦ κ, ψ(P κ ) ⊆ Qλ might be called open, in view of the fact that a map f between kernel spaces is open (i.e., images of open sets are open) iﬀ the lifted power map ψ = f is an open morphism in the above sense. In contrast to the classical situation of (topological) closure and kernel spaces, where the properties “continuous” and “open” are rather unrelated, we have learned from Proposition 4.8 that the category of (boolean, complete) closure algebras and continuous morphisms is dual to the category of (boolean, complete) kernel algebras and open morphisms. But there is yet another obvious dualization process suggested by set complementation: the involution ι of a boolean algebra A, sending each element to its complement, transforms each closure operation γ into a closure operation γ ι = ι◦κ◦ι of Then, assigning to each boolean closure algebra (A, γ) the dual boolean closure A. γ ι ), one obtains an automorphism (but not a duality!) of the category algebra (A, of boolean closure algebras and doubly residuated morphisms. In order to characterize initial and ﬁnal structures in the categories CA and CCA, we consider for any residuated map ϕ : P → Q and any closure operation γ on P the right shift along ϕ γϕ = ϕ ◦ γ ◦ ϕ : Q → Q, and for any closure operation δ on Q the left shift along ϕ ϕ

δ=ϕ ◦ δ ◦ ϕ : P → P.

Recall that a residuated map ϕ : P → Q is a promorphism iﬀ it satisﬁes the equivalent conditions γ ϕ ≤ δ and γ ≤ ϕ δ. We are now in a position to describe initial closure structures in our general order-theoretical setting.

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Proposition 4.12. Let (C, γ) and (D, δ) be complete closure algebras, and suppose ϕ : C → D is a residuated (= join-preserving) map. Then: (1) ϕ δ is the unique closure operation on C with range ϕ(D δ ). (2) ϕ : (C, γ) → (D, δ) is an initial morphism in CCA iﬀ γ = ϕ δ. Proof. (1). For c ∈ C, the image ϕ δ(c) = ϕ(δ(ϕ(c))) belongs to ϕ(D δ ), and c ≤ ϕ(ϕ(c)) ≤ ϕ δ(c). For d ∈ Dδ , we get the implications c ≤ ϕ(d) ⇒ ϕ(c) ≤ d ⇒ ϕ ◦ ϕ ◦ δ ◦ ϕ(c) ≤ δ ◦ ϕ(c) ≤ d ⇒ ϕ δ(c) ≤ ϕ(d). Thus, ϕ(D δ ) is a closure range and ϕ δ is the associated closure operation. (2). Suppose ϕ : (C, γ) → (D, δ) is an initial morphism in CCA. Then ϕ (Dδ ) is contained in Cγ , by Corollary 4.11. For the reverse inclusion, note that the composite morphism ϕ ◦ idC : (C, ϕ δ) → (C, γ) → (D, δ) is continuous (as ϕ(D δ ) = Cϕ δ by the ﬁrst part), and by initiality, so the same holds for idC : (C, ϕ δ) → (C, γ), whence Cγ ⊆ Cϕ δ = ϕ(D δ ). Since a closure operation is determined by its range, the equation Cϕ δ = ϕ(D δ ) proves the equivalence of the equations γ = ϕ δ and Cγ = ϕ(D δ ). On the other hand, the latter equation implies that ϕ : (C, γ) → (D, δ) is an initial morphism: for any closure algebra (B, β) and any residuated map ρ : B → C such that ϕ ◦ ρ : (B, β) → (D, δ) becomes a continuous morphism, we conclude that ρ : (B, β) → (C, γ) must be continuous, too: ρ(Cγ ) = ρ(ϕ(D δ )) = ϕ ◦ ρ(Dδ ) ⊆ Bβ . The proof carries over verbally to the concrete category CA over Pos∨ , but we have formulated the result for the concrete category CCA over CL∨ , because it comes closer to the classical theory of closure spaces. Since embeddings are initial with underlying monomorphisms, and monomorphisms in CL∨ are injective, we obtain the abstract counterpart to the known description of embedding structures in topology: Corollary 4.13. A map ϕ between complete closure algebras (C, γ) and (D, δ) is a CCA-embedding iﬀ it is injective, preserves joins, and γ = ϕ δ. Recall that ﬁnality is deﬁned dually to initiality, and that a quotient morphism is a ﬁnal morphism with underlying epimorphism. At ﬁrst glance, one might guess that the previous results may be dualized in a straightforward manner — but there is an obstacle: while the right shifts γ ϕ = ϕ ◦ γ ◦ ϕ of closure operations γ are always isotone, they need neither be extensive nor idempotent in general! However, surjectivity of ϕ entails extensivity of γ ϕ , and injectivity of ϕ entails idempotency of γ ϕ . Hence, for isomorphisms ϕ, the right shift γ ϕ is again a closure operation. Example 4.14. On the ﬁnite chain 4 = {0, 1, 2, 3}, consider the residuated map ϕ deﬁned by ϕ(0) = 0, ϕ(1) = ϕ(2) = 1, ϕ(3) = 2, whose upper adjoint is given by ϕ(0) = 0, ϕ(1) = 2, ϕ(2) = ϕ(3) = 3,

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γ

γ

◦ •O TTTT ϕ TTTT TTTT )◦ ◦ TTTT ϕ TTTT TTTT ϕ /) ◦ •O ◦

ϕ

/◦

Figure 16. ϕ of Example 4.14 and the following closure operation: γ(0) = γ(1) = 1, γ(2) = γ(3) = 3. For γ = ϕ ◦ γ ◦ ϕ, one computes ϕ

γ ϕ (0) = ϕ(γ(0)) = 1, γ ϕ (1) = ϕ(γ(2)) = 2, γ ϕ (2) = ϕ(γ(3)) = 2. Hence, γ ϕ is not idempotent. For the surjective corestriction ϕo : 4 → 3, the right shift γ ϕo is not idempotent either. However, one easily checks that in this example, there is a unique closure operation above γ ϕo . Generally, we use the fact that for every isotone self-map ϕ on a complete lattice there is a least closure operation ϕ above ϕ. From that, we deduce the following dualized analogue of Proposition 4.12: Proposition 4.15. Let (C, γ) and (D, δ) be complete closure algebras, and suppose ϕ : C → D is a residuated, i.e., join-preserving map. Then: (1) γ ϕ is the unique closure operation on D with range ϕ (Cγ ). (2) ϕ : (C, γ) → (D, δ) is a quotient morphism iﬀ δ = γ ϕ . Proof. For b ∈ D, we get the equivalences ≤ ϕ(b) ⇔ ϕ ◦ γ ◦ ϕ(b) ≤ b ⇔ γ ϕ (b) ≤ b b∈ϕ (Cγ ) ⇔ γ(ϕ(b)) which show that (∗)

Dγ ϕ = Dγ ϕ = ϕ (Cγ )

is a closure range with associated closure operation γ ϕ . Suppose ϕ : (C, γ) → (D, δ) is a ﬁnal CCA-morphism. Then Dδ is contained in ϕ (Cγ ), again by Corollary 4.11. For the reverse inclusion, consider this time the composite map idD ◦ϕ : (C, γ) → (D, δ) → (D, γ ϕ ) which is a CCA-morphism since ϕ(D γ ϕ ) ⊆ Cγ by (∗). The ﬁnality of ϕ now assures ϕ that idD : (D, δ) → (D, γ ) is a CCA-morphism, too, and again by (∗), ϕ (Cγ ) = ϕ Dγ ϕ ⊆ Dδ . But the equation Dγ ϕ = Dδ is equivalent to δ = γ . On the other hand, the latter equation implies that ϕ : (C, γ) → (D, δ) is a ﬁnal CCA-morphism, by arguments dual to those for Proposition 4.12: given any CCA-object (E, ε) and any CL∨ -morphism ρ : D → E such that ρ ◦ ϕ : (C, γ) → (E, ε) is a CCA-morphism, we conclude that ρ : (D, δ) → (E, ε) must be a CCA-morphism, too, because Eε ⊆ ρ (ϕ (Cγ )) = ρ (Dδ ).

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Example 4.16 (Z-initial and Z-ﬁnal closure operations). Special instances of initial and ﬁnal closure structures are obtained by looking at functorial subset selections Z (see Section 2.3) and the join maps Z ν = C : Z ∧ C → C, A → A which have the upper adjoint Z ν = ηC : C → Z ∧ C, a → ↓ a

and, in case C is Z-continuous (see Section 3.4), also the lower adjoint ∧ Z λ = ⇓Z C : C → Z C, a → ⇓ a .

Each closure operation γ : C → C lifts to a closure operation Γ = Z ∧ γ : Z ∧ C → Z ∧ C, A → ↓ γ(A), (since A ⊆ ↓ γ(A) = ↓ γ(γ(A)) = ↓ γ(↓ γ(A))). By the equation γ(a) = γ(↓ a) = (ν ◦ Γ ◦ ν)(a) = Γν (a), γ is the ﬁnal closure structure of Z ∧ γ along the join map ν. On the other hand, if C is Z-continuous and γ Z is the Z-modiﬁcation of γ (i.e., the greatest Z-join preserving closure map below γ, see Corollary 3.46), then we know from Theorem 3.54 that the lower adjoint λ of ν satisﬁes the equation ◦ Γ ◦ λ)(a) = λ Γa, γ Z (a) = γ(⇓Z a) = (λ which shows that γ Z is the initial closure structure of Z ∧ γ along the Z-below map λ =⇓Z C. After having settled the case of single initial morphisms, it is not hard to verify that the unique initial lift to CCA of an arbitrary “structured source” (ϕi : C → (Di , δi ))i∈I is obtained by endowing C with the closure operation γ = {ϕi ◦ δi ◦ ϕi : i ∈ I} where the meet is formed pointwise (this would not work in CA!) Similarly, the unique ﬁnal lift to CCA of a “structured sink” (ϕi : (Ci , γi ) → D)i∈I is obtained by taking on D the closure operation δ associated with the -closed subset {ϕi (Cγi ) : i ∈ I} which may also be obtained as the hull δ = ϕ where ϕ is the pointwise join ϕ = {ϕi ◦ δi ◦ ϕi : i ∈ I}. In all, this establishes the claimed main property of CCA: Theorem 4.17. The category CCA of complete closure algebras is topological over the self-dual category CL∨ of complete lattices. Obviously, the category CCA is ﬁber small (there is only a set, not a class of closure operations on a complete lattice), has as indiscrete objects the constant closure operations, and as discrete objects the identity maps. On account of the previous results, it is now possible to translate many notions and facts from the theory of closure spaces into the extended framework of closure algebras, and to derive analogous (sometimes easier) conclusions. For example, the category ZCSp of Z-closure spaces is equivalent to the category of ABC lattices with closure operations preserving Z-joins and doubly residuated promorphisms. Hence,

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ZCSp embeds (non-fully) in the category ZCCA of complete Z-closure algebras, whose closure operations preserve Z-joins, and (fully) in the categories ZCCA∨ and ABCCA∨ . 4.4. Complete closure algebras over closure spaces. While in the previous section the emphasis was put on the topological character of the category CCA of complete closure algebras, viewed as a category over the algebraic construct CL∨ of complete lattices, we turn now to some typically algebraic aspects of the category CCA, regarded (in a suitable sense) as a concrete category over the “classical” topological construct CSp of closure spaces. In fact, we shall show that CCA is, up to a categorical isomorphism, the category of algebras for a suitable regular monad over CSp — the favorite description of “good algebraic” categories. To that aim, we ﬁrst have to deﬁne a forgetful functor from CCA to CSp that has a left adjoint. Starting with a complete closure algebra (C, γ), we pass to the initial closure algebra along the residuated join map ν : AC → C, A → A with upper adjoint ν : C → AC, a → ↓ a and obtain a closure operation ν

γ = ν ◦ γ ◦ ν : AC → AC, A → ↓ γ( A),

hence a set-theoretical closure operator ν γ on the underlying set X of C. Thus, any CCA-object (C, γ) gives rise to a closure space Gc (C, γ) = (X, ν γ), and moreover, any CCA-morphism ϕ : (C, γ) → (D, δ) induces a continuous map, hence a CSp-morphism Gc ϕ = ϕ : Gc (C, γ) → Gc (D, δ). Indeed, for arbitrary subsets A of C, the morphism properties of ϕ yield ϕ(ν γ(A)) ⊆ ↓ ϕ(γ( A)) ⊆ ↓ δ(ϕ( A)) = ↓ δ( ϕ(A)) = ν δ(ϕ(A)). Thus, we have in fact a faithful functor Gc : CCA → CSp. In the opposite direction, we deﬁne a functor Pc : CSp → CCA by sending any continuous map f between closure spaces (X, Γ) and (Y, ∆) to the lifted map Pc f = f : PX → PY which is then, by deﬁnition, a CCA-morphism between the complete closure algebras Pc (X, Γ) = (PX, Γ) and Pc (Y, ∆) = (PY, ∆). The so deﬁned faithful functor Pc will turn out to be left adjoint to Gc . As in the case of the power set monad, the unit of the adjunction is given by the CSp-morphisms η(X,Γ) : (X, Γ) → Gc ◦ Pc (X, Γ) = (PX, ∪ Γ), x → {x}, where, by deﬁnition, ∪ Γ(Y) = P Γ( Y). For continuity of η(X,Γ) , note that η(X,Γ) (A) = {{a} : a ∈ A}, η(X,Γ) (Γ(A)) = {{x} : x ∈ Γ(A)} ⊆ P Γ(A) = ∪ Γ(η(X,Γ) (A)). Concerning the universal property required for adjointness, suppose that f : (X, Γ) → Gc (D, δ) = (Y, ν δ) is any CSp-morphism, and deﬁne ϕ : Pc (X, Γ) = (PX, Γ) → (D, δ) by ϕ(A) =

f (A).

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Then ϕ is lower adjoint to the map ψ : D → PX with ψ(b) = f (↓ b): ϕ(A) ≤ b ⇔ f (A) ≤ b ⇔ f (A) ⊆ ↓ b ⇔ A ⊆ f (↓ b) = ψ(b). Hence ϕ preserves arbitrary joins. Moreover, ϕ is a CCA-morphism, since for A ⊆ X, we have f (Γ(A)) ⊆ ν δ(f (A)) = ↓ δ( f (A)) and therefore ϕ(Γ(A)) = f (Γ(A)) ≤ δ( f (A)) = δ(ϕ(A)). By deﬁnition, the map Gc ϕ : Gc ◦ Pc (X, Γ) → Gc (D, δ) satisﬁes Gc ϕ(A) = ϕ(A) = f (A), hence Gc ϕ ◦ η(X,Γ) (x) =

f ({x}) = f (x),

and any join-preserving map ϕ satisfying Gc ϕ ◦ η(X,Γ) = f is uniquely determined by that equation. In all, this shows: Proposition 4.18. The functor Pc from the category CCA of complete closure algebras to the category CSp of closure spaces is left adjoint to the forgetful functor Gc in the opposite direction. As with any adjunction, the counit ε is uniquely determined by the equation Gc ε ◦ ηGc = idGc . Hence, in our concrete case it is given by ε(C,γ) : Pc ◦ Gc (C, γ) = (AC, ν γ) → (C, γ), A → A. By general categorical principles, the counit has the following universal property: for any complete closure algebra (D, δ), any closure space (X, Γ) and any CCAmorphism ϕ : Pc (X, Γ) → (D, δ), there is aunique continuous map f : (X, Γ) → Gc (D, δ) with ϕ = ε(D,δ) ◦ Pc f , i.e., ϕ(A) = f (A). Explicitly, that CSp-morphism is given by f = Gc ϕ ◦ η(X,Γ) , or elementwise by f (x) = ϕ({x}) (see Section 4.1). One could also invoke the so-called Taut Lift Theorem (see Wyler [93] and [2, 21.28]) in order to show that the adjoint situation between CL∨ and Set lifts to one between CCA and CSp, but we found it instructive to check the universal properties of the unit and the counit directly. The previously established adjunction between closure spaces and complete closure algebras gives rise, in a natural way, to a monad ˘ η, µ) P˘ = (P, which will be referred to as the power space monad. Here, P˘ is the composite functor Gc ◦ Pc : CSp → CSp, sending (X, Γ) to(PX, ∪ Γ), where ∪ Γ is the initial closure operator on PX (along the union map : P 2 X → PX): ∪ Γ(Y) = P Γ( Y), and any continuous map h : (X, Γ) → (Y, ∆) to h : (PX, ∪ Γ) → (PY, ∪ ∆). The unit η and the multiplication µ are deﬁned as for the power set monad: η(X,Γ) (x) = {x}, µ(X,Γ) (Y) = Y. What are the algebras of that power space monad? As we shall see, they are, up to a concrete isomorphism, just the complete closure algebras! By deﬁnition, a

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˘ is a pair ((X, Γ), f ) consisting of a closure space (X, Γ) and a continuous P-algebra ˘ map f from P(X, Γ) = (PX, ∪ Γ) to (X, Γ) so that f ◦ η(X,Γ) = id(X,Γ) i.e., f ({x}) = x, and ˘ i.e., f (Y) = f (f (Y)). f ◦ µ(X,Γ) = f ◦ Pf, From our preliminary considerations on the power set monad (Section 4.1) we know that these conditions force f to be the join map ν = C of a unique complete lattice C = (X, ≤). Now, it remains to ﬁnd, in a uniform manner, closure operations γ on C that uniquely correspond to the closure operators Γ on X so that the CCA-morphisms between the resulting closure algebras are just the homomorphisms between the ˘ corresponding P-algebras ((X, Γ), f ) and ((Y, ∆), g), i.e., the continuous maps ˘ ϕ : (X, Γ) → (Y, ∆) satisfying ϕ ◦ f = g ◦ Pϕ. The “right” idea is to take for γ : C → C the right shift of Γ, γ = Γν = ν ◦ Γ ◦ ν with γ(a) = Γ(↓ a) along the join map ν : AC → C. This gives in fact a closure operation having the desired properties: ˘ Proposition 4.19. For any P-algebra ((X, Γ), f ) there is a unique complete closure algebra (C, γ) such that f = ν = C and Γ = ν γ, viz. γ = Γν . Conversely, if (C, γ) is any complete closure algebra then Γ = ν γ is the unique closure operator ˘ with γ = Γν . such that ((X, Γ), ν) is a P-algebra ˘ Proof. Suppose ﬁrst that ((X, Γ), f ) is an arbitrary P-algebra and C = (X, ≤) is the associated complete lattice with f = ν = C . (1) A ⊆ Γ( Y) implies A ∈ Γ({ B : B ∈ Y}), for any Y ⊆ PX. This is just a reformulation of the continuity condition for f = C . Looking at the special case Y = {{b} : b ∈ B}, we see that (2) A ⊆ Γ(B) implies A ∈ Γ(B), which means that each closed set has a greatest element. On the other hand, taking Y = {B}, we deduce from (1): (3) Γ(B) ⊆ Γ({ B}), since a ∈ Γ(B) = Γ( {B}) implies a = {a} ∈ Γ({ B}). Conversely, setting A = B in (2) yields { B} ⊆ Γ(B) and consequently (4) Γ(B) = Γ({ B}). Next, we prove that (5) a ≤ b and b ∈ Γ(B) imply a ∈ Γ(B), which means that all closed sets are downsets. Indeed, by (4), a ∈ {a, b} ⊆ Γ({a, b}) = Γ({ {a, b}}) = Γ({b}) ⊆ Γ(B). Combining (2) with (5), we conclude that (6) each Γ-closed set is a principal ideal in C. Now, it is easy to show that (7) γ(a) = Γ(↓ a)

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deﬁnes a closure operation γ = Γν on C = (X, ≤). Indeed, by (6), a ≤ γ(b) ⇔ ↓ a ⊆ Γ(↓ b) ⇔ Γ(↓ a) ⊆ Γ(↓ b) ⇔ γ(a) ≤ γ(b). From (4), (6) and (7), we deduce Γ(A) = Γ({ A}) = ↓ Γ({ A}) = ↓ Γ(↓ A) = ↓ γ( A), i.e., Γ = ν γ. ν Concerning δ be any uniqueness, let closure operation on C with Γ = δ, i.e., Γ(A) = ↓ δ( A). Then δ(a) = ↓ δ( ↓ a) = Γ(↓ a) = γ(a). Now to the second part. Let (C, γ) be an arbitrary complete closure algebra. We have already remarked that(X, Γ) = (X, ν γ) is a closure space, so it remains to see why the join map f = C : (PX, ∪ Γ) → (X, Γ) is a CSp-morphism mak˘ ing ((X, Γ), f ) a P-algebra. Since f preserves joins, it suﬃces to show that f is continuous. For Y ⊆ PX and A = { Y : Y ∈ Y}, we get Γ( Y) = ↓ γ( Y) = ↓ γ( A) = ν γ(A) = Γ(A), hence

f (∪ Γ(Y)) = {

B : B ⊆ Γ(

Y) = Γ(A)} ⊆ Γ(A) = Γ(f (Y)).

˘ The P-algebra equations ˘ f ◦ η(X,Γ) = id(X,Γ) and f ◦ µ(X,Γ) = f ◦ Pf have already been veriﬁed at the levelof thepower set monad. The identity Γν (a) = Γ(↓ a) = ↓ γ( ↓ a) = γ(a) and the remark that the uniqueness claim follows from the ﬁrst part conclude the proof. The next proposition conﬁrms that we have chosen the right morphisms, not only from the topological, but also from the algebraic point of view: Proposition 4.20. Let C = (X, ≤) and D = (Y, ≤ ) be complete lattices. A map ϕ is a morphism between the closure algebras (C, γ) and (D, δ) iﬀ it is a ˘ homomorphism between the P-algebras ((X, ν γ), C ) and ((Y, ν δ), D ). Proof. That ϕ is a CCA-morphism means that (1) ϕ( A) = ϕ(A) for A ⊆ X, and (2) ϕ ◦ γ ≤ δ ◦ ϕ, i.e., ϕ(γ(x)) ≤ δ(ϕ(x)) for x ∈ X. ˘ for P-algebra ˘ Condition (1) is merely the characteristic identity ϕ ◦ C = D ◦Pϕ homomorphisms, while the second condition turns out to be equivalent to continuity of ϕ : (X, ν γ) → (Y, ν δ), i.e., (2’) ϕ(ν γ(A)) ⊆ ν δ(ϕ(A)) for A ⊆ X. Indeed, under hypothesis (1), condition (2) implies (2’): ϕ(↓ γ( A)) ⊆ ↓ ϕ(γ( A)) ⊆ ↓ δ(ϕ( A)) = ↓ δ( ϕ(A)) = ν δ(ϕ(A)), and conversely, (2’) implies (2): ↓ ϕ(γ( {x})) = ↓ ϕ(ν γ({x})) ⊆ ↓ ν δ({ϕ(x)}) = ↓ δ( {ϕ(x)}) = ↓ δ(ϕ(x)).

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In accordance with the corresponding general categorical construction, the comparison functor K (see Section 4.1 and [2, 20.37]) from the category CCA to the ˘ ˘ category CSpP of P-algebras is given by K((X, ≤), γ) = ((X, ν γ), ν). And we have seen that this is a concrete functorial isomorphism in the present situation. In other words: Theorem 4.21. The forgetful functor Gc from the category CCA of complete closure algebras to the category CSp of closure spaces is monadic; that is, the com˘ η, µ) is an isomorphism between parison functor of the power space monad P˘ = (P, ˘ CCA and the category of P-algebras. Moreover, P˘ is a regular monad, meaning that (i) the base category CSp has regular factorizations (which is the case because CSp is a topological construct) and (ii) Gc preserves regular epimorphisms (see [2, 20.21 and 20.32]). But the latter condition requires that regular epimorphisms in CCA are surjective (since the regular epimorphisms in CSp are the surjective continuous maps); and this is true as well, because CCA is a topological category over CL∨ , epimorphisms in CL∨ are surjective, and topological functors preserve (regular) epimorphisms (see [2, 21.13]). Thus, the category CCA has not only very good topological properties as a concrete category over the algebraic construct CL∨ , but also very good algebraic properties as a concrete category over the topological construct CSp, so that both topological and algebraic tools apply quite eﬀectively to this category. For example, the following properties are now immediate consequences of the general categorical machinery (see [2, Ch. 21]): Corollary 4.22. The category CCA • has as mono-sources the point-separating sources and as epi-sinks the jointly dense sinks, • is complete and cocomplete; in particular, it has arbitrary products and coproducts, both with underlying cartesian products, • is ﬁber-small, ﬁber-complete, well-powered and co-well-powered, • has regular factorizations. The following morphism classes coincide in CCA: • extremal monomorphisms = regular monomorphisms = embeddings, • extremal epimorphisms = regular epimorphisms = quotient maps. Finally, note that in CCA, • a source (ϕi : C → Ci )I is a mono-source iﬀ idC = {ϕi ◦ ϕi : i ∈ I} and • a sink (ϕi : Ci → C)I is an epi-sink iﬀ idC = {ϕi ◦ ϕi : i ∈ I}.

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, Logic, semantics, metamathematics. Papers from 1923 to 1938, Clarendon Press, Oxford, 1956, Translated by J. H. Woodger. MR0078296 (17,1171a) W. J. Thron, Lattice-equivalence of topological spaces, Duke Math. J. 29 (1962), 671–679. MR0146787 (26 #4307) P. Venugopalan, Z-continuous posets, Houston J. Math. 12 (1986), no. 2, 275–294. MR862043 (88b:06013) S. Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5, Cambridge University Press, Cambridge, 1989. MR1002193 (90j:03110) H. Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), no. 1, 112–126. MR1503392 J. B. Wright, E. G. Wagner, and J. W. Thatcher, A uniform approach to inductive posets and inductive closure, Theoret. Comput. Sci. 7 (1978), no. 1, 57–77. MR0480224 (58 #404) O. Wyler, On the categories of general topology and topological algebra, Arch. Math. (Basel) 22 (1971), 7–17. MR0287563 (44 #4767) E. Zermelo, Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904), no. 4, 514–516. MR1511281 , Neuer Beweis f¨ ur die Wohlordnung, Math. Ann. 65 (1908), 181–198.

Leibniz University, Faculty for Mathematics and Physics, Welfengarten 1, 30167 Hannover, Germany E-mail address: [email protected]

Contemporary Mathematics Volume 486, 2009

An Introduction to Quasi-uniform Spaces Hans-Peter A. K¨ unzi Abstract. It is the aim of this chapter to introduce the reader to the theory of quasi-uniform spaces. Besides discussing various bitopological aspects of that theory, we treat the important concepts of transitivity, precompactness and (bi)completeness. In the second part of the chapter we shall deal with several applications of quasi-uniformities to topological algebra and functional analysis. In particular examples from the theories of function spaces, isomorphism groups, hyperspaces and topological ordered spaces will be presented.

Contents 1. Introduction 2. Basic concepts and results 2.1. Quasi-uniform and quasi-pseudometric spaces 2.2. Transitive quasi-uniform and topological spaces 2.3. Bitopological spaces 2.4. Precompactness and its variants 2.5. Quasi-proximities 2.6. Various kinds of completeness 2.7. Bicompleteness 3. Some applications of quasi-uniformities 3.1. Paratopological groups 3.2. Uniform convergence 3.3. Isomorphism groups and equicontinuity 3.4. The Hausdorﬀ quasi-uniformity 3.5. Topological ordered spaces 3.6. Asymmetrically normed linear spaces References

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2000 Mathematics Subject Classiﬁcation. Primary 54E15, 54E05; Secondary 54E55, 54-02. Key words and phrases. quasi-uniformity, quasi-proximity, quasi-pseudometric, totally bounded, bicomplete, precompact, transitive, initial quasi-uniformity, ﬁnal quasi-uniformity, paratopological group, quasi-uniform function space, Hausdorﬀ quasi-uniformity, asymmetric norm, quasi-uniform isomorphism group, completely regularly ordered space. The author was supported in part by the South African NRF Grant FA2006022300009. c 2009 2008 American Mathematical Society

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1. Introduction This section brieﬂy sketches the historic background of the theory that will be discussed in the following. It also contains some hints to more advanced readings on the subject. In 1937 Weil [154] published his booklet on (entourage) uniformities. Three years later Tukey [150] suggested an approach to uniformities via uniform coverings. The study of quasi-uniformities started in 1948 with Nachbin’s investigations (see e.g., [127]) on uniform preordered spaces, that is, those topological preordered spaces whose preorder is given by the intersection of the entourages of a (ﬁlter) quasi-uniformity U and whose topology is induced by the associated supremum uniformity U ∨ U −1 . In a certain sense, explained below, uniformities can be identiﬁed with families of pseudometrics on a set. Similarly, quasi-uniformities can be understood as families of quasi-pseudometrics [133]. The asymmetric distance functions called quasi-pseudometrics naturally ﬁrst appeared in the deﬁnition of the Hausdorﬀ metric of a metric space [45]. Each quasi-uniformity induces a topology. The ﬁlter U −1 of inverse relations of a quasi-uniformity U is also a quasi-uniformity. Therefore quasi-uniformities are intrinsically linked to the theory of bitopological spaces (in the sense of Kelly [62]). The concept of a quasi-proximity [130, 147] turned out to be categorically equivalent to the concept of a totally bounded quasi-uniformity. Cs´asz´ar [14] developed the theory of the bicompletion for quasi-uniform spaces, which was subsequently popularized by Fletcher, Lindgren and Salbany [29, 143]. It generalized the theory of the completion from the metric and uniform setting to the asymmetric context. Since its underlying idea is symmetric, many further attempts were made to ﬁnd other asymmetric completion theories (see for instance the contributions of De´ ak and Doitchinov [18, 23]). The work of Fox, Junnila and Kofner established that for the class of completely regular spaces the three concepts of a γ-space (= a topological space admitting a local quasi-uniformity with a countable base), a quasi-pseudometrizable space and a non-archimedeanly quasipseudometrizable space are distinct [34, 63]; furthermore the ﬁnest compatible quasi-uniformities of metrizable and suborderable (= generalized ordered) spaces were shown to have a base consisting of transitive entourages [58, 65]. Using quasi-pseudometrics several authors tried to ﬁnd a common generalization of the established theories of metric spaces and (partially) ordered sets in order to unify common classical results like ﬁxed point theorems and completions. Recently such investigations were often motivated by problems from theoretical computer science [146, 148]. In connection with those investigations also the idea to replace the reals in the deﬁnition of a quasi-pseudometric by some more general structure was thoroughly investigated (e.g., in studies about Kopperman’s continuity spaces, see [68]). In such theories a quasi-uniformity is interpreted as a kind of generalized quasi-pseudometric. Quasi-uniform structures were also studied in algebraic structures (see for instance the work of Romaguera and his collaborators [40, 122]). In particular the study of paratopological groups and asymmetrically normed linear spaces with the help of quasi-uniformities is well known. Important classical results about quasi-uniformities were extended to fuzzy topology. Based on Lowen’s theory of approach spaces [121], the concept of an approach quasiuniformity was introduced and investigated. In recent years many results about quasi-uniformities were generalized to a pointfree setting (see for instance the work of Picado and his colleagues [26]).

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In this chapter we wish to introduce the reader to many of the aforementioned ideas. Obviously, for the more sophisticated topics and results we refer him/her to the pertinent literature. In this connection let us mention ﬁrst two survey articles that mainly deal with some historic aspects of our topic [5, 87]. Then we would like to list the two published monographs about quasi-uniform spaces [30, 126] and several more specialized survey articles [8, 15, 18, 64, 70, 80, 81, 90]. Of course, also many well-known books on topology contain much information about uniform structures [25, 51, 54, 56]. The short articles on (quasi-)uniformities in the recent Encyclopedia of General Topology [44, 37, 93] should also be useful to the interested reader. Throughout we shall make use of the entourage approach to the theory of (quasi-)uniformities. While the theory of covering uniformities [54] is often easier to handle than the equivalent theory of entourage uniformities, a theory of quasi-uniformities based on (pair)covers is possible [36], but often rather cumbersome. Nevertheless it should be stressed that the latter approach has its merits in pointfree topology and bitopology. 2. Basic concepts and results This section contains the basic deﬁnitions of the theory, illustrated by some facts and examples. Our exposition assumes familiarity with elementary concepts from general topology, but otherwise it is nearly self-contained. In particular the reader need not possess prior knowledge of the theory of uniform spaces. Our notation is standard. For a subset A of a topological space the closure of A is denoted by cl A or A, and the interior of A by int A. 2.1. Quasi-uniform and quasi-pseudometric spaces. It is well known that for topological spaces there is no obvious way to compare the sizes of neighborhoods of distinct points. Hence important concepts from the theory of metric spaces like total boundedness and uniform continuity cannot be formulated for topological spaces. Of course, one natural way to overcome this problem is to choose for a given topological space X some suﬃciently large index set I and to consider for each point x ∈ X an indexed neighborhood base {Ni (x) : i ∈ I} at x. However this is not yet what we want. For instance certainly there should also be some order structure ≥ on the index set I so that for instance if i1 , i2 ∈ I with i1 ≥ i2 , then we have Ni1 (x) ⊆ Ni2 (x) whenever x ∈ X. In fact, it turns out to be useful to put even more structure on the set I in order to be able to formulate at least a halving condition for the chosen families of neighborhoods, which is reminiscent of the triangle inequality. We are then naturally led to the fairly simple, but nevertheless very useful concept of a quasi-uniform space. Definition 2.1.1. A quasi-uniformity U on a set X is a ﬁlter on X × X such that (1) Each member U of U contains the diagonal ∆X = {(x, x) : x ∈ X} of X; (2) For each U ∈ U there is V ∈ U such that V 2 ⊆ U . (Here V 2 = V ◦ V = {(x, z) ∈ X × X : there is y ∈ X such that (x, y) ∈ V and (y, z) ∈ V }. Hence ◦ is the usual composition of binary relations.) The members U ∈ U are called the entourages of U. The elements of X are called points. The pair (X, U) is called a quasi-uniform space.

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Note that the ﬁlter on X × X generated by a (ﬁlter)base B of reﬂexive binary relations of X is a quasi-uniformity if and only if for each B ∈ B there is C ∈ B such that C 2 ⊆ B. Example 2.1.2. Let T be a reﬂexive and transitive relation, that is, a preorder, on a set X. Then the ﬁlter generated by the base {T } on X ×X is a quasi-uniformity on X. Furthermore for any quasi-uniformity U on X, U is a preorder on X. Definition 2.1.3. Let [0, ∞) be the set of nonnegative reals. A quasi-pseudometric d on a set X is a function d : X ×X → [0, ∞) such that d(x, x) = 0 whenever x ∈ X, and d(x, z) ≤ d(x, y)+d(y, z) whenever x, y, z ∈ X. The quasi-pseudometric d−1 : X × X → [0, ∞) deﬁned by d−1 (x, y) = d(y, x) whenever x, y ∈ X is called the quasi-pseudometric conjugate to d. Symmetrizing d, we shall set ds = max{d, d−1 }. As usual, a quasi-pseudometric d on X satisfying d(x, y) = d(y, x) whenever x, y ∈ X is called a pseudometric. Of course, for any quasi-pseudometric d on a set X, ds is a pseudometric on X. A quasi-pseudometric d on a set X is called non-archimedean if d satisﬁes a strong form of the triangle inequality, namely d(x, z) ≤ max{d(x, y), d(y, z)} whenever x, y, z ∈ X. Let d be a quasi-pseudometric on a set X. For each > 0 set U = {(x, y) ∈ 2 X ×X : d(x, y) < }.1 Since for each > 0, U/2 ⊆ U , the ﬁlter on X ×X generated by the base {U : > 0} is a quasi-uniformity and is called the quasi-pseudometric quasi-uniformity Ud induced by d on X. Definition 2.1.4. Each quasi-uniformity U on a set X induces a topology τ (U) as follows: For each x ∈ X and U ∈ U set U (x) = {y ∈ X : (x, y) ∈ U }. A subset G of X belongs to τ (U) if and only if for each x ∈ G there exists U ∈ U such that U (x) ⊆ G. The following simple calculations determine the interior operator of the induced topology τ (U) of a quasi-uniform space (X, U): For any A ⊆ X we set o(A) = {y ∈ A : there is W ∈ U such that W (y) ⊆ A}. We ﬁrst note that o(A) is τ (U)-open: Let y ∈ o(A). By condition (2) there is H ∈ U such that H 2 (y) ⊆ A. Thus H(y) ⊆ o(A) and therefore indeed o(A) ∈ τ (U). On the other hand, let y ∈ intτ (U) A. Then there exists some G ∈ τ (U) such that y ∈ G ⊆ A. Therefore there is W ∈ U such that W (y) ⊆ G ⊆ A. Consequently y ∈ o(A) and intτ (U) A ⊆ o(A). Altogether we conclude that o(A) = intτ (U) A. In particular x ∈ intτ (U) U (x) whenever x ∈ X and U ∈ U. It follows that the neighborhood ﬁlter at x ∈ X with respect to the topology τ (U) is given by U(x) = {U (x) : U ∈ U}. Let (X, U) be a quasi-uniform space. Then τ (U) is a T0 -topology if and only if U is a partial order; furthermore τ (U) is a T1 -topology if and only if U is equal to the diagonal of X (compare [30, Proposition 1.9]). Given a quasi-pseudometric d on a set X, then τ (Ud ) is the standard quasipseudometric topology τ (d) on X having all the open balls Bd (x, ) = {y ∈ X : d(x, y) < } (x ∈ X, ∈ (0, ∞)) as a base. Observe that τ (d) is a T0 -topology if and only if for all x, y ∈ X, d(x, y) = d(y, x) = 0 implies that x = y. In this chapter a quasi-pseudometric d that satisﬁes the latter condition will be called a T0 -quasi-pseudometric. Similarly, note that τ (d) is a T1 -topology if and only if for 1Note that this notation does not show the dependence of U on d.

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all x, y ∈ X, d(x, y) = 0 implies that x = y. Quasi-pseudometrics satisfying that condition are traditionally called quasi-metrics in mathematics. In this chapter we shall adopt this convention, but remark that in computer science the term “quasimetric” is often used for the concept that we here call “T0 -quasi-pseudometric”. For an arbitrary quasi-pseudometric d an interesting simple connection between the topologies τ (d) and τ (d−1 ) is given by the following result. Remark 2.1.5 ([73, Theorem 4]). Let d be a quasi-pseudometric on a set X. If τ (d) has a base B of (inﬁnite) cardinality κ, then τ (d−1 ) has a base of cardinality κ. Proof. For each B ∈ B let SB,n = {x ∈ B : x ∈ B ⊆ Bd (x, 2−n )}. Set S = {SB,n : B ∈ B, n ∈ ω}. Furthermore let B = {Bd−1 (S, 2−n ) : S ∈ S, n ∈ ω}.2 Note that the cardinality of B is not larger than the cardinality of B. Observe also that SB,n × SB,n ⊆ U2−n whenever n ∈ ω and B ∈ B. Fix n ∈ ω and x ∈ X. Since B is a base for τ (d), there is B ∈ B such that x ∈ B ⊆ Bd (x, 2−(n+1) ). Consequently x ∈ SB,n+1 ⊆ Bd−1 (SB,n+1 , 2−(n+1) ) ⊆ Bd−1 (x, 2−n ), since SB,n+1 ⊆ Bd−1 (x, 2−(n+1) ). We have shown that B is a base for τ (d−1 ).

Definition 2.1.6. A map f : (X, U) → (Y, V) between two quasi-uniform spaces (X, U) and (Y, V) is called uniformly continuous provided that for each V ∈ V there is U ∈ U such that (f × f )(U ) ⊆ V . Here f × f is the product map from X × X into Y × Y deﬁned by (f × f )(x1 , x2 ) = (f (x1 ), f (x2 )) (x1 , x2 ∈ X). Of course the identity map on any quasi-uniform space into itself is always uniformly continuous. Furthermore the composition of two uniformly continuous maps between quasi-uniform spaces is readily seen to be uniformly continuous. Two quasi-uniform spaces are called isomorphic provided that there is a uniformly continuous bijection between them whose inverse map is uniformly continuous, too. It is obvious from the deﬁnitions that each uniformly continuous map between quasi-uniform spaces is continuous with respect to the induced topologies. Somewhat surprisingly, each topological space (X, τ ) is quasi-uniformizable (compare [129]), that is, there is a quasi-uniformity U on X such that τ (U) = τ . If the latter condition is satisﬁed, we shall say that U is compatible with the topology τ or that (X, τ ) admits U. In fact for each subset A of X set SA = [(X \ A) × X] ∪ [X × A]. Note that SA is a preorder on X.3 We conclude that the collection {SG : G ∈ τ } yields a subbase of a quasi-uniformity on X. This quasi-uniformity is called the Pervin quasi-uniformity P(τ ) of the topological space (X, τ ). One checks that indeed it induces the topology τ on X. A collection C of open sets of a topological space X is called interior preserving provided that E is open whenever E ⊆ C. Observe that if T is a preorder on a topological space X such that T (x) is open whenever x ∈ X, then {T (x) : x ∈ X} is interior preserving. We note that for the following construction some authors assume that each collection C is also a cover of X. Obviously this is merely a matter of taste, since it can always be achieved by just adding the set X to C. 2Of course, we use the standard convention that U (S) = S s∈S U (s) for any S ⊆ X and

> 0.

3Let us mention that some authors write equivalently S = [(X \ A) × X] ∪ [A × A]. A

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Example 2.1.7 (The Fletcher construction). Let D be a collectionof interiorpreserving open collections C of a given topological space X such that C∈D C is a subbase of X. For each C ∈ D set T = S . Observe that for each x ∈ X, C C C∈C TC (x) = {C : x ∈ C ∈ C}. Since each TC is a preorder, it is readily checked that {TC : C ∈ D} yields a subbase for a compatible quasi-uniformity UD on X. In fact if D consists of the collection of all ﬁnite open collections of X, then UD is the Pervin quasi-uniformity of X. We recall that a family L of subsets of a topological space (X, τ ) is called well-monotone provided that the partial order ⊆ of set-inclusion is a well-order on L. The quasi-uniformity UD where D is the collection of all well-monotone open collections of X is called the well-monotone quasi-uniformity M(τ ) of X. If U is a quasi-uniformity on a set X, then the ﬁlter U −1 = {U −1 : U ∈ U} on X × X is also a quasi-uniformity on X. (Here and in the following, R−1 = {(x, y) ∈ X × X : (y, x) ∈ R} is the inverse of the binary relation R on X. A binary relation R is called symmetric if R = R−1 .) The quasi-uniformity U −1 is called the conjugate of U. A quasi-uniformity that is equal to its conjugate is called a uniformity. As in the case of quasi-pseudometrics there exist intriguing connections between the conjugate topologies induced by a quasi-uniformity. For instance it has been shown that a metrizable topological space X is completely metrizable if and only if it admits a quasi-uniformity U such that the topology induced by the conjugate quasi-uniformity U −1 on X is compact (see [91]). If U1 and U2 are two quasi-uniformities on a set X, then we say that U1 is coarser than U2 (or U2 is ﬁner than U1 ) provided that U1 ⊆ U2 . Observe that for any topological space (X, τ ), M(τ ) is a compatible quasi-uniformity on X that is ﬁner than P(τ ). The set of all quasi-uniformities on a set X ordered by set-theoretic inclusion is a complete lattice. The smallest element of this lattice is the indiscrete uniformity I = {X × X}. The largest element is the discrete uniformity D which consists of all reﬂexive binary relations on X. Obviously the supremum i∈IUi of a family (Ui )i∈I of quasi-uniformities ona set X is generated by the subbase i∈I Ui on X × X. It is easy to see that τ ( i∈I Ui ) = i∈I τ (Ui ). It follows that each topological space (X, τ ) admits a ﬁnest (compatible) quasi-uniformity, which will be called the ﬁne quasi-uniformity of X and denoted by FN (τ ) in the following. In particular the union of a quasi-uniformity U and its conjugate U −1 yields a subbase of the coarsest uniformity ﬁner than U. It will be denoted by U s in this chapter and is called the symmetrization of U. We note that if a function f : (X, U) → (Y, V) between quasi-uniform spaces (X, U) and (Y, V) is uniformly continuous, then f : (X, U −1 ) → (Y, V −1 ) and f : (X, U s ) → (Y, V s ) are also uniformly continuous. The following observation motivates the concept of a functorial quasi-uniformity (see [8]): If f : (X, τ ) → (Y, ρ) is a continuous map between topological spaces (X, τ ) and (Y, ρ), then f : (X, P(τ )) → (Y, P(ρ)) is uniformly continuous. It is not diﬃcult to see that analogous statements hold for the well-monotone and the ﬁne quasi-uniformity. Hence one deﬁnes: Let T denote the (obvious) forgetful functor from the category QU of quasi-uniform spaces and uniformly continuous maps to the category Top of topological spaces and continuous maps. A functorial quasiuniformity is a functor F : Top → QU such that T F = 1, i.e., F is a T -section.

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Definition 2.1.8. A quasi-uniform space (X, U) is said to have the Lebesgue property provided that for each open cover C of (X, τ (U)) there is U ∈ U such that {U (x) : x ∈ X} is a reﬁnement of C. The following result is well known for compact metric spaces. Proposition 2.1.9 (see e.g., [30, Corollary 5.1]). If (X, U) is a quasi-uniform space such that τ (U) is compact, then (X, U) has the Lebesgue property. Proof. Let C be an open cover of (X, τ (U)). For each x ∈ X there are C ∈ C and Vx ∈ U such that Vx2 (x) ⊆ C.The open cover {intτ (U) Vx (x) : x ∈ X} of X has a ﬁnite subcover. Hence X = x∈F Vx (x) for some ﬁnite subset F of X. Set V = x∈F Vx . Clearly V ∈ U. Fix x ∈ X. There is y ∈ F such that x ∈ Vy (y). Thus V (x) ⊆ Vy2 (y). We have shown that (X, U) has the Lebesgue property. Definition 2.1.10 (compare e.g., [102, Lemma 4]). A quasi-uniform space (X, U) is called small-set symmetric provided that τ (U −1 ) ⊆ τ (U). (The conjugate quasi-uniformity U −1 is then called point-symmetric [30, Proposition 2.21].) Proposition 2.1.11. Let (X, U) be a quasi-uniform space such that τ (U) is a compact T1 -topology. Then U is point-symmetric (see e.g., [30, p. 38]). Moreover each countably compact quasi-metric space is compact (compare [128]). Proof. Let y ∈ X. In order to reach a contradiction, suppose that there is V ∈ U such that {U −1 (y) \ V (y) : U ∈ U} is a base for a ﬁlter F on X. Since (X, τ (U)) is compact, F has a cluster point x in (X, τ (U)). Hence for each U ∈ U, U (x) ∩ U −1 (y) = ∅. Since τ (U) is a T1 -topology, it follows that x = y. But then V (x) ∩ (X \ V (x)) = ∅. We have reached a contradiction and conclude that τ (U) ⊆ τ (U −1 ). For the second statement we ﬁrst use that in a quasi-pseudometric space X the neighborhood ﬁlter of a point y ∈ X has a countable base. So we see with the help of the preceding argument that Ud is point-symmetric, since τ (d) is countably compact. Then let C be any open cover of (X, τ (d)). For each x ∈ X there are Cx ∈ C and (by point-symmetry of Ud ) nx ∈ ω such that Bd−1 (x, 2−nx ) ⊆ Cx . Suppose that C does not possess a ﬁnite subcover. By countable compactness of (X, τ (d)) this means that C does not have a countable subcover. Hence we can inductively choose a transﬁnite sequence (xα )α<ω1 of points in X as follows: For each α < ω1 ﬁnd xα ∈ X \ β<α Cxβ . There is k ∈ ω such that Bk = {xα : α < ω1 and nxα = k} is uncountable. Find a strictly increasing sequence of ordinals (αn )n∈ω such that each point xαn belongs to this Bk . Let δ = sup{αn : n ∈ ω}. By countable compactness of (X, τ (d)) there is x ∈ k∈ω {xαn : n ≥ k, n ∈ ω}. Clearly by construction of the sequence (xαn )n∈ω the point x cannot belong to −k C ) whenever n ∈ ω. Hence xαn ∈ α<δ xα . It follows that x ∈ Bd−1 (xαn , 2 −k Bd (x, 2 ) whenever n ∈ ω — a contradiction. We conclude that (X, τ (d)) is compact. We remark that the preceding result does not hold if we omit the T1 -condition: Let the ﬁrst uncountable ordinal X = ω1 be endowed with the quasi-pseudometric d on X where d(x, y) = 0 if x ≥ y and d(x, y) = 1 otherwise. Evidently τ (d) is countably compact, but not compact. A quasi-uniform space (X, U) is called locally symmetric [30, p. 37] provided that for all U ∈ U and x ∈ X, there is V ∈ U such that V −1 (V (x)) ⊆ U (x).

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Obviously each locally symmetric quasi-uniform space is point-symmetric. It is readily seen that a topological space X admits a locally symmetric quasi-uniformity if and only if X is regular: To this end note that for any quasi-uniform space (X, U) and any subset A ⊆ X we have clτ (U) A = U∈U U −1 (A). In particular the Pervin quasi-uniformity of a regular topological space is locally symmetric. It is known that the following result requires some symmetry condition on the codomain (compare e.g., [82, end of p. 88], where the identity function between the spaces given by the coarsest and the ﬁnest compatible quasi-uniformities on the real unit interval equipped with the euclidean topology is considered). Proposition 2.1.12 ([123, Proposition 2.1]). Let (X, U) be a quasi-uniform space satisfying the Lebesgue property, let (Y, V) be a small-set symmetric quasiuniform space and let f : (X, τ (U)) → (Y, τ (V)) be a continuous map. Then f : (X, U) → (Y, V) is uniformly continuous. (Indeed, f : (X, U) → (Y, V s ) is uniformly continuous.) Proof. Let V ∈ V. Choose W ∈ V such that W 2 ⊆ V . Then for each x ∈ X there is Wx ∈ V such that Wx ⊆ W and Wx (f (x)) ⊆ W −1 (f (x)) by small-set symmetry of (Y, V). By continuity of f the collection C = {intτ (U) f −1 [Wx (f (x))] : x ∈ X} is an open cover of X. There is U ∈ U such that {U (x) : x ∈ X} reﬁnes C by the Lebesgue property of (X, U). Fix x ∈ X. Then there exists y ∈ X such that U (x) ⊆ f −1 Wy (f (y)). Thus f (U (x)) ⊆ Wy (f (y)) ⊆ W −1 (f (y)) and consequently f (y) ∈ W (f (x)). Therefore f (U (x)) ⊆ Wy (f (y)) ⊆ W 2 (f (x)) ⊆ V (f (x)). Hence f : (X, U) → (Y, V) is uniformly continuous. The remark in parentheses immediately follows from this result, since by small-set symmetry of V the map f : (X, τ (U)) → (Y, τ (V s )) is also continuous and V s , as any uniformity, is small-set symmetric. Corollary 2.1.13. A small-set symmetric quasi-uniformity U on a set X that induces a compact topology τ (U) is a uniformity. Proof. Consider the identity map idX : (X, U) → (X, U s ). By Propositions 2.1.9 and 2.1.12 it is uniformly continuous. Therefore U s ⊆ U and thus U is a uniformity. It is rather delicate (see e.g., [153, Theorem 2.2], where the analogous uniform case is dealt with) to explicitly describe a (sub)base of the inﬁmum of an arbitrary family (Ui )i∈I of quasi-uniformities on a set in terms of the entourages of the quasiuniformities Ui . In general the inﬁmum of two compatible quasi-uniformities on a topological space is not compatible (see e.g., [99, Proposition 1]). Nevertheless it is easy to see that conjugation commutes with arbitrary inﬁma and suprema in the lattice of quasi-uniformities (compare [16, Introduction]); in particular for any quasi-uniformity U on a set, U ∧ U −1 is a uniformity. A map f : (X, U) → (Y, V) between quasi-uniform spaces (X, U) and (Y, V) is called uniformly open provided that for each U ∈ U there is V ∈ V such that V (f (x)) ⊆ f (U (x)) whenever x ∈ X. Proposition 2.1.14 (compare e.g., [85, Proposition 3.1]). Let (X, U) be a compact uniform space and (Y, V) an arbitrary quasi-uniform space. Then any map f : (X, U) → (Y, V) that is open and continuous is uniformly open.

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Proof. Let U ∈ U. There is P ∈ U such that P 2 ⊆ U . Because f is open, for each a ∈ X we ﬁnd Wa ∈ V such that Wa2 (f (a)) ⊆ f (P (a)). By continuity of f and since U is a uniformity, we can consider the open cover {intτ (U) (P −1 (a) ∩ a ∈ X} of X. Since X is compact, there is a ﬁnite subset F f −1 Wa (f (a))) : of X such that a∈F intτ (U) (P −1 (a) ∩ f −1 Wa (f (a))) = X. Set W = a∈F Wa and note that W ∈ V. Consider any x ∈ X. There is b ∈ F such that x ∈ P −1 (b) ∩ f −1 Wb (f (b)). Therefore f (x) ∈ Wb (f (b)) and W (f (x)) ⊆ Wb2 (f (b)) ⊆ f (P (b)) ⊆ f (P 2 (x)) ⊆ f (U (x)). We have proved that f is uniformly open. Definition 2.1.15. Let X and I be sets and suppose that for each i ∈ I there is given a map fi : X → Yi into a quasi-uniform space (Yi , Vi ). Then there exists a coarsest quasi-uniformity V on X such that all maps fi are uniformly continuous, the so-called initial quasi-uniformity. It is generated by the subbase {(fi × fi )−1 (V ) : i ∈ I, V ∈ Vi } on X × X. One checks that the initial quasi-uniformity V induces the coarsest topology on X such that all maps fi : X → (Yi , τ (Vi )) (i ∈ I) are continuous, that is, the corresponding initial topology. Note also that the initial quasi-uniformity V is a uniformity if each quasi-uniformity Vi is a uniformity. We next give two well-known applications of the concept of the initial quasi-uniformity. Let ((Xi , Ui ))i∈I be any (nonempty) family of quasi-uniform spaces. Let i∈I Xi be the set-theoretic product of the family (Xi )i∈I and let πj : i∈I Xi → Xj (j ∈ I) be the projection. Then the coarsest quasi-uniformity on i∈I Xi that makes all those projections uniformly continuous is called the product quasi-uniformity. It induces the product topology of the topological spaces (Xi , τ (Ui ))i∈I . Let (X, U) be a quasi-uniform space and let A be any subset of X. The coarsest quasi-uniformity on A such that the inclusion map e : A → X is uniformly continuous is called the subspace quasi-uniformity U|A : It consists of the entourages U |A := U ∩ (A × A) with U ∈ U and induces the subspace topology (τ (U))|A of (X, τ (U)) on A. As an exercise the reader may want to check the following facts. Let (X, τ ) be a topological space and let A be any subset of X. Then (P(τ ))|A = P(τ |A ) and (M(τ ))|A = M(τ |A ). The corresponding result does not hold for the ﬁne quasiuniformity (compare e.g., [30, Theorem 2.20]). Indeed the restriction of the ﬁne quasi-uniformity of the usual euclidean topology on the set R of the reals to the set Q of the rationals does not yield the ﬁne quasi-uniformity on the subspace Q of R. We ﬁnish this subsection with two results that show the power of uniform conditions. The following proposition extends to asymmetric topology a known result of Vidossich [151, Theorem] about uniformly continuous maps from a subspace of a product of uniform spaces into a metric space. It is important to note that corresponding results of merely topological character require additional assumptions on the factor spaces (see e.g., [25, Problem 2.7.12(d)]). Proposition 2.1.16 ([24, Theorem 3.1]). Let Z be any subset of the product of an arbitrary family of quasi-uniform spaces (Xi , Ui ) (i ∈ I). Then every uniformly continuous map f : Z → (Y, d) into a T0 -quasi-pseudometric space (Y, d) has the form g ◦ (π|Z ) where for some countable C ⊆ I, π : i∈I Xi → i∈C Xi is the projection deﬁned by π((xi )i∈I ) = (xi )i∈C and g : π(Z) → Y is a uniformly continuous map. (One says that f depends on the countably many coordinates in C.)

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Proof. By uniform continuity of f , for each n ∈ ω there are a ﬁnite subset In 1 of I and entourages Ui ∈ Ui where i ∈ In such that d(f (x), f (y)) < n+1 whenever −1 (x, y) ∈ [ i∈In (πi × πi ) Ui ] ∩ (Z × Z). Put C = n∈ω In and, as stated above, let π be the projection given by (xi )i∈I → (xi )i∈C . For every x ∈ π(Z), let zx be a point of Z ∩ π −1 ({x}). We deﬁne the map g : π(Z) → Y by the rule x → f (zx ). 1 If z , z ∈ Z have the same image under π, then d(f (z ), f (z )) < n+1 whenever n ∈ ω, since In ⊆ C. This implies by symmetry that d(f (z ), f (z )) = 0 = d(f (z ), f (z )), that is, f (z ) = f (z ). It follows that f = g ◦ (π|Z ). Let n ∈ ω. Suppose that z, z ∈ Z are such that (π(z)i , π(z )i ) ∈ Ui whenever i ∈ In . Then 1 d(g(π(z)), g(π(z ))) = d(f (z), f (z )) < n+1 by assumption. Thus g is uniformly continuous. Let Y and I be sets and suppose that for each i ∈ I there is given a map fi : Xi → Y from a quasi-uniform space (Xi , Ui ) to Y . Then the ﬁnest quasiuniformity on Y ×Y coarser than the ﬁlter {U : ∆Y ⊆ U ⊆ Y ×Y and (fi ×fi )−1 U ∈ Ui for each i ∈ I} is the ﬁnest quasi-uniformity V on Y such that all maps fi are uniformly continuous, the so-called ﬁnal quasi-uniformity. Observe that V is a uniformity if all quasi-uniformities Ui are uniformities. In particular we call a uniformly continuous surjection f : X → Y between quasiuniform spaces X and Y quotient if Y carries the ﬁnest quasi-uniformity that makes f uniformly continuous (compare [12, 49]).4 Simple examples show that the ﬁnal quasi-uniformity V need not be compatible with the ﬁnal topology on Y that is determined by the family of all maps fi : (Xi , τ (Ui )) → Y (i ∈ I). For instance the set R equipped with its usual metric uniformity, together with the partition R/∼ = {(−∞, 0], (0, ∞)} of R and the corresponding quotient map q : R → R/∼ witness this fact. Example 2.1.17. If (X, U) is a quasi-uniform space, then U ∩ ( U)−1 is an equivalence relation ∼ on X. The ﬁnal quasi-uniformity on the quotient set X/∼ with respect to the quotient map q : X → X/∼ is called the T0 -quotient of U. In the category Top of topological spaces and continuous maps products of quotient maps are not necessarily quotient maps [25, Example 2.4.20]. However the situation is better in the quasi-uniform setting as we show next. Our proof follows the presentation in [24], which is based on the proof of [53] given for the analogous result in the category of uniform spaces. During the argument for a quasi-uniform space X the quasi-uniformity of the space X will be denoted by UX . Furthermore for any quasi-uniform space X, DX will denote the underlying set of X equipped with the discrete uniformity. Lemma 2.1.18. Let X and Y be quasi-uniform spaces. Then the product quasiuniformity UX×Y is the ﬁnest quasi-uniformity that is coarser than both UX×DY and UDX×Y . Proof. We have to show that UDX×Y ∧UX×DY = UX×Y . In order to prove the nontrivial inclusion, let W, Z ∈ UDX×Y ∧UX×DY be such that W 2 ⊆ Z. Then there are U ∈ UX and V ∈ UY such that U (x)×{y} ⊆ W (x, y) and {x}×V (y) ⊆ W (x, y) whenever x ∈ X and y ∈ Y . Fix x ∈ X and y ∈ Y . Consider any (a, b) ∈ 4It is well known and easy to see that an onto uniformly continuous map q : X → Y between quasi-uniform spaces X and Y is quotient if and only if the facts that h : Y → Z is any map into a quasi-uniform space Z and h ◦ q is uniformly continuous imply that h is uniformly continuous.

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U (x)×V (y). Then (a, b) ∈ U (x)×{b} ⊆ W (x, b) and (x, b) ∈ {x}×V (y) ⊆ W (x, y). Thus (a, b) ∈ W 2 (x, y) ⊆ Z(x, y). Therefore U (x)×V (y) ⊆ Z(x, y) whenever x ∈ X and y ∈ Y , and consequently Z ∈ UX×Y . The stated equality is established. Lemma 2.1.19. Let f : X → Y be a quotient map between quasi-uniform spaces X and Y . Moreover let g : D → E be any surjection, where D and E are uniform spaces carrying the discrete uniformity. Then f × g : X × D → Y × E is a quotient map between quasi-uniform spaces. Proof. Let Z be a quasi-uniform space and let h : Y × E → Z be any map between the carrier sets of Y ×E and Z such that h◦(f ×g) is uniformly continuous. 2 Let (Zn )n∈ω be a sequence of entourages belonging to UZ such that Zn+1 ⊆ Zn −1 whenever n ∈ ω. For each n ∈ ω set Wn = (h × h) Zn . For each n ∈ ω, Wn is 2 reﬂexive and we see that Wn+1 ⊆ Wn . Furthermore ((f ×g)×(f ×g))−1 Wn ∈ UX×D whenever n ∈ ω, since h ◦ (f × g) is uniformly continuous. For each n ∈ ω set Vn = {(y1 , y2 ) ∈ Y × Y : ((y1 , e), (y2 , e)) ∈ Wn whenever e ∈ E}. 2 For each n ∈ ω, Vn is reﬂexive and Vn+1 ⊆ Vn whenever n ∈ ω: Indeed ﬁx n ∈ ω. Suppose that y ∈ Y . For any e ∈ E we have ((y, e), (y, e)) ∈ Wn . Therefore Vn is indeed reﬂexive. Suppose that ((y1 , e), (y2 , e)), ((y2 , e), (y3 , e)) ∈ Wn+1 whenever 2 ⊆ Vn . e ∈ E; then ((y1 , e), (y3 , e)) ∈ Wn whenever e ∈ E. Thus Vn+1 Observe also that if Vn ∈ UY whenever n ∈ ω, then Wn ∈ UY ×E whenever n ∈ ω, since {((y1 , e), (y2 , e)) : (y1 , y2 ) ∈ Vn , e ∈ E} ∈ UY ×E and by the deﬁnition of Vn , {((y1 , e), (y2 , e)) : (y1 , y2 ) ∈ Vn , e ∈ E} ⊆ Wn . Consequently we shall conclude that h is uniformly continuous, and thus f × g is a quotient map, if we can verify that Vn ∈ UY whenever n ∈ ω. In order to see that each Vn belongs to UY , we only need to show that (f × f )−1 (Vn ) ∈ UX whenever n ∈ ω, because f is a quotient map. Fix n ∈ ω. Since ((f × g) × (f × g))−1 Wn ∈ UX×D , there is Mn ∈ UX such that {((f (m1 ), g(d)), (f (m2 ), g(d))) : (m1 , m2 ) ∈ Mn and d ∈ D} ⊆ Wn . Let (m1 , m2 ) ∈ Mn . Consider any e ∈ E. Since g is surjective, there exists d ∈ D such that g(d) = e. We deduce that (f (m1 ), f (m2 )) ∈ Vn by the deﬁnition of Vn and hence Mn ⊆ (f × f )−1 Vn . Therefore we ﬁnally conclude that each Vn belongs to UY and hence h is uniformly continuous. Lemma 2.1.20. The product f1 × f2 of two quotient maps f1 : X1 → Y1 and f2 : X2 → Y2 between quasi-uniform spaces X1 and Y1 , and X2 and Y2 , respectively, is a quotient map. Proof. Assume that h ◦ (f1 × f2 ) : X1 × X2 → Z is uniformly continuous for a map h : Y1 × Y2 → Z into an arbitrary quasi-uniform space Z. By Lemma 2.1.19 both f1 ×f2 : X1 ×DX2 → Y1 ×DY2 and f1 ×f2 : DX1 ×X2 → DY1 ×Y2 are quotient maps. Considering the uniformly continuous maps h ◦ (f1 × f2 ) : X1 × DX2 → Z and h ◦ (f1 × f2 ) : DX1 × X2 → Z, we conclude that h : Y1 × DY2 → Z and h : DY1 ×Y2 → Z are uniformly continuous. Hence by Lemma 2.1.18 h : Y1 ×Y2 → Z is uniformly continuous. Proposition 2.1.21. The product i∈I fi of any family (fi )i∈I of quotient maps fi : Xi → Yi (where i ∈ I) in QU is a quotient map. Proof. From Lemma 2.1.20 we obtain the result forﬁnite I by induction. For the case of inﬁnite I, suppose that k = h ◦ ( i∈I fi ) : i∈I Xi → Z is uniformly continuous where h : i∈I Yi → Z is any map into a quasi-uniform space Z.

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Let R ∈ UZ . Because themap k is uniformly continuous, there exists a ﬁnite set J ⊆ I such that if x, y ∈ i∈I Xi and πJX (x) = πJX (y), then (k(x), k(y)) ∈ R. Here πJX : i∈I Xi → i∈J Xi denotes the obvious projection. Similarly we deﬁne Y that ais a (ﬁxed) point in i∈I Xi . Let b = π J : i∈I Yi → i∈J Yi . Suppose ( i∈I fi )(a). We deﬁne ea : i∈J Xi → i∈I Xi by (ea (p)) i = ai , if i ∈ J, and (ea (p))j = pj if j ∈ J. Similarly, we deﬁne eb : i∈J Yi → i∈I Yi by (eb (q))i = bi , if i ∈ J, and (eb (q))j = qj if j ∈ J. Note that the maps ea and eb are uniformly continuous and that ( i∈I fi ) ◦ ea = eb ◦ ( i∈J fi ). By the ﬁrst observation in this proof, i∈J fi : i∈J Xi → i∈J Yi is a quotient map. Since the map h ◦ ( i∈I fi ) ◦ ea = (h ◦ eb ) ◦ i∈J fi is uniformly continuous, it then follows that h ◦ eb , and hence he := h ◦ eb ◦ πJY : i∈I Yi → Z are uniformly continuous. Because the map he is uniformly continuous, there is V ∈ UQi∈I Yi such that (he (p), he (q)) ∈ R whenever (p, q) ∈ V . Consider now any (p, q) ∈ V . Choose p , q ∈ i∈I Xi such that ( i∈I fi )(p ) = p and ( i∈I fi )(q ) = q. We wish to prove that (h(p), h(q)) ∈ R3 . observe that (h(p), he (p)) = (h ◦ ( i∈I fi ))(p ), (h ◦ eb ◦ πJY )(p) = First k(p ), (k ◦ ea ◦ πJX )(p ) ∈ R by the continuity property of k mentioned above and since ( i∈I fi )◦ea ◦πJX = eb ◦πJY ◦( i∈I fi ). We already know that (he (p), he (q)) ∈ Y R. Finally, (he (q), h(q)) = (h ◦ eb ◦ πJ )(q), (h ◦ ( i∈I fi ))(q ) = (k ◦ ea ◦ πJX )(q ), k(q ) ∈ R. Consequently, (h(p), h(q)) ∈ R3 and we conclude that h is uniformly continuous. Hence the proof is complete. 2.2. Transitive quasi-uniform and topological spaces. A quasi-uniform space (X, U) is called transitive provided that it has a base consisting of transitive entourages (that is, preorders). Such a base is called a transitive base. Note that all quasi-uniformities obtained by the Fletcher construction are transitive. Furthermore observe that if d is a non-archimedean quasi-pseudometric on a set X, then the quasi-pseudometric quasi-uniformity Ud has a countable transitive base. The following converse can also be readily veriﬁed by noticing that q(x, y) = inf{2−n : (x, y) ∈ Un } whenever x, y ∈ X is a quasi-pseudometric satisfying all its stated conditions:5 Let U be a quasi-uniformity having a countable base of transitive entourages on a set X. Then U is non-archimedeanly quasi-pseudometrizable, that is, there is a non-archimedean quasi-pseudometric d on X such that U = Ud . We also remark that a topological space admits a transitive uniformity if and only if it is zero-dimensional. Corollary 2.2.1 (compare [30, Theorem 7.1]). The topology of a topological space can be induced by a non-archimedean quasi-pseudometric if and only if it has a σ-interior preserving base (that is, a base of open sets that can be written as the union of countably many interior-preserving collections). Proof. The statement is a consequence of preceding assertions and a straightforward application of the Fletcher construction. Recall that each pseudometrizable space has a σ-interior preserving base, since it has a σ-locally ﬁnite base by the Nagata–Smirnov Metrization Theorem. It is easy to see that the supremum of any family of transitive quasi-uniformities is transitive. On the other hand the inﬁmum of a transitive quasi-uniformity with 5Here we assume without loss of generality that U = X ×X and {U : n ∈ ω} is a decreasing n 0 base of transitive entourages of U.

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its conjugate need not be transitive. For instance for the real unit interval [0, 1] equipped with its usual euclidean topology τ the inﬁmum of the Pervin quasiuniformity P(τ ) and its conjugate (P(τ ))−1 is the usual (nontransitive) uniformity on [0, 1] (compare [17, Example 2]). The proof of the following result is crucial for the theory of quasi-uniform spaces, but is well known (see e.g., [30, Lemma 1.5] and [25, Theorem 8.1.10]) and thus omitted in this chapter. Lemma 2.2.2 (Quasi-Pseudometrization Lemma). Let (Hn )n∈ω be a sequence of binary reﬂexive relations on a set X such that for each n ∈ ω, Hn+1 ◦Hn+1 ◦Hn+1 ⊆ Hn . Then there is a quasi-pseudometric d on X such that Hn+1 ⊆ {(x, y) ∈ X ×X : d(x, y) < 2−n } ⊆ Hn whenever n ∈ ω. If each Hn is a symmetric relation, then d can be constructed to be a pseudometric. Corollary 2.2.3. Each (quasi-)uniformity U on a set X with a countable base is (quasi-)pseudometrizable (that is, there is a (quasi-)pseudometric d on X such that Ud = U). Corollary 2.2.4. Each (quasi-)uniformity can be written as the supremum of a family of (quasi-)pseudometric quasi-uniformities. Corollary 2.2.5. A topological space is completely regular if and only if its topology is uniformizable, that is, can be induced by a uniformity. Proof. Let (X, τ ) be a topological space that is uniformizable by some uniformity U. Then by Corollary 2.2.4 U can be written as the supremum of pseudometric uniformities. Hence τ is the supremum of pseudometric topologies and thus is completely regular. In order to prove the converse, assume that (X, τ ) is completely regular. Then X carries the initial topology with respect to the family of all continuous maps from X into the real unit interval ([0, 1], τ (m)) where m denotes the usual metric m(x, y) = |x − y| (x, y ∈ [0, 1]) on [0, 1]. Consider the initial (quasi-)uniformity C ∗ (τ ) on X determined by the family of all those maps into the uniform space ([0, 1], Um ). Since C ∗ (τ ) induces τ , we have shown that (X, τ ) is uniformizable. Corollary 2.2.6 (compare [30, p. 99]). Let V be a compatible quasi-uniformity on a completely regular space X. If V has the Lebesgue property, then it is ﬁner than the ﬁnest compatible uniformity U on X. Proof. The continuous identity map idX : (X, V) → (X, U) is uniformly continuous by Proposition 2.1.12. Corollary 2.2.7 (compare [30, Proposition 1.47]). Let (X, τ ) be a compact regular space, let U be a compatible uniformity and let V be a compatible quasiuniformity on X. Then U ⊆ V. In particular C ∗ (τ ) is the unique compatible uniformity on X and the coarsest compatible quasi-uniformity on X. Proof. Consider the identity map idX : (X, τ (U)) → (X, τ (V)). It is continuous and open. Hence by Proposition 2.1.14 the map idX : (X, U) → (X, V) is uniformly open. Consequently U ⊆ V. Definition 2.2.8. Let X be a set. A ﬁlter U on X × X consisting of reﬂexive relations is called a local quasi-uniformity provided that for each U ∈ U and each x ∈ X there is V ∈ U such that V 2 (x) ⊆ U (x). A local quasi-uniformity U on a set

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X such that U = U −1 is called a local uniformity. A local quasi-uniformity such that U −1 is a local quasi-uniformity will be called a pairwise-local quasi-uniformity. In the same way as in the case of a quasi-uniformity, a local quasi-uniformity U induces a topology τ (U). A topological space X is called a γ-space provided that its topology can be induced by a local quasi-uniformity on X having a countable base. Since for each U ∈ U and x ∈ X, there is V ∈ U such that V 2 (x) ⊆ U (x), the argument following Deﬁnition 2.1.4 shows the following fact: Let (X, U) be a local quasi-uniform space and x ∈ X. Then U(x) = {U (x) : U ∈ U} is the neighborhood ﬁlter at x with respect to the topology τ (U). It can be readily shown [156, Theorem 1.4] that a topological space (X, τ ) admits a local uniformity if and only if its topology τ is regular. Example 2.2.9. A method to build γ-spaces that are not quasi-pseudometrizable was found by Fox [34], answering a question that originated in work of Ribeiro [135]. In fact Fox and Kofner [35] constructed a γ-space with a Tychonoﬀ topology that is not quasi-pseudometrizable. The problem of whether a γ-space satisfying some additional property is quasi-pseudometrizable often turns out to be quite delicate. A binary relation N on a topological space X is called a neighbornet of X if for each x ∈ X, N (x) is a neighborhood at x. Surgery on neighbornets is one of the most powerful techniques in the topological theory of quasi-uniform spaces which is often useful to deal with such problems. Junnila [57] showed that each developable γ-space is quasi-metrizable. Recall that a topological T1 -space X is called developable if there is a sequence (Dn )n∈ω of open covers of X such that for each x ∈ X, {st(x, Dn ) : n ∈ ω} is a neighborhood base at x. As usual here st(x, Dn ) = {D : x ∈ D ∈ Dn } whenever n ∈ ω and x ∈ X. The sequence (Dn )n∈ω is called a development for X. The following proof of Junnila’s aforementioned result is due to Fox [32]. Lemma 2.2.10. Let U be a neighbornet of a developable γ-space X. Then there exists a neighbornet W of X such that W 4 ⊆ U 2 . Proof. Let (Dn )n∈ω be a development for X where we can assume without loss of generality that Dn+1 reﬁnes Dn whenever n ∈ ω. Let {Vn : n ∈ ω} be a decreasing base for a compatible local quasi-uniformity on X. Note that for each x ∈ X, {Vn5 (x) : n ∈ ω} is a neighborhood base at x. For each x ∈ X deﬁne m(x) = min{n ∈ ω : Vn5 (x) ⊆ U (x)}, k(x) = min{n ≥ m(x) : st(x, Dn ) ⊆ Vm(x) (x)}, l(x) = max{min{k(y) : y ∈ N } : N is a neighborhood of x}. Let us note that l(x) ≤ k(x) whenever x ∈ X. Set C−1 = ∅ and for each n ∈ ω, let Cn = {x ∈ X : k(x) ≤ n}. Then Cn = {x ∈ X : l(x) ≤ n} whenever n ∈ ω. Thus obviously, x ∈ Cl(x)−1 whenever x ∈ X. Deﬁne the neighbornet W of X by W (x) = Vk(x) (x) \ Cl(x)−1 whenever x ∈ X. Now suppose that x4 ∈ W 4 (x0 ), and ﬁnd x1 , x2 , x3 ∈ X such that xi+1 ∈ W (xi ) for i ≤ 3. Since xi+1 ∈ Cl(xi )−1 , we have l(x0 ) ≤ l(xi ) for all i. There exists y ∈ U (x0 ) ∩ st(x0 , Dl(x0 ) ) such that k(y) = l(x0 ). Therefore m(y) ≤ k(y) = l(x0 ) ≤ l(xi ) ≤ k(xi ) for all i, and consequently xi+1 ∈ W (xi ) ⊆ Vk(xi ) (xi ) ⊆

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4 Vm(y) (xi ) for all i ≤ 3. Thus x4 ∈ Vm(y) (x0 ), and as y ∈ st(x0 , Dl(x0 ) ), then 5 x0 ∈ st(y, Dl(x0 ) ) = st(y, Dk(y) ) ⊆ Vm(y) (y); hence x4 ∈ Vm(y) (y) ⊆ U (y). But 2 y ∈ U (x0 ), thus x4 ∈ U (x0 ) as stated.

Proposition 2.2.11. Every developable γ-space is quasi-metrizable. Proof. Let X be a developable γ-space and let {Vn : n ∈ ω} be a base for a compatible local quasi-uniformity on X. Using Lemma 2.2.10, inductively deﬁne a 4 sequence (Wn )n∈ω of neighbornets of X such that W04 ⊆ V02 and Wn+1 ⊆ (Wn ∩ 2 2 Vn+1 ) whenever n ∈ ω. Then {Wn : n ∈ ω} is a countable base for a compatible quasi-uniformity on X and hence X is quasi-metrizable by Corollary 2.2.3. The following result is not diﬃcult, but the idea of proof is often useful. Proposition 2.2.12 (compare [33, Theorem 9] and [73, Theorem 5]). A topological space X is quasi-pseudometrizable if and only if its topology can be induced by a pairwise-local quasi-uniformity with a countable base. Proof. Necessity of the condition is obvious. In order to prove suﬃciency let U be a local quasi-uniformity with a countable base {Un : n ∈ ω} for X such that U −1 is a local quasi-uniformity. Without loss of generality we may assume that Un+1 ⊆ Un whenever n ∈ ω. Inductively we deﬁne a sequence (Vn )n∈ω of τ (U −1 ) × τ (U)-neighborhoods of the diagonal ∆X as follows. If n = 0, for each −1 4 4 (x) ⊆ U0 (x) and (Un(x) ) (x) ⊆ U0−1 (x) x ∈ X choose an n(x) ∈ ω such that Un(x) and set −1 V0 = {Un(x) (x) × Un(x) (x) : x ∈ X}. If n > 0 and Vn−1 is already deﬁned, for each x ∈ X ﬁnd an n(x) ∈ ω such that −1 8 8 (x) ⊆ (Vn−1 ∩ Un )(x) and (Un(x) ) (x) ⊆ (Vn−1 ∩ Un )−1 (x) and set Un(x) −1 (x) × Un(x) (x) : x ∈ X}. Vn = {Un(x) 2 ∩ Un2 whenever An easy computation shows that V02 ⊆ U02 and Vn4 ⊆ Vn−1 2 n > 0. Hence the quasi-uniformity V on X generated by {Vn : n ∈ ω} satisﬁes τ (V) = τ (U) and τ (V −1 ) = τ (U −1 ). The assertion follows from Corollary 2.2.3.

Corollary 2.2.13 ([156, Corollary 2.6]). A topological space is pseudometrizable if and only if its topology can be induced by a local uniformity possessing a countable base. Problem 2.2.14 (for research; compare [30, Problem O]). Much work has gone into the problem of characterizing topologically those topological spaces that admit a compatible quasi-pseudometric.6 Various solutions to this problem have been suggested and discussed in the literature (see in particular the pertinent articles of Hung and Kopperman [52, 69]). In particular in order to obtain an asymmetric variant of Proposition 2.2.12 Hung suggested to work under the following strengthening of the concept of a local quasi-uniformity. He required that the crucial condition in the deﬁnition of a local quasi-uniformity U (see Deﬁnition 2.2.8) does not only hold for singletons {x}, but also for some larger subsets of X (see [52, Theorem] for more details). Under such conditions U −1 may turn out to be a pairwise-local quasi-uniformity: Indeed let 6The quasi-pseudometrization problem for bitopological spaces will be considered in the next subsection.

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U be a local quasi-uniformity on the set X. Furthermore let x ∈ X and U ∈ U. Suppose that all subsets of the form X \ U −1 (x) satisfy the crucial condition in the deﬁnition of a local quasi-uniformity. Hence given U ∈ U we can ﬁnd V ∈ U such that V 2 (X \ U −1 (x)) ⊆ U (X \ U −1 (x)). From the latter inclusion we immediately deduce that (V −1 )2 (x) ⊆ U −1 (x) and thus that U −1 is a local quasi-uniformity, since otherwise there is a ∈ (V −1 )2 (x) such that a ∈ X \ U −1 (x), which implies that x ∈ V 2 (a) ⊆ U (X \ U −1 (x)) — a contradiction. The search however for new conditions that characterize quasi-pseudometrizability of topological spaces is still going on. Although apparently a ﬁniteness condition in the spirit of the well-known characterizations of paracompactness is not available in the quasi-pseudometric context, it should be said that the attractive Nagata–Smirnov Theorem from metrization theory (see e.g., [46, 47]) motivated many of these asymmetric investigations and results (compare [3]). As a more modest exercise the reader may want to establish that a compact topological T1 -space is quasi-metrizable if and only if it has a countable base. Each quasi-uniformity U contains a ﬁnest transitive quasi-uniformity coarser than U, which is often denoted by Ut . Of course, (Ut )−1 = (U −1 )t . Note that Ut and U may induce distinct topologies, since for instance on any uniform space (X, U) with a connected topology τ (U), the indiscrete uniformity on X is the only transitive quasi-uniformity coarser than U. For any topological space (X, τ ) we shall call (FN (τ ))t the ﬁne transitive quasi-uniformity for τ and denote it by FT (τ ) (compare [30, p. 30]). It is functorial and obtained by applying the Fletcher construction to all interior-preserving open covers of (X, τ ). Indeed since each topological space admits a transitive quasi-uniformity, namely its Pervin quasiuniformity, FT (τ ) is compatible with τ . A topological space is called transitive if its ﬁne quasi-uniformity is transitive. In the following example each n ∈ ω is as usual identiﬁed with its set of predecessors, that is n = {0, . . . , n − 1}. Example 2.2.15. [114] Let X be the set Qω (of sequences from ω to the set Q of the rationals). For each n ∈ ω and x ∈ X deﬁne the set Un (x) as the set of all x ∈ X which (1) agree with x on the initial segment n, and (2) for which the ﬁrst coordinate ∆ := ∆(x, x ) in which x and x diﬀer satisﬁes x(∆) ≤ x (∆) ≤ x(∆) + 2−n . We next verify that the ﬁlter U on X × X generated by the base {Un : n ∈ ω} where Un = x∈X ({x} × Un (x)) (n ∈ ω) is a quasi-uniformity: It will suﬃce to show that for any n ∈ ω and any x, x , x ∈ X, we have that x ∈ Un+1 (x) and x ∈ Un+1 (x ) imply that x ∈ Un (x). Since the sequence (Un )n∈ω is decreasing, it suﬃces to consider the case that x = x and x = x . Note ﬁrst that under our hypothesis x, x , x agree on n + 1. Set ∆0 := ∆(x , x) and ∆1 := ∆(x , x ). We now distinguish three cases. Case 1 : If ∆1 < ∆0 , then ∆1 = ∆(x , x), x (∆1 ) = x(∆1 ), x (∆1 ) ≤ x (∆1 ) ≤ x (∆1 ) + 2−(n+1) , and thus x ∈ Un+1 (x). Case 2 : If ∆1 > ∆0 , then ∆0 = ∆(x , x), x (∆0 ) = x (∆0 ), x(∆0 ) ≤ x (∆0 ) ≤ x(∆0 ) + 2−(n+1) , and thus x ∈ Un+1 (x). Case 3 : If ∆1 = ∆0 , then ∆0 = ∆(x , x), x(∆0 ) ≤ x (∆0 ) ≤ x(∆0 ) + 2−(n+1) and x (∆0 ) ≤ x (∆0 ) ≤ x (∆0 ) + 2−(n+1) , and thus x ∈ Un (x). Let us note that τ (U) is a T1 -topology, since n∈ω Un (x) = {x} whenever x ∈ X.

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Remark 2.2.16 ([114]). The space (X, τ (U)) described in the preceding example is not transitive. Proof. Indeed we shall show that τ (U) is not non-archimedeanly quasi-pseudometrizable. In order to reach a contradiction, we assume that it admits a quasiuniformity with a countable decreasing base {Tn : n ∈ ω} consisting of transitive entourages. Then for each x ∈ X, choose k(x) ∈ ω such that Tk(x) (x) ⊆ U1 (x); furthermore ﬁx nx ∈ ω such that Unx (x) ⊆ Tk(x) (x). Viewing the rationals as having the discrete topology and working in the resulting product space Qω , for each m, k ∈ ω we set Am,k = {x ∈ Qω : nx = m and k(x) = k}. With the help of the Baire Category Theorem we ﬁnd m, k ∈ ω such that int Am,k = ∅ in the space Qω . Hence there are n ∈ ω, σ ∈ Qn and a dense D ⊆ [σ] such that, for all x ∈ D, k(x) = k and nx ≤ n. (Here, [σ] = {x ∈ Qω : x |n = σ}.) Construct x1 , . . . , x2n +1 ∈ D such that for each i = 1, . . . , 2n + 1, xi restricted to n equals σ and such that for each i = 1, . . . , 2n , xi (n) + 2−n = xi+1 (n). We see that for each i = 1, . . . , 2n , we have xi+1 ∈ Tk (xi ), because xi+1 ∈ Un (xi ) ⊆ Unxi(xi ). By transitivity of Tk we conclude that x2n +1 ∈ Tk (x1 ) ⊆ U1 (x1 ). But now x2n +1 ∈ U1 (x1 ) requires x2n +1 (∆) ≤ x1 (∆) + 2−1 where ∆ := ∆(x1 , x2n +1 ) = n. However x1 (∆) + 1 = x1 (∆) + 2−n (2n ) = x2n +1 (∆). This contradiction completes the proof. Example 2.2.17 (The Kofner plane). The ﬁrst example of a topological space that was quasi-pseudometrizable, but not non-archimedeanly quasi-pseudometrizable was constructed by Kofner and is now called the Kofner plane [63]: Let X = R2 . For each x ∈ X and > 0 let C(x, ) be the usual closed disk of radius lying above the horizontal line through x and tangent to this line at x. For x, y ∈ X deﬁne d(x, y) as follows: Set d(x, y) = 1 if y ∈ / C(x, 1), set d(x, y) = r if r ≤ 1, y ∈ C(x, r) and y ∈ / C(x, s) for all s < r, and set d(x, y) = 0 if x = y. An application of the Baire Category Theorem similar to the one just given shows that the quasi-metric space (X, d) is not non-archimedeanly quasipseudometrizable. Hence it cannot be transitive, since this would imply that it admits a quasi-uniformity with a countable base of transitive relations. We omit the proof that can be found in [30, p. 147]. Problem 2.2.18 (for research; compare with [30, Problem R]). It is not known whether each (regular) quasi-pseudometric space (X, d) such that τ (d) ⊆ τ (d−1 ) is non-archimedeanly quasi-pseudometrizable. Problem 2.2.19 (for research; compare with [30, Problem P]). No non-archimedeanly quasi-pseudometrizable (Tychonoﬀ) space seems to be known that is not transitive. Example 2.2.20 ([76, Example 1]). There exists a transitive space that is the union of two subspaces homeomorphic to the (nontransitive) Kofner plane. Let X = (R2 × {1}) ∪ (R2 × {−1}). For each r ∈ R2 and each n ∈ ω let Sn (r) be the open disk of radius 2−n lying above the horizontal line through r and tangent to this line at r. Similarly, Sn−1 (r) will denote the open disk of radius 2−n lying below the horizontal line through r and tangent to this line at r. Construct a base for a topology on X by deﬁning for each r ∈ R2 , n ∈ ω and i ∈ {1, −1} basic open

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neighborhoods Kn (r, i) at the point (r, i), where Kn (r, 1) = [(Sn (r) ∪ {r}) × {1}] ∪ [Sn (r) × {−1}], Kn (r, −1) = [Sn−1 (r) × {1}] ∪ [(Sn−1 (r) ∪ {r}) × {−1}]. Obviously, both the subspaces R2 × {1} and R2 × {−1} of X are homeomorphic to the Kofner plane. However X can be shown to be transitive, by considering the transitive topology K on R2 with basic neighborhoods Kn (r) = Sn−1 (r)∪{r}∪Sn(r) (n ∈ ω) at each point r ∈ R2 (see [76] for details). Hence transitivity is not a hereditary topological property. Example 2.2.21 ([76, Example 2]). We show that the product of two transitive topological spaces need not be transitive. Let S denote the Sorgenfrey topology on R (compare with Example 3.1.3 below). Furthermore let K denote the topology on R2 deﬁned in Example 2.2.20. Since generalized ordered spaces are transitive (see e.g., [30, Theorem 6.30]), (R, S) is transitive. In the discussion of the preceding example we stated that (R2 , K ) is transitive. We now argue that (R3 , S × K ) is not transitive. The plane P = {(x, y, z) ∈ R3 : x = z} is closed in (R3 , S × K ). However, as a subspace of (R3 , S × K ), P is homeomorphic to the Kofner plane: If T denotes the yz-plane equipped with the topology of the Kofner plane, then the projection T → P in the direction of the x-axis is a homeomorphism. Since transitivity is a closed-hereditary property [30, Proposition 6.14], we conclude that (R3 , S × K ) is not transitive. Many other interesting results about transitive spaces are discussed in [30, Chapter 6]. For instance it is shown that countable unions of closed (respectively open) transitive topological spaces are transitive [30, Theorems 6.15 and 6.17], and that transitivity is preserved under continuous closed surjections [30, Proposition 6.20]. It should be mentioned that in general it is not easy to decide whether a given topological space is transitive. Next we shall present a representative proof of a positive result of this type. Proposition 2.2.22 ([60]). Each hereditarily metacompact compact regular space is transitive. We recall that a topological space X is called metacompact if each open cover C of X has a point-ﬁnite open reﬁnement R (that is, each point x ∈ X is contained in ﬁnitely many sets of R). A topological space X is called hereditarily metacompact if each subspace of X is metacompact. The preceding proposition is an immediate consequence of the following much stronger result. Lemma 2.2.23 ([60, Proposition 4]). Let O be a neighbornet of a hereditarily metacompact compact regular topological space X. Then there exists a point-ﬁnite open family V of X such that TV ⊆ O 3 . Proof. Without loss of generality we suppose that O(x) is open whenever x ∈ X. Set E = {F ⊆ X : F is closed and there exists x ∈ X such that x ∈ F ⊆ O(x)}. For every F ∈ E, we deﬁne a point pF and closed subset s(F ) of F as follows: For pF we pick any point x satisfying x ∈ F ⊆ O(x) and then we set s(F ) = F ∩ O −1 (pF ). Claim 1. There is no inﬁnite sequence (Fn )n∈ω of members of E such that we have Fn+1 ⊆ Fn \ s(Fn ) whenever n ∈ ω.

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Proof of Claim 1. Assume that (Fn )n∈ω is such a sequence. Let K = n∈ω Fn . The set O(K) is a neighborhood of the set K and it follows by compactness of X, since (Fn )n∈ω is a decreasing sequence of closed sets, that there exists k ∈ ω such that Fk ⊆ O(K). We now have that pFk ∈ O(K), in other words, that O −1 (pFk ) ∩ K = ∅. Since K ⊆ Fk , it follows from the foregoing that s(Fk ) ∩ K = ∅; however, from this it further follows, since Fk+1 ∩ s(Fk ) = ∅, that K ⊆ Fk+1 , and this is a contradiction. Claim 2. Set G = {V ⊆ X : V is open and V ∈ E}. To every open subset U of X there corresponds a ﬁnite family G(U ) ⊆ G such that G(U ) = U . of x such Proof of Claim 2. For every x ∈ U , let Vx be an open neighborhood that Vx ⊆ O(x). Let A be a ﬁnite subset of U such that U ⊆ x∈A Vx and let G(U ) = {Vx ∩ U : x ∈ A}. Claim 3. There exists a point–ﬁnite family U ⊆ G such that the family H = {U ∩ s(U ) : U ∈ U} covers X. Proof of Claim 3. Let U ∈ G. By regularity and hereditary metacompactness of X there exists a point-ﬁnite family V(U ) of open subsets of X such that we have {V : V ∈ V(U )} = {V : V ∈ V(U )} = U \ s(U ). Note then that, by Claim 2, we may assume that V(U ) ⊆ G: If this were not already the case, we could replace V(U ) by {G(V ) : V ∈ V(U )}. We deﬁne inductively point-ﬁnite families Un ⊆ G by setting U0 = G(X) and Un+1= {V(U ) : U ∈ Un }. We shall show that Claim 3 holds for the family U = n∈ω Un : Suppose that U is not point-ﬁnite at some point x ∈ X. By K¨ onig’s 7 Lemma there is a sequence (Un )n∈ω such that x ∈ n∈ω Un and for each n ∈ ω, Un+1 ∈ V(Un ). Thus Un+1 ⊆ Un \ s(Un ) whenever n ∈ ω — a contradiction to Claim 1, because Un ∈ E for every n ∈ ω. We conclude that U is point-ﬁnite. Assume that some point x ∈ X is not covered by H. Then x is not in any set U ∩ s(U ) where U ∈ U. We construct inductively a sequence (Vn )n∈ω such that we have x ∈ Vn ∈ Un whenever n ∈ ω and Vn ∈ V(Vn−1 ) whenever n > 0. Since U0 = G(X) = X, there exists V0 ∈ U0 such that x ∈ V0 . Suppose that n > 0 and that Vn−1 be such that x ∈ Vn−1 ∈ Un−1 has already been chosen. By our assumption, we have that x ∈ s(Vn−1 ) and it follows that there exists Vn ∈ V(Vn−1 ) such that x ∈ Vn . This concludes the induction. As above, we deduce that Vn+1 ⊆ Vn \ s(Vn ) whenever n ∈ ω — a contradiction to Claim 1. To complete the proof of the lemma, let V = {U ∩ O(pU ) : U ∈ U}. Observe that V is a point-ﬁnite family of open subsets of X. We show that TV ⊆ O 3 . Let x ∈ X. There exists U ∈ U such that x ∈ U ∩ s(U ). Since U ∈ E, we note that U ⊆ O(pU ) and hence that x ∈ s(U) ⊆ O(pU ) ∩ O −1 (pU ). Furthermore, we deduce that x ∈ U ∩ O(pU ) ∈ V and hence that TV (x) = {V ∈ V : x ∈ V } ⊆ O(pU ). Since we have that x ∈ O −1 (pU ), there exists a point y ∈ O(x) ∩ O −1 (pU ). We now conclude that y ∈ O(x) and pU ∈ O(y). It follows that pU ∈ O 2 (x) and, further, that O(pU ) ⊆ O 3 (x). As a consequence, we see that TV (x) ⊆ O(pU ) ⊆ O 3 (x). 7It says in its original formulation: Let (E ) n n∈ω be a sequence of ﬁnite nonempty sets and let R be a binary relation such that for each element xn+1 ∈ En+1 (n ∈ ω) there is at least one element xn ∈ En such that (xn , xn+1 ) ∈ R. Then there is a sequence (an )n∈ω such that an ∈ En and (an , an+1 ) ∈ R whenever n ∈ ω [66].

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We remark that with additional work the condition of compactness of X in Lemma 2.2.23 can be replaced by considerably weaker conditions (e.g., that X is a β-space, see [92, Theorem 2.2]). 2.3. Bitopological spaces. Definition 2.3.1. A bitopological space (X, τ1 , τ2 ) consists of a set X equipped with two topologies τ1 and τ2 . A map f : X → Y between two bitopological spaces (X, τ1 , τ2 ) → (Y, ρ1 , ρ2 ) is called bicontinuous provided that both f : (X, τ1 ) → (Y, ρ1 ) and f : (X, τ2 ) → (Y, ρ2 ) are continuous. In this context it is natural to consider each topological space (X, τ ) as the bitopological space (X, τ, τ ). Much work has been done in extending concepts and results about topological spaces to bitopological spaces. As expected, while in some cases such generalizations are more or less straightforward, in other cases they are quite demanding and often highly controversial. Sometimes they also lead to completely unsatisfactory results, since concepts that are equivalent in the case of topological spaces may extend to concepts that are no longer equivalent in the bitopological setting. In particular in the light of the quasi-pseudometrization problem for topological spaces, it is natural to attempt a characterization of those bitopological spaces of the form (X, τ (d), τ (d−1)), where d is a quasi-pseudometric on the given set X. We shall next present such a characterization due to Fox. Let X be a set and let A, B subsets of X. We let TA,B denote the binary relation (X × X) \ (A × B) = [(X \ A) × X] ∪ [X × (X \ B)]. In particular note that SA = TA,X\A where SA was introduced in the discussion of −1 the Pervin quasi-uniformity after Deﬁnition 2.1.6. Evidently TA,B = TB,A . In the following suppose that ξ is a collection of pairs (A, B) of subsets of a set X such that A ⊆ B. A pairbase for a topological space X is a collection ξ as above satisfying (i) if (A, B) ∈ ξ, then A is open, and (ii) if C is a neighborhood of x ∈ X, then there exists (A, B) ∈ ξ with x ∈ A ⊆ B ⊆ C. If for a topology τ on X and for each ζ ⊆ ξ, we have clτ {A : (A, B) ∈ ζ} ⊆ {B : (A, B) ∈ ζ}, then ξ is called (τ -)cushioned. If for a topology τ on X and for each ζ ⊆ ξ, we have {A : (A, B) ∈ ζ} ⊆ intτ {B : (A, B) ∈ ζ}, then ξ is called (τ -)cocushioned. If ξ = n∈ω ξn where each ξn is τ -cushioned (resp. τ -cocushioned), then ξ is called σ-τ -cushioned (resp. σ-τ -cocushioned ). Remark 2.3.2. Let X be a topological space. Then a collection of pairs (A, B) of subsets of X is cushioned if and only if the collection of the corresponding pairs (X \ B, X \ A) is cocushioned. Proposition 2.3.3 (compare [33, Theorem 8]). A bitopological space (X, τ1 , τ2 ) is quasi-pseudometrizable8 if and only if τ1 has both a σ-τ2 -cushioned pairbase and a σ-τ1 -cocushioned pairbase, and τ2 has both a σ-τ1 -cushioned pairbase and a σ-τ2 cocushioned pairbase. Proof. We ﬁrst show that the stated conditions are necessary. Let (X, d) be a quasi-pseudometric space. For each n ∈ ω let βn = {(Bd (x, 2−(n+1) ), Bd (x, 2−n )) : x ∈ X} and δn = {(Bd−1 (x, 2−(n+1) ), Bd−1 (x, 2−n )) : x ∈ X}. Then each βn is both 8This means that there is a quasi-pseudometric d on X such that τ (d) = τ and τ (d−1 ) = τ . 1 2

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τ (d−1 )-cushioned and τ (d)-cocushioned, and β = n∈ω βn is a pairbase for τ (d). Similarly each δn is both τ (d)-cushioned and τ (d−1 )-cocushioned, and δ = n∈ω δn is a pairbase for τ (d−1 ). In order to prove suﬃciency we shall make use of the following result due to Fox: Lemma 2.3.4 (compare [33, Corollary 6.1 and Theorem 6]). If (X, τ1 , τ2 ) is a bitopological space where τ1 has a σ-τ2 -cushioned pairbase µ and a σ-τ1 -cocushioned pairbase ξ, and where τ2 has a σ-τ1 -cushioned pairbase λ and a σ-τ2 -cocushioned pairbase ζ, then (X, τ1 , τ2 ) possesses a compatible pairwise-local quasi-uniformity with a countable base. Proof. Let β = {(A ∩ C, B ∪ D) : (A, B) ∈ ξ and (C, D) ∈ µ} and δ = {(A ∩ C, B ∪ D) : (A, B) ∈ ζ and (C, D) ∈ λ}. Note that we can write β = n∈ω βn and δ = n∈ω δn , where for each n ∈ ω, βn ⊆ βn+1 and δn ⊆ δn+1 , and where each βn is both τ2 -cushioned and τ1 -cocushioned, and each δn is both τ1 -cushioned and τ2 -cocushioned. In particular β is a σ-τ2 -cushioned, σ-τ1 -cocushioned pairbase for τ1 , and δ is a σ-τ1 -cushioned, σ-τ2 -cocushioned pairbase for τ2 . For any n ∈ ω we deﬁne Vn = {TA,X\B : (A, B) ∈ βn } ∩ {(TA,X\B )−1 : (A, B) ∈ δn }. Fix n ∈ ω and x ∈ X. Then Vn (x) is a τ1 -neighborhood of x, since βn is τ1 -cocushioned and δn is τ1 -cushioned. Also Vn−1 (x) is a τ2 -neighborhood of x, as βn is τ2 -cushioned and δn is τ2 -cocushioned. Given any τ1 -neighborhood S of a point x ∈ X, as β is a pairbase for τ1 , we ﬁnd n ∈ ω and (A, B), (C, D) ∈ βn such that x ∈ A ⊆ B ⊆ C ⊆ D ⊆ S. Then x ∈ A implies Vn (x) ⊆ B ⊆ C, which implies Vn2 (x) ⊆ D ⊆ S. Similarly, given any τ2 -neighborhood R of x, we ﬁnd n ∈ ω such that Vn−2 (x) ⊆ R. It follows that {Vn : n ∈ ω} is a countable base for a compatible pairwise-local quasi-uniformity on (X, τ1 , τ2 ). By the arguments just given and with the help of the proof of Proposition 2.2.12 we conclude that the condition stated in Proposition 2.3.3 is also suﬃcient. Remark 2.3.5. Using the method above (compare [30, p. 163] or [33, Proposition 1]) it can be shown that a topological space (X, τ ) admits a local quasiuniformity with a countable base (that is, is a γ-space) if and only if it possesses a σ-τ -cocushioned pairbase. The idea mentioned in Remark 2.3.2 was further developed by Kopperman in order to obtain with the help of Proposition 2.3.3 a characterization of quasipseudometrizable topological spaces (see [69]). Corollary 2.3.6 ([62, Theorem 2.8]). A bitopological space (X, τ1 , τ2 ) is quasipseudometrizable if X is pairwise regular (that is, each point of X has a τi -neighborhood base consisting of τj -closed sets (i, j ∈ {1, 2}, i = j)) and both topologies τ1 and τ2 have a countable base. Proof. Let B = {Bn : n ∈ ω} be a countable base for the topology τ1 . For each n ∈ ω set βn = {(Bn , clτ2 Bn )}. Then n∈ω βn is a σ-τ2 -cushioned, σ-τ1 cocushioned pairbase for τ1 . Interchanging the topologies τ1 and τ2 yields the result by Proposition 2.3.3. Corollary 2.3.7. A topological space is pseudometrizable if and only if it is a γ-space that has a σ-cushioned pairbase. (A topological T1 -space that has a σ-cushioned pairbase is called stratiﬁable or an M3 -space in the literature.)

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Proof. Let (X, τ ) be a γ-space that has a σ-cushioned pairbase. By Proposition 2.3.3 there exists a compatible quasi-pseudometric d on (X, τ, τ ). Then τ is pseudometrizable by ds . Necessity of the given condition follows from Proposition 2.3.3. Remark 2.3.8 (compare [30, Theorem 2.32]). Let (X, U) be a locally symmetric quasi-uniform space with a countable base {Un : n ∈ ω}. By Corollary 2.3.7 it follows that τ (U) is pseudometrizable, since τ (U) has the σ-cushioned pairbase m,n∈ω βm,n , where for each m, n ∈ ω, βm,n = {(intτ (U) Um (x), Un (x)) : −1 (Um (x)) ⊆ Un (x), x ∈ X}. Um The obvious question of which bitopological spaces can be induced by a quasiuniformity U on a given set X via the construction (X, τ (U), τ (U −1)) has the expected answer, as we show next. Example 2.3.9. Let [0, 1] be the real unit interval. For all x, y ∈ [0, 1] set r(x, y) = max{0, x − y}. Then r is a T0 -quasi-pseudometric on [0, 1] and τ (r) is the so-called upper topology {(a, 1] : a ∈ [0, 1)} ∪ {∅, [0, 1]} on [0, 1]. Similarly τ (r −1 ) is the lower topology {[0, a) : a ∈ (0, 1]} ∪ {∅, [0, 1]} on [0, 1]. Of course, rs = m (compare with the proof of Corollary 2.2.5). A bitopological space (X, τ1 , τ2 ) is called pairwise completely regular (compare [116]) if it carries exactly the coarsest bitopology that makes all the bicontinuous maps f : (X, τ1 , τ2 ) → ([0, 1], τ (r), τ (r −1)) bicontinuous (that is, τ1 and τ2 are both initial with respect to that collection of maps and the topologies τ (r) and τ (r −1 ), respectively). Proposition 2.3.10 ([116, Theorem 4.2]). A bitopological space (X, τ1 , τ2 ) is quasi-uniformizable if and only if it is pairwise completely regular. Proof. The proof is completely analogous to the proof of Corollary 2.2.5. Of course, here we have to equip [0, 1] with the quasi-uniformity Ur and its induced bitopology, and consider bicontinuous maps instead of continuous ones. 2.4. Precompactness and its variants. A quasi-uniform space (X, U) is called precompact (resp. totally bounded) if for each U ∈ U, the cover {U (x) : x ∈ X} of X has a ﬁnite subcover (resp. U s is precompact). It is called hereditarily precompact if each subspace of (X, U) is precompact. Note that a quasi-uniformity is totally bounded if and only if its conjugate is totally bounded. Furthermore for a uniformity total boundedness and precompactness are equivalent. Remark 2.4.1 ([30, p. 12 and p. 51]). A quasi-uniformity U on a set X is totally bounded if and only if for each U ∈ U there is a ﬁnite cover C of X such that C × C ⊆ U whenever C ∈ C. In particular each subspace of a totally bounded quasi-uniform space is totally bounded. For arbitrary quasi-uniformities, total boundedness implies hereditary precompactness, and hereditary precompactness implies precompactness. The converses do not hold. Example 2.4.2 ([102, Example 7]). Let X = N ∪ {∞} be equipped with the usual order ≤. For each x, y ∈ X, set d(x, y) = 0 if x = y, d(x, y) = n1 if x = n, n ∈ N and x < y, and d(x, y) = 1 otherwise. Then d is a quasi-metric on X. The metric max{d, d−1 } is discrete although Ud is hereditarily precompact. Furthermore τ (d−1 ) is compact.

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Recall that a preorder ≤ on a set X is called a well-quasi-order provided that for every inﬁnite sequence (xn )n∈ω of elements in X there are i, j ∈ ω such that i < j and xi ≤ xj (compare [72, 108]). The reader may ﬁnd it helpful to verify the following facts: Remark 2.4.3. (a) (compare [108, Lemma 1]). Let T be a preorder on a set X. The uniformity generated by the base {T } is hereditarily precompact only if T is a well-quasi-order. (b) (compare [30, Proposition 7.2]). Let X be a topological space. X is second countable if and only if X admits a totally bounded uniformity with a countable base. (c) ([42]). The Pervin quasi-uniformity of a topological space X is pseudometrizable if and only if the topology of X is countable.

quasiif and Then quasiquasi-

The supremum of two precompact quasi-uniformities need not be precompact [115], while the supremum of an arbitrary family of totally bounded quasi-uniformities is clearly totally bounded. Hence each quasi-uniformity U on a set X contains a ﬁnest totally bounded quasi-uniformity Uω coarser than U, which will be discussed below. For any topological space (X, τ ), the Pervin quasi-uniformity P(τ ) is the ﬁnest totally bounded quasi-uniformity that the topological space X admits [30, p. 28]. In particular (FN (τ ))ω = P(τ ) where FN (τ ) denotes the ﬁnest quasiuniformity that τ admits (compare with the remarks in front of Deﬁnition 2.1.8). Furthermore for any quasi-uniformity U, the equation Uω = U ∧ P(τ (U)) holds (see [120, Theorem 2.2]). With the help of the next result it can be shown that hereditary precompactness is preserved under arbitrary suprema of quasiuniformities, too. Proposition 2.4.4 (compare [102, Corollary 8]). Let U and V be hereditarily precompact quasi-uniformities on a set X. Then U ∨ V is hereditarily precompact. Proof. In order to reach a contradiction, suppose that there are U ∈ U, V ∈ V and a sequence (xn )n∈ω of points of X such that xn ∈ (U ∩ V )(xk ) whenever n, k ∈ ω and n > k. Set S = {xn : n ∈ ω}. Deﬁne a map h : [S]2 → {0, 1} as follows. Here, [S]2 denotes the two element subsets of S. For each n, k ∈ ω such that n > k set h({xk , xn }) = 1 if xn ∈ U (xk ), and set h({xk , xn }) = 0 if xn ∈ U (xk ) (and, thus, xn ∈ V (xk )). By Ramsey’s Theorem9 there is an inﬁnite subset Y of S such that the map h : [S]2 → {0, 1} is constant on [Y ]2 . However because the subspace (Y, U|Y ) of (X, U) is precompact, h cannot be equal to 1 on [Y ]2 . Similarly, because the subspace (Y, V|Y ) of (X, V) is precompact, h cannot be equal to 0 on [Y ]2 . We have reached a contradiction and conclude that (X, U ∨ V) is hereditarily precompact. Corollary 2.4.5. A quasi-uniform space (X, U) is totally bounded if and only if both U and U −1 are hereditarily precompact.

9We use the following basic result from Ramsey theory: Let G be a graph with inﬁnitely many vertices. Then either G or the complement graph of G contains a complete subgraph with inﬁnitely many vertices [132].

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Remark 2.4.6. The following facts are readily veriﬁed: (a) ([102, Theorem 5]). A nonempty product of quasi-uniform spaces is hereditarily precompact if and only if each factor space is hereditarily precompact. (b) ([81, p. 327]). Let (X, τ ) be a topological space. The conjugate M(τ )−1 of the well-monotone quasi-uniformity M(τ ) is hereditarily precompact. Furthermore P(τ ) = M(τ ) if and only if X is hereditarily compact (see [96, Remark 1]). A ﬁlter F on a quasi-uniform space (X, U) is called U-stable provided that F ∈F U (F ) ∈ F whenever U ∈ U. Proposition 2.4.7 ([98, Proposition 2.5]). Let (X, U) be a quasi-uniform space. Then each (ultra)ﬁlter on (X, U) is stable if and only if (X, U −1 ) is hereditarily precompact. Proof. Let A be a subspace of (X, U −1 ) that is not precompact. Then there exists U ∈ U such that A \ U −1 (E) = ∅ for any ﬁnite subset E of A. Let F be a(n ultra)ﬁlter on X containing {A \ U −1 (E) : E ⊆ A, E ﬁnite}. Then F ∈F U (F ) ⊆ X \ A and thus F ∈F U (F ) ∈ F. Hence F is not stable. For the converse assume that F is a ﬁlter on (X, U) which is not stable. Hence there is U ∈ U such that F ∈F U (F ) ∈ F. We deﬁne a sequence (xn )n∈ω of points of X as follows: For each n ∈ ω choose inductively Fn ∈ F such that Fn \ U (Fn+1 ) = ∅ and Fn+1 ⊆ Fn . Let xn ∈ Fn \ U (Fn+1 ) whenever n ∈ ω. Consider k, m ∈ ω with m > k. Then xm ∈ Fm ⊆ Fk+1 , but xk ∈ Fk \ U (Fk+1 ). Thus xm ∈ U −1 (xk ). Consequently the subspace {xn : n ∈ ω} of (X, U −1 ) is not precompact. 2.5. Quasi-proximities. We shall now brieﬂy discuss the concept of a quasiproximity. Since the material is standard and well presented in [30, Chapter 1], we shall refer the reader repeatedly to that book in the following. We also stress that analogous results can be proved in a symmetric setting about (totally bounded) uniformities and proximities, which in general will not be explicitly stated in this subsection. For each quasi-uniformity U on a set X we deﬁne a binary relation δU 10 on the power set P(X) of X as follows: For A, B ⊆ X, we set A δU B if for all V ∈ U we have V (A) ∩ B = ∅. One checks that indeed the induced quasi-proximity δU satisﬁes the conditions listed in the following deﬁnition. Definition 2.5.1. A relation δ on the power set P(X) of a set X is called a quasi-proximity on X if it has the following properties (here A δ B means (A, B) ∈ δ, and A δ B means (A, B) ∈ δ): (i) X δ ∅ and ∅ δ X, (ii) C δ (A ∪ B) if and only if C δ A or C δ B, and (A ∪ B) δ C if and only if A δ C or B δ C, (iii) {x} δ {x} for each x ∈ X, (iv) if A δ B, there exists C ⊆ X such that A δ C and (X \ C) δ B. If δ is a quasi-proximity on X, then δ −1 is a quasi-proximity on X. A proximity is a quasi-proximity δ satisfying the additional symmetry condition δ = δ −1 . If δ is a (quasi-)proximity on X, then (X, δ) is called a (quasi-)proximity 10We say that δ is the quasi-proximity induced by U on X. U

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space. The topology τ (δ) induced by the quasi-proximity δ on X is given via its closure operator clτ (δ) deﬁned by x ∈ clτ (δ) A if and only if x δ A. If A and B are subsets of X in the quasi-proximity space (X, δ), then A is said to be near B provided that A δ B and A is said to be far from B provided that A δ B. A map between f : (X, δ) → (Y, δ ) between quasi-proximity spaces (X, δ) and (Y, δ ) is called proximally continuous if A δ B implies that f A δ f B whenever A, B ∈ P(X). Note that each proximally continuous map between quasi-proximity spaces is continuous with respect to the induced topologies. Observe also that if U is a uniformity, then the induced quasi-proximity δU is a proximity. Quasi-uniformities inducing proximities are called proximally symmetric or Smyth symmetric. Remark 2.5.2. One readily veriﬁes the following statement: Let (X, δ) be a quasi-proximity space and let δ be the relation on P(X) deﬁned by A δ B if and only if A δ X \ B. Then δ (usually called the strong inclusion of δ) satisﬁes the following conditions: (a) X δ X and ∅ δ ∅. (b) If A δ B, then A ⊆ B. (c) If A ⊆ B δ C ⊆ D, then A δ D. (d) If A δ B1 and A δ B2 , then A δ B1 ∩ B2 . (e) If A1 δ B and A2 δ B, then A1 ∪ A2 δ B. (f) If A δ B, then there exists C ⊆ X such that A δ C δ B. Remark 2.5.3. On the power set P(X) of a set X let a relation be deﬁned satisfying the conditions (a)–(f) of Remark 2.5.2. One readily checks that the relation δ deﬁned by A δ B provided that A X \ B is a quasi-proximity on X and is equal to the strong inclusion of δ. In this sense quasi-proximities and strong inclusions correspond to each other. Clearly for strong inclusions the symmetry condition equivalent to δ = δ −1 can be formulated as follows: A δ B implies X \ B δ X \ A. Given a quasi-proximity δ on a set X we shall denote the set of all quasiuniformities inducing δ by π(δ) and call it the quasi-proximity class of δ.11 Proposition 2.5.4. Let (X, δ) be a quasi-proximity space. Then the collection S of all sets of the form TA,B (compare Subsection 2.3) where A δ B, is subbase for a totally bounded quasi-uniformity Uδ on X, which is compatible with δ. Moreover Uδ is the coarsest quasi-uniformity in π(δ) and is the only totally bounded member of π(δ). Proof. We check that S is a subbase for a quasi-uniformity Uδ on X: Let TA,B ∈ S. Obviously TA,B is reﬂexive. Since A δ B, there exists a subset C of X such that A δ C and X \ C δ B. It follows that (TA,C ∩ TX\C,B )2 ⊆ TA,B : Assume that (a, b), (b, c) ∈ TA,C ∩ TX\C,B , but (a, c) ∈ TA,B . Then a ∈ A and c ∈ B, and b ∈ C ∩ (X \ C) — a contradiction. Hence the stated inclusion holds. Moreover Uδ is clearly totally bounded. Let U ∈ π(δ). If A δ B, there exists U ∈ U such that (A × B) ∩ U = ∅. Thus U ⊆ TA,B , and therefore Uδ ⊆ U. For the well-known arguments that Uδ induces δ and that Uδ is the unique totally bounded quasiuniformity inducing δ we refer the reader to the proofs in [30, Theorem 1.33]. 11A quasi-uniformity U on X is often called compatible with a quasi-proximity δ if δ = δ . U

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Remark 2.5.5. (a) If (X, δ) is a quasi-proximity space, then obviously τ (δ) = τ (Uδ ). (b) Given a quasi-uniformity U on a set X, we have Uω = UδU . Furthermore U and Uω induce the same bitopological space on X. (c) For any quasi-uniformity U, the equality (Uω )−1 = (U −1 )ω [30, p. 17] holds, and Uω is transitive (that is, U is proximally transitive) if U is transitive [30, Lemma 6.3]. (d) If the map f : (X, U) → (Y, V) between quasi-uniform spaces (X, U) and (Y, V) is uniformly continuous, then f : (X, Uω ) → (Y, Vω ) is uniformly continuous. A map f : (X, δ) → (Y, δ ) between quasi-proximity spaces (X, δ) and (Y, δ ) is proximally continuous if and only if f : (X, Uδ ) → (Y, Uδ ) is uniformly continuous. The following well-known result implies the important fact that a map between two pseudometric spaces is uniformly continuous if and only if it is proximally continuous. Proposition 2.5.6 (compare [30, Proposition 1.60]). Let (X, U) be a quasiuniform space with a countable decreasing base {Un : n ∈ ω} and let f : (X, U) → (Y, V) be a map into a uniform space (Y, V). Then f is uniformly continuous provided that f : (X, δU ) → (Y, δV ) is proximally continuous. Proof. Suppose otherwise. There is a symmetric entourage W ∈ V such that (f × f )Un ⊆ W 3 whenever n ∈ ω. For each n ∈ ω choose (xn , yn ) ∈ Un such that (f (xn ), f (yn )) ∈ W 3 . Set An = {m ∈ ω : (f (xn ), f (ym )) ∈ W } and Bn = {m ∈ ω : (f (xm ), f (yn )) ∈ W } whenever n ∈ ω. Assume that some An is inﬁnite. Since {xm : m ∈ An } δU {ym : m ∈ An } and f is proximally continuous, {f (xm ) : m ∈ An } δV {f (ym ) : m ∈ An }. Thus there are s, t ∈ An such that (f (xs ), f (yt )) ∈ W . Therefore (f (yt ), f (xn )) ∈ W, (f (xn ), f (ys )) ∈ W and consequently (f (xs ), f (ys )) ∈ W 3 — a contradiction. Hence each An is ﬁnite. Analogously one shows that each Bn is ﬁnite. Let n0 = 0 and for each j ∈ ω choose nj+1 ∈ ω greater than nj and every nj (Ai ∪ Bi ). Since f is proximally continuous, {f (xnj ) : j ∈ ω} δV member of i=0 {f (ynj ) : j ∈ ω}. Therefore there are j, k ∈ ω such that (f (xnj ), f (ynk )) ∈ W . But nj = nk , nj > nk , and nj < nk are all impossible. We have reached a contradiction and conclude that f is uniformly continuous. Corollary 2.5.7 (compare [30, Theorem 1.59]). Let U be a quasi-uniformity with a countable base on a set X. Then U is ﬁner than any uniformity V of π(δU ). Proof. Consider the identity map idX : (X, U) → (X, V). Since Uω = Vω , the map idX is proximally continuous. We conclude by Proposition 2.5.6 that it is uniformly continuous. Hence V ⊆ U. It is known that the supremum of two quasi-uniformities belonging to the same quasi-proximity class need not belong to that quasi-proximity class [30, Example 1.58]. On the other hand for any nonempty family (Ui )i∈I of quasi-uniformities on a set X, we have ( i∈I Ui )ω = i∈I (Ui )ω according to [17, Lemma 1]. An important instance where the operation (·)ω is compatible with the operation ∨ is given in the next proposition.

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Proposition 2.5.8 ([30, Proposition 1.40]). Let U be a totally bounded quasiuniformity and let V be an arbitrary quasi-uniformity on a set X. Then (U ∨ V)ω = U ∨ Vω . Proof. It is easy to see that U ∨ Vω ⊆ (U ∨ V)ω . In order to prove the second inclusion, let A ⊆ X, U ∈ U and V ∈ V. It will suﬃce to show that A δ X \ (U ∩ V )(A) where δ is the quasi-proximity on X induced by U ∨Vω . Choose W ∈ U such that W 3 ⊆ U . By total boundedness of U there is a ﬁnite subset F of A such that A ⊆ f ∈F (W ∩ W −1 )(f ). Fix f ∈ F . Set Df = A ∩ (W ∩ W −1 )(f ). Then Df δU W 2 (f ) and Df δVω V (Df ). Therefore Df δ X \ (W 2 (f ) ∩ V (Df )) by the deﬁnition ofδ. Moreover A = f ∈F Df and hence A δ X \ f ∈F (W 2 (f ) ∩ V (Df )). Rewriting the latter union we also get W 2 (f ) ∩ W 3 (a) ∩ V (a) ⊆ (U ∩ V )(A), V (a) ⊆ f ∈F

a∈Df

a∈A

by the deﬁnition of Df (f ∈ F ). Hence the assertion is veriﬁed.

Remark 2.5.9. (a) (compare [30, Proposition 1.43]). Let (X, U) be a quasi-uniform space and let K ⊆ G where K is τ (U)-compact and G is τ (U)-open in X. Then there is V ∈ U such that V (K) ⊆ G: Indeed for each x ∈ K choose Vx ∈ U such that Vx2 (x) ⊆ G. Since K is compact, there is a ﬁnite subset F of K such that {Vf (f ) : f ∈ F } covers K. Set V = f ∈F Vf . Let x ∈ K. Then there is f ∈ F such that x ∈ Vf (f ) and thus V (x) ⊆ Vf2 (f ) ⊆ G. Consequently V (K) ⊆ G. Therefore Kδ U X \ G. It follows from this observation that the Pervin quasi-uniformity is the unique compatible totally bounded quasi-uniformity on a hereditarily compact space. (b) A topological space X is called locally compact if each point has a neighborhood base consisting of compact subsets of X. If in a locally compact space X there are given a compact set K and an open set G satisfying K ⊆ G, then by local compactness of X there exists a compact subset K of X such that K ⊆ int K ⊆ K ⊆ G. With the help of part (a) we can then deduce that a locally compact space X admits a coarsest quasiuniformity Q0 . Indeed it is generated by the subbase {TK,X\G : K ⊆ G where K is compact and G is open in X} (compare with the proof of Proposition 2.5.4). Of course, Q0 is totally bounded. Often τ ((Q0 )−1 ) is called the cocompact topology of X. (c) While a topological space that admits a coarsest quasi-uniformity need not be locally compact12, completely regular spaces admitting a coarsest uniformity U0 are characterized by the property of local compactness (see [25, Problem 8.5.9]). The coarsest compatible quasi-uniformity Q0 of a locally compact completely regular space X coincides with the coarsest compatible uniformity U0 on X if and only if X is compact (compare with Corollary 2.2.7 and [30, Proposition 1.47]). 12We shall show below that a weakening of the condition of core-compactness yields a characterization of those topological spaces that admit a coarsest quasi-uniformity.

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As an exercise the reader may want to verify the following astonishing fact: Each locally compact zero-dimensional γ-space is non-archimedeanly quasi-pseudometrizable (compare [78, Remark]). It is known that a completely regular topological space X admits a unique uniformity if and only if for any pair of closed subsets of X that are completely separated, at least one of them is compact [25, Problems 8.5.11 and 3.12.16]. We next study the analogous problem for quasi-uniformities. Proposition 2.5.10 ([78]). A topological space (X, τ ) admits a unique quasiuniformity if and only if (1) X is hereditarily compact and (2) X does not possess a strictly decreasing sequence (Gn )n∈ω of open sets with an open intersection. Proof. Suppose that the given two conditions are satisﬁed. By hereditary compactness all compatible quasi-uniformities on X belong to π(δP(τ ) ) (see Remark 2.5.9(a)). Hence it suﬃces to show that the ﬁne quasi-uniformity FN (τ ) of X is totally bounded. Since by hereditary compactness of X, clearly FN (τ ) is hereditarily precompact, by Corollary 2.4.5 it remains to be shown that (FN (τ ))−1 is hereditarily precompact. Suppose otherwise. Then there are V ∈ FN (τ ) and a sequence (xn )n∈ω in X such that xn ∈ (V −1 )2 (xk ) whenever n, k ∈ ω and n > k. Set A = clτ {xn : n ∈ ω} and put Gk = (X \ A) ∪ (X \ clτ V −1 ({xs : s < k, s ∈ ω})) whenever k ∈ ω. Consequently k∈ω Gk = X \ A and xk ∈ Gk , but xk ∈ Gk+1 whenever k ∈ ω. However this contradicts the second condition, because then (Gk )k∈ω is a strictly decreasing sequence of open sets with an open intersection. Hence FN (τ ) is totally bounded and we conclude that (X, τ ) admits a unique quasi-uniformity. In order to prove the converse suppose that (X, τ ) admits a unique quasiuniformity. We ﬁrst use the Fletcher construction to see that X is hereditarily compact: Otherwise X possesses a strictly increasing sequence (Gn )n∈ω of open sets and, thus, the ﬁne transitive quasi-uniformity FT (τ ) is not hereditarily precompact and so is distinct from P(τ ) — a contradiction. Similarly, condition (2) is satisﬁed: Otherwise by the Fletcher construction (FT (τ ))−1 would not be hereditarily precompact and thus FT (τ ) would be distinct from P(τ ) — another contradiction. Hence X satisﬁes both conditions. From the preceding result one can deduce that a topological space admits a unique quasi-uniformity if and only if all its interior-preserving open collections are ﬁnite (compare [7]). In the light of the fact (see [118, Corollary 3.5]) that a topological space admits a unique quasi-uniformity if and only if its Pervin quasiproximity class possesses a unique member, Proposition 2.5.10 can be considered a special case of a more general result that characterizes those quasi-proximity classes possessing a unique member (see [24]). The coﬁnite topology on an uncountable set is a typical example of an inﬁnite topology that admits a unique quasi-uniformity. Recall that a nonempty topological space is called irreducible if each pair of nonempty open sets intersects, and that a topological space is called quasi-sober if each closed irreducible subspace is equal to the closure of a singleton. A T0 -space that is quasi-sober is called sober. It is known that each hereditarily compact quasi-sober space admits a unique quasi-uniformity

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(compare [95, Proposition 3]), which turns out to be bicomplete (compare Proposition 2.7.12). A ﬁlter on a topological space is called primitive if the set of its cluster points coincides with the set of its limit points. It is well known that each ultraﬁlter on a topological space has this property. After some preparations we shall next characterize the classes of topological spaces that admit a unique quasi-proximity and a coarsest (totally bounded) quasi-uniformity. Lemma 2.5.11 ([77, Lemma 3]). Let δ be a compatible quasi-proximity on a topological space (X, τ ) and let F be a primitive ﬁlter on X. Then S(δ, F) = {TA,B : A, B ⊆ X, A δ B, and intτ A contains a limit point of F or B ∈ F} is a subbase for a compatible totally bounded quasi-uniformity V on X. Furthermore V ⊆ Uδ . Proof. Similarly as in the ﬁrst part of the proof of Proposition 2.5.4 one proves that S(δ, F) is a subbase for a totally bounded quasi-uniformity V on X. Obviously V ⊆ Uδ . It remains to show that V is compatible. Assume that x ∈ X is (resp. is not) a limit point of F. Let N be a neighborhood of x. Then there is a neighborhood G of x such that G ⊆ N and such that (the complement of) G is contained in F. There is a subset H of X such that x δ X \ H and H δ X \ G. Then TH,X\G ∈ S(δ, F). We conclude that V is compatible. Definition 2.5.12 ([77, p. 184]). Let A, B be subsets of a topological space X such that A ⊆ B. Then A is called handy in B with respect to X (in symbols A <X B) provided that each ultraﬁlter G on X with A ∈ G contains a ﬁnite (possibly empty) subcollection M of open sets in X such that M ⊆ B and each member M of M contains a limit point of G. Lemma 2.5.13 ([77, Lemma 2]). Let A and B be subsets of a topological space (X, τ ). Then A <X B if and only if A δ B for all strong inclusions δ of compatible quasi-proximities δ on (X, τ ). Proof. Obviously, the assertion is true if A = ∅ or B = X. Therefore, we suppose that A = ∅ and B = X in the following proof. Assume that A <X B, but that A δ X \ B for some compatible quasi-proximity δ on (X, τ ). Set F = {B \ C : C δ B} ∪ {A}. Since A δ X \ B, F has the ﬁnite intersection property. Let G be an ultraﬁlter on X that contains F. Because A ∈ G and A <X B, there is a ﬁnite collection M of open subsets of X such that each member of M contains a limit point of G and such that M ⊆ B. Since B = X, we have that M = ∅. If M ∈ M, then for some limit point x of G in M we choose an open subset N (M ) of X such x δ N (M ) and N (M ) δ M . that Set N = {N (M ) : M ∈ M}. Then N M. Hence B \ N ∈ F ⊆ G, δ but N ∈ G — a contradiction. We conclude that A δ B for each compatible quasi-proximity δ on (X, τ ) if A <X B. Consider now the converse. Let δ = δP(τ ) . Assume that A, B are subsets of X such that A β B for each compatible quasi-proximity β on (X, τ ). Let G be an ultraﬁlter on X that contains A and let α be the quasi-proximity on X that is induced by the quasi-uniformity W generated by the subbase S(δ, G), as deﬁned in Lemma 2.5.11. Since A α B, there are sets nUi = TAi ,Bi ∈ S(δ, G) (where i ∈ {1, n . . . , n} and n is a positive integer) such that [ i=1 Ui ](A)∩(X \B) = ∅. Because [ i=1 Ui ](A) ∈ G and B = X, there exists a nonempty subcollection K of {X \ Bi : i ∈ {1, . . . , n}}

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such that K ∈ G and K ⊆ B. Let K = {intτ Ai : X \ Bi ∈ K, i ∈ {1, . . . , n}}. We conclude that each member of K contains a limit point of G. Moreover K ⊆ K. We have shown that A <X B. Proposition 2.5.14 ([74]). A topological space (X, τ ) admits a unique (compatible) quasi-proximity if and only if its topology is the only base of open sets that is closed under ﬁnite unions and ﬁnite intersections. Proof. Let B be an open base for τ that is closed under ﬁnite unions and ﬁnite intersections. Then {SB : B ∈ B} is a subbase for a compatible transitive totally bounded quasi-uniformity U on X. Observe that SG ∈ U for some G ∈ τ means that H∈M SH ⊆ SG for some ﬁnite collection M ⊆ B, which implies that G can be written as a ﬁnite union of ﬁnite intersections of elements of B and, thus, G ∈ B. Consequently each topological space (X, τ ) that admits a unique quasi-proximity has a unique base of open sets that is closed under ﬁnite unions and ﬁnite intersections, namely the topology τ itself. For the second part of the proof assume that U is a compatible (possibly nontransitive) totally bounded quasiuniformity distinct from the Pervin quasi-uniformity P(τ ) on the topological space (X, τ ). Then there is an open subset G of X such that G δU G does not hold. By Lemma 2.5.13 therefore G <X G. Hence there is an ultraﬁlter G on X such that G ∈ G, but there does not exist a ﬁnite open collection M such that M ⊆ G and each member of M contains a limit point of G. Then according to Lemma 2.5.11 S(δP(τ ) , G) generates a compatible totally bounded quasi-uniformity V on (X, τ ), which is transitive, since P(τ ) is transitive. Obviously B := {H ⊆ X : H δV X \ H} is a base of τ that is closed under ﬁnite unions and ﬁnite intersections. deﬁnition of S(δP(τ ) , G) imply that But G δV X \ G together with G ∈ G and the there is a ﬁnite open collection M such that M ⊆ G and each element of M contains a limit point of G (compare with the end of the proof of Lemma 2.5.13). We have reached a contradiction and conclude that G δV X \ G and thus G ∈ B. Therefore B = τ . In particular we note that while a completely regular space admits a unique uniformity if and only if it admits a unique proximity (compare [25, Problems 8.5.11 and 3.12.16]), a topological space that admits a unique quasi-proximity need not admit a unique quasi-uniformity. The question of whether the property described in Proposition 2.5.14 implies hereditary compactness in T1 -spaces attracted some attention and was ﬁnally settled negatively in [113]. Example 2.5.15 ([74, Example 2]). The topology τ on X = ω + 2 deﬁned by {[0, n] : n ∈ ω} ∪ {∅, ω, ω + 1, X \ {ω}} has a unique compatible quasi-proximity, while this is not true for the subspace X \ {ω}. Remark 2.5.16 ([99, Propositions 3 and 4]). For every non-zero cardinal number κ there exists a topological space that admits exactly κ compatible quasiproximities. Proposition 2.5.17 ([77, Proposition 2]). A topological space X admits a coarsest (totally bounded) quasi-uniformity if and only if every open subset G of X is the union of open sets H such that H <X G. Proof. If a topological space X possesses a coarsest compatible quasi-proximity, then by Lemma 2.5.13 its strong inclusion must coincide with the handy relation

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<X . Hence in each topological space admitting a coarsest quasi-uniformity, the condition stated in Proposition 2.5.17 is obviously satisﬁed. Suppose now that X is a topological space such that <X satisﬁes that condition. It is easy to see that for any topological space Y ,
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Observe that V ⊆ V ⊆ V 3 . We conclude that card({V : V belongs to V and V (x) is open whenever x ∈ X}) ≤ (card(τ × τ ))nw(X) ≤ 2nw(X) . Since by [30, p. 3] V has a base B such that B(x) is open whenever x ∈ X and B ∈ B, we conclude that V has a base of cardinality ≤ 2nw(X) . Corollary 2.5.20 ([84, Corollary 1]). Let X be a topological space of network weight nw(X). Then the number of transitive neighbornets of X is ≤ 2nw(X) . Proof. For every transitive neighbornet V of X we have V = V 3 and V (x) is open whenever x ∈ X. The assertion is a consequence of the preceding argument. Proposition 2.5.21 ([84, Corollary 2]). Let X be a topological space of network weight nw(X). Then the number of compatible quasi-uniformities on X is nw(X) ≤ 22 . Proof. We shall use the notation explained in the proof of Lemma 2.5.19. If V is a compatible quasi-uniformity on X, then {V : V ∈ V and V (x) is open whenever x ∈ X} yields a base for V. The assertion follows from the proof of Lemma 2.5.19. Remark 2.5.22. (a) In particular, systematic eﬀorts were undertaken to construct large families of nontransitive compatible quasi-uniformities on topological spaces. For instance it is known that each inﬁnite Tychonoﬀ space admits at least ℵ0 ℵ0 22 nontransitive (as well as at least 22 transitive) totally bounded quasi-uniformities (compare [41]). Similarly, if a topological space admits more than one quasi-uniformity, then the Pervin quasi-proximity class ℵ0 ℵ0 contains at least 22 nontransitive (as well as at least 22 transitive) quasi-uniformities (compare [88]). The latter construction of nontransitive quasi-uniformities was based on the fact that there exist nontransitive quasi-uniformities which cannot be made transitive by taking the supremum with any totally bounded quasi-uniformity. (b) In [86] it was also shown that a quasi-proximity class that contains more ℵ0 than one member contains at least 22 quasi-uniformities. Problem 2.5.23 (Losonczi; for research, [119, Problem 3]). Given quasiproximities δ1 and δ2 on a set X, is there any connection between card π(δ1 ) and card π(δ2 ) if δ2 ⊆ δ1 ? Is it true that card π(δ1 ) ≤ card π(δ2 ) in this case? 2.6. Various kinds of completeness. Definition 2.6.1. A ﬁlter F on a quasi-uniform space (X, U) is called a U s Cauchy ﬁlter if for each U ∈ U there is an F ∈ F such that F × F ⊆ U . A quasi-uniform space (X, U) is called bicomplete provided that each U s -Cauchy ﬁlter converges with respect to the topology τ (U s ) in X. Usually bicomplete uniform spaces are called complete. Hence a quasi-uniformity U is bicomplete if and only if U s is a complete uniformity. Bicompleteness of a quasi-uniform space (X, U) could equivalently have been deﬁned by requesting τ (U s )-convergence of U s -Cauchy nets, that is, nets (xd )d∈D in X satisfying the

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condition that for each U ∈ U there is dU ∈ D such that d1 , d2 ∈ D and d1 , d2 ≥ dU imply that (xd1 , xd2 ) ∈ U . It is well known that for a quasi-pseudometric space (X, d) each (Ud )s -Cauchy ﬁlter τ (ds )-converges if and only if each ds -Cauchy sequence τ (ds )-converges (compare Proposition 2.6.13). Hence bicompleteness and sequential bicompleteness are equivalent in quasi-pseudometric spaces. Beginning in Subsection 2.7 we shall show that the bicompletion is well behaved in the category of quasi-uniform spaces. In this subsection we look at completion theories that are based on more asymmetric concepts of completeness. It is very reasonable to look for such completion theories, since fairly many asymmetric spaces turn out to be bicomplete. Remark 2.6.2. In the beginning of the theory of quasi-uniform spaces the following concept of a Cauchy ﬁlter was mainly considered: A ﬁlter F on a quasiuniform space (X, U) is called a Pervin–Sieber - or P S-Cauchy ﬁlter (compare [145]) if for each U ∈ U there is x ∈ X such that U (x) ∈ F. An obvious advantage of this deﬁnition is that every τ (U)-convergent ﬁlter possesses this property. A quasi-uniform space (X, U) is called P S-complete provided that each P SCauchy ﬁlter has a cluster point. It is known [30, Corollary 3.9] that for locally symmetric quasi-uniform spaces (but not in general) that property is equivalent to convergence completeness, i.e., to the property that each P S-Cauchy ﬁlter converges. It is readily checked that each quasi-uniformity with the Lebesgue property is convergence complete (see [30, Proposition 5.7]). Unfortunately a ﬁlter on a subspace that is the restriction of a P S-Cauchy ﬁlter deﬁned on a quasi-uniform space is not necessarily a P S-Cauchy ﬁlter, which makes the concept of P S-completeness diﬃcult to handle. In the historic development of the theory of quasi-uniform spaces the problem of whether the ﬁne quasi-uniformity of an arbitrary (regular) topological space is P S-complete was open for some time.14 The answer to this problem turned out to be negative. In [97] it was shown that any nontrivial Tychonoﬀ Σ-product yields a counterexample. While we do not have the space here to discuss the details of these investigations, we can describe one approach to establish that the ﬁne quasi-uniformity of some topological spaces is not P S-complete. Recall that a topological space is called ω-bounded provided that the closure of each countable subspace is compact. We also say that a topological space satisﬁes the Souslin property if every collection of pairwise disjoint open subsets of X is countable. An open ﬁlter is a ﬁlter with a ﬁlter base of open sets. A maximal open ﬁlter is called an open ultraﬁlter. Lemma 2.6.3 ([97, Proposition 3.3]). If the ﬁne quasi-uniformity FN (τ ) of a regular topological space (X, τ ) satisfying the Souslin property is P S-complete, then the closure of each ω-bounded subspace of X is compact. Proof. The proof is by contradiction. Let B be an ω-bounded subspace of X such that B is not compact. By regularity of X there is an open cover C of B that has no ﬁnite subfamily covering B. Let F be the open ﬁlter generated by {G : G is 14The ﬁne quasi-uniformity of each weakly orthocompact topological space is known to be P S-complete (compare [30, p. 116 and Corollary 5.32]). Of course, by deﬁnition each completely regular Dieudonn´e complete topological space admits a complete uniformity [25, Problem 8.5.13] and hence its ﬁne quasi-uniformity is also P S-complete.

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an open set in X and there exists C ∈ C such that B \ C ⊆ G} and let G be an open ultraﬁlter on X containing F. Since X is regular, G has no cluster point in X. The contradiction is established by showing that G is a P S-Cauchy ﬁlter on (X, FN (τ )). Let U ∈ FN (τ ) and let V be a neighbornet of X such that V 2 ⊆ U and V (x) is open whenever x ∈ X. Because V (B) is an open subspace of a space with the Souslin property, there is a countable set D of B such that V (B) ⊆ V (D). Since D is compact, there is a ﬁnite subset E of X such that D ⊆ V (E). Thus V (B) ⊆ V (D) ⊆ V 2 (E) ⊆ U (E). Because V (B) ∈ G, we have U (E) ∈ G by maximality of G. Since E is ﬁnite, it follows that G is a P S-Cauchy ﬁlter on (X, FN (τ )). Hence we have reached a contradiction. When looking for an asymmetric concept of completeness suitable for quasiuniform spaces, it is natural to consider the following asymmetric generalizations of the concept of a U s -Cauchy net (compare with Kelly’s article [62]): A net (xd )d∈D in a quasi-uniform space (X, U) is called left K-Cauchy (resp. right K-Cauchy) if for any U ∈ U there is dU ∈ D such that d2 , d1 ∈ D and d2 ≥ d1 ≥ dU imply that (xd1 , xd2 ) ∈ U (resp. (xd2 , xd1 ) ∈ U ). Corresponding concepts for ﬁlters were suggested by Romaguera [137, 139]. A ﬁlter F on a quasi-uniform space (X, U) is said to be left (resp. right) K-Cauchy if for each U ∈ U there is F ∈ F such that U (x) ∈ F (resp. U −1 (x) ∈ F) whenever x ∈ F . A quasi-uniformity U on a set X is called left (resp. right) K-complete provided that each left (resp. right) K-Cauchy ﬁlter converges in (X, U). Observe that P S-completeness implies left K-completeness in quasi-uniform spaces by Proposition 2.6.5(b) below. As we shall show on the next pages, K-concepts in a quasi-uniform space (X, U) have a rich combinatorial structure, which makes them interesting and useful, although they carry the obvious defect that in general τ (U)-convergence is rather unrelated to these properties. In fact a regular quasi-metric space in which each convergent sequence has a left K-Cauchy subsequence is metrizable (compare with the proof of [102, Proposition 4]). Lemma 2.6.4 (see e.g., [111, Lemma 5]). On a quasi-uniform space (X, U) each right K-Cauchy ﬁlter is stable. Proof. Let F be a right K-Cauchy ﬁlter on (X, U) and U ∈ U. Then there is FU ∈ F such that for any y ∈ FU we have U −1 (y) ∈ F. Consider any x ∈ (x) ∩ F = ∅ whenever F ∈ F. Thus x ∈ F ∈F U (F ) and, FU . Note that U −1 consequently, FU ⊆ F ∈F U (F ). We conclude that F ∈F U (F ) ∈ F. Therefore, F is U-stable. Proposition 2.6.5. (a) ([139, Proposition 1]). An ultraﬁlter on a quasi-uniform space (X, U) is right K-Cauchy if and only if it is stable. (b) ([139, Lemma 1]). Let (X, U) be a quasi-uniform space and let F be a right K-Cauchy ﬁlter (resp. left K-Cauchy ﬁlter) on (X, U). If x is a τ (U)-cluster point in X, then F τ (U)-converges to it. (X, U) and let U ∈ U. Thus Proof. (a). Let F be a stable ultraﬁlter on −1 U (F ) ∈ F. Let x ∈ U (F ). Then U (x) ∩ F = ∅ whenever F ∈ F F ∈F F ∈F and therefore U −1 (x) ∈ F, because F is an ultraﬁlter. Thus F is a right K-Cauchy ﬁlter. The converse follows from Lemma 2.6.4.

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(b). Let U ∈ U. Choose V ∈ U such that V 2 ⊆ U . By assumption x is a τ (U)-cluster point of F on X. There is FV ∈ F such that V −1 (y) ∈ F whenever y ∈ FV . Let y ∈ FV . Consequently V (x) ∩ V −1 (y) = ∅. Thus y ∈ V 2 (x) and hence FV ⊆ V 2 (x). Therefore U (x) ∈ F and F converges to x in (X, U). For the second part, consider U ∈ U. Choose V ∈ U such that V 2 ⊆ U . There is FV ∈ F such that V (y) ∈ F whenever y ∈ FV . Since by assumption x ∈ clτ (U) FV , there is y ∈ V (x) ∩ FV . But then V (y) ∈ F and V (y) ⊆ V 2 (x). We have shown that U (x) ∈ F and F τ (U)-converges to x. Corollary 2.6.6 ([111, Corollary 5]). Let (X, U) be a quasi-uniform space in which each U-stable ﬁlter has a τ (U)-cluster point. Then (X, U) is right K-complete. Proof. We use Lemma 2.6.4 and Proposition 2.6.5(b).

Lemma 2.6.7 ([81, p. 320]). Let (X, U) be a quasi-uniform space. Then a ﬁlter F on X is a U s -Cauchy ﬁlter if and only if it is stable and left K-Cauchy on (X, U). Proof. Let F be a U s -Cauchy ﬁlter on X. For each U ∈ U there is F ∈ F such that F × F ⊆ U . Clearly F is then a left K-Cauchy ﬁlter and a right KCauchy ﬁlter, because (U ∩ U −1 )(x) ∈ F whenever x ∈ F . Thus F is also stable by Lemma 2.6.4. In order to prove the converse, suppose that the ﬁlter F is stable and left K-Cauchy on (X, U). Let U ∈ U. Choose V ∈ U such that V 2 ⊆ U and ﬁnd FV ∈ F such that V (x) ∈ F whenever x ∈ FV . Consider now arbitrary points x1 , x2 ∈ FV ∩ F ∈F V (F ). Then V (x1 ) ∈ F and thus x2 ∈ V (V (x1 )) ⊆ U (x1 ). Since FV ∩ F ∈F V (F ) ∈ F, we conclude that F is a U s -Cauchy ﬁlter on X. Corollary 2.6.8 ([139, Corollary 1.1]). A quasi-uniform space (X, U) is hereditarily precompact if and only if each ultraﬁlter on X is a left K-Cauchy ﬁlter. Proof. The assertion is a consequence of Propositions 2.4.7 and 2.6.5(a). Similarly one proves that a quasi-pseudometric space is hereditarily precompact if and only if each sequence has a left K-Cauchy subsequence (see [102, Theorem 3]). Next we formulate a symmetrized version of Corollary 2.6.8. Corollary 2.6.9 ([30, Proposition 3.14]). A quasi-uniform space (X, U) is totally bounded if and only if every ultraﬁlter on X is a U s -Cauchy ﬁlter. Proof. For instance: U is totally bounded ⇔ U and U −1 are hereditarily precompact ⇔ each ultraﬁlter is left K-Cauchy ﬁlter and stable in (X, U) ⇔ each ultraﬁlter is a U s -Cauchy ﬁlter. The equivalences follow from Corollary 2.4.5, Corollary 2.6.8 and Proposition 2.4.7, and Lemma 2.6.7. A direct proof of the statement is sketched in [30, Proposition 3.14]. Proposition 2.6.10. A quasi-uniform space (X, U) is totally bounded and bicomplete if and only if τ (U s ) is compact.15 Proof. Suppose that τ (U s ) is compact. Then U s is precompact and hence U is totally bounded, because U s is a uniformity. Furthermore by compactness of τ (U s ) each U s -Cauchy ﬁlter has a τ (U s )-cluster point and hence τ (U s )-converges as a consequence of Proposition 2.6.5(b). Thus U is bicomplete. On the other hand suppose that (X, U) is totally bounded and bicomplete. Let G be an ultraﬁlter on 15Because of the bitopological space that it induces a quasi-uniformity U is sometimes called joincompact provided that τ (U s ) is compact (compare [70]).

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X. Then by Corollary 2.6.9 G is a U s -Cauchy ﬁlter and thus τ (U s )-converges by bicompleteness of (X, U). Hence the topology τ (U s ) is compact. Lemma 2.6.11 ([81, p. 325]). A quasi-uniform space (X, U) is left K-complete if and only if each left K-Cauchy net converges in (X, U). Proof. Suppose that (X, U) is left K-complete and that (xd )d∈D is a left KCauchy net in (X, U). Then the ﬁlter F generated by the base {{xd : d ≥ d, d ∈ D} : d ∈ D} is a left K-Cauchy ﬁlter on (X, U). By assumption F converges to some x and thus the net (xd )d∈D converges to x in (X, U). In order to prove the converse, suppose that each left K-Cauchy net converges in (X, U). Let F be an arbitrary left K-Cauchy ﬁlter on (X, U) and for each Z ∈ U let MZ be a set belonging to F that witnesses the left K-Cauchy property of F with respect to Z. For any U ∈ U and F ∈ F ﬁx xU,F ∈ F such that U (xU,F ) ∈ F. Using an idea due to S¨ underhauf (see [149, proof of Theorem 3]) we deﬁne a directed relation ≤ on D = U ×F by setting (U, A) ≤ (V, B) if and only if (1) (U, A) = (V, B) or (2) V ⊆ U and B ⊆ U (xU,A ) ∩ A. Since (xU,F )(U,F )∈D is a left K-Cauchy net on (X, U), it converges to some point x in (X, U). Let P ∈ U. Choose Z ∈ U such that Z 2 ⊆ P . There is (H0 , F0 ) ∈ D such that xH,F ∈ Z(x) whenever (H, F ) ∈ D with (H, F ) ≥ (H0 , F0 ). Set A = F0 ∩ MZ ∩ H0 (xH0 ,F0 ). Then (H0 ∩ Z, A) ≥ (H0 , F0 ). Therefore xH0 ∩Z,A ∈ Z(x) and Z(xH0 ∩Z,A ) ⊆ Z 2 (x) ⊆ P (x). Since xH0 ∩Z,A ∈ A ⊆ MZ , we have that Z(xH0 ∩Z,A ) ∈ F. We conclude that P (x) ∈ F and that F converges to x in (X, U). Hence (X, U) is left K-complete. Remark 2.6.12. It is also true that a quasi-uniform space (X, U) is right Kcomplete if and only if each right K-Cauchy net converges in (X, U) (see [107, Lemma 1]). Proposition 2.6.13 ([111, p. 169]). The following statements are equivalent for a quasi-pseudometric space (X, d): (a) Each Ud−1 -stable ﬁlter has a cluster point in (X, d). (b) Each left K-Cauchy ﬁlter converges in (X, d). (c) Each left K-Cauchy sequence converges in (X, d). Proof. (a) ⇒ (b). A left K-Cauchy ﬁlter on (X, Ud ) is Ud−1 -stable by Lemma 2.6.4 and converges to its cluster points in (X, d) by Proposition 2.6.5(b). Hence the assertion is obvious. (b) ⇒ (c). The statement follows in the same way as in the ﬁrst part of the proof of Lemma 2.6.11. Let F be a Ud−1 -stable ﬁlter on X. For each n ∈ ω set Fn := (c) ⇒ (a). −n ). Choose x0 ∈ F0 arbitrarily. Suppose that for some n ∈ F ∈F Bd−1 (F, 2 ω, the point xn ∈ Fn is deﬁned. Then we ﬁnd xn+1 ∈ Fn+1 such that xn ∈ Bd−1 (xn+1 , 2−n ). We see with the help of the triangle inequality and the geomet∞ ric series k=0 2−k that (xn )n∈ω is a left K-Cauchy sequence in (X, d). By our assumption there exists a limit x of (xn )n∈ω in (X, d). In order to reach a contradiction, suppose that there are F ∈ F and n ∈ ω such that Bd (x, 2−n ) ∩ F = ∅. Then Bd (x, 2−(n+1) ) ∩ Bd−1 (F, 2−(n+1) ) = ∅, but xk+1 ∈ Fk+1 ⊆ Bd−1 (F, 2−(n+1) ) whenever k ∈ ω such that k ≥ n, which contradicts the fact that (xn )n∈ω converges to x. We conclude that x is a τ (d)-cluster point of F.

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Results similar to Proposition 2.6.13 for the corresponding right K-concepts require additional conditions (compare [2, 111]). Next we observe that the Baire Category Theorem holds in regular left K-complete quasi-pseudometric spaces. Remark 2.6.14 ([4, 43]). Each regular left K-complete quasi-pseudometric space (X, d) is a Baire space. Proof. Let (Gn )n∈ω be a sequence of dense open sets in (X, d) and let G be any nonempty open subset of (X, d). Choose x0 ∈ X and k0 ∈ ω such that Bd (x0 , 2−k0 ) ⊆ G0 ∩ G. Furthermore inductively for each n ∈ ω ﬁnd xn+1 ∈ X and kn+1 ∈ ω such that Bd (xn+1 , 2−kn+1 ) ⊆ Gn+1 ∩ Bd (xn , 2−kn ) and kn+1 > kn . Then (xn )n∈ω is a left K-Cauchy sequence and thus converges to some x ∈ Bd (xn , 2−kn ) whenever n ∈ ω, we conclude that x ∈ x ∈ X. Since G ∩ n∈ω Gn and thus n∈ω Gn is dense in (X, d). In connection with the next result observe that in a quasi-uniform space the property that each co-stable ﬁlter16 clusters implies left K-completeness and for uniform spaces it is equivalent to supercompleteness (that is, completeness of the Hausdorﬀ uniformity) of the uniform space (compare Proposition 3.4.9). Proposition 2.6.15 ([85, Proposition 3.3]). Let f be a uniformly open continuous map from a quasi-uniform space (X, U) in which each U −1 -stable ﬁlter F has a cluster point onto a quasi-uniform space (Y, V). Then each V −1 -stable ﬁlter on Y has a cluster point in (Y, V). Proof. Let F be a ﬁlter that is stable on (Y, V −1 ) and ﬁx U ∈ U. Since f is uniformly open, there is V ∈ V such that V (f (x)) ⊆ f (U (x)) whenever x ∈ X. the ﬁlter F is stable on (Y, V −1 ), there 0 ∈ F such that F0 ⊆ is F−1 Because −1 −1 V (F ). We want to show that f (F ) ⊆ U (f −1 (F )): Let F ∈ F 0 F ∈F F ∈F and a ∈ f −1 (F0 ). Hence f (a) = f0 for some f0 ∈ F0 . Thus f0 ∈ V −1 (e) for some e ∈ F . Then e ∈ V (f0 ) ⊆ f (U (a)). Therefore e = f (c) for some c ∈ U (a). It follows that a ∈ U −1 (c) and c ∈ f −1 (F ). We have shown that a ∈ U −1 (f −1 (F )). We conclude that f −1 (F0 ) ⊆ F ∈F U −1 (f −1 (F )). Hence the ﬁlter f −1 F generated on X by the ﬁlter base {f −1 (F ) : F ∈ F} is stable on (X, U −1 ). Suppose now that x is a cluster point of f −1 F in (X, U). By continuity of f, we see that f (x) is a cluster point of F in (Y, V). We next obtain an asymmetric version of the “uniform” Proposition 2.6.10. Remark 2.6.16. (a) ([81, Proposition 13]). A quasi-uniform space (X, U) is compact if and only if it is left K-complete and precompact. (b) ([102, Theorem 2]). A quasi-pseudometric space (X, d) carries a compact topology τ (d) if and only if (X, Ud ) is left K-complete and precompact. Proof. (a). For a proof of the ﬁrst statement we refer the reader to [81, Proposition 13]. We only present an argument for the quasi-pseudometric variant of the result (see part (b) below). 16Given a quasi-uniform space (X, U), a U −1 -stable ﬁlter is often called a co-stable ﬁlter.

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(b). Suppose that Ud is precompact and left K-complete. For each k ∈ ω there is a ﬁnite subset Fk ⊆ X such that x∈Fk Bd (x, 2−k ) = X. Let F be any ultraﬁlter on X. For each k ∈ ω we can choose a point bk ∈ Fk such that Bd (bk , 2−k ) ∈ F. Furthermore, for each k ∈ ω let us set Ek = {x ∈ Fk : there are arbitrarily large indices m ∈ ωsuch that bm ∈ Bd (x, 2−(k−1) )}; then Ek = ∅. We deﬁne a binary relation R on n∈ω En in the following way. For each k ∈ ω, s ∈ Ek and t ∈ Ek+1 we put s R t if d(s, t) < 2−k . We shall next prove that the hypothesis of the Lemma of K¨onig (see the proof of Lemma 2.2.23) is satisﬁed. Consider t ∈ Ek+1 where k ∈ ω. Then there exists x ∈ Fk such that t ∈ Bd (x, 2−k ). Consequently Bd (t, 2−k ) ⊆ Bd (x, 2−(k−1) ). Since t ∈ Ek+1 , we deduce that x ∈ Ek , that is, x R t. It follows from K¨onig’s Lemma that there exists a sequence (ak )k∈ω in X such that ak ∈ Ek and d(ak , ak+1 ) < 2−k whenever k ∈ ω. The triangle inequality immediately implies that (ak )k∈ω is a left K-Cauchy sequence. Hence it converges to a point x in (X, d). Let > 0. For an m ∈ ω suﬃciently large, again by the triangle inequality, we have Bd (am , 2−(m−2) ) ⊆ Bd (x, ). Since am ∈ Em , we ﬁnd an index s ∈ ω suﬃciently large such that bs ∈ Bd (am , 2−(m−1) ) and further Bd (bs , 2−s ) ⊆ Bd (am , 2−(m−2) ) ⊆ Bd (x, ). Therefore Bd (x, ) ∈ F and the ﬁlter F converges to x in (X, d). We conclude that τ (d) is compact. For the converse, suppose that τ (d) is compact. Evidently Ud is precompact. It is also left K-complete, since by compactness each left K-Cauchy ﬁlter on (X, Ud ) has a cluster point, and thus a limit point by Proposition 2.6.5(b). A quasi-uniform space (X, U) is called Smyth-complete provided that each left K-Cauchy ﬁlter is τ (U s )-convergent. It is called Smyth-completable if each left K-Cauchy ﬁlter is stable and thus a U s -Cauchy ﬁlter (recall Lemma 2.6.7). These weak symmetry properties for quasi-uniform spaces were studied in connection with the so-called Smyth–S¨ underhauf-completion (which can be considered a kind of Kcompletion) and its connections to the bicompletion [146, 148]. Note that evidently each Smyth-complete quasi-uniform space is left K-complete and bicomplete. We ﬁnish our discussion of K-concepts by presenting a characterization of those quasi-metrizable spaces that admit a left K-complete quasi-metric (compare also with Proposition 2.7.16(a)). To this end we recall the following concept due to Wicke and Worrell (see e.g., [155]): A topological T1 -space X has a λ-base if it possesses a sequence of bases (Bn )n∈ω such that every decreasing ﬁlter base {Bn : n ∈ ω} with Bn ∈ Bn whenever n ∈ ω converges to some x ∈ X (see e.g., [89]). Such a sequence of bases will be called a λ-base sequence in the following. Proposition 2.6.17 ([89, Theorem 1]). A quasi-metrizable space possesses a λ-base if and only if it admits a left K-complete quasi-metric. Proof. Suppose that the quasi-metric space (X, d) is left K-complete. Then, using left K-completeness of the quasi-metric d, it is easy to check that (Gn )n∈ω , where Gn = {Bd (x, 2−k ) : x ∈ X, k ≥ n, k ∈ ω} whenever n ∈ ω, is a λ-base sequence. So each left K-complete quasi-metric space has a λ-base. For the converse suppose that (X, d) is a quasi-metric space that possesses a λ-base. Let (Bn )n∈ω be a λ-base sequence for X. Inductively, for each n ∈ ω we well-order Bn as follows: Choose an arbitrary well-ordering on B0 , say B0 = {Bα : α < γ}. Suppose now that for some n ∈ ω, Bn is well-ordered, say Bn = {Bα : α < β}. For each Bα ∈ Bn let Bn+1,α = {B ∈ Bn+1 : B ⊆ Bα }. Furthermore let

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Bn+1,β = Bn+1 \ α<β Bn+1,α . We well-order these collections arbitrarily, say Bn+1,α = {B(α,δ) : δ < βα } for each Bα ∈ Bn , and Bn+1,β = {B(β,δ) : δ < ββ }. Consider now the following obvious sum well-ordering on Bn+1 , where the elements of Bn+1 are indexed as above with the possibility of repetitions: Set B(α,δ) ≤ B(α ,δ ) if α < α , or if α = α and δ ≤ δ . In the following each collection Bn is always equipped with the well-ordering deﬁned in this way. For each x ∈ X and n ∈ ω let Bα(n,x) be minimal among the B satisfying x ∈ B ∈ Bn with respect to the deﬁned well-ordering on Bn . Then for each n ∈ ω and x ∈ X set Tn (x) = {Bα ∈ Bn : α ≤ α(n, x)}. Note that each Tn is a transitive neighbornet of X. Let V be the compatible quasi-metrizable quasiuniformity generated by the subbase Ud ∪ {Tn : n ∈ ω} on X. By Lemma 2.2.2 there is a quasi-metric e on X inducing V. We are going to show that (X, e) is left K-complete. To this end, let F be a left K-Cauchy ﬁlter on (X, Ue ). For each n ∈ ω we inductively deﬁne a sequence of points (xn )n∈ω in X such that (Bα(n,xn ) )n∈ω is a decreasing sequence of subsets of X, where each Bα(n,xn ) ∈ Bn , and each Hn := Bα(n,xn ) \ {Bα ∈ Bn : α < α(n, xn )} belongs to F. Set H−1 = X. Let n ∈ ω and suppose that the construction has been carried out for any k ∈ ω with k < n. Consider µ(n) = min An where An = {α(n, y) : y ∈ Hn−1 , Tn (y) ∈ F}. Note that the latter set is nonempty, since F is a left K-Cauchy ﬁlter on (X, Ue ) containing Hn−1 as an element. Choose some point y ∈ An for which the minimum µ(n) is attained. Call it xn . Since F is a left K-Cauchy ﬁlter on (X, Ue ), we can ﬁnd F ∈ F such that y ∈ F implies that Tn (y) ∈ F. Without loss of generality we suppose that F ⊆ Tn (xn ) ∩ Hn−1 . We next argue that y ∈ F implies that α(n, y) = α(n, xn ): The conditions Tn (y) ∈ F and y ∈ Hn−1 imply that α(n, y) ≥ α(n, xn ) by the choice of xn ; moreover by the deﬁnition of the set Tn (xn ) it follows from y ∈ Tn (xn ) that α(n, y) ≤ α(n, xn ). Hence the claim is veriﬁed. We conclude that F ⊆ Hn and Hn ∈ F. Note that if n ∈ ω we have α(n + 1, xn+1 ) = (α(n, xn ), δ) for some δ < βα(n,xn ) by deﬁnition of the well-ordering on the base Bn+1 , because xn+1 ∈ Hn , and so xn+1 does not belong to the elements of Bn+1,α where α < α(n, xn ). It follows that Bα(n+1,xn+1 ) ⊆ Bα(n,xn ) whenever n ∈ ω. This completes the inductive construction. Now (Bn )n∈ω is a λ-base sequence on X, each Bα(n,xn ) belongs to Bn ∩ F (n ∈ ω) and the constructed sequence (Bα(n,xn ) )n∈ω is decreasing. Therefore the ﬁlter F converges in (X, d) and thus in (X, e). We have shown that (X, Ue ) is left K-complete. We end this subsection with some remarks on a completion theory of quasiuniform spaces due to Doitchinov. A pair of ﬁlters (F, G) on a quasi-uniform space (X, U) is called a Cauchy ﬁlter pair provided that for each U ∈ U there are F ∈ F and G ∈ G such that F × G ⊆ U . In particular, in this case we call G a DCauchy ﬁlter. We say that a quasi-uniform space (X, U) is D-complete provided that each D-Cauchy ﬁlter converges in (X, U). Note that each U s -Cauchy ﬁlter is a D-Cauchy ﬁlter on (X, U). Doitchinov showed that a categorically satisfactory (and conjugate invariant) theory of a D-completion can be developed for so-called quiet quasi-uniform spaces. Indeed [22, 23] he called a quasi-uniform space (X, U) quiet if for each U ∈ U there is V ∈ U such that for each Cauchy ﬁlter pair (F, G) on (X, U) and all x, y ∈ X

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with V −1 (x) ∈ F and V (y) ∈ G we have (y, x) ∈ U . Of course, each uniformity is quiet, and if U is quiet, then U −1 is quiet, too. Each quiet quasi-uniform space (X, U) is known to be uniformly regular, that is, for each U ∈ U there is V ∈ U such that clτ (U) V (x) ⊆ U (x) whenever x ∈ X (see for instance [87, p. 891]). This fact and the observation that each totally bounded quiet quasi-uniform space is a uniform space [28, 79] show that quietness is a rather restrictive property of quasi-uniform spaces. Doitchinov proved that each quiet quasi-uniform T0 -space (X, U) has a standard D-completion, now called its Doitchinov-completion, i.e., there is an (up to quasiuniform isomorphism) uniquely determined quiet D-complete quasi-uniform T0 space (X † , U † ) in which X is dense with respect to both topologies τ (U † ) and τ ((U † )−1 ); moreover X † has the reﬂection property for uniformly continuous maps into quiet D-complete quasi-uniform T0 -spaces. Each quiet D-complete space is bicomplete, which implies that the Doitchinov completion of a quiet quasi-uniform T0 -space contains its bicompletion as a subspace. In [21] for quasi-pseudometric spaces Doitchinov developed a similar (quasipseudometric) completion theory by means of a quasi-pseudometric concept slightly stronger than quietness, which he called “balancedness”. Remarkably, the latter theory can even be generalized to arbitrary T0 -quasi-pseudometric spaces (see [100]). Quasi-uniformities for which each D-Cauchy ﬁlter is stable have been called stable by Doitchinov. Each uniformity is of this kind, as is every quasi-uniformity whose conjugate is hereditarily precompact (see Proposition 2.4.7). Stability of quasi-uniformities turns out to be a strong condition, in particular for quasi-pseudometric quasi-uniformities. We refer the reader to the literature (see [98]) and mention here only the following auxiliary result as an example. Lemma 2.6.18 ([98, Lemma 3.7]). Let (X, U) be a stable quasi-uniform space, let x ∈ X and let P, U ∈ U. Then there is a ﬁnite subset F ⊆ P (x) such that x ∈ intτ (U) U −1 (F ). Proof. Assume the contrary. Since by our assumption the ﬁlter F generated on X by {V (x) \ U −1 (a) : a ∈ P (x), V ∈ U} is stable, we have F ∈F U (F ) ∈ F, but F ∈F U (F ) ∩ P (x) = ∅ — a contradiction. Hence F does not exist and there is a ﬁnite subset F ⊆ P (x) such that x ∈ intτ (U) U −1 (F ). 2.7. Bicompleteness. The bicompletion of a quasi-uniform T0 -space is well explained in the monograph [30, Chapter 3] so that we shall omit several substantial proofs in this subsection and mainly concentrate on illustrating a few applications of the concept of bicompleteness. Generalizing the construction of the completion of metric spaces one proves that each T0 -quasi-pseudometric space X has a T0 -quasi-pseudometric bicompletion, which is unique up to isometries that leave the points of X ﬁxed: Example 2.7.1 (compare [19, 143]). Let (X, d) be a T0 -quasi-pseudometric n )n∈ω , (yn )n∈ω ) = be the set of all ds -Cauchy sequences. Set d((x space. Let X limn→∞ d(xn , yn ) where (xn )n∈ω , (yn )n∈ω ∈ X. Then d is a bicomplete quasi Let ∼ be We next build the T0 -quotient of the space (X, d): pseudometric on X. s the equivalence relation on X deﬁned by x ∼ y if x, y ∈ X and (d) (x, y) = 0. Let X be the set of the corresponding equivalence classes [x] where x ∈ X. Furthermore set

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y) whenever x, y ∈ X. is a well-deﬁned bicomplete T0 Then (X, d) d([x], [y]) = d(x, quasi-pseudometric space. For each x ∈ X let x be the constant sequence equal to Furthermore d). x. Then x → [x] yields an isometric embedding i of (X, d) into (X, s s s ˜ i(X) is τ ((d) )-dense in X. Observe that (d) = (d ). The T0 -quasi-pseudometric is called the quasi-pseudometric bicompletion of (X, d). d) space (X, Similarly, each quasi-uniform T0 -space (X, U) has an essentially unique quasiuniform T0 -bicompletion. We shall describe its construction with the help of minimal U s -Cauchy ﬁlters so that we can avoid the T0 -quotient construction. Here we make use of the following lemma which can be found with proof in [30, Proposition 3.30]. Lemma 2.7.2. Let F be a U s -Cauchy ﬁlter on a quasi-uniform space (X, U). There is exactly one minimal U s -Cauchy ﬁlter on X that is coarser than F. Furthermore if B is any base on X for F, then B0 = {U (B) : B ∈ B and U is symmetric member of U s } is a base for the minimal U s -Cauchy ﬁlter coarser than F. Proposition 2.7.3 (see [30, Theorem 3.33]). Every quasi-uniform T0 -space has a T0 -bicompletion. Proof. We shall only outline a construction of the bicompletion without giving be the set of all the details. Let (X, U) be a quasi-uniform T0 -space and let X s = {(F, G) ∈ X ×X : there minimal U -Cauchy ﬁlters on X. For each U ∈ U, let U exist F ∈ F and G ∈ G such that F × G ⊆ U }. One ﬁrst shows that {U : U ∈ U} As every member of X is a U s -Cauchy is a base for a quasi-uniformity U on X. ﬁlter, for each U ∈ U and for each F ∈ X, (F, F) ∈ U . Let V1 , V2 ∈ U. Then 1 ∩ V 2 . Given U ∈ U let V ∈ U be such that V 2 ⊆ U . V = V1 ∩ V2 ∈ U and V = V Let (F, G), (G, H) ∈ V ; then there exist F ∈ F, G1 ∈ G such that F × G1 ⊆ V and G2 ∈ G, H ∈ H such that G2 × H ⊆ V . Since G1 ∩ G2 = ∅, F × H ⊆ V 2 ; thus . (V )2 ⊆ U s and U s . Let i : X → X be the function We also note that we can identify (U) s deﬁned by i(x) = U (x). One checks that i is a uniform embedding. For the rest of the proof we refer the reader to [30, Theorem 3.33]. In particular one shows that the ﬁlter base i(F) on X converges to F ∈ X with respect to the for each F ∈ X, s s topology τ (U ). Thus i(X) is τ (U )-dense in X. Uniqueness of the bicompletion can be established with the help of the following extension theorem for uniformly continuous maps. Proposition 2.7.4. Let (X, U) be a quasi-uniform space, let (Y, V) be a bicomplete quasi-uniform T0 -space, let D be a dense subset of (X, τ (U s )), and let f : (D, U|D ) → (Y, V) be a uniformly continuous map. Then there is a unique continuous extension g : (X, τ (U s )) → (Y, τ (V s )) of f , and g : (X, U) → (Y, V) is uniformly continuous. Proof. For the details of the argument we refer the reader to [30, Theorem 3.29], but we like to recall how the extension map g is deﬁned. Let x ∈ X. Then U s (x) is a U s -Cauchy ﬁlter so that Fx = {D ∩ R : R ∈ U s (x)} is a Cauchy ﬁlter on (D, U s |D ). Since f is uniformly continuous, we see that {f (F ) : F ∈ Fx } is a base for a V s -Cauchy ﬁlter Gx . The space (Y, V) is bicomplete and (Y, τ (V s )) is a Hausdorﬀ space so that Gx has a unique limit in this space, denoted by g(x).

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It remains to be shown that g : X → Y, deﬁned in this manner, is the required (unique) uniformly continuous extension of f . Remark 2.7.5. Let (X, U) be a quasi-uniform space that is a τ (V s )-dense subspace of a bicomplete quasi-uniform T0 -space (Y, V). Then (Y, V) is isomorphic to the bicompletion of (X, U) under an isomorphism that keeps the points of X pointwise ﬁxed. When this identiﬁcation is made, the minimal U s -Cauchy ﬁlters on X are the traces of the τ (V s )-neighborhood ﬁlters of the points of Y . Proposition 2.7.6 ([30, Theorem 3.35]). Let (X, U) be a uniform Hausdorﬀ U) is a complete uniform Hausdorﬀ space, and any complete unispace. Then (X, form Hausdorﬀ space that has a dense subspace isomorphic to (X, U) is isomorphic U). to (X, Proposition 2.7.7 (see e.g., [30, Proposition 3.36]). The bicompletion of a quasi-uniform T0 -space (X, U) is totally bounded if and only if (X, U) is totally bounded. Proof. Let (X, U) be totally bounded. Let U ∈ U. Choose V ∈ U such that V 3 ⊆ U . By Remark 2.4.1 there is a ﬁnite cover C of X such that A × A ⊆ V since X is τ (Us )whenever A ∈ C. Note that {clτ (U s ) A : A ∈ C} is a cover of X, Then cl es A × cl es A ⊆ V 3 ⊆ U . We conclude that (X, U) is dense in X. τ (U ) τ (U ) totally bounded. The converse is immediate, since total boundedness is a hereditary property of quasi-uniform spaces. τ (Q 0 s )), Example 2.7.8 ([75, Proposition 4]). The compact Hausdorﬀ space (X, 0 is the bicompletion of the coarsest compatible quasi-uniformity Q0 of a where Q locally compact T0 -space X is known under the name of the Fell compactiﬁcation. It is often constructed as a subspace of a hyperspace. Lemma 2.7.9. Let U be a quasi-uniformity on a set X. If F is a ﬁlter on X that τ (U s )-converges to some x ∈ X, then adhτ (U) F = limτ (U) F = clτ (U) {x}. Proof. Denote the set of the τ (U)-cluster points of F by F . Let U ∈ U. Since x is a τ (U s )-limit point of F, (U ∩ U −1 )(x) ∈ F. Hence F ⊆ clτ (U) (U ∩ U −1 )(x) ⊆ (U −1 )2 (x). Thus F ⊆ {U −2 (x) : U ∈ U} = clτ (U) {x}. Furthermore, because τ (U) ⊆ τ (U s ), x is a τ (U)-limit point of F. Thus clτ (U) {x} = F , since F is τ (U)-closed. Clearly, F is also the set of the τ (U)-limit points of F (compare also Proposition 2.6.5(b)). A super(quasi-)sober space X that is compact is called strongly (quasi-)sober. Strong quasisobriety of a topological space (X, τ ) is therefore characterized by the property that for the convergence set limτ G of any ultraﬁlter G on X there is a point x ∈ X such that limτ G = {x}. Results like the following one explain why the notion of a strongly sober locally compact space is considered a reasonable generalization of the concept of a compact T2 -space to the category of topological T0 -spaces. Proposition 2.7.10 (see [95, Proposition 2]). (a) Let X be a strongly quasi-sober and locally compact topological space and let Q0 be the coarsest compatible quasi-uniformity on X. Then Q0 is totally bounded and bicomplete, thus joincompact (compare Proposition 2.6.10).

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(b) Let (X, U) be a totally bounded and bicomplete quasi-uniform space. Then τ (U) is locally compact and strongly quasi-sober. Furthermore U is the coarsest quasi-uniformity that (X, τ (U)) admits. Proof. (a). Since X is locally compact, it admits a coarsest quasi-uniformity Q0 , which is necessarily totally bounded (see Remark 2.5.9(b)). Therefore it only remains to be seen that Q0 is bicomplete: To verify this, we check that τ (Q0 s ) is compact (compare Proposition 2.6.10). Let H be an ultraﬁlter on X. Then there is an x ∈ X such that clτ (Q0 ) {x} is the τ (Q0 )-convergence set of H, because X is strongly quasi-sober. We show that H converges to x with respect to the topology τ (Q0 −1 ). Hence we have to prove that if K is compact, G is open, K ⊆ G, and U = [(X \ K) × X] ∪ [X × G], then U −1 (x) ∈ H. If x ∈ G, this is obvious. If x ∈ G, then U −1 (x) = X \ K and thus K ∩ clτ (Q0 ) {x} = ∅. Since K is compact, we conclude that K ∈ H. Hence X \ K = U −1 (x) ∈ H. Therefore H converges to x with respect to the topology τ (Q0 s ) and thus Q0 is bicomplete. (b). Let x ∈ X. Since τ (U s ) is compact, {clτ (U −1 ) U (x) : U ∈ U} is a τ (U)neighborhood base of x that consists of τ (U)-compact sets. Thus τ (U) is locally compact. Let G be an ultraﬁlter on X. Since τ (U s ) is compact, G has a τ (U s )-limit point x ∈ X. By Lemma 2.7.9 clτ (U) {x} is its set of τ (U)-limit points. Hence (X, τ (U)) is strongly quasi-sober. By the proof of part (a) the coarsest compatible quasi-uniformity Q0 on (X, τ (U)) is (totally bounded and) bicomplete. s s Then Q0 ⊆ U and thus τ (Q0 ) ⊆ τ (U ). Moreover τ (U) = τ (Q0 ) and thus U = Q0 . By a well-known result due to Nachbin about compact T2 -ordered spaces (compare with Proposition 3.5.1 in the next section) we conclude that the T0 quotient quasi-uniformities of U and Q0 are the same, since comparable compact Hausdorﬀ topologies are equal. Hence U and Q0 are the same quasi-uniformity on X. Numerous results can be found in the literature that deal with bicomplete quasi-uniformities. We are going to give some examples next. Lemma 2.7.11 (see [96, Proposition 1]). Let (X, τ ) be a topological space and let W be a compatible quasi-uniformity on X that is ﬁner than the well-monotone quasi-uniformity of X. Furthermore let F be a W s -Cauchy ﬁlter on X. Then adhτ F ∈ F and adhτ F is τ -irreducible. Proof. In order to reach a contradiction, assume that adhτ F ∈ F. Then there exists a minimal (inﬁnite) cardinal number m so that there is a subcollection E of F consisting of τ -closed subsets of X such that card(E) = m and E ∈ F. We can assume that E = {Fα : α < m}. For each β < m let Eβ = {Fα : α < β}. (In particular let E0 = X.) Set C = {X \ Eβ : β < m}. We have TC ∈ W and Eβ ∈ F for each β < m. Since F is a W s -Cauchy ﬁlter on X, there is an x ∈ X such that (TC ∩ TC−1 )(x) ∈ F. If x ∈ E, then TC−1 (x) = E ∈ F — a contradiction. Therefore x ∈ E. Hence there is β < m such that x ∈ (X \ Eβ ). Then TC (x) ⊆ (X \ Eβ ) and (X \ Eβ ) ∩ Eβ ∈ F. We have reached another contradiction and conclude that adhτ F ∈ F. By Proposition 2.6.5(b) each point of adhτ F is a τ -limit point of F. Because adhτ F ∈ F, it follows that adhτ F is τ -irreducible.

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Proposition 2.7.12 (compare [95]). The (totally bounded) Pervin quasi-uniformity P(τ ) of a topological space (X, τ ) is bicomplete if and only if (X, τ ) is hereditarily compact and quasi-sober. Proof. First note that τ (P(τ )s ) has as subbase τ ∪ {C : C is τ -closed in X}.17 Let F be a P(τ )s -Cauchy ﬁlter on a hereditarily compact and quasi-sober space (X, τ ). Observe that for each G ∈ τ we have G ∈ F or X \ G ∈ F. Set A = adhτ F = {clτ F : F ∈ F}. By hereditary compactness of X, each strictly increasing sequence of open sets is ﬁnite, and thus A ∈ F. Similarly as in the preceding proof we see that A is τ -irreducible, since G ∈ τ and G∩A = ∅ imply that G ∈ F. In particular by quasisobriety of X there is x ∈ A such that clτ {x} = A. Let G be an open set containing x. Then G ∈ F. Let F be a closed set containing x. Then clτ {x} ⊆ F . Thus F ∈ F and F converges to x with respect to the Skula topology of (X, τ ). Hence P(τ ) is bicomplete. For the converse suppose that τ (P(τ )s ) is compact (compare with Proposition 2.6.10). Let (Gi )i∈I be an open collection in (X, τ ). Set B = i∈I Gi . Since B is τ (P(τ )s )-closed and thus compact, the τ (P(τ )s )-open cover {Gi : i ∈ I} of B has a ﬁnite subcover. Thus (X, τ ) is hereditarily compact. Let A be closed and irreducible in (X, τ ). Consider the ﬁlter G on X generated by the base {G ∩ A : G ∩ A = ∅, G ∈ τ }. By compactness of τ (P(τ )s ) the ﬁlter G has a τ (P(τ )s )-cluster point x ∈ X. Thus x ∈ G and clτ {x} = A. Therefore (X, τ ) is quasi-sober. Proposition 2.7.13 ([96, Proposition 4]). The well-monotone quasi-uniformity M(τ ) of a topological space (X, τ ) is bicomplete if and only if (X, τ ) is quasi-sober. Proof. Suppose that (X, τ ) is quasi-sober. Let F be an M(τ )s -Cauchy ﬁlter on X. Then adhτ F ∈ F and adhτ F is τ -irreducible by Lemma 2.7.11. Thus there is x ∈ X such that adhτ F = clτ {x}. In the light of Proposition 2.6.5(b) for each U ∈ M(τ ) we conclude that (U ∩U −1 )(x) ∈ F. Then F converges to x with respect to the Skula topology τ (M(τ )s ) = τ (P(τ )s ) of (X, τ ). Hence M(τ ) is bicomplete. For the converse suppose that M(τ ) is bicomplete. Let F be closed and irreducible in (X, τ ). Then the ﬁlter H generated by {F ∩ G : F ∩ G = ∅ and G is τ -open in X} is an M(τ )s -Cauchy ﬁlter on X: Indeed let (Ci )i∈F be a ﬁnite family of well-monotone open covers of (X, τ ). If for each i ∈ F, Ci is the ﬁrst element of Ci such that F ∩ Ci = ∅ and x ∈ i∈F Ci ∩ F, then i∈F (TCi ∩ TC−1 )(x) belongs to the i ﬁlter H. By our assumption it converges to some point x in X with respect to the Skula topology τ (M(τ )s ). Then F = adhτ F = clτ {x} by Lemma 2.7.9 and hence we conclude that (X, τ ) is quasi-sober. Proposition 2.7.14 ([96, Corollary 1]). A topological space admits a bicomplete quasi-uniformity if and only if its ﬁne quasi-uniformity is bicomplete. Proof. Let (X, τ ) be a topological space admitting a bicomplete quasi-uniformity U. Let F be an (FN (τ ))s -Cauchy ﬁlter on X. Clearly F is a U s -Cauchy ﬁlter on X. Moreover limτ F = adhτ F = clτ {x} for some x ∈ X by Lemma 2.7.9, since U is bicomplete. Since adhτ F ∈ F by Lemma 2.7.11, we conclude that U (x) ∩ U −1 (x) ∈ F whenever U ∈ FN (τ ). Thus F converges to x with respect to 17The topology τ (P(τ )s ) is often called the Skula topology of (X, τ ).

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the Skula topology τ ((FN (τ ))s ) of (X, τ ). Therefore FN (τ ) is bicomplete. The converse is evident. Proposition 2.7.15. (a) ([140, Theorem 2.1]). A quasi-pseudometrizable bitopological space (X, τ, ρ) admits a bicomplete quasi-pseudometric if and only if its supremum topology τ ∨ ρ is completely pseudometrizable. (b) ([59, Theorem 1]). A metrizable topological space admits a bicomplete quasi-metric if and only if it is an Fσδ -subset in each metric space in which it is embedded. (c) ([96, Proposition 3]). The ﬁne quasi-uniformity of any quasi-pseudometrizable topological space is bicomplete. Proof. (a). If d is a bicomplete compatible quasi-pseudometric on (X, τ, ρ), then ds is a complete pseudometric compatible with the topology τ ∨ ρ. Conversely assume that d is a compatible quasi-pseudometric on (X, τ, ρ) and e is a compatible complete pseudometric on (X, τ ∨ ρ). For each integer n ∈ ω, s the set Ln = {(x, y) ∈ X × X : e(x, y) < 2−n } is τ (ds ) × τ (d )-open. Thus for each x ∈ X and n ∈ ω there is rn (x) > 0 such that Un = {Bds (x, rn (x)) × Bds (x, rn (x)) : x ∈ X} ⊆ Ln . Clearly, inductively each rn (x) can be chosen such that 5rn+1 (x) < rn (x) < 2−n . For each n ∈ ω put now Vn = {Bd−1 (x, rn (x)/3) × Bd (x, rn (x)/3) : x ∈ X}. Let n ∈ ω. We want to show that Vn ∩ Vn−1 ⊆ Un . Consider (x, y) ∈ Vn ∩ Vn−1 . Then there exist a, b ∈ X such that d(x, a) < rn (a)/3,

d(a, y) < rn (a)/3,

d(b, x) < rn (b)/3,

d(y, b) < rn (b)/3.

Assume, without loss of generality that rn (a) ≤ rn (b). Then, d(x, b) ≤ d(x, a) + d(a, y) + d(y, b) < rn (b). Since d(b, x) < rn (b), it follows that ds (x, b) < rn (b). Analogously, ds (b, y) < rn (b). Hence (x, y) ∈ Un . 3 A similar argument shows that Vn+1 ⊆ Vn whenever n ∈ ω (see proof of Proposition 2.2.12). By the Quasi-Pseudometrization Lemma 2.2.2, there is a quasipseudometric q on X such that Vn+1 ⊆ {(x, y) ∈ X × X : q(x, y) < 2−n } ⊆ Vn . For each n ∈ ω and x ∈ X we have Bd (x, rn (x)/3) ⊆ Vn (x) and Bd−1 (x, rn (x)/3) ⊆ Vn−1 (x), so that τ (q) ⊆ τ (d) and τ (q −1 ) ⊆ τ (d−1 ). Now ﬁx x ∈ X and n ∈ ω. Then Vn (x) = {Bd (y, rn (y)/3) : x ∈ Bd−1 (y, rn (y)/3), y ∈ X}, where x ∈ Bd−1 (y, rn (y)/3) means that d(x, y) < rn (y)/3. It follows that Vn (x) ⊆ Bd (x, 2−n ). Therefore τ (d) = τ (q) and similarly, τ (d−1 ) = τ (q −1 ). Hence q is a compatible quasi-pseudometric on (X, τ, ρ). Finally consider any q s -Cauchy sequence (xn )n∈ω in X. Then for each n ∈ ω there is kn ∈ ω such that for any s, t ∈ ω with s, t ≥ kn the pair (xs , xt ) belongs to Vn ∩ Vn−1 , hence to Un , and thus to Ln . We conclude that (xn )n∈ω is an e-Cauchy sequence. By assumption, there is x ∈ X such that (xn )n∈ω is τ (ds )-convergent to x. Therefore q is a bicomplete compatible quasi-pseudometric on the bitopological space (X, τ, ρ).

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(b). For a proof we refer the reader to [59]. (c). Let (X, τ ) be a topological space the topology of which is induced by a quasi-pseudometric d on X. For each n ∈ ω set Vn = {(x, y) ∈ X × X : d(x, y) < 2−n }. Let F be an (FN (τ ))s -Cauchy ﬁlter on X. Then adhτ F ∈ F and adhτ F is τ -irreducible by Lemma 2.7.11. Consider any U ∈ FN (τ ). Choose V ∈ FN (τ ) such that V 3 ⊆ U . There is z ∈ X such that (V ∩ V −1 )(z) ∈ F. Fix y ∈ V (z) ∩ adhτ F. Then adhτ F ⊆ clτ V −1 (z) ⊆ V −3 (y) ⊆ U −1 (y). In particular we conclude that for each n ∈ ω −1 there is yn ∈ adhτ F such that adhτ F ⊆ Vn (yn ). Assume next that n∈ω Vn (yn ) ∩ adhτ F = ∅. Then the open collection C = {( k∈n Vk (yk )) ∪ (X \ adhτ F) : n ∈ ω} of X is interior preserving. Hence TC ∈ FN (τ ), however there is no y ∈ adhτ F such that adhτ F ⊆ TC−1 (y). We have reached a contradiction to a statement proved in the preceding paragraph and deduce that there is x ∈ n∈ω Vn (yn ) ∩ adhτ F. It follows that clτ {x} ⊆ adhτ F ⊆ −2 n∈ω Vn (x) = clτ {x}. We conclude that F converges to x with respect to the Skula topology τ ((FN (τ ))s ), since each τ -cluster point of F is a τ -limit point of F by Proposition 2.6.5(b) and clτ {x} ∈ F. Thus FN (τ ) is bicomplete. In connection with the preceding result (c) let us mention (compare [141]) that the ﬁnest compatible quasi-uniformity on each quasi-pseudometrizable bitopological space is known to be bicomplete; on the other hand an arbitrary pairwise completely regular bitopological space possessing a supremum topology which admits a complete uniformity need not admit a bicomplete quasi-uniformity, as the coﬁnite and the discrete topology on an uncountable set show. It is an interesting problem to investigate which functorial quasi-uniformities on Top are preserved under the bicompletion. For instance Proposition 2.7.13 can be modiﬁed as follows. Let (X, τ ) be a T0 -space and let M(τ ) be the well-monotone ))) underlying the bicom τ (M(τ quasi-uniformity of X. The topological space (X, )) yields the sobriﬁcation of (X, τ ) and M(τ ) is the well-monotone M(τ pletion (X, quasi-uniformity for that space [96, Proposition 5 and Corollary 4]. A far reaching extension of this result was furthermore obtained in [71]. Functorial quasi-uniformities that are preserved under the bicompletion functor were called bicompletion-true by Br¨ ummer [8, 9]. Besides the well-monotone quasiuniformity for instance the ﬁne quasi-uniformity is bicompletion-true [96, Corollary 5], while the Pervin quasi-uniformity is not [95, Theorem 3]. It may be worthwhile to mention that other completeness properties of the ﬁne quasi-uniformity of a topological space have been studied. We give two examples of such results. Proposition 2.7.16. (a) ([105, Corollary 1]). The ﬁne quasi-uniformity FN (τ ) of an arbitrary topological space (X, τ ) is left K-complete. (b) ([111, Example 5]). The ordinal ω1 (equipped with its usual order topology) does not admit any right K-complete quasi-uniformity. Proof. (a). Clearly FN (τ ) is left K-complete if we can show that even the coarser compatible quasi-uniformity M(τ ) on X is left K-complete. Let F be a left K-Cauchy ﬁlter on (X, M(τ )). Since (M(τ ))−1 is hereditarily precompact by Remark 2.4.6(b), the ﬁlter F is M(τ )-stable by Proposition 2.4.7, and thus an

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M(τ )s -Cauchy ﬁlter by Lemma 2.6.7. Thus adhτ F ∈ F by Lemma 2.7.11. In particular F has a cluster point and thus a limit point in (X, τ ) by Proposition 2.6.5(b). Hence M(τ ) is left K-complete. (b). Let U be a compatible quasi-uniformity on X = ω1 and let F be the ﬁlter generated by the base {F ⊆ X : F is closed and unbounded} on X. Of course, F does not have a cluster point. We prove that F is a right K-Cauchy ﬁlter on (X, U): Suppose that U ∈ U and V ∈ U such that V 2 ⊆ U . For any x ∈ ω1 \ {0} there is βx ∈ ω1 such that βx < x and (βx , x] ⊆ V (x). By the Pressing-Down Lemma18 there are β ∈ ω1 and an uncountable subset S of ω1 such that βx < β whenever x ∈ S. Consider an arbitrary α ∈ ω1 such that α ≥ β. Then S ∩ [α, →) ⊆ V −1 (α). Thus S ∩ [α, →) ⊆ V −2 (α) ⊆ U −1 (α). Since S ∩ [α, →) ∈ F, we see that U −1 (α) ∈ F. Because [β, →) ∈ F, we have shown that F is a right K-Cauchy ﬁlter on (X, U). 3. Some applications of quasi-uniformities In this section we brieﬂy discuss several constructions that yield important examples of quasi-uniform spaces. 3.1. Paratopological groups. Literature: [110, 122, 124, 142]. The literature about (para)topological groups is vast. We can only recall here some basic facts that are relevant to our purpose. A paratopological group is a triple (G, ·, τ ) where (G, ·) is a group and τ is a topology on G such that the group operation · : (G × G, τ × τ ) → (G, τ ) deﬁned by ·(x, y) = xy whenever x, y ∈ G is continuous. If besides that operation also the inversion (G, τ ) → (G, τ ) deﬁned by x → x−1 whenever x ∈ G is continuous, then (G, ·, τ ) is called a topological group. If (G, ·, τ ) is a paratopological group, then so is (G, ·, τ −1 ) where τ −1 = {A ⊆ G : A−1 ∈ τ }; τ −1 is called the conjugate topology of τ and (G, ·, τ, τ −1 ) is called a parabitopological group. Note that τ = τ −1 if and only if G is a topological group. Remark 3.1.1 (compare [122, p. 401]). Let (G, ·, τ ) be a paratopological group and let e be the identity element of (G, ·). Then the neighborhood ﬁlter η(e) of e satisﬁes the following conditions: (a) V ∈ η(e) implies the existence of W ∈ η(e) such that W · W ⊆ V . (b) For every a ∈ G, {aV a−1 : V ∈ η(e)} = η(e). Moreover, for every a ∈ G, the neighborhood ﬁlter η(a) is given by η(a) = {aV : V ∈ η(e)} = {V a : V ∈ η(e)}. Let (G, ·) be a group and let η be a ﬁlter on G satisfying properties (a) and (b). Then there exists exactly one topology τ on G such that η is the neighborhood ﬁlter of the identity e ∈ G and such that (G, ·, τ ) is a paratopological group. Example 3.1.2 (compare Example 2.3.9). Let R be the set of the reals. For each x, y ∈ R set r(x, y) = max{0, x − y}. Then r is a T0 -quasi-pseudometric on R and τ (r) is the so-called upper topology {(a, ∞) : a ∈ R} ∪ {∅, R} on R. Similarly τ (r −1 ) is the lower topology {(−∞, a) : a ∈ R} ∪ {∅, R} on R. Observe 18The following weak version of that result is suﬃcient here: A function f : ω → ω is said 1 1 to be regressive if for every x > 0 we have f (x) < x. Then every regressive function f : ω1 → ω1 is constant on some unbounded set (compare e.g., [117, p. 153]).

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that (R, τ (r), τ (r −1)) is a parabitopological group under the usual addition on R. Note that r + r −1 = r s is the usual metric on R. Example 3.1.3. Let R be the set of the reals equipped with the quasi-metric s on R deﬁned by s(x, y) = x − y if x ≥ y and s(x, y) = 1 otherwise. It is readily checked that τ (s) is the so-called Sorgenfrey topology on R, which is generated by the base {(a, b] : a, b ∈ R, a < b} on R. Similarly, τ (s−1 ) is generated by the base {[a, b) : a, b ∈ R, a < b}. Indeed (R, τ (s), τ (s−1)) is a parabitopological group under the usual addition on R. Let η(e) denote the neighborhood ﬁlter of the identity element e of a paratopological group G. For each U ∈ η(e) put UL = {(x, y) ∈ G × G : y ∈ xU }. One checks that {UL : U ∈ η(e)} is a base for a quasi-uniformity UL on G. Moreover for each U ∈ η(e), put UR = {(x, y) ∈ G × G : y ∈ U x}. Then {UR : U ∈ η(e)} is also a base for a quasi-uniformity UR on G. Of course, these quasi-uniformities coincide for commutative groups. Furthermore they indeed are uniformities in the case of topological groups. We have that τ (UL ) = τ and τ ((UL )−1 ) = τ −1 ; similarly, τ (UR ) = τ and τ ((UR )−1 ) = τ −1 . The quasi-uniformities UL and UR are called the left quasiuniformity and the right quasi-uniformity for (G, ·, τ, τ −1 ). The quasi-uniformity UB = UL ∨UR is called the two-sided (or bilateral) quasi-uniformity for (G, ·, τ, τ −1 ). Every parabitopological group (G, ·, τ, τ −1 ) generates a topological group (G, ·, τ ∨ ) with τ ∨ = τ ∨ τ −1 . Note that (UL )s , (UR )s and (UB )s are the left uniformity, the right uniformity and the bilateral uniformity for (G, ·, τ ∨ ), respectively. Let us illustrate some of these concepts with a simple result. Proposition 3.1.4 ([110, Theorem 2]). Let G be a regular paratopological group possessing V ∈ η(e) that is hereditarily precompact in (G, UL ). Then G is a topological group. Proof. Let U ∈ η(e) be arbitrary. Then e ∈ (V \ U −1 )U . Choose W ∈ η(e) such that W ⊆ U . By our assumption on V there is a ﬁnite set F ⊆ (V \ U −1 ) such that V \ U −1 ⊆ F W . Then F W ⊆ F U ⊆ (V \ U −1 )U . Consequently e ∈ V \ F W ⊆ U −1 . We conclude that G is a topological group. Without proof we next formulate an algebraic version of the Quasi-Pseudometrization Lemma 2.2.2. To this end recall the following concept. Let G be a group and d a quasi-pseudometric on G. Then d is called left invariant if for all a, x, y ∈ G, d(ax, ay) = d(x, y). Proposition 3.1.5 (compare [122, Theorem 2 and Proposition 7]). Each ﬁrstcountable parabitopological group (G, τ, τ −1 ) admits a (compatible) left invariant quasi-pseudometric. It induces UL . Remark 3.1.6. The reader may wish to verify the following simple, but useful statements: (a) The topology of a topological group is completely regular. (b) Each ﬁrst-countable Hausdorﬀ topological group is metrizable. (c) In a compact Hausdorﬀ topological group we have UL = UR . Remark 3.1.7. In recent years many well-known results about the canonical uniformities of topological groups were generalized to analogous results about the

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canonical quasi-uniformities of paratopological groups. Let us give two examples, again without proofs: Example 3.1.8 ([122, Theorem 3]). The ground set of the bicompletion of the two-sided quasi-uniformity of a paratopological T0 -group naturally carries the structure of a paratopological T0 -group; moreover the quasi-uniformity of that bicompletion yields the two-sided quasi-uniformity of the constructed paratopological group. Example 3.1.9 ([124, Theorems 2.3 and 2.7]). The ground set of the bicompletion of the left quasi-uniformity of a paratopological T0 -group carries the structure of a topological semigroup. Those paratopological groups G for which the semigroup is a group H are characterized by the condition that each ﬁlter which is Cauchy with respect to the left uniformity is Cauchy with respect to the right uniformity of the associated topological group on G; under the latter condition the bicompleted left quasi-uniformity of G yields the left quasi-uniformity of the paratopological group H. Some of these results can be (partially) generalized further to topological monoids and semigroups. For instance Kopperman [67, Lemma 7] observed that if (X, τ ) is a topological monoid and UL (resp. UR ) is its left (resp. right) quasi-uniformity deﬁned as above, then τ = τ (UL ) (resp. τ = τ (UR )) if and only if the left translations (resp. the right translations) are open (compare [101, Remark 1.8]). In the theory of paratopological groups also other asymmetric completions have turned out to be useful, for instance the Doitchinov completion, as our next observation indicates. Proposition 3.1.10 ([110, Theorem 3]). The bilateral quasi-uniformity UB of a regular paratopological group is quiet. Proof. For U ∈ η(e), choose L ∈ η(e) such that L ⊆ U . Furthermore let V ∈ η(e) be such that V V ⊆ L. Suppose that VL−1 (x) ∩ VR−1 (x) ∈ F, (F, G) is a Cauchy ﬁlter pair with respect to UB and VL (y) ∩ VR (y) ∈ G. For each H ∈ η(e) choose FH ∈ F and GH ∈ G such that FH × GH ⊆ HL ∩ HR . For each H ∈ η(e) ﬁnd fH ∈ FH ∩ VL−1 (x) ∩ VR−1 (x) and gH ∈ GH ∩ VL (y) ∩ VR (y). −1 −1 x ∈ V, y −1 gH ∈ V, xfH ∈ V and gH y −1 ∈ V . It follows that for each H ∈ η(e), fH −1 −1 −1 −1 Furthermore fH gH → e and gH fH → e. Hence y gH fH x ∈ V V ⊆ L whenever −1 )x → y −1 x. Therefore y −1 x ∈ L ⊆ U and thus (y, x) ∈ H ∈ η(e) and y −1 (gH fH −1 −1 −1 UL . Similarly, xfH gH y ∈ V V ⊆ L whenever H ∈ η(e) and xfH gH y −1 → xy −1 , −1 and thus xy ∈ U and (y, x) ∈ UR . We have shown that UB is quiet. Unfortunately, in general the product of two Cauchy ﬁlter pairs with respect to the bilateral quasi-uniformity UB of a regular paratopological group need not be such a Cauchy ﬁlter pair. Hence the Doitchinov completion of the bilateral quasiuniformity of a regular paratopological group does not necessarily carry a natural group structure. Fortunately this diﬃculty cannot occur in the realm of abelian regular paratopological groups (compare [110, Theorem 4 and Example 3]). We ﬁnally formulate an interesting open problem for paratopological groups, which is open even for topological groups. In the latter context it is known under the name of “the Itzkowitz Problem” (compare [55]).

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Problem 3.1.11 (for research; compare also [94]). If the left and the right quasi-uniformity of a paratopological group induce the same quasi-proximity, are they necessarily equal? 3.2. Uniform convergence. Literature: [10, 82, 104, 107, 134, 138]. Again we can only present a few typical results. Definition 3.2.1. Let (X, τ ) be a topological space and (Y, V) a quasi-uniform space. Furthermore let D be a family of maps from X to Y . If A is a family of subsets of X we denote by VA the quasi-uniformity on D which has as a subbase the family of all relations of the form (A, U ) = {(f, g) ∈ D × D : (f (x), g(x)) ∈ U for all x ∈ A} whenever A ∈ A and U ∈ V. The quasi-uniformity VA is called the quasi-uniformity of uniform convergence of A.19 Example 3.2.2. If A = {X}, VA is called the quasi-uniformity of uniform convergence and is denoted by VX . If A = {K ⊆ X : K is a compact subset of (X, τ )}, VA is called the quasi-uniformity of compact convergence and is denoted by VK , and if A = {F ⊆ X : F is a ﬁnite subset of X}, then VA is called the quasi-uniformity of pointwise convergence and is denoted by Vp . Note that on the product set Y X , Vp agrees with the product quasi-uniformity. Let X be a topological space and (Y, V) a quasi-uniform space, and let DC be a subcollection of the collection C(X, Y ) of all continuous maps from X to (Y, τ (V)). For each compact set K in X and open set U of (Y, τ (V)) let [K, U ] = {f ∈ DC : f (K) ⊆ U }. The topology generated by the subbase of all sets [K, U ] is called the compact-open topology on DC . (Of course this topology remains unchanged if we choose another quasi-uniformity V inducing τ (V).) It is known [82, p. 88] that the following result does not hold for arbitrary quasi-uniformities. Indeed the second part of the proof is based on the given additional assumption. Proposition 3.2.3 ([82, Proposition 2]). Let X be a topological space and let (Y, V) be a small-set symmetric quasi-uniform space. Then the quasi-uniformity of compact convergence induces the compact-open topology on any collection DC . Proof. Let f ∈ [K, U ] where K is compact in X and U is τ (V)-open in Y . Then f (K) is compact by continuity of f . Furthermore f (K) ⊆ U . Since V is a compatible quasi-uniformity on Y and U is open, there is V ∈ V such that V (f (K)) ⊆ U by Remark 2.5.9(a). Therefore {g ∈ DC : (f (x), g(x)) ∈ V whenever x ∈ K} ⊆ [K, U ]. Suppose now that the quasi-uniform space (Y, V) is small-set symmetric, K is a (nonempty) compact set in X, V ∈ V and f ∈ DC . Choose W ∈ V such that W 2 ⊆ V . For each x ∈ X we ﬁnd Vx ∈ V such that Vx Vx (f (x)) ⊆ W (f (x)) ∩ W −1 (f (x)). (This is possible, because τ (V −1 ) ⊆ τ (V) and thus τ (V) is regun lar.) Then f (K) ⊆ i=1 Vxi (f (xi )) for some ﬁnite subset {x1 , . . . , xn } of K. −1 For each i ∈ {1, . . . , n} deﬁne a compact set K Vxi (f (xi )) in X. i = K ∩ f n n Clearly f ∈ i=1 [Ki , intτ (V) W (f (xi ))]. Let g ∈ i=1 [Ki , intτ (V) W (f (xi ))]. We want to show that (f (x), g(x)) ∈ V whenever x ∈ K. Let x ∈ K. Then x ∈ Ki ⊆ f −1 Vxi (f (xi )) ⊆ f −1 W −1 (f (xi )) and thus (f (x), f (xi )) ∈ W for some i ∈ 19Note that the same notation V is used for diﬀerent ground sets D. A

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{1, . . . , n}; observe also that (f (xi ), g(x)) ∈ W . We conclude that (f (x), g(x)) ∈ V . Hence the two topologies are equal. It was observed in [107, Examples] that the condition of local symmetry cannot be omitted in the following proposition. Proposition 3.2.4 ([107, Lemma 2]). Let (X, τ ) be a topological space and (Y, V) a locally symmetric quasi-uniform space. Let A be a family of subsets of X and CA (X, Y ) the family of all functions from X to (Y, τ (V)) which are continuous on each member of A. Then CA (X, Y ) is closed in Y X with respect to the topology τ (VA ). Proof. We shall prove that Y X \ CA (X, Y ) belongs to the topology τ (VA ) on Y X . Let f ∈ Y X \ CA (X, Y ). Then f |A is not continuous for some A ∈ A. Hence, there are x ∈ A and U ∈ V such that (f |A )−1 (U (f (x)) contains no neighborhood of x in τ |A . Choose V ∈ V with V −1 (V (f (x))) ⊆ U (f (x)). Then ﬁnd W ∈ V with W 2 ⊆ V . We shall prove that (A, W )(f ) ∩ CA (X, Y ) = ∅. In fact, given g ∈ (A, W )(f ) let y ∈ (g|A )−1 (W (g(x))); then (g(x), g(y)) ∈ W, and since (f (x), g(x)) ∈ W we obtain (f (x), g(y)) ∈ V . Furthermore (f (y), g(y)) ∈ W because y ∈ A. Hence f (y) ∈ V −1 (V (f (x))) ⊆ U (f (x)), which implies that y ∈ (f |A )−1 (U (f (x))). We conclude that (g|A )−1 (W (g(x))) ⊆ (f |A)−1 (U (f (x))), so g ∈ Y X \ CA (X, Y ). The proof is complete. Corollary 3.2.5. Let X be a topological space and let (Y, V) be a locally symmetric quasi-uniform space. Then the set C(X, Y ) of all continuous functions from X to (Y, τ (V)) is closed in Y X where Y X is equipped with the topology induced by the quasi-uniformity of uniform convergence. Proposition 3.2.6 ([107, Proposition 1]). Let (X, τ ) be a topological space and (Y, V) a right K-complete quasi-uniform space. Then for each family A of subsets of X which covers X the quasi-uniformity of quasi-uniform convergence of A is right K-complete on Y X . Proof. We make use of Remark 2.6.12. Let A be a family of subsets of X which covers X and let (fd )d∈D be a right K-Cauchy net in (Y X , VA ). For each U ∈ V and each A ∈ A there is d0 ∈ D such that (fd , fd ) ∈ (A, U ) whenever d0 ≤ d ≤ d. Fix x ∈ X. Then x is in some Ax ∈ A. It follows that (fd (x))d∈D is a right K-Cauchy net in (Y, V). Hence, it is τ (V)-convergent to a point f (x) ∈ Y . Thus we have deﬁned a function f from X to Y . We shall prove that the net (fd )d∈D converges to f with respect to the topology τ (VA ). Given U ∈ V and A ∈ A choose V ∈ V with V 2 ⊆ U . There is d0 ∈ D such that (fd , fd ) ∈ (A, V ) whenever d0 ≤ d ≤ d . We want to show that (f, fd ) ∈ (A, U ) for all d ∈ D with d0 ≤ d. In fact, for such d and any x ∈ A there is d(x) ∈ D with d ≤ d(x) and (f (x), fd(x) (x)) ∈ V . Since (fd(x) (x), fd (x)) ∈ V, we conclude that (f (x), fd (x)) ∈ U . Therefore VA is right K-complete. A result analogous to Proposition 3.2.6 can also be proved for bicompleteness [107, Proposition 5]; but it does not hold for left K-completeness (see [104, Example 3.2]).

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3.3. Isomorphism groups and equicontinuity. Literature: [20, 27, 83, 144]. This subsection is mainly based on our survey article [83] to which we refer the reader for additional information and references. Proposition 3.3.1 ([83, Proposition 1]). Let (X, U) be a quasi-uniform space and let Q(X) be the group of all uniform self-isomorphisms of (X, U) under composition of maps as operation. Equip Q(X) with the quasi-uniformity of uniform convergence that in this case we shall denote by UU . (For simplicity in the following, for V ∈ U we shall denote the entourage (X, V ) by V .) Then (Q(X), ◦, τ (UU )) is a paratopological group and UU is the right quasi-uniformity of that paratopological group. The left quasi-uniformity of the paratopological group (Q(X), ◦, τ (UU )) is generated by the entourages {(f, g) ∈ Q(X) × Q(X) : (g −1 (x), f −1 (x)) ∈ U whenever x ∈ X} where U ∈ U. Proof. Let f, g ∈ Q(X) and U ∈ U. Choose W ∈ U such that W 2 ⊆ U . Since f is uniformly continuous, there exists V ∈ U such that (f × f )(V ) ⊆ W . Let h1 ∈ W (f ), h2 ∈ V (g) and x ∈ X. Then h2 (x) ∈ V (g(x)) and therefore (f ◦ h2 )(x) ∈ W ((f ◦ g)(x)). Since h1 ∈ W (f ), it follows that (h1 ◦ h2 )(x) ∈ W ((f ◦ h2 )(x)) ⊆ W 2 ((f ◦ g)(x)) ⊆ U ((f ◦ g)(x)). Thus W (f ) ◦ V (g) ⊆ U (f ◦ g). Hence ◦ : Q(X)×Q(X) → Q(X) is continuous and so (Q(X), ◦, τ (UU )) is a paratopological group. Let U ∈ U. Then UR = {(f, g) ∈ Q(X) × Q(X) : g ◦ f −1 ∈ U (idX )} = {(f, g) ∈ Q(X) × Q(X) : (x, (g ◦ f −1 )(x)) ∈ U whenever x ∈ X} = {(f, g) ∈ Q(X) × Q(X) : (f (x), g(x)) ∈ U whenever x ∈ X}. Similarly, UL = {(f, g) ∈ Q(X) × Q(X) : f −1 ◦ g ∈ U (idX )} = {(f, g) ∈ Q(X) × Q(X) : (x, (f −1 ◦ g)(x)) ∈ U whenever x ∈ X} = {(f, g) ∈ Q(X) × Q(X) : (g −1 (x), f −1 (x)) ∈ U whenever x ∈ X}. Hence we are done.

Observe that for a topological space X endowed with some quasi-uniformity such that each continuous self-map is uniformly continuous (see e.g., the discussion in front of Deﬁnition 2.1.8) Q(X) consists of all the self-homeomorphisms of X. In general the paratopological group described in Proposition 3.3.1 is not a topological group if U is not a uniformity.20 However it turns out to be a topological group under surprising conditions, for instance if U is the ﬁne quasi-uniformity of a completely metrizable space (compare [83, Proposition 2 and Example 1]). Remark 3.3.2. In [83, Proposition 5] it is observed that the group of all uniform self-isomorphisms of a bicomplete quasi-uniform T0 -space equipped with the topology induced by the quasi-uniformity of uniform convergence yields a paratopological group whose two-sided quasi-uniformity is bicomplete. 20Note that for a symmetric entourage U and a uniform self-isomorphism f we have (x, f (x)) ∈ U whenever x ∈ X if and only if (x, f −1 (x)) ∈ U whenever x ∈ X, so that for a uniformity U we indeed obtain a topological group.

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Other types of quasi-uniformities on function spaces have been used to analyze special groups of self-homeomorphisms of topological spaces. We here still present some results about the quasi-uniformity of pointwise convergence. Definition 3.3.3. If D is a collection of maps from a topological space X to a quasi-uniform space (Y, V) and x ∈ X, then D is said to be equicontinuous at x provided that for each V ∈ V there exists a neighborhood N of x such that for all f ∈ D, f (N ) ⊆ V (f (x)); furthermore D is called equicontinuous on X provided that D is equicontinuous at each x ∈ X. Similarly, a collection D of maps from a quasi-uniform space (X, U) to a quasi-uniform space (Y, V) is said to be uniformly equicontinuous provided that for each V ∈ V there is U ∈ U such that (f ×f )U ⊆ V whenever f ∈ D. Of course, uniformly equicontinuous families of maps between quasi-uniform spaces are equicontinuous. First we mention a condition under which the converse obtains (compare with Proposition 2.1.12). Proposition 3.3.4 ([83, Proposition 9]). Let (X, U) be a quasi-uniform space having the Lebesgue property and let D be an equicontinuous family of maps from (X, τ (U)) into a uniform space (Y, V). Then D is uniformly equicontinuous. Proof. Indeed, let V ∈ V and W ∈ V be such that W 2 ⊆ V . By equicontinuity of D, {intτ (U) ( f ∈D f −1 (W ∩ W −1 )(f (x))) : x ∈ X} is an open cover C of X. Since (X, U) possesses the Lebesgue property, there is H ∈ U such that {H(x) : x ∈ X} reﬁnes the cover C. Let x ∈ X. There is y ∈ X such that H(x) ⊆ f −1 (W ∩ W −1 )(f (y)) whenever f ∈ D; therefore f (H(x)) ⊆ W (f (y)) and f (x) ∈ W −1 (f (y)). It follows that (f × f )H ⊆ W 2 ⊆ V whenever f ∈ D. Hence D is uniformly equicontinuous. Proposition 3.3.5 (compare [144] and [83, Remark 8]). Let G be a group of (topological) self-homeomorphisms of a quasi-uniform space (X, U) that is equicontinuous with respect to U. Then (G, ◦, τ (Up )) is a paratopological group; furthermore (G, ◦, τ (Up )) is a topological group if U is point-symmetric. Proof. Suppose that (fd )d∈D and (gd )d ∈D are nets in G such that (fd )d∈D → f pointwise in G and (gd )d ∈D → g pointwise in G. Fix x ∈ X. Let V ∈ U and W ∈ U be such that W 2 ⊆ V . By equicontinuity of G we can choose a neighborhood N of g(x) such that fd (N ) ⊆ W (fd (g(x))) whenever d ∈ D. Since (gd (x))d ∈D → g(x) and (fd (g(x)))d∈D → f (g(x)), we deduce that (fd ◦ gd )(x) ∈ W (fd (g(x))) ⊆ W 2 ((f ◦ g)(x)) ⊆ V ((f ◦ g)(x)) provided that d ∈ D and d ∈ D are large enough, because the elements of the net (gd (x))d ∈D belong to N eventually. Thus (fd ◦ gd ) → f ◦ g pointwise in G. Hence (G, ◦, τ (Up )) is a paratopological group. Suppose now that G is not a topological group. Without loss of generality, in G there exists a net (fd )d∈D converging pointwise to idX , but for some x ∈ X, (fd−1 (x))d∈D → x. If U is point-symmetric, there exists V ∈ U such that for each (x) ∈ V −1 (x). Since {fd−1 : d ∈ D} is d ∈ D there is d ∈ D with d ≥ d and f−1 d equicontinuous at x, there is a neighborhood N of x such that fd−1 (N ) ⊆ V (fd−1 (x)) whenever d ∈ D. Then the net (fd (x))d∈D converges to x and thus x = (f−1 ◦ d −1 fd )(x) ∈ V (fd (x)) for some large d ∈ D — a contradiction. Hence G is a topological group provided that U is point-symmetric.

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Proposition 3.3.6 ([83, Proposition 11]). Let G be a uniformly equicontinuous group of uniform self-isomorphisms of a quasi-uniform space (X, U). Then the left quasi-uniformity of the paratopological group (G, ◦, τ (Up )) is equal to Up ; furthermore the right quasi-uniformity of that paratopological group is generated by the entourages {(f, g) ∈ G × G : (g −1 (x), f −1 (x)) ∈ U } where U ∈ U and x ∈ X. Proof. Let U ∈ U. By uniform equicontinuity of G there is H ∈ U such that (f × f )H ⊆ U whenever f ∈ G. Let x ∈ X and f, g ∈ G. Then by our assumption (x, (f −1 ◦ g)(x)) ∈ H implies that (f (x), g(x)) ∈ U . Similarly, g(x) ∈ H(f (x)) implies that (f −1 ◦ g)(x) ∈ U ((f −1 ◦ f )(x)) = U (x). We conclude that the left quasi-uniformity of G is equal to Up . The assertion about the right quasi-uniformity of G is proved analogously. 3.4. The Hausdorﬀ quasi-uniformity. Literature: [6, 11, 111, 136]. Observe that in the following deﬁnition U− and U+ need not be uniformities, even if U is a uniformity. Definition 3.4.1. Let (X, U) be a quasi-uniform space and let P0 (X) be the set of nonempty subsets of X. For any U ∈ U let U+ = {(A, B) ∈ P0 (X) × P0 (X) : B ⊆ U (A)}, U− = {(A, B) ∈ P0 (X) × P0 (X) : A ⊆ U −1 (B)}. Furthermore set UH = (U− ) ∩ (U+ ) whenever U ∈ U. Then {U− : U ∈ U} is a base for the lower quasi-uniformity U− on P0 (X) and {U+ : U ∈ U} is a base for the upper quasi-uniformity U+ on P0 (X). Moreover UH = U+ ∨ U− is the so-called Hausdorﬀ or Bourbaki quasi-uniformity of (X, U) (see [6]). Lemma 3.4.2 ([111, Lemma 1]). • Let (X, U) be a quasi-uniform space. Then x → {x} is a uniform embedding of (X, U) into (P0 (X), UH ). • Let (X, U) and (Y, V) be quasi-uniform spaces and let f : (X, U) → (Y, V) be a uniformly continuous map. Then the map f : (P0 (X), UH ) → (P0 (Y ), VH ) deﬁned by f (A) := {f (a) : a ∈ A} is uniformly continuous, too. Proof. (a). The assertion is readily veriﬁed. (b). If (f × f )(U ) ⊆ V where U ∈ U and V ∈ V, then (f × f )(UH ) ⊆ VH .

For a uniform space (X, U), it is usually the T0 -quotient of (P0 (X), UH ), rather than (P0 (X), UH ) itself, which is studied. It is well known that this quotient can be described as follows: Each nonempty subset A of X is identiﬁed with its closure A and then the Hausdorﬀ uniformity is restricted to the collection of nonempty closed subsets of X. Similarly, our next lemma can be used to show that for a quasi-uniform space (X, U) the T0 -quotient of (P0 (X), UH ) can be described as follows: Identify an arbitrary nonempty subset A of X with clτ (U) A ∩ clτ (U −1 ) A and study the Hausdorﬀ quasi-uniformity restricted to the collection of all subsets of X obtained in that way. Lemma 3.4.3 ([111, Lemma 2]). Let (X, U) be a quasi-uniform space and A, B ∈ P0 (X). We have (A, B) ∈ UH ∩ ( UH )−1 if and only if clτ (U) A = cl B and clτ (U −1 ) A = clτ (U −1 ) B. In particular for any A ∈ P0 (X), (A, C) ∈ τ (U) UH ∩ ( UH )−1 where C = clτ (U) A ∩ clτ (U −1 ) A.

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Proof. See [111, Lemma 2].

Remark 3.4.4. Let d be a bounded quasi-pseudometric on a set X. Then the Hausdorﬀ quasi-pseudometric on P0 (X) deﬁned by dH (A, B) = max{sup d(A, b), sup d(a, B)} b∈B

a∈A

whenever A, B ∈ P0 (X) induces on P0 (X) the Hausdorﬀ quasi-uniformity of the quasi-uniform space (X, Ud ). In the literature many results of the following type have been established: A quasi-uniform space (X, U) has property P if and only if (P0 (X), UH ) has property P . In some cases P can be chosen equal to P as in the case of total boundedness [111, Corollary 2], precompactness [111, Proposition 1] or joincompactness [111, Corollary 9]. We here present a proof for the result concerning precompactness. Proposition 3.4.5 ([111, Proposition 1]). Let (X, U) be a quasi-uniform space. Then (P0 (X), UH ) is precompact if and only if (X, U) is precompact. Proof. Let (X, U) be precompact and let V ∈ UH . There is U ∈ U such that X such that UH ⊆ V . Since U is precompact, there exists a ﬁnite set F ⊆ f ∈F U (f ) = X. Set M = P0 (F ). We want to show that P0 (X) = E∈M UH (E): Consider an arbitrary B ∈ P0 (X). Set FB = {f ∈ F : B ∩ U (f ) = ∅}. Thus FB ⊆ U −1 (B) and therefore B ∈ U− (FB ). Furthermore B ∈ U+ (FB ), because B ⊆ f ∈FB U (f ). Hence B ∈ UH (FB ). We conclude that (P0 (X), UH ) is precompact. On the other hand, suppose that (P0 (X), UH ) and thus (P0 (X), U− ) is precompact. Let V ∈ U. Then there is a ﬁnite subcollection A of P0 (X) such that for each B ∈ P0 (X) there is A∈ A with A ⊆ V −1 (B). Choose for each A ∈ A some xA ∈ A. Then B = X \ A∈A V (xA ) is necessarily empty. Therefore (X, U) is precompact. On the other hand if P is equal to hereditary precompactness, an appropriate property P was described in [108], where also the connection of this result to the theories of well- and better-quasi-orderings was brieﬂy discussed. As the following example taken from [111, Example 2] implies, in this case P has to be strictly stronger than hereditary precompactness. Example 3.4.6. Let X = ω × ω. For any n ∈ ω set An = (n × n) ∪ ({n} × ω). Furthermore let C = {X \ An : n ∈ ω} ∪ {X}. Consider the quasi-uniformity U generated by the base {TC } on X. We ﬁrst show that (X, U) is hereditarily precompact. Note that for each x ∈ X, there is j0 ∈ ω such that x ∈ Aj for all j ∈ ω with j ≥ j0 ; thus X \ TC (x) = {An : x ∈ / An , n ∈ ω} is equal to the union of ﬁnitely many sets Aj . Suppose that (X, U) is not hereditarily precompact. Then there is a sequence (zn )n∈ω of points of X such that zj ∈ / TC (zi ) whenever i, j ∈ ω and i < j. Since ∅ = X \ TC (z0 ) consists of the union of ﬁnitely many sets Aj and since the elements (zi )i∈ω are pairwise distinct, we conclude that there exists n ∈ ω such that zj ∈ {n} × ω for inﬁnitely many j ∈ ω. Thus there are m1 , m2 ∈ ω such that m1 < m2 and (n, m2 ) ∈ / TC ((n, m1 )). Clearly this is impossible, since (n, m2 ) ∈ Aj ⊆ X \ TC ((n, m1 )) for some j ∈ ω implies that (n, m1 ) ∈ Aj . We conclude that (X, U) is hereditarily precompact. Finally we prove that (P0 (X), U− ) is not hereditarily precompact. In fact we show that (E, U− |E ) is not precompact where E = {An : n ∈ ω}. Observe ﬁrst that

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TC−1 (An ) = An for each n ∈ ω. If (E, U− |E ) is precompact, then there are i, j ∈ ω such that i < j and Ai ⊆ TC−1 (Aj ). But Ai ⊆ Aj clearly does not hold, since {i} × ω ⊆ Aj . We have shown that E is not precompact in (P0 (X), U− ). While the Hausdorﬀ uniformity of a complete metric uniformity is always complete (compare [25, Problem 4.5.23(c)] and [111, Proposition 5]; see also Propositions 2.6.13 and 3.4.9), the Hausdorﬀ uniformity of an arbitrary complete uniform space need not be complete [54, p. 31]. K¨ unzi and Ryser [111, Proposition 8] characterized those quasi-uniform spaces possessing a bicomplete Hausdorﬀ quasiuniformity (compare also [109]). In particular the Hausdorﬀ quasi-uniformity of a bicomplete quasi-metric quasi-uniformity need not be bicomplete: Example 3.4.7. Equip the set Q of the rationals with the restriction e = s|(Q × Q) of the Sorgenfrey quasi-metric s (see Example 3.1.3). Then (Q, Ue ) is obviously bicomplete, but (P0 (Q), (Ue )H ) is not bicomplete (for a proof compare [111, Example 7]). Below we shall present an elegant characterization of those quasi-uniformities for which the Hausdorﬀ quasi-uniformity is right K-complete. This result generalizes the Isbell–Burdick theorem about supercompleteness from the theory of uniform spaces. An analogous quasi-uniform result does not hold for left Kcompleteness (see [106]). Lemma 3.4.8 ([111, Lemma 6]). Suppose that (X, U) is a quasi-uniform space in which each stable ﬁlter has a τ (U)-cluster point. Let F be a stable ﬁlter on (X, U) and let C be its set of τ (U)-cluster points. Then for each U ∈ U there is F ∈ F such that F ⊆ U (C). Proof. Suppose the contrary. Hence there is U0 ∈ U such that E \ U02 (C) = ∅ whenever E ∈ F. For each U ∈ U and E ∈ F set HUE = {a ∈ X : there is V ∈ U such that V 2 ⊆ U, V −2 (a) ∩ U0 (C) is empty and a ∈ F ∈F V (F ) ∩ E}. We verify that each such set HUE = ∅: To this end choose V ∈ U such that V 2 ⊆ U0 ∩ U . Then any a ∈ ( F ∈F V (F ) ∩ E) \ U02 (C) belongs to HUE . Note also that for any U1 , U2 ∈ U such that U1 ⊆ U2 and any E1 , E2 ∈ F such that E1 ⊆ E2 we have that HU1 E1 ⊆ HU2 E2 . Thus {HUE : U ∈ U, E ∈ F} is a base for a ﬁlter H on X. In order to show that H is stable on (X, U), we verify that for any U, V ∈ U and E ∈ F we have HUX ⊆ U (HV E ): Let a ∈ HUX . Then there exists W ∈ U such that W 2 ⊆ U, W −2 (a) ∩ U0 (C) = ∅ and a ∈ F ∈F W (F ). Choose Z ∈ U such that Z 2 ⊆ V ∩ W . There is y ∈ [E ∩ F ∈F Z(F )] ∩ W −1 (a), because a ∈ F ∈F W (F ) and E ∩ F ∈F Z(F ) ∈ F. Furthermore Z −2 (y) ⊆ W −1 (y) ⊆ W −2 (a) and thus Z −2 (y) ∩ U0 (C) = ∅. We conclude that y ∈ HV E and a ∈ W (y) ⊆ U (y). Therefore HUX ⊆ U (HV E ). We have shown that H is stable on (X, U). Hence it has a τ (U)cluster point x ∈ X. Since HX×XF ⊆ F whenever F ∈ F, it follows that x ∈ C. But HX×XX ∩ intτ (U) U0 (C) = ∅ and x ∈ C ⊆ intτ (U) U0 (C). We have obtained a contradiction and deduce that our initial assumption was wrong. Proposition 3.4.9 ([111, Proposition 6]). Let (X, U) be a quasi-uniform space. Then (P0 (X), UH ) is right K-complete if and only if each stable ﬁlter on (X, U) has a τ (U)-cluster point.

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Proof. Suppose that (P0 (X), UH ) is right K-complete. Let F be a stable ﬁlter on (X, U). Consider the net (F )F ∈(F ,⊇) on P0 (X). Let U ∈ U. Since F is stable, there is FU ∈ F such that FU ⊆ U (F ) whenever F ∈ F. Thus for any F1 , F2 ∈ F such that F1 ⊆ F2 ⊆ FU , we have that F2 ⊆ U (F1 ) and F1 ⊆ U −1 (F2 ). Therefore (F )F ∈F is a right K-Cauchy net in (P0 (X), UH ). Since (P0 (X), UH ) is right K-complete, (F )F ∈F converges to some C in (P0 (X), UH ) by Remark 2.6.12. Fix x ∈ C. It follows that x is a cluster point of F in (X, U). Hence each stable ﬁlter on (X, U) has a τ (U)-cluster point. In order to prove the converse, suppose that each stable ﬁlter on (X, U) has a τ (U)-cluster point. Consider any right K-Cauchy net (Fd )d∈D on (P0 (X), UH ). For each U ∈ U there is dU ∈ D such that for any d1 , d2 ∈ D satisfying d1 ≥ d2 ≥ dU we have that Fd2 ⊆ U (Fd1 ) and Fd1 ⊆ U −1 (Fd2 ). Let F be the ﬁlter generated by {Ee : e ∈ D} on X where Ee = d∈D,d≥e Fd whenever e ∈ D. We verify that for each U ∈ U we have EdU ⊆ d∈D U (Ed ): Let x ∈ EdU and d ∈ D. Then x ∈ Fd0 for some d0 ∈ D such that d0 ≥ dU . Choose h ∈ D such that h ≥ d0 , d. Observe that x ∈ Fd0 ⊆ U (Fh ) ⊆ U (Ed ). We have shown that EdU ⊆ U (Ed ) and that F is a stable ﬁlter on (X, U). Let C be the (nonempty) set of cluster points of F in (X, U). Take any U ∈ U. Choose W ∈ U such that W 2 ⊆ U . We wish to show that C ⊆ U −1 (Fd ) whenever d ∈ D and d ≥ dW : Let x ∈ C and d ∈ D such that d ≥ dW . Then W (x) ∩ Ed = ∅. Hence W (x) ∩ Fp = ∅ for some p ∈ D with p ≥ d. Thus x ∈ W −1 (Fp ) ⊆ W −1 (W −1 (Fd )). We conclude that C ⊆ U −1 (Fd ) whenever d ∈ D and d ≥ dW , as we have stated above. By Lemma 3.4.8 for each U ∈ U there exists e ∈ D such that d∈D,d≥e Fd ⊆ U (C). We deduce that (Fd )d∈D converges in (P0 (X), UH ) to C. Consequently (P0 (X), UH ) is right K-complete (see Remark 2.6.12). Let (X, τ ) be a topological space. Recall that the upper Vietoris topology of (X, τ ) is the topology τV+ on P0 (X) generated by all sets of the form G+ = {A ∈ P0 (X) : A ⊆ G} where G ∈ τ , and the lower Vietoris topology of (X, τ ) is the topology τV− on P0 (X) generated by all sets of the form G− = {A ∈ P0 (X) : A∩G = ∅} where G ∈ τ . The topology τV+ ∨ τV− on P0 (X) is called the Vietoris topology of (X, τ ) and is denoted by τV . It is known that given a uniform space (X, U) the Hausdorﬀ quasi-uniformity UH induces the Vietoris topology on the set K0 (X) of nonempty compact subsets of (X, U) (compare [125]).21 Our next proposition explains why this result does no longer hold for arbitrary quasi-uniform spaces. Proposition 3.4.10 ([136, Theorem 5]). Let (X, U) be a quasi-uniform space and let A be a subfamily of the family K0 (X) of nonempty τ (U)-compact subsets of X containing all ﬁnite nonempty subsets of X. Then τ (UH ) is equal to the Vietoris topology on A if and only if for each K ∈ A, U −1 |K is precompact. Proof. We ﬁrst show that (τ (U))V ⊆ τ (UH ) on K0 (X). Let A ∈ K0 (X). Assume that A ∈ G+ for some G ∈ τ (U). Then A ⊆ G. Thus by Remark 2.5.9(a) there is U ∈ U such that U (A) ⊆ G, because A is compact. Consequently U+ (A) ⊆ G+ . Suppose next that A ∈ G− for some G ∈ τ (U). Find x ∈ A ∩ G and choose Ux ∈ U with Ux (x) ⊆ G. Therefore (Ux )− (A) ⊆ G− , since B ∈ (Ux )− (A) implies that Ux (x) ∩ B = ∅ and thus G ∩ B = ∅. Hence the claim is veriﬁed. 21Note that here and in the following our notation does not indicate that for instance U is H indeed restricted to a subspace of P0 (X).

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Assume now that τ (UH ) = (τ (U))V on A. Let K ∈ A and U ∈ U. By assumption we can ﬁnd τ (U)-open sets G, H1 , . . . , Hn such that K ∈ G+ ∩ ( nj=1 Hj− ) ⊆ . . . , n} there is kj ∈ K ∩ Hj . Set UH (K) where n is a positive integer. For j ∈ {1, n B = {k1 , . . . , kn }. Then B ∈ A and B ∈ G+ ∩ ( j=1 Hj− ). So K ⊆ U −1 (B). We have shown that U −1 |K is precompact. Conversely, by the ﬁrst paragraph of the proof it will suﬃce to show that τ (UH ) ⊆ (τ (U))V on A. Let K ∈ A and U ∈ U. Choose V ∈ U such that V 2 ⊆ U . By our assumption there is a ﬁnite subset K0 of K such that K ⊆ V −1 (K0 ). Put G = intτ (U) U (K). Then K ∈ G+ . Consider the (τ (U))V -neighborhood G+ ∩ k∈K0 (intτ (U) V (k))− of K and let B ∈ A be such that B ∈ G+ ∩ k∈K0 (intτ (U) V (k))− . Then B ⊆ U (K). Let k ∈ K. There is k0 ∈ K0 such that k ∈ V −1 (k0 ). Furthermore there is b ∈ B such that b ∈ V (k0 ) ∩ B. Then k ∈ V −2 (b) ⊆ U −1 (b). Consequently K ⊆ U −1 (B). We conclude that B ∈ UH (K) and the proof is ﬁnished. Using similar methods the reader should have no problems to establish the following interesting result. Proposition 3.4.11 (compare [136, Theorem 1]). Let (X, τ ) be a topological space. Then both (P(τ ))H and (M(τ ))H induce the Vietoris topology on P0 (X). Problem 3.4.12 (for research; compare [94] and the references cited there). Two quasi-uniformities U and V on a set X are called QH-equivalent provided that τ (UH ) = τ (VH ) on P0 (X). It is known that QH-equivalence of U and V implies that Uω = Vω , but no useful characterization of QH-equivalence of quasi-uniformities has been found yet. (However such a description was provided by Ward (see [152]) in the case of uniformities.) 3.5. Topological ordered spaces. Literature: [30, 112, 127]. In this subsection we shall outline the use of quasi-uniformities in the theory of ordered compactiﬁcations. Further details about this application of quasiuniformities can be found in [30, 127]. A topological preordered space (X, τ, ≤) is a topological space (X, τ ) equipped with a preorder ≤. A continuous map between two topological preordered spaces f : (X, τ, ≤1 ) → (Y, ρ, ≤2 ) is called increasing (resp. decreasing) if f (x) ≤2 f (y) (resp. f (x) ≥2 f (y)) whenever x, y ∈ X and x ≤1 y. A subset A of a topological preordered space X is called increasing provided that x ∈ A, y ∈ X and x ≤ y imply that y ∈ A. Dually, a subset A of a topological preordered space X is called decreasing provided that x ∈ A, y ∈ X and y ≤ x imply that y ∈ A. Note that the set of all increasing open sets yields a topology, which is called the upper topology τ of τ (compare Examples 2.3.9 and 3.1.2). Similarly, the set of all decreasing open sets is called the lower topology τ of τ . We remark that increasing sets are often called upper sets and decreasing sets are also called lower sets. A topological preordered space (X, τ, ≤) is said to be determined by a quasiuniformity U on X 22 if there is a quasi-uniformity U on X such that τ (U s ) = τ and U = ≤. Note that for such topological preordered spaces ≤ is closed in X × X with respect to the topology τ × τ . A topological ordered space with a closed order is called T2 -ordered. If a topological preordered space X is determined by the quasi-uniformity U, then X is also determined by the quasi-uniformity Uω . 22These are Nachbin’s uniform preordered spaces mentioned in the introduction.

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Furthermore observe that always τ (U) ⊆ τ and τ (U −1 ) ⊆ τ , where we cannot expect equalities to hold in general. Let U be a totally bounded quasi-uniformity on a set X such that U is a par U) be the bicompletion of (X, U). Then (X, τ ((U) s )) tial order. Furthermore let (X, is obviously a Hausdorﬀ compactiﬁcation of (X, τ (U s )) and U is a closed partial that extends the order U. One says that (X, τ ((U) s ), U) is a T2 order on X ordered compactiﬁcation of (X, τ (U s ), U). Among other things the next proposition shows that conversely T2 -ordered compactiﬁcations of a topological ordered space X give rise to totally bounded quasi-uniformities that determine the space X (compare also with Proposition 2.7.10). Proposition 3.5.1 ([127] and [30, Chapter 4]). Let (X, τ ) be a compact Hausdorﬀ space and let ≤ be a closed partial order on X. Then there is exactly one (necessarily totally bounded and bicomplete) quasi-uniformity U on X that determines (X, τ, ≤). It consists of all τ × τ -neigborhoods of ≤. Furthermore τ (U) is the upper topology of τ and τ (U −1 ) is the lower topology of τ . On the other hand each bicomplete totally bounded quasi-uniformity inducing a T0 -topology determines a compact T2 -ordered space. ([95, Proposition 1]). The compact T2 -ordered spaces which are determined by a transitive quasi-uniformity are exactly the totally order disconnected spaces, which occur in Priestley’s representation theory of distributive lattices [131]. Proof. We refer the reader to [30, Theorem 4.21 and Proposition 4.22] and [95, Proposition 1]. Corollary 3.5.2. Each compact Hausdorﬀ space (X, τ ) has a unique compatible uniformity consisting of all the τ × τ -neighborhoods of the diagonal. Naturally topological ordered spaces X that are determined by quasi-uniformities are called completely regularly (T2 -)ordered spaces. Their topology and order are given in the expected way by the family of all continuous increasing functions from X into the real unit interval [0, 1]. Indeed they are exactly those topological ordered spaces that satisfy the following two conditions (see e.g., [30, p. 77 and Theorem 4.18] and [127, p. 53]): (i) If a, b ∈ X such that f (a) ≤ f (b) whenever f : X → [0, 1] is continuous and increasing, then a ≤ b. (ii) For any point a ∈ X and neighborhood V of a there are two maps f, g : X → [0, 1] such that f is continuous and increasing, g is continuous and decreasing, f (a) = g(a) = 1 and min{f (x), g(x)} = 0 for any x∈X \V. Let us note that condition (i) implies T2 -orderedness (see e.g., [103, p. 186]). Our next example demonstrates that for a completely regularly ordered space (X, τ, ≤) the bitopological space (X, τ , τ ) need not be pairwise completely regular (compare also [112]): Example 3.5.3 ([50, 2.2.3(3)], compare [103, Example 6]). Let X = [0, 1] × [0, 1] be equipped with its usual compact topology and ordered by (x1 , y1 ) ≤ (x2 , y2 ) provided that x1 ≤ x2 and y1 = y2 . Then the partial order ≤ is closed, and since compact T2 -ordered spaces are completely regularly ordered, we have deﬁned a completely regularly ordered space. Consider the subspace E = ([0, 1) × (Q ∩ [0, 1])) ∪ F where F = {1} × ([0, 1] \ Q) of X. Of course, E is also a completely

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regularly ordered space. Evidently now in E there does not exist an open increasing set containing (0, 0) and an open disjoint decreasing set that contains the closed decreasing set F . Hence the bitopological space associated with the topological ordered space E is not pairwise (completely) regular. Furthermore a topological T2 -ordered space (X, τ, ≤) with a completely regular topology τ need not be completely regularly ordered, even if τ is convex 23, as our next example shows. Example 3.5.4 ([103, Example 1]). Let X = (ω1 + 1) × {0} ⊕ [(ω1 + 1) × (ω0 + 1) \ {(ω1 , ω0 )}] × {1} ⊕ (ω0 + 1) × {2}; that is, X is the topological sum of copies of ω1 + 1, ω0 + 1 and the (deleted) Tychonoﬀ plank, all equipped with their usual topology. As a partial order on X choose ≤ = ∆X ∪ {((α, 0), (α, ω0 , 1)) : α ∈ ω1 } ∪ {((ω1 , n, 1), (n, 2)) : n ∈ ω0 }. Obviously ≤ is closed so that (X, ≤) is T2 -ordered. Note that each point has a neighborhood base consisting of open increasing sets only or of open decreasing sets only. Hence the topology of X is convex. Since the unique Hausdorﬀ compactiﬁcation of the Tychonoﬀ plank is the one-point-compactiﬁcation, we obtain the unique compactiﬁcation βX of X as the one-point-compactiﬁcation by adding the point (ω1 , ω0 , 1). Suppose that ) is a closed order on βX extending the order ≤ on X. Then clearly ((ω1 , 0), (ω1 , ω0 , 1)) and ((ω1 , ω0 , 1), (ω0 , 2)) belong to ), and by transitivity, we have reached the contradiction that (ω1 , 0) ≤ (w0 , 2). We conclude that (X, ≤) does not have a T2 -ordered compactiﬁcation. Hence X is not completely regularly ordered, although the completely regular topology of X is convex. 3.6. Asymmetrically normed linear spaces. Literature: [1, 38, 39, 40, 61]. Let E be a linear space over the ﬁeld of the real numbers R. We say that a function *·| : E → [0, ∞) is an asymmetric norm on E if for all x, y ∈ E and a ∈ [0, ∞): (i) *x| = *−x| = 0 if and only if x = 0, (ii) *ax| = a *x|, and (iii) *x + y| ≤ *x| + *y|. Asymmetric norms have also been called quasi-norms. The conjugate function |x* := *−x| whenever x ∈ E is then also an asymmetric norm on E. The symmetrized function *·* = max{*·| , |·*} is a norm on E. An asymmetric norm *·| induces a T0 -quasi-pseudometric d·| by means of the formula d·| (x, y) = *x − y| where x, y ∈ E. If d·| is a bicomplete quasi-pseudometric on E, then (E, *·|) is called a biBanach space. Similarly, asymmetric locally convex spaces have been deﬁned [13]. Over the past years numerous important results from the theories of normed spaces and locally convex spaces were generalized to these asymmetric settings. Surprisingly, many of these investigations were motivated by problems in theoretical computer science. We have to refer the reader to the pertinent literature mentioned above and the references cited in these articles.

23The topology τ of a topological ordered space is called convex provided that τ ∪ τ is a subbase for τ (compare e.g., [103, p. 186] or [70, p. 28]). Each completely regularly ordered space possesses a convex topology.

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[137] S. Romaguera, Left K-completeness in quasi-metric spaces, Math. Nachr. 157 (1992), 15–23. MR1233044 (94i:54068) , An example of quasi-uniform convergence and quiet spaces, Questions Answers [138] Gen. Topology 13 (1995), no. 2, 169–171. MR1350235 (96i:54022) [139] , On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar. 73 (1996), no. 1–2, 159–178. MR1415928 (97h:54030) [140] S. Romaguera and S. Salbany, On bicomplete quasi-pseudometrizability, Topology Appl. 50 (1993), no. 3, 283–289. MR1227555 (94f:54065) , Dieudonn´ e complete bispaces, Studia Sci. Math. Hungar. 36 (2000), no. 3–4, 407– [141] 421. MR1798747 (2001j:54035) [142] S. Romaguera, M. Sanchis, and M. Tkachenko, Free paratopological groups, Topology Proc. 27 (2003), no. 2, 613–640. MR2078531 (2005f:22003) [143] S. Salbany, Bitopological spaces, compactiﬁcations and completions, Mathematical Monographs of the University of Cape Town, vol. 1, Department of Mathematics, University of Cape Town, Cape Town, 1974. MR0425919 (54 #13869) [144] M. Seyedin, Quasi-uniform spaces and topological homeomorphism groups, Canad. Math. Bull. 17 (1974), 97–98. MR0355982 (50 #8455) [145] J. L. Sieber and W. J. Pervin, Completeness in quasi-uniform spaces, Math. Ann. 158 (1965), 79–81. MR0172229 (30 #2449) [146] M. B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc. (2) 49 (1994), no. 2, 385–400. MR1260120 (95c:54045) [147] E. F. Steiner, The relation between quasi-proximities and topological spaces, Math. Ann. 155 (1964), 194–195. MR0163278 (29 #581) [148] Ph. S¨ underhauf, The Smyth-completion of a quasi-uniform space, Semantics of programming languages and model theory (Schloß Dagstuhl, 1991) (Droste M. and Y. Gurevich, eds.), Algebra, Logic and Applications, vol. 5, Gordon and Breach Science Publishers, Amsterdam, 1993, pp. 189–212. MR1253816 (94k:54056) , Quasi-uniform completeness in terms of Cauchy nets, Acta Math. Hungar. 69 [149] (1995), no. 1–2, 47–54. MR1339570 (96f:54038) [150] J. W. Tukey, Convergence and uniformity in topology, Annals of Mathematics Studies, vol. 2, Princeton University Press, Princeton, NJ, 1940. MR0002515 (2,67a) [151] G. Vidossich, Two remarks on A. Gleason’s factorization theorem, Bull. Amer. Math. Soc. 76 (1970), 370–371. MR0256365 (41 #1021) [152] A. J. Ward, On H-equivalence of uniformities: The Isbell–Smith problem, Paciﬁc J. Math. 22 (1967), 189–196. MR0215283 (35 #6125) [153] H. Weber, On lattices of uniformities, Order 24 (2007), no. 4, 249–276. MR2377916 [154] A. Weil, Sur les espaces a ` structure uniforme et sur la topologie g´ en´ erale, Hermann, Paris, 1937. [155] H. H. Wicke and J. M. Worrell, Jr., Topological completeness of ﬁrst countable Hausdorﬀ spaces. I, Fund. Math. 75 (1972), no. 3, 209–222. MR0309078 (46 #8189) [156] J. Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435–469. MR0296891 (45 #5950) Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa E-mail address: [email protected]

Contemporary Mathematics Volume 486, 2009

Approach Theory Robert Lowen and Christophe Van Olmen

Contents 1. Basic deﬁnitions 1.1. Gauges 1.2. Distances 1.3. Limit operators 2. Examples 2.1. Initially dense object 2.2. Spaces of measures 2.3. Function spaces 2.4. Hyperspaces 2.5. Functional analysis 3. Compactness 3.1. Measure of compactness 3.2. Measure of compactness 0 and Kuratowski–Mr´ owka 4. Approach frames 4.1. Deﬁnitions 4.2. Sobriety and spatiality 4.3. Regularity 4.4. Compactness 4.5. Normality 5. Hulls 5.1. Convergence approach spaces 5.2. The extensional topological hull of Ap 5.3. The cartesian closed topological hull of PrAp 5.4. The topological quasi-topos hull of PrAp and Ap 5.5. The cartesian closed hull of Ap References

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306 306 308 310 310 311 311 311 312 312 314 314 315 318 318 320 322 323 324 324 324 325 326 327 327 329

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There are many papers and a basic reference work [46] on approach theory and we would like to refer the interested reader to those (see the references) for more indepth information. In this survey we would like to focus our attention on three aspects of the theory. First, in sections 1–3 we give the basic concepts of (local) approach theory. We look at several examples paying particular attention to applications in functional analysis in section 2.5. As far as properties in approach spaces are concerned we restrict ourselves to highlighting the all important concept of compactness in section 3. Second, we look at the recent developments in the area of pointless approach theory, so-called approach frames, in section 4. Third, in section 5, we collect the various results on hulls of the category of approach spaces, which are somewhat scattered over the literature. The topological theory of approach spaces was ﬁrst introduced in [45] and it gives a setting in which both topological and metric spaces can be studied as objects on equal footing. Precisely this means it provides us with a category in which topological spaces (with continuous maps) and metric spaces (with contractions) can be embedded as full subcategories. The theory is such that in this new category the problem of the non-metrizability of general initial structures of metrizable topological spaces is solved. The typical example where it goes wrong in classical theory is products: the countable product of metrizable spaces is metrizable, but there is no canonical metric for the product topology, and for uncountable products there simply is no metric at all for the product topology. 1. Basic deﬁnitions 1.1. Gauges. [46, 47] Recall that a pseudo-quasi-metric on a set X (shortened to pq-metrics for the rest of the chapter) is a map d : X ×X → [0, ∞) satisfying d(x, x) = 0 for all x ∈ X, and the triangular inequality. A pseudo-metric (p-metric from now on) additionally satisﬁes d(x, y) = d(y, x) for all x and y in X. Approach spaces can be introduced in various ways, here we will start with a characterization using gauges, i.e., ideals of pseudo-quasi-metrics satisfying a saturation condition. Note that in the present work we do not suppose that pseudo-quasi-metrics are ﬁnite. Definition 1.1. An approach space is a pair (X, G) with G an ideal of pqmetrics that is locally saturated ideal, meaning that whenever e is a pq-metric such that ∀x ∈ X, ∀ > 0, ∀ω < ∞, ∃d ∈ G : e(x, ·) ∧ ω ≤ d(x, ·) + then e ∈ G. It is clear that this condition is local. A global counterpart is obtained by dropping the condition “for all x”. We say that an ideal G consisting of p-metrics is globally (or uniformly) saturated if, whenever e is a p-metric such that ∀ε > 0, ∀ω < ∞, ∃d ∈ G : e ∧ ω ≤ d + ε it follows that e ∈ G. Such an ideal is called a uniform gauge. A pair (X, G) where G is a uniform gauge is referred to as a uniform gauge space. Uniform gauge spaces were introduced in [59], although under a diﬀerent name. In this text we will not be dealing with the uniform aspects of approach theory but only with the local side of it.

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Definition 1.2. Let (X, GX ) and (Y, GY ) be approach spaces and let f : X → Y be a function, then f is called a contraction if ∀d ∈ GY : d ◦ (f × f ) ∈ GX . For practical reasons, one often works with a gauge basis instead of with the entire gauge. By deﬁnition this is nothing more than an ideal basis. We then say that such a basis H generates a gauge G if saturating H according to the saturation condition gives the entire collection G. Approach spaces together with contractions form a topological construct (see [1]) and we will denote this category as Ap. The relation among this category and several well-known other categories is depicted in the following diagram: c / UAp o ? _ CReg pMet _ _ r r < _ z z z z zz c zzz r r r zz zz z zz . zz c c / Ap o ? _ Top pqMet r

We mention the following salient facts about the above diagram. (1) All functors are concrete. (2) If (X, T ) is a topological space then it is embedded into Ap by associating with it the approach space (X, G(T )) where G(T ) is the gauge consisting of all pq-metrics which generate topologies coarser than T , i.e., G(T ) := {e pq-metric | Te ⊂ Td } (3) Top is embedded in Ap simultaneously bireﬂectively and bicoreﬂectively. If (X, G) is an approach space, then the Top coreﬂection is determined by the closure operator x ∈ A ⇐⇒ ∀ > 0, ∀d ∈ G, ∃y ∈ A : d(x, y) < . The reﬂection is slightly more cumbersome but also less interesting hence we refrain from giving an explicit description. (4) If (X, d) is a pq-metric space then it is embedded in Ap by associating with it the approach space (X, G(d)) where the gauge G(d) is generated by {d}. In this case this is simply the principal ideal G(d) := {e pq-metric | e ≤ d}. (5) pqMet (the category of pq-metric spaces and non-expansive maps) is bicoreﬂectively embedded in Ap however it is not stable under the formation of inﬁnite products. If (X, G) is an approach space, then the pqMet coreﬂection is determined by the pq-metric d(x, y) := sup e(x, y). e∈G

(6) The category UAp of so-called uniform approach spaces is the full subcategory of Ap with objects those spaces which have a gauge generated by p-metrics. Hence it is the epireﬂective hull of pMet in Ap. It is also easily seen that UAp is actually bireﬂective in Ap analogously to the way Creg is embedded in Top.

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(7) For the top row in the diagram the same facts hold as given in 2–4 above modulo the replacement of pq-metrics by p-metrics. In this uniﬁed setting where approach spaces, topological spaces and (pq)metric spaces can be considered as objects of the same type, there are objects which are at the same time in UAp and in Top and these are exactly the completely regular spaces. So with a mild abuse of notation we might write UAp ∩ Top = CReg. Similarly the “intersection in Ap” of pMet and Top is the category formed by coproducts of indiscrete spaces, and replacing pMet by pqMet, we get ﬁnitely generated spaces and limiting further to Met, we get the discrete spaces Dis. Let us now have a look at the product problem. We ﬁrst give the description of initial structures in Ap. Proposition 1.3. Given approach spaces (Xi , Gi )i∈I and the source (fi : X → (Xi , Gi ))i∈I in Ap, if, for all i ∈ I, Hi is a basis for Gi , then a basis for the initial gauge on X is given by H = {sup dk ◦ (fk × fk ) | K ∈ 2(I) , ∀k ∈ K : dk ∈ Hk }. k∈K

Let now (Xi , Ti )i∈I be a collection of topological spaces, where each Ti is generated by a pq-metric di . Then we obtain two diﬀerent collections of approach spaces (Xi , G(Ti ))i∈I and (Xi , G(di ))i∈I . If we now take the products of both collections then respectively we obtain (Xi , G(Ti )) = Xi , G( Ti ) i∈I

and

i∈I

i∈I

(Xi , G(di )) = Xi , G i∈I

i∈I

where G is the gauge generated by {sup dk ◦ (prk × prk ) | K ⊂ I ﬁnite} k∈K

from which it immediately followsthat the topological coreﬂection of ( i∈I Xi , G) is the product topological space ( i∈I Xi , i∈I Ti ). Hence any product of metrizable topological spaces can be endowed, maybe not with a metric, but with an approach structure, the underlying topology of which is the product topology. The following section sheds further light on this result and emphasizes its naturality. 1.2. Distances. [46] As is the case in topology, there are alternative characterizations of approach spaces which are frequently used. A natural concept in metric spaces is the notion of point-set distance given by δ(x, A) = inf d(x, a). a∈A

This concept can be generalised to approach spaces, thus extending the notion for metric spaces but also having a natural meaning for topological spaces. Definition 1.4. A distance on a set X is a function δ : X × 2X → [0, ∞] with the following axioms:

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(D1) (D2) (D3) (D4)

309

∀x ∈ X : δ(x, {x}) = 0, ∀x ∈ X : δ(x, ∅) = ∞, ∀x ∈ X, ∀A, B ∈ 2X : δ(x, A ∪ B) = min{δ(x, A), δ(x, B)}, ∀x ∈ X, ∀A ∈ 2X , ∀ ∈ [0, ∞] : δ(x, A) ≤ δ(x, A() ) + with A() = {x | δ(x, A) ≤ }.

The transition from a gauge to the associated distance is given by δ(x, A) = sup inf d(x, a) d∈G a∈A

and conversely from a distance to a gauge G = {d | ∀A ⊂ X : inf d(·, a) ≤ δ(·, A)}, a∈A

these associations being of course inverse to each other. In terms of distances, a function f : (X, δX ) → (Y, δY ) is a contraction if ∀x ∈ x, ∀A ⊂ X : δY (f (x), f (A)) ≤ δX (x, A). The embedding of Top in Ap becomes especially elegant when characterised with distances. For a topological space, we have that the associated distance is either 0 or ∞, more precisely 0 x ∈ A, δ(x, A) = ∞ otherwise. It is now also very easy to express the pq-metric coreﬂection: given an arbitrary approach space (X, δ) it is simply determined by the (pq)-metric dδ (x, y) = δ(x, {y}). The topological coreﬂection is given by the closure operator clδ (A) = {x ∈ X | δ(x, A) = 0} and we will denote this topology as Tδ . The topological reﬂection on the other hand is slightly more complicated, since we only have a pre-closure clrδ (A) = {x | δ(x, A) < ∞} which then has to be transformed in the usual way to a closure operator by taking its Top-reﬂection. Now let us return to the product problem for a moment. In view of the comments given in the foregoing section and the transition from gauges to distances, again given a collection of topological spaces (Xi , Ti )i∈I , whereeach Ti is generated by a pq-metric di , we see that the distance of the product i∈I (Xi , G(di )) = ( i∈I Xi , G) is given by δ(x, A) =

sup

inf sup dk (xk , ak ).

K⊂I, ﬁnite a∈A k∈K

In the case of a ﬁnite product it is the distance associated with the supremum metric (which in that case actually metrizes the product topology). For any inﬁnite product, countable or not, it is a genuine non-metric and non-topological distance which distancizes the product topology.

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1.3. Limit operators. [42, 46] We will also require a characterization of approach spaces with limit operators of ﬁlters. The intuition here is that a notion of limit operator should somehow express how far a point is from being a limit of a ﬁlter. This notion will be used to study completeness and compactness in Ap and we will also need it (and subsequent generalizations) to describe various hulls of Ap. Given a subset A ⊂ X, we will use the notation F(A) (resp. U(A)) for the collection of all ﬁlters (resp. ultraﬁlters) on X containing the set A, and x˙ for the principal ﬁlter {A ⊂ X | x ∈ A}. Definition 1.5. A function λ : F(X) → [0, ∞]X is called a limit operator if it satisﬁes (L1) ∀x ∈ X : λx(x) ˙ = 0. (L2) For any family (Fi )i∈I of ﬁlters on X, λ( i∈I Fi ) = supi∈I λFi . (L3) For any F ∈ F(X) and any selection of ﬁlters (S(x))x∈X on X, λ(D(S, F)) ≤ λF + sup λ(S(x))(x). x∈X

In condition (L3) D(S, F) stands for the so-called diagonal ﬁlter S(y). D(S, F) := F ∈F y∈F

A particularly important transition is the one between limits and distances: λF(x) = sup δ(x, A), A∈sec F

with sec F := {A ⊂ X | ∀F ∈ F : F ∩ A = ∅}, and conversely δ(x, A) =

inf

U∈U(A)

λU(x),

the transitions again being inverse to each other. A function f : (X, λX ) → (Y, λY ) is a contraction if ∀F ∈ F(X) : λY (stack(f (F))) ◦ f ≤ λF, where stack associates to each family A of subsets of a set X the family stack(A) := {B ⊆ X | ∃A ∈ A : A ⊆ B}. In an approach space X, a ﬁlter F is said to be Cauchy if inf x∈X λF(x) = 0. For metric spaces it is easily seen that this notion coincides with the usual notion of a Cauchy ﬁlter, and for topological spaces this simply means the ﬁlter is convergent. Furthermore, it allows us to build a completeness theory for uniform approach spaces which extends the completeness theory for p-metric spaces (see section refsec:approach frames on approach frames). Another notion involving ﬁlters which appears alongside the notion of limit operator is that of the adherence operator α, αF(x) = sup δ(x, A), A∈F

We will use this notion to describe compactness of approach spaces further on. 2. Examples We have already seen that both topological spaces and pq-metric spaces can be viewed as approach spaces. Natural examples of approach spaces turn up in measure spaces, function spaces and hyperspaces.

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2.1. Initially dense object. [46] A very important basic example of an approach space which is neither topological nor (pq)-metric is the following. Consider the set [0, ∞] equipped with the distance δP determined by δP (x, A) = (x − sup A) ∨ 0 for all A = ∅ (with the understanding that ∞−∞ = 0) and δP (x, ∅) = ∞ for all x. An investigation of this space shows that, it is pq-metric in nature in all ﬁnite points and topological in nature at ∞. The fundamental importance of this space stems from the fact that it is an initially dense object in Ap in precisely the same way as the two-point Sierpi´ nski space is for Top. Its underlying topology which we will denote by TP is the so-called lower semicontinuous topology {]α, ∞] | α ∈ [0, ∞[} ∪ {∅, [0, ∞]}. 2.2. Spaces of measures. [46] Let (X, T ) be a separable metrizable topological space and let P be the set of all probability measures on X. One of the most important and widely used structures on P is the so-called weak topology. We can introduce a gauge on P by taking as a basis all pq-metrics dG0 : P × P → [0, ∞] : (P, Q) → sup (P (G) − Q(G)) ∨ 0 G∈G

where G is a ﬁnite collection of open sets. This gauge determines what we can call the weak approach structure δw on P. This approach structure can also be obtained by other natural bases for the gauge. (1) With p-metrics determined by continuous functions. For any ﬁnite set H of continuous functions on X, we take H d1 : P × P → [0, ∞] : (P, Q) → sup f dP − f dQ . f ∈H Clearly this is a p-metric and {dH 1 | H ⊂ C(X, [0, 1]), ﬁnite} too generates the weak approach structure. (2) With p-metrics determined by uniformly continuous functions. Analogously, precisely the same collection as before but restricted to ﬁnite sets of uniformly continuous maps does the job. The topological coreﬂection is precisely the weak topology on P. Better yet, recall that the weak topology is the initial structure for the source (ωG : P → ([0, ∞], TP ) : P → P (G))G∈T . This is generalised in our situation: the “weak approach structure” is initial for the source (ωG : P → P : P → P (G))G∈T . Furthermore, we can easily derive a Portmanteau theorem similar to the classical one and we have that the pMet-coreﬂection of (P, δw ) is the total variation metric d(P, Q) = supB∈B |P (B) − Q(B)|, where B denotes the family of Borel sets. 2.3. Function spaces. [46] In the setting of function spaces, we obtain the so-called distance of S-convergence. Recall the classical concept: take a set X and a metric space (Y, d). Consider the function space Y X and a collection S of subsets of X which covers X and is closed under the formation of ﬁnite unions, then the collection of entourages W (A, ) = {(f, g) ∈ Y X × Y X | ∀x ∈ A : d(f (x), g(x)) < } with A ∈ S and > 0 determines the uniformity of uniform convergence on S-sets.

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However, rather than determining entourages we can deﬁne for any subset A of X the pseudometric dA : Y X × Y X → R : (f, g) → sup d(f (x), g(x)). x∈A

For a collection S as above, we then have that DS = {dA | A ∈ S} is closed under formation of ﬁnite suprema and thus a base for a gauge. The distance generated by this gauge is denoted δS . The topological coreﬂection is exactly the topology of S-convergence and the pseudometric coreﬂection is the uniform metric. Furthermore, for S the system of all ﬁnite subsets of X, we have that the source (evx : Y X → (Y, δd ))x∈X is initial and that the map (Y, δd ) → (Y X , δS ) : y → y˜, with y˜ the constant function with value y, is an embedding. Note ﬁnally that for S the collection of all ﬁnite subsets of X the topological coreﬂection of (Y X , δS ) is the topology of pointwise convergence. This topology is not metrizable in general, but here we have a natural notion of distance on it. 2.4. Hyperspaces. [49, 50, 78] In the setting of hyperspaces, we can look at the Hausdorﬀ metric and the Wijsman topology. Take a metric space (X, d) and let CL(X) be the set of all nonempty closed subsets of X. The Hausdorﬀ metric on CL(X) is deﬁned as hd (A, B) = supx∈X |δd (x, A) − δd (x, B)|. The Wijsman topology is deﬁned as the initial topology on CL(X) for the source (CL(X) → R+ : A → δd (x, A))x∈X . Now consider, for each ﬁnite subset F of X, the metric dF on CL(X) by dF (A, B) = sup |δd (x, A) − δd (x, B)|. x∈F

Then the collection of pseudometrics DWd = {dF | F ⊂X, F ﬁnite} is closed under formation of ﬁnite suprema and hence is the basis for a gauge. The distance obtained from this gauge is δWd : CL(X) × 2CL(X) → [0, ∞] : (A, A) →

sup

inf dF (A, B).

F ⊂X,ﬁnite B∈A

This is called the Wijsman distance. This is justiﬁed as the topological coreﬂection is exactly the Wijsman topology on CL(X). The metric coreﬂection on the other hand is exactly the Hausdorﬀ metric hd . Again we can show that the Wijsman distance is the initial structure on CL(X) for the source (CL(X) → R+ : A → δd (x, A))x∈X + where R is equipped with the usual Euclidean metric. We also have that the function (X, δd ) → (CL(X), δWd ) : x → {x} is an embedding. Moreover, classically we have that for a totally bounded metric d the Wijsman topology is metrized by the Hausdorﬀ metric. For approach spaces, this relation becomes even stronger: for a totally bounded metric space (X, d) we have δWd = δhd . 2.5. Functional analysis. [57, 81, 80, 79] Finally, we will look at an example in the setting of functional analysis. In order to do this, we need to introduce the notion of an approach vector space and of a locally convex approach space. For the ﬁrst notion, we recall prenorms. These are sub-additive functions ϕ : X → [0, ∞] such that (1) for all λ with |λ| ≤ 1 we have ϕ(λx) ≤ ϕ(x) (balanced),

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(2) for all x ∈ X and > 0 there exists δ > 0 such that for all |λ| ≤ δ we have ϕ(λx) ≤ (absorbing). An approach vector space then is a pair (X, N ) where X is a vector space and N an ideal of prenorms satisfying an adapted saturation condition: if a prenorm η is such that ∀ > 0, ∀ω < ∞, ∃µ ∈ N : η ∧ ω ≤ µ + then η ∈ N . Morphisms obviously are linear maps f : (X, NX ) → (Y, NY ) such that for all ϕ ∈ NY , ϕ ◦ f ∈ NX . For the notion of locally convex approach spaces, we add that the ideal of prenorms has a basis of convex prenorms. But we can also show that this is equivalent to another, more elegant deﬁnition, namely: a locally convex approach space is a pair (X, M), consisting of a vector space X and a Minkowski system M. This is a saturated ideal in the lattice of seminorms on X. Saturation here means that if η is a seminorm on X such that, for any > 0, there exists µ ∈ M such that η ≤ (1 + )µ, then η is in M. These characterizations relate to another way of describing approach spaces, namely approach systems, but we can also see a connection in terms of gauges: given a vector space X the approach space (X, G) is an approach vector space if and only if G has a basis of vector pseudometrics, i.e., translation invariant metrics d such that d(0, ·) is a prenorm. For locally convex approach spaces this is strengthened to seminorms. The category of locally convex approach spaces and linear contractions is denoted lcApVec. With the deﬁnition of these categories, we get several important categorical interactions with known categories and extensions of known theorems. For example, we know that in sNorm1 , the category of seminormed spaces with contractions as morphisms the projective tensor product functor −⊗X has a right adjoint for every X. For locally convex approach spaces, one can prove that this functor has a right adjoint if and only if X is a seminormed space (note that seminormed spaces form an obvious subcategory of lcApVec). To underline the relation with various important categories in functional analysis, consider the following commutative (!) categorical diagram. / lcApVec sNorm 4 4l VVVVVV 7 7m 1 WWWWWWW WWWWW VVVV 77 WWWWW44 VVVV 77 W44WWWW VVVV WWWWW VVVV 77 4 + 4 / lcTopVec 77 /* UAp 44 sNormTopVec pMet

7m l V ? C W 4 7 V W 4 V W 77 4 77 WWWWWW 4V4 VVVVV WWWWW44 V 77 77 V 4 V VVVV W4W4WWW 4 77 77 V 44 WWW44W+ VVVV*/ 77 77 4 44 CReg 44pMetTop 77 77 C 44 44 ? 77 77 44 44 77 77 44 4 77 7 44 77 / 44 ApVec 77 pMetVec 44 77 44 77 4 7 / TopVec pMetTopVec

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The downward arrows send spaces with a numerical structure to the underlying spaces with only a topological structure. Many of these arrows have adjoints. Using locally convex approach spaces, we also have excellent duality results. We will shortly look at the weak structure and the Mackey structure. Denote by K(X, M) the set of linear contractions on (X, M), i.e., linear functionals ϕ such that |ϕ| ∈ M. We deﬁne the weak approach structure as the seminorm saturation of the collection n

σ(M) := {sup |ϕi | | n ∈ N0 , ϕi ∈ K(X, M)}. i=1

This structure is a generalization of the classical notion: the weak structure of the underlying topology of a locally convex approach spaces and the underlying topology of the weak approach structure are the same and for topological locally convex approach spaces we have that the weak structure is again topological. Furthermore, the weak approach structure has the same seminorm coreﬂection as the original space and we have that K(X, M) = K(X, σ(M)), in fact it is the smallest Minkowski system on X with this property. Now, consider K a weak∗ -compact subset of K(X, M), then the assignment x → pK (x) = supϕ∈K |ϕ(x)| deﬁnes a seminorm on X. Since compactness is stable for ﬁnite unions, the collection of all pK forms an ideal basis. The Minkowski system generated by this collection is denoted τ (M) and is called the Mackey structure of (X, M). This notion extends the classical concept in several ways: the topological coreﬂection of the Mackey structure is equal to the Mackey topology of the coreﬂection of the original structure. Further the Mackey structure of a topological locally convex approach space is topological and hence equal to the Mackey topology. Finally, from the Hahn–Banach theorem, a seminorm equals its own Mackey structure, a numerical counterpart of the fact that the Mackey structure of a seminormable topology is invariant. Finally, τ (M) is the largest Minkowski system on X such that K(X, M) = K(X, τ (M)). 3. Compactness There are of course many topological and metric properties which have interesting extensions or counterparts in approach theory, such as lower and higher separation properties, connectedness and completeness. Since it would lead us too far to consider them all we will however restrict ourselves in this section to highlighting only the important notion of compactness. We will brieﬂy mention completeness in the section about approach frames. 3.1. Measure of compactness. [44] Definition 3.1. Given an approach space X its measure of compactness is deﬁned as µc (X) :=

sup

inf λU(x)

U∈U(X) x∈X

= sup

inf αF(x).

F ∈F(X) x∈X

An approach space X is called 0-compact if µc (X) = 0. For the important subcategories Top and pMet we ﬁnd the following result.

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Proposition 3.2. A topological approach space is 0-compact if and only if it is compact, and a pseudometric approach space is 0-compact if and only if it is totally bounded. In general, however, compactness of the topological coreﬂection is strictly stronger and we refer to this property as being compact. This follows from the fact that the adherence operator of a ﬁlter F takes value 0 in a point x only if the ﬁlter F has x as an adherence point in the topological coreﬂection. An evident example of this is (Q ∩ [0, 1], δdE ), the rationals in the unit interval with the Euclidean distance. One of the very useful properties of compactness in topology is that the continuous image of a compact space is again compact, here we ﬁnd an analogous property: Proposition 3.3. If X and Y are approach spaces and the function f : X → Y is a surjective contraction then µc (Y ) ≤ µc (X). As an immediate corollary, we see the property for 0-compactness. Corollary 3.4. If f : X → Y is a surjective contraction and X is 0-compact then Y is 0-compact. It was shown in [44] that the measure of compactness itself satisﬁes a general form of the Tychonoﬀ theorem. Theorem 3.5 (Tychonoﬀ). For a collection of nonempty approach spaces (Xj )j∈J the following formula holds: µc (

j∈J

Xj ) = sup µc (Xj ). j∈J

As a consequence of Threoem 3.5 a Tychonoﬀ theorem of course also holds for 0-compactness. Corollary 3.6 (Tychonoﬀ). The product of a family of nonempty approach spaces is 0-compact if and only if each member of the family is 0-compact. 3.2. Measure of compactness 0 and Kuratowski–Mr´ owka. [23] In what follows, we will use some concepts presented in Clementino, Giuli and Tholen [18] and hence we will brieﬂy recall these. Given a category with a factorization structure (E, M) that satisﬁes properties (F0)–(F2) and given an extra class F of closed morphisms satisfying stability conditions (F3)–(F5), one can describe a variety of topological notions such as compactness and Hausdorﬀ, and prove in a general categorical setting a number of classical topological results. The conditions in [18] are: (F0) M is a class of monomorphisms and E is a class of epimorphisms and both are closed under composition with isomorphisms. (F1) Every morphism f decomposes as f = m ◦ e with m ∈ M and e ∈ E. (F2) Every e ∈ E is orthogonal to every m ∈ M. That is, given any morphisms u and v such that m ◦ u = v ◦ e there exists a unique morphism w making

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the following diagram commutative. /· @ e w m · v /· ·

u

(F3) F contains all isomorphisms and is closed under composition. (F4) F ∩ M is stable under pullbacks. (F5) Whenever g ◦ f ∈ F and f ∈ E then g ∈ F. For the case of approach spaces, the factorization structure (E,M) considered consists of the pair with E the class of epimorphisms (i.e., surjective contractions) and M the class of extremal monomorphisms (i.e., embeddings). This factorization structure satisﬁes the aforementioned conditions (F0)–(F2). In order to deﬁne the class F, we will introduce closed expansions. Recall that if A ⊂ X then θA denotes the indicator of A, i.e., the [0, ∞]valued function which takes on the value 0 on A and ∞ outside of A. We also recall that given a function f : X → Y , µ ∈ [0, ∞]X and ν ∈ [0, ∞]Y the image of µ is deﬁned as f (µ)(y) := inf x∈f −1 (y) µ(x) and the preimage of ν is deﬁned as f −1 (ν) := ν ◦ f . The functions f −1 (ν) and f (µ) deﬁne a pair of adjoint mappings and so one is completely determined by the other via the relation which says that for all µ ∈ [0, ∞]X and ν ∈ [0, ∞]Y : ν ≤ f (µ) ⇔ f −1 (ν) ≤ µ. Given the deﬁnition of contractions in approach spaces, it is not surprising that a concept of closed map in Ap should involve a form of expansiveness, as opposed to contractiveness. Note that the characterization of contractions can also be written as f −1 (δf (A) ) ≤ δA or, equivalently δf (A) ≤ f (δA ). Definition 3.7. Given approach spaces X and Y , a function f : X → Y is called closed-expansive (or a closed expansion) if for all A ⊂ X we have f (δA ) ≤ δf (A) . Proposition 3.8. If X and Y are topological approach spaces then a map f : X → Y is closed-expansive if and only if it is closed (in the topological sense). With this in mind, we take F the class of all closed-expansive contractions. These are functions f : X → Y such that δf (A) = f (δA ) for all A ⊂ X. That isomorphisms are closed expansions and that closed expansions are stable under composition is evident from the deﬁnition. In order to show that F ∩ M is stable under pullbacks, the following characterization of injective closed-expansive contractions is useful. Proposition 3.9. Given approach spaces X and Y and a function f : X → Y the following are equivalent: (1) f is an injective closed-expansive contraction, (2) f is an embedding such that δf (X) = θf (X) . It can easily be veriﬁed that the class F satisﬁes the extra conditions (F3)–(F5). Definition 3.10. A contraction f : X → Y is called a proper contraction if for each approach space Z the map f × 1Z : X × Z → Y × Z is a closed expansion. This deﬁnition coincides with what in [18] are called the F-proper maps. From now on, the class of proper contractions will be denoted by F ∗ . From the deﬁnition,

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it is immediately clear that F ∗ ⊂ F. The following two results are also immediate consequences of the general results proved in [18]. Proposition 3.11. A contraction f : A → C is a proper contraction if and only if it belongs stably to F, i.e., whenever P

f

g

g

A

/B

f

/C

is a pullback diagram, f ∈ F. Proposition 3.12. The class of proper contractions fulﬁlls the following stability properties: (1) Proper contractions are stable under composition. (2) Closed expansive embeddings are proper contractions. (3) F ∗ is the largest pullback-stable subclass of F. (4) If g ◦ f is a proper contraction and g is an injective contraction, then f is a proper contraction. In order to prove the Kuratowski–Mr´ owka theorem, we need to know that the 0-compactness notion in Ap coincides with the compactness notion deﬁned in [18]. The deﬁnition there is that an object X is F-compact if and only if the unique morphism to the terminal object is F-proper. Proposition 3.13. If for a one-point space P the unique morphism π : X → P is a proper contraction then X is 0-compact. Proof. We need to show that an arbitrary ultraﬁlter U on X has inf λU(x) = 0.

x∈X

In order to do this we consider the following ultraﬁlter spaces: take ω ∈ X, let XU := X ∪ {ω} and put Uω the ultraﬁlter on XU generated by U. The following deﬁnes a topology on XU ; the only convergent ultraﬁlters are the principal ultraﬁlters, which converge to their deﬁning points and the ﬁlter Uω which converges to ω. The approach space generated by this topology has as limit operator (on ultraﬁlters) 0 (V = stack x and x ∈ X) or (V = Uω and x = ω), λU V(x) := ∞ all other cases, and as distance

δ U (x, A) :=

0 ∞

x ∈ A or (A ∈ Uω and x = ω), all other cases.

Considering the diagram below, where i is the evident isomorphism: pr2

/ XU : uu u u π×!XU u uu uu i P × XU X × XU

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since π is proper, π×!XU is closed and hence so is pr2 . Thus (U)

(U) (ω) 2 (∆)

pr2 (δ∆ )(ω) ≤ δpr

= 0,

where ∆ = {(x, x) | x ∈ X}. (U) With some calculations we have pr2 (δ∆ )(ω) = inf x∈X λU(x) and so the desired result follows. Proposition 3.14. For approach spaces X and Y and f : X → Y the following are equivalent: (1) f is a proper contraction, (2) f is a closed-expansive contraction and for each y ∈ Y , f −1 (y) is 0compact, (3) for each U ∈ U(X) : f (λU) = λf (U). Proposition 3.15. If X and Y are topological approach spaces then a map f : X → Y is a proper contraction if and only if it is proper in the topological sense. Theorem 3.16. An approach space X is 0-compact if and only if for any onepoint space P the unique morphism π : X → P is a proper contraction. Proof. Since π is always a closed-expansive contraction, the result follows from Propositions 3.13 and 3.14. Finally we obtain the desired characterization. Theorem 3.17 (Kuratowski–Mr´ owka). An approach space X is 0-compact if and only if for any approach space Z the projection prZ : X × Z → Z is a closed expansion. Proof. Take P the one-point object and consider the following pullback diagram. prZ /Z X ×Z prX

X

π

/P

π

4. Approach frames 4.1. Deﬁnitions. [7, 83] A pointfree counterpart of approach spaces, approach frames, was introduced in 1999 by B. Banaschewski in order to describe a notion of sobriety for approach spaces. This notion turns out to be a generalization of the topological notion of sobriety and has interesting connections with completeness, especially for approach spaces. Furthermore, with this “approach” pointfree setting, we ﬁnd an interaction between frames and approach frames analogous to the one between topological spaces and approach spaces. In the topological case the lattice of open sets stands as model for the deﬁnition of a frame. For approach spaces it is an abstraction of the notion of a regular function frame which we have to seek. Regular function frames are just another characterization of approach spaces. A regular function frame is a set of functions from X to [0, ∞] that satisﬁes the following conditions:

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(R1) ∀S ⊂ R : S ∈ R, (R2) ∀µ, ν ∈ R : µ ∧ ν ∈ R, (R3) ∀µ ∈ R, ∀α ∈ [0, ∞] : µ + α ∈ R, (R4) ∀µ ∈ R, ∀α ∈ [0, ∞] : (µ − α) ∨ 0 ∈ R. Given an approach space, the regular function frame can be characterised as the set of all contractions from the approach space to the initially dense object P. Also, the distance functionals δ(·, A) are given by sup{µ | µ ∈ R, µ|A = 0}. On the algebraic side, this set of functions is a frame with additional operations given by the addition and subtraction of constants. Hence, as deﬁnition for an approach frame we have the following. Definition 4.1. An approach frame L is a frame with two additional families of unary operations, the addition by λ (notation Aλ ) and the subtraction by λ (notation Sλ ) for each λ ∈ [0, ∞]. These operations satisfy all identities valid for addition and truncated subtraction on the frame [0, ∞]. As usual in frame theory we denote bottom and top element by ⊥ and + respectively. From the deﬁnition, we can immediately conclude that there is a special role for the elements Aλ ⊥. So we will use the notation λ for them. We also see that the set [0, ∞] with the obvious approach frame structure is the simplest non-trivial approach frame and we will denote it by J. The morphisms between approach frames are frame homomorphisms (i.e., commuting with arbitrary joins and ﬁnite meets, hence also preserving bottom and top element) which moreover commute with the addition and subtraction operations. This deﬁnes the category of approach frames, denoted AFrm. It can be shown that this category is complete and co-complete, and actually monadic over Set (see [1] for details). Furthermore, we can obtain a complete set of conditions on addition and subtraction operations which axiomatically determines the precise structure of an approach frame. ax(a) Aα ◦ Aβ = Aα+β , ax(b) Sα Aα a = a for all α < ∞, ax(c) S∞ a = ⊥, ax(d) Aα Sα a = a ∨ Aα ⊥, ax(e) A α (a ∨ b) = Aα a ∨ Aα b, ax(f) i∈I Aαi a = Asupi∈I αi a, I = ∅. Here ax(b), ax(d) and ax(e) give us that the addition and subtraction operations are order-preserving (even commuting with non-empty ﬁnite joins) and ax(b) and ax(d) then moreover deﬁne a Galois adjunction between addition and subtraction. It is clear that we have a functor R from Ap to AFrm which sends an approach space X to its regular function frame RX. This functor is contravariant since a contraction f : X → Y is a map for which the composition ϕ ◦ f , with a regular function ϕ of Y , is a regular function of X. This is analogous to the situation with frames and topological spaces, where a topological space is mapped to its lattice of open sets and continuous functions are mapped to the inverse mapping on open sets. The adjoint, called the spectrum, is obtained as follows. First we map an approach frame L to the set ΣL of all approach frame homomorphisms from L to J. Next, we consider the approach structure (on the set of points we have

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which is the set of all obtained) determined by the regular function frame L, a ˆ : ΣL → [0, ∞] : ξ → ξ(a) with a ∈ L. It is easy to see that this is indeed a regular function frame. In fact we have that a ˆ ∧ ˆb = a ∧ b, a ˆ+α = A α a and analogous results hold for the supremum and subtraction. For approach frame homomorphisms h : L → M , we ﬁnd a contraction Σh : ΣM → ΣL : ξ → ξ ◦ h. With these two functors deﬁned, we can deﬁne natural transformations L : L → ˜ with x ˜ : RX → J : f → f (x) the RΣL : a → a ˆ and ηX : X → ΣRX : x → x evaluation in x. That L is an approach frame homomorphism is implied by the ˆ determines an approach structure. arguments used to show that L Theorem 4.2. Σ and R are adjoint on the right. 4.2. Sobriety and spatiality. [7, 83] The natural transformations also allow us to deﬁne the notions of spatiality for approach frames and sobriety for approach spaces. Definition 4.3. An approach frame L is spatial if L is an isomorphism. An approach space X is sober if ηX is an isomorphism. Note that it is easily seen that spatiality is equivalent to L being injective. In order to make the study of the notion of sobriety more intuitive, we will introduce an alternative view on the spectrum that is easier to work with. An approach frame homomorphism ξ : L → J is completely determined by ξ ∗ (0) = {a ∈ L | ξ(a) = 0}. Elements of this type are approach prime elements: i.e., prime elements a, in the sense that if b ∧ c = a then either b = a or c = a, which moreover satisfy the condition that if λ ≤ a, then λ = 0. Given an approach prime element a this determines an approach frame homomorphism ξa : L → J : x → inf{λ | x ≤ Aλ a} and we have that ξa∗ (0) = a and ξξ∗ (0) = ξ. This alternative description of the spectrum allows for a nicer characterization of sobriety, since the approach prime element associated with x ˜ is precisely δ{x} . Hence we have the following result. Lemma 4.4. An approach space X is sober if and only if every approach prime element of RX is of the form δ{x} for a unique x. As sobriety for approach spaces is deﬁned in an analogous fashion to the classical notion of sobriety, one would expect to ﬁnd some analogous results. Indeed, we ﬁnd that for any approach frame L the spectrum ΣL is sober and that sobriety is a reﬂective property in Ap. It is natural to wonder whether there is a relation between sobriety in the AFrm-context and the classical notion of sobriety. The next proposition sheds light on this relation. Proposition 4.5. If an approach space X is sober then its topological coreﬂection is sober and for topological approach spaces, sobriety is equivalent to sobriety of the underlying topological space. Proof. If we take A closed, join-irreducible in the lattice of closed sets, we ﬁnd that δA is approach prime in RX. By sobriety of X there exists a unique x ∈ X

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such that δA = δ{x} and hence A = {x}. For the case of a topological approach space, an approach prime element f ∈ RX is in fact of the form δ{f =0} and then this set must necessarily also be join-irreducible. In fact, we have an even stronger relation between the two notions of sobriety: Corollary 4.6. The sobriﬁcation of a topological approach space is again topological. However, the converse is not true as the example (]0, 1], δE ) shows. From [77] we recall that an approach space X is complemented if and only if for every pair of points x, y and every ﬁlter F on X we have that δ(x, {y}) ≤ λF(x) + λF(y). With this deﬁnition we ﬁnd a ﬁrst direct connection to completeness. Proposition 4.7. A sober, complemented approach space is complete. Proof. This follows from the fact that the so-called supertight maps, maps ϕ : X → [0, ∞] with (1) ϕ : X → ([0, ∞], δP ) is a contraction, (2) inf x∈X ϕ(x) = 0, (3) ∀x, y ∈ X : δ(x, {y}) ≤ ϕ(x) + ϕ(y), are approach prime elements. Obviously limit operators of Cauchy ﬁlters are supertight in complemented spaces, hence approach prime and thus converging to a unique point x by sobriety. Proposition 4.8. For a T0 complemented approach space sobriety is equivalent to completeness. Proof. In a complete T0 complemented approach space, the limit operator of a Cauchy ﬁlter is exactly of the form δ{x} , so every approach prime element has this form and hence we have sobriety. Proposition 4.9. If X is uniform then so is its sober reﬂection and the sobriﬁcation of a complemented approach space is complemented. Corollary 4.10. For X a T0 complemented (uniform or pseudometric) approach space, ΣRX is isomorphic to the completion of X in the category of complemented (uniform or pseudometric) approach spaces. To further enhance the naturality of the introduced notion of sobriety, we mention that Sob is ﬁrmly embedded in, Ap0 , the category of T0 approach spaces, with regard to the class U of epimorphic embeddings. Let us ﬁrst recall the notion of ﬁrmness, introduced in [16]. Definition 4.11. Let X be a category and U a class of morphisms of X, which is closed under composition and closed under composition with isomorphisms. The class U is said to be ﬁrm if there exists a U-reﬂective subcategory S of X, such that U = L(S), with L(S) = {f ∈ Mor X | Sf is an isomorphism} and where S denotes the reﬂection functor. Then S is called ﬁrmly U-reﬂective in X.ﬁrmly U-reﬂective subcategory We will ﬁrst need to study the class U of epimorphic embeddings in Ap0 .

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Proposition 4.12. Let X, Y T0 be approach spaces and let f : X → Y be a contraction. (1) f is an epimorphism in Ap0 if and only if Rf is injective, (2) f is an embedding in Ap0 if and only if Rf is surjective. Using this characterization, we can then easily prove the proposition. Proposition 4.13. Sob is U-reﬂective in Ap0 . The next result shows that the sobriﬁcation of a T0 approach space can be considered as a unique completion. Theorem 4.14. A contraction f : X → Y is an epimorphic embedding in Ap0 if and only if ΣRf is an isomorphism. This implies that U = L(Sob). Combined with previous proposition, this gives us the main result of this section. Theorem 4.15. Sob is ﬁrmly U-reﬂective in Ap0 . Restrictions of this result can give us some classical results, by using the following proposition. Proposition 4.16. Let X be a full subconstruct of Ap0 . Suppose that ηX (X) ⊂ X, where ηX denotes the restriction of the functor η : Ap0 → Sob to X. Then we have that X ∩ Sob is ﬁrmly U ∩ Mor X-reﬂective in X. We will use ﬁrmly reﬂective in the sense of ﬁrmly reﬂective with regard to the epimorphic embeddings.ﬁrmly relective category Proposition 4.17. Sober topological spaces are ﬁrmly reﬂective in Top0 and complete complemented (uniform or pseudometric) approach spaces are ﬁrmly reﬂective in the category of complemented (uniform or pseudometric) T0 approach spaces. 4.3. Regularity. [8, 83] There exist interesting separation axioms on approach frames. The deﬁnitions are inspired mainly by their counterparts in frames and moreover they correspond well with known counterparts in approach spaces. First, let us have a short look at regularity. We say a frame is regular if every element a is the join of all elements rather below a. The relation rather below (notation ≺) in frames is deﬁned by a ≺ b ⇐⇒ b ∨ a∗ = + with a∗ = {c | c ∧ a = ⊥} the pseudocomplement of a. We see that the top element plays an important role in this deﬁnition. In approach frames this role will be taken over by a ﬁlter, namely NL := {a | ∃λ > 0 : λ ≤ a}. Then we say that in an approach frame L the element a is θ-rather below b, denoted a ≺θ b if a ≤ b and Sθ a∗ ∨ b ∈ NL . Definition 4.18. An approach frame is called pre-regular if every element a is the join of all elements rather below a and it is called regular if for every element a and every λ ≥ 0 we have a ≤ Aλ {b | b ≺λ a}.

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It is clear that regular implies pre-regular and the two notions are not equivalent. With θ equal to 0, we see a clear generalization of the rather below notion of frames, however we need to introduce the extra demand that a ≤ b which is automatic in the frame case. Note further that for a ∈ NL we have that all elements smaller than a are θ-rather below a for all θ. For approach spaces this notion of regularity can be simpliﬁed in the sense that for any θ ∈ [0, ∞], in order to have f ≤ {g | g ≺θ f } + θ for all f ∈ RX it suﬃces that δA ≤ {δB | δB ≺θ δA } + θ for all A ⊂ X. Since knowledge of the functions of the form δA with A ⊂ X suﬃces to characterise regularity, this provides a means to see that the formerly deﬁned notion of regularity for approach spaces is in fact equivalent with saying that RX is regular. We recall that an approach space is said to be regular if for every ﬁlter F we have λF ≤ λF () + with F () the ﬁlter generated by {F () | F ∈ F}. For the notion of pre-regularity in approach frames, we also have a known counterpart in approach spaces. It is equivalent to saying that every ﬁlter F satisﬁes λF = λF ∗ where F ∗ = {F () | > 0, F ∈ F}. This immediately gives us an example of a pre-regular, non-regular approach frame, namely the regular function frame of (R, δd ) with |x − y| if x ≤ y, d(x, y) = 2|x − y| if y ≤ x. On the pointfree side, analogously to the situation in frames, we ﬁnd that the category of (pre-)regular approach frames is monocoreﬂective in AFrm and that they are complete and co-complete categories. 4.4. Compactness. [6, 83] Recall that a frame L is said to be compact if for all A ⊂ L with A = + there exists a ﬁnite subset B of A such that B = +. Evidently we have that the frame of open sets of a topology is compact if and only if the topology is compact. We could express this concept also for approach frames, but it would not give us the desired property. Rather we will consider another concept which is related to the concept of compactness in approach spaces. Definition 4.19. I ⊂ L is a zero-ideal if I is a lattice ideal in L and λ ∈ I for all λ > 0. An approach frame L is said to be compact if for all zero-ideals I we have that λ ≤ I for all λ > 0. Note that in frames we can also deﬁne compactness using ideals. A frame is compact if for all ideals I we have that I = + implies that + ∈ I. So again we see that NL , and in particular the elements λ with λ > 0, take over the role of the top element. Recall that an approach space X is called 0-compact if for all ﬁlters F we have that inf x∈X αF(x) = 0. Zero-ideals are the right concept to generalise the adherence operator of a ﬁlter and using this, we ﬁnd the equivalence between compactness of X and compactness of RX. We also ﬁnd some results that are familiar from frame theory (and topology), namely that a sub-approach frame of a compact approach frame is compact, a counterpart that topologies coarser than a compact topology are compact and that

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a closed sublocale of a compact approach frame L is compact. This last fact is a triviality in frames, but this is no longer the case here. 4.5. Normality. [83] As a ﬁnal notion, we will look at normality for approach frames and hence also approach spaces. Recall that a frame L is normal if and only if ∀a, b ∈ L : a ∨ b = +, ∃u, v ∈ L : u ∧ v = ⊥, a ∨ u = b ∨ v = +. We have that the frame of open subsets of a topology is normal if and only if the topology is normal. Now to introduce this notion for approach frames, we again replace the top element by NL , obtaining the following deﬁnition: Definition 4.20. An approach frame L is said to be normal if ∀a, b ∈ L : a ∨ b ∈ NL ⇒ ∃u, v ∈ L : u ∧ v = ⊥, u ∨ a, v ∨ b ∈ NL . For approach spaces, normality is then deﬁned as normality of the associated regular function frame of the space. As a ﬁrst indication that this notion is interesting we ﬁnd that a frame is normal if and only if its embedding in approach frames is normal. This then implies that for topological spaces X the approach space notion of normality is equivalent with the topological notion. In general however, normality of an approach space does not imply normality of the topological coreﬂection. For arbitrary approach spaces we can simplify the condition for normality, and this is possible using the following proposition. Proposition 4.21. For any approach space X, RX is normal if the normality condition is satisﬁed for elements of type δA ∈ RX. As a consequence, for approach spaces, we can express normality as follows: Corollary 4.22. An approach space X is normal if and only if for all A, B ⊂ X for which there exists and > 0 such that A() ∩ B () = ∅ we have that ∃C ⊂ X, ∃δ > 0 : C (δ) ∩ A(δ) = ∅ and (X \ C)(δ) ∩ B (δ) = ∅. With this notion we can for instance show the desired results that a pseudometric approach space is normal and that a compact pre-regular approach space is normal. 5. Hulls In this ﬁnal section, we will look at hulls of the category Ap. Speciﬁcally, we will look at the extensional topological hull ETH(Ap), the quasi-topos (or topological universe) hull QTH(Ap) and the cartesian closed topological hull CCTH(Ap). 5.1. Convergence approach spaces. [42] In order to deﬁne all these hulls, we will use the category of convergence approach spaces, denoted CAp. Definition 5.1. A function λ : F(X) → [0, ∞]X is called a convergence-approach limit if it satisﬁes the conditions (L1) ∀x : λx(x) ˙ = 0,

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(CL) ∀F, G ∈ F(X) : λ(F ∩ G) = λF ∨ λG. The pair (X, λ) is called a convergence approach space. Morphisms between convergence approach spaces are deﬁned in the same way as contractions in approach spaces. It can easily be seen that Ap is a full subcategory of the topological construct CAp and the existence of all the hulls described above then follows from the following results shown in [43] as hulls are supercategories with some desired property and quasi-topoi have all these properties. Theorem 5.2. Ap is ﬁnally dense in CAp. Theorem 5.3. CAp is a topological quasi-topos. 5.2. The extensional topological hull of Ap. [43] Recall that we need to ﬁnd a supercategory of Ap in which so-called #-objects exist. These are one-point extensions Y # = Y ∪ {∞Y } of objects Y such that for any subobject A of X and any morphism f : A → Y we have an extension f # : X → Y # such that f # (X \ A) = {∞Y }. It is clear that Ap itself does not have this property, just as Top does not. Definition 5.4. A pair (X, λ) is called a pre-approach space if λ : F(X) → [0, ∞]X (called a pre-limit operator) satisﬁes properties (L1) and (L3) of Deﬁnition 1.5. The category of pre-approach spaces, denoted PrAp, has as objects pre-approach spaces and morphisms deﬁned in the same way as before, hence we also call these contractions. Alternatively, we can deﬁne this type of space also by means of a pre-distance δ, a distance function which satisﬁes (D1), (D2) and (D3) of Deﬁntion 1.4. It can be shown that the same formulas that provide the transformation of the limit operator into the distance and vice versa hold for pre-limit operators and pre-distances. Theorem 5.5. PrAp is extensional. Further, one can show that PrAp is a bireﬂective subcategory of CAp and it evidently contains Ap. Recall that PrTop is the extensional topological hull of Top. One way of seeing this was given by Herrlich [36] who showed that the one-point extension (in PrTop) of the Sierpi´ nski space in Top is an initially dense object in PrTop. An analogous situation presents itself in the approach case. Let P# := P ∪ {p} and let λ# P be determined by λP (F|X )(y) if y ∈ X, # λP (F)(y) = 0 if y = p. Equivalently the structure of this space is determined by the pre-distance δP# which extends δP and fulﬁlls δP# (x, A) := 0 if p ∈ {x} ∪ A and A = ∅.

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Theorem 5.6. ETH(Ap) = PrAp. Proof. This is done by showing that (P# , δP# ) is initially dense in PrAp. This gives us the desired result since PrAp is extensional and Ap is ﬁnally dense in PrAp by [36]. 5.3. The cartesian closed topological hull of PrAp. [43] We can deﬁne a suitable cartesian closed topological category in the following way. See also Bourdaud [15]. Definition 5.7. The structure Hom(X, Y ) with X, Y ∈ |CAp| is deﬁned as follows: given Ψ ∈ F(Hom(X, Y )) and f ∈ Hom(X, Y ), take L(Ψ, f ) = {α ∈ [0, ∞] | ∀F ∈ F(X) : λY (Ψ(F)) ◦ f ≤ λX F ∨ α} with Ψ(F) = {{g(y) | g ∈ ψ, y ∈ F } | ψ ∈ Ψ, F ∈ F}. The limit of Ψ in f is then inf L(Ψ, f ). This limit can be shown to satisfy the necessary properties to make (Hom(X, Y ), λ) into a CAp-space. Definition 5.8. Let C(P# ) be the full subcategory of CAp with objects those spaces X which carry the initial structure of the source f : X → Hom(Hom(X, P# ), P# ) : x → (f → f (x)). It can be shown that C(P# ) is bireﬂective in CAp. Furthermore, we have that P ∈ | C(P# )|. This combined with the fact that PrAp is bireﬂective in CAp and that P# is initially dense in PrAp gives us #

Proposition 5.9. PrAp is a subcategory of C(P# ) closed under formation of ﬁnite products in C(P# ). In fact, we have an even better result: Proposition 5.10. C(P# ) is the cartesian closed hull of PrAp. Proof. Since Ap is ﬁnally dense in CAp, PrAP is ﬁnally dense in C(P# ). Further, because Hom(−, P# ) : CAp → CAP transforms ﬁnal epi-sinks into initial sources, we have from the same result that powers of PrAp-objects are initially dense in C(P# ). However, a more elegant description of this category is possible: Definition 5.11. We deﬁne and denote by PsAp the full subcategory of CAp the objects (X, λ) of which satisfy the following supplementary condition (PsAL)

∀F ∈ F(X) : λF = sup λU U∈U(F )

where U(F) is the collection of all ultraﬁlters ﬁner than F. An object in PsAp is called a pseudo-approach space and λ is called a pseudo-limit operator. Proposition 5.12. If (X, λX ) ∈ |CAp| and (Y, λY ) ∈ |PsAp| then Hom(X, Y ) equipped with its natural limit is a PsAp-object. Theorem 5.13. C(P# ) = PsAp. Proof. Using the previous proposition and the fact that PsAp is bireﬂective in CAp, we have that C(P# ) is a subcategory of PsAp. The converse follows by calculation. Corollary 5.14. PsAp is the cartesian closed topological hull of PrAp.

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5.4. The topological quasi-topos hull of PrAp and Ap. [42, 43] Proposition 5.15. PsAp is extensional. Since PsAp is extensional and the cartesian closed hull of PrAp, we immediately have that Theorem 5.16. PsAp is the quasi-topos hull of PrAp. Theorem 5.17. QTH(Ap) = PsAp. Proof. This follows from the fact that Ap is ﬁnally dense in PsAp and PrAp is the extensional hull of Ap. Combining this with the previous theorem, we have that PsAp is the quasi-topos hull of Ap. 5.5. The cartesian closed hull of Ap. [60, 66] We will identify the cartesian closed hull of Ap with a subcategory of PsAp. Recall that Ap is bireﬂective in PsAp. We will denote the reﬂection of (X, λ) ¯ in Ap as (X, λ). Definition 5.18. Given (X, λ) ∈ PsAp, we deﬁne F˙ ρ := {y ∈ X | ∃x ∈ F : δλ¯ (x, {y}) ≤ ρ}. Given (X, λ) ∈ PsAp and ﬁlters F and G on X, we deﬁne dX : F(X) × F(X) → [0, ∞] : (F, G) → dX (F, G) := inf{ρ ≥ 0 | G˙ ρ ⊂ F}, where G˙ ρ := stack({G˙ ρ | G ∈ G}). Proposition 5.19. Given (X, λ) ∈ PsAp, (F(X), dX ) is an extended pseudoquasi-metric space. Given an approach space (X, λ), we know that the function λF : X → P is a contraction for every F ∈ F(X). However, with the deﬁnition of a pq-metric on F(X) we can also consider the function λ in two variables and consider contraction or continuity properties of this function of two variables. Definition 5.20. Deﬁne EpiAp to be the full subconstruct of PsAp whose objects (X, λ) satisfy the following condition: (C)

λ : (U(X), TdX ) × (X, Tλ¯ ) → ([0, ∞], TP ) is a continuous map.

The following illustrates why we could restrict ourselves to ultraﬁlters (as usual) without causing any diﬃculties. Proposition 5.21. Given (X, λ) ∈ PsAp, the following are equivalent: (1) (X, λ) ∈ EpiAp. (2) (X, λ) satisﬁes the following condition: (C )

λ : (F(X), TdX ) × (X, Tλ¯ ) → ([0, ∞], TP ) is a continuous map. With this in mind, we go over to the main result.

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Theorem 5.22. CCTH(Ap) = EpiAp Proof. This is done in several steps. Ap ⊂ EpiAp. This involves showing that for an approach space (X, λ) the function λ(−)(x) : F(X) → [0, ∞] is a contraction for all x ∈ X and using this fact in a construction of an approach structure on F(X) × X such that λ : F(X) × X → P actually becomes a contraction. Since the topological coreﬂection of this approach structure gives TdX × Tλ , we conclude that Ap ⊂ EpiAp. EpiAp is a cartesian closed topological construct. First, one has to show that EpiAp is bireﬂective in PsAp by proving that it is initially closed. Then one has to show that EpiAp is closed under formation of power-objects in PsAp. Better yet, we actually have that Hom(X, Y ) with X ∈ |PsAp| and Y ∈ |EpiAp| is an EpiAp-object. Hence we have that EpiAp is cartesian closed. The necessary density arguments are satisﬁed. Namely, we have that Ap is ﬁnally dense in EpiAp. Furthermore, one can show that for an EpiAp-object X the morphism j : X → Hom(Hom(X, P), P) : x → j(x) with j(x)(f ) = f (x) is an initial contraction. Finally, we also need initial density of H := {Hom(X, Y ) | X, Y ∈ Ap} in EpiAp. Combining the initiality of j and the fact that the functor [−, P] : EpiAp → EpiAp transforms ﬁnal epi-sinks into initial sources (see [37, Lemma 6]) with the ﬁnal density of Ap in EpiAp, we have the desired density of H. Finally, we want to make some remarks concerning the hulls of Top and their relation with the hulls of Ap. Speciﬁcally, we look at EpiTop = CCTH(Top) and EpiAp. To this end, we ﬁrst recall some facts regarding EpiTop from G. Bourdaud ([14]). Definition 5.23. Let (X, q) be a pseudotopological space. We denote its Topbireﬂection by (X, q¯) and deﬁne the point-operator (with respect to (X, q)) as •

: 2X → 2X : A → A• := {x ∈ X | clq¯ ({x}) ∩ A = ∅}.

Definition 5.24. A pseudotopological space X is called an Antoine space or epi-topological space if and only if it satisﬁes the following conditions (where F • is the ﬁlter generated by {F • | F ∈ F}): (1) ∀F ∈ F(X) : lim F is closed in (X, q¯) (closed-domainedness), (2) ∀F ∈ F(X) : lim F = lim F • (point-regularity). The full subconstruct of PsTop consisting of Antoine spaces is denoted by EpiTop and it was shown by work of A. Machado ([62]) and G. Bourdaud ([14]) that EpiTop = CCTH(Top). Proposition 5.25. Let (X, q) ∈ PsTop, then (X, λq¯) = (X, λ¯q ). Proposition 5.26. EpiAp ∩ PsTop = EpiTop. Proposition 5.27. EpiTop is both bireﬂective and bicoreﬂective in EpiAp.

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Finally, the foregoing results and analogous results for the other hulls are combined in the following diagram: Conv O

r c

r

/ CAp O r

r

PsTop c O dII II r r I II r II c EpiTop PrTop O : uu u u u r uu r uu r Top c

/ PsAp O cHH r HH r r HHH HH c ' ' PrAp EpiAp O v; v v r vv vv r v v / Ap

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University of Antwerp, Department of Mathematics and Computer Science, 2020 Antwerp, Belgium E-mail address: [email protected] University of Antwerp, Department of Mathematics and Computer Science, 2020 Antwerp, Belgium E-mail address: [email protected]

Contemporary Mathematics Volume 486, 2009

Semiuniform Convergence Spaces and Filter Spaces Gerhard Preuß Abstract. Semiuniform convergence spaces are mainly studied in the realm of Convenient Topology, a new foundation of Topology, in which several deﬁciencies of topological and uniform spaces are remedied. They generalize not only uniform convergence structures such as uniform structures and uniform limit space structures but also convergence structures such as (symmetric) topological structures and (symmetric) limit space structures. Furthermore, ﬁlter spaces introduced by M. Katˇetov form the natural link between convergence structures and uniform convergence structures, and the construct Fil of ﬁlter spaces can be nicely embedded into the construct SUConv of semiuniform convergence spaces. The reason for studying semiuniform convergence spaces and ﬁlter spaces results from the fact that they are better behaved w.r.t. the formation of function spaces as well as subspaces and quotient spaces, in other words: they are more ‘convenient’. Finally, a simple completion of semiuniform convergence spaces is studied from which the usual Hausdorﬀ completion of a (separated) uniform space can be derived, e.g., the completion of the rationals to the reals.

Contents 1. Introduction 334 2. Some important topological constructs and the relations between them 339 2.1. Deﬁnition of topological constructs 339 2.2. Examples of topological constructs 340 2.3. Remarks on topological constructs 343 2.4. Bireﬂective and bicoreﬂective subconstructs 343 2.5. Applications to convergence structures 345 2.6. Applications to uniform convergence structures 347 2.7. Fil as a link between convergence structures and uniform convergence structures 348 2.8. Filter spaces and merotopic spaces 350 3. Convenient properties of topological constructs 351 3.1. Existence of natural function spaces 351 3.2. Examples of natural function spaces 352 3.3. The role of natural function spaces in SUConv 353 2000 Mathematics Subject Classiﬁcation. 54A05, 54A20, 54B05, 54B15, 54C35, 54D05, 54D18, 54E15, 54E52, 18A40, 18D15. c 2009 2008 American Mathematical Society

333

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3.4. Existence of one-point extensions 3.5. Examples of strong topological universes 3.6. The aim of Convenient Topology 4. The structural behaviour of SUConv-invariants 4.1. Deﬁnitions 4.2. Structural survey of SUConv invariants 4.3. Better behaviour of subspaces and quotients in SUConv 5. Completions 5.1. The simple completion 5.2. The Hausdorﬀ completion 6. Preuniform convergence spaces References

354 356 357 358 358 360 360 364 364 366 368 368

1. Introduction General topology arose from the desire to study convergence and continuity known from analysis in a general context. At ﬁrst M. Fr´echet [26] introduced metric spaces, a very fruitful concept for many purposes in analysis. But unfortunately the class of metric spaces is not big enough in order to describe pointwise convergence in function spaces as the following example shows: On the set X of all maps from the set R of real numbers into itself there is no metric d describing pointwise convergence, i.e., such that for each f ∈ X and each sequence (fn ) in X the following are equivalent: (a) (fn ) converges pointwise to f , i.e., (fn (x)) converges to f (x) for each x ∈ X with respect to the usual (= Euclidean) metric on R. (b) (fn ) converges to f in (X, d). (cf. e.g. [39, 6.7.1]). The bigger class of topological spaces, introduced by F. Hausdorﬀ [35] (nowadays called Hausdorﬀ spaces) and (in the usual meaning) by C. Kuratowski [59] solves the problem by means of the product topology, which was ﬁrst observed by A. Tychonoﬀ [105]. Another important concept of convergence in function spaces is the notion of continuous convergence which was introduced by H. Hahn [33] in his book entitled ‘Theorie der reellen Funktionen’. Since in function theory there is no diﬀerence between uniform convergence on compacta and continuous convergence (cf. e.g. [88, 6.1.32]). C. Carath´eodory [15] proposed already in 1929 to substitute uniform convergence (on compacta) by continuous convergence because it is easier to handle. In the realm of topological spaces continuous convergence can be deﬁned as follows: If C(X, Y ) denotes the set of all continuous maps from a topological space X into a topological space Y , then a ﬁlter F on C(X, Y ) converges continuously to f ∈ C(X, Y ) whenever for each x ∈ X and each ﬁlter G on X converging to x in X the ﬁlter eX,Y (G × F) converges to f (x) in Y , where eX,Y : X × C(X, Y ) → Y denotes the evaluation map (i.e., eX,Y (x, f ) = f (x) for each (x, f ) ∈ X × C(X, Y )) and eX,Y (G × F) is generated by the ﬁlter base {eX,Y [G × F ] : F ∈ F, G ∈ G}. But for any Hausdorﬀ space X, there does not exist for each topological space Y a topology Z on C(X, Y ) describing continuous convergence, i.e., such that convergence with respect to Z means continuous convergence, (equivalently: there

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is no coarsest topology Z on each C(X, Y ) such that the evaluation map eX,Y : X × (C(X, Y ), Z) → Y is continuous) unless X is locally compact (cf. e.g. [88, 6.1.31]). For example on the set of all real-valued continuous maps on the (real) Hilbert space H there is no topology describing continuous convergence which follows from results of R. Arens [3] in 1946 (cf. also [88, p. XIV]. Thus, for inﬁnite-dimensional analysis continuous convergence cannot be described in the framework of topological spaces. This result can be reformulated as follows: (1) The category Top of topological spaces (and continuous maps) is not cartesian closed. This means that the following is not satisﬁed: For any pair (X, Y ) of topological spaces the set C(X, Y ) can be endowed with the topology of a topological space denoted by Y X (and called power object or natural function space) such that the following are satisﬁed: (a) The evaluation map eX,Y : X × Y X → Y is continuous. (b) For each topological space Z, and each continuous map f : X × Z → Y , the map f¯: Z → Y X deﬁned by f¯(z)(x) = f (x, z) is continuous. (It is easily proved that if Top were cartesian closed, then for any pair (X, Y ) of topological spaces the topology of Y X would be the coarsest topology Z on C(X, Y ) such that the evaluation map eX,Y : X ×(C(X, Y ), Z) → Y is continuous. But then Z is a topology describing continuous convergence. Conversely, if the topology of continuous convergence would always exist, (a) and (b) would be satisﬁed with Y X carrying this topology, which is easy to check, i.e., Top would be cartesian closed.) Now the question arises, whether it is possible to enlarge the concept of a topological space to some kind of space such that continuous convergence can be described in the realm of these spaces . The category PsTop of pseudotopological spaces (and continuous maps), introduced by G. Choquet [17], as well as the bigger category Lim of limit spaces (and continuous maps), independently introduced by H.-J. Kowalsky [58] and H. R. Fischer [23], solve the problem: PsTop and Lim are cartesian closed, and if X and Y are pseudotopological spaces (resp. limit spaces) the structure of the pseudotopological space (resp. limit space) Y X is the structure of continuous convergence (cf. e.g. [43, 3.7] for PsTop and [88, 3.1.9.6.b] for Lim). Next, let us consider the behaviour of quotients in Top. Let X be a Hausdorﬀ space which is a Fr´echet space (a Fr´echet space is a topological space X such that for every A ⊂ X and every point x belonging to the closure of A, there is a sequence (xn ) of points of A converging to x), e.g., a ﬁrst countable Hausdorﬀ space, in particular a metric space. If f : X → Y is a quotient map in Top between X and a Hausdorﬀ space Y (i.e., f : X → Y is surjective and O ⊂ Y is open iﬀ f −1 [O] is open in X), then Y is a Fr´echet space iﬀ f is hereditary, i.e., if A is a subspace of Y , B a subspace of X with underlying set f −1 [A] and g : B → A is the corresponding restriction of f , then g : B → A is a quotient map in Top, too (cf. A. V. Arhangel skii [4], who characterized hereditary quotient maps in Top as pseudo-open maps, where a surjective, continuous map f : X → Y between topological spaces X and Y is called pseudo-open if, for each point y in Y and each neighborhood U of f −1 (y) in X, y belongs to the interior of f [U ]). Unfortunately, the following is valid: (2) In Top quotient maps are not hereditary.

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This is proved by means of the following example, where X and Y are topological spaces with subspaces B and A respectively, and the non-empty open sets are marked by rectangles: B = f −1 [A]

0

g=(f |f −1 [A] )

A

1 ? 01

-

02 31

X

? -

021

Y

Then f is a quotient map, but g is not a quotient map. Again, the question arises, whether Top can be substituted by a better behaved supercategory. One candidate is the category PrTop of pretopological spaces (and continuous maps). A topological space can be described by means of a closure operator satisfying Kuratowski’s axioms, cf. [59], whereas in the deﬁnition of a pretopological space the requirement that the closure operator is idempotent is ˇ omitted, cf. E. Cech [16], where pretopological spaces are called closure spaces. It has been proved by D. C. Kent ([52, Theorem 4]) that the pseudo-open maps between topological spaces are exactly the quotient maps between these spaces (regarded as pretopological spaces) in PrTop. Thus, Arhangel skii’s result from above reads now as follows: Let X and Y be Hausdorﬀ spaces (regarded as pretopological spaces), and let X be a Fr´echet space. If f : X → Y is a quotient map in PrTop, then Y is a Fr´echet space, and vice versa. Since in PrTop quotient maps are hereditary (cf. [8, Theorem 26]), it is better behaved than Top w.r.t. this property. Another deﬁciency of quotient maps in Top is the following: (3) In Top quotient maps are not productive. : Xi → Yi )i∈I is any is, if (fi family of quotient maps in Top, then That fi ((xi )) = (fi (xi )), is a surjective, confi : i∈I Xi → i∈I Yi deﬁned by tinuous map but no quotient map in general as the following example shows: Let π be an equivalence relation on the usual topological space R of real numbers deﬁned by x1 π x2 iﬀ x1 = x2 or {x1 , x2 } ⊂ Z, where Z denotes the set of integers. The natural map ω : R → R/π and the identity map 1Q : Q → Q on the usual topological space of rational numbers are quotient maps in Top, but their product ω × 1Q : R × Q → (R/π) × Q is no quotient map in Top. If it were a quotient map, it would map closed saturated sets to closed sets (note: if f : X → Y is a quotient map, then A ⊂ X is saturated iﬀ A = f −1 [B] for some B ⊂ Y ). Now let N be the set of positive integers and (an )n∈N a sequence of irrational numbers converging to 0. For each n ∈ N, let (rn,m )m∈N be a sequence of rational numbers converging to an . Put 1 A= n+ m , rn,m : n, m ∈ N, m > 1 . A is closed and saturated in R × Q but A = (ω × 1Q )[A] is not closed in (R/π) × Q (note: (ω(0), 0) ∈ / A but each neighborhood of (ω(0), 0) in (R/π) × Q meets A). By the way, the above example can be used to give another proof of the fact that Top is not cartesian closed, since otherwise quotient maps in Top would be ﬁnitely productive (cf. e.g. [88, 3.1.3]). In contrast to the situation for metric spaces an additional inconvenience occurs in the framework of topological spaces:

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(4) Uniform concepts, such as uniform continuity, uniform convergence, Cauchy sequences (or Cauchy ﬁlters) and completeness are not available in Top. Let us consider for example the deﬁnition for a Cauchy sequence in analysis: A sequence (xn )n∈N of real numbers is called a Cauchy sequence iﬀ for each ε > 0 there is some N (ε) ∈ N such that (∗)

|xn − xm | < ε

whenever n, m ≥ N (ε). (∗) may be interpreted as follows: xn belongs to an ε-neighborhood of xm (= ε-sphere about xm ) and xm belongs to an ε-neighborhood of xn , but xn and xm are district elements of R is general. ε-neighborhoods of distinct points may be considered to have the same size. In a topological space X there is assigned a neighborhood system U(x) to each x ∈ X such that certain axioms are satisﬁed (analogously to Hausdorﬀ’s original deﬁnition of a topological space) but there is no possibility to compare neighborhoods of diﬀerent points with respect to their size. Thus, whenever it is possible to compare neighborhoods of diﬀerent points with respect to their size (e.g., in metric spaces), Cauchy sequences can be deﬁned. The same argument may be used to explain why uniform continuity and uniform convergence are not available in Top. Finally, we cannot deﬁne completeness in Top because of the absence of the concept of Cauchy ﬁlter. The question whether there is a bigger class than the class of metric spaces such that uniform concepts are available has been solved by A. Weil [107] in 1937. He introduced so-called uniform spaces and constructed a completion for each uniform space generalizing Hausdorﬀ’s completion of metric spaces [35] which is a generalization of Cantor’s construction of the real numbers from the rational numbers. Though the category Unif of uniform spaces fulﬁlls the property that quotient maps are productive (cf. [46]), it is not cartesian closed, and in Unif quotients are not hereditary (cf. [88, 3.1.9.2 and 3.2.7.2]). The well-known fact that the quotient space of a topological space obtained by the decomposition of X into its components is totally disconnected (i.e., the component of each of its points is a singleton) does not have an analogue in the realm of uniform spaces (cf. [88, 5.2.7] for an example). If in Unif quotients were hereditary, this strange behaviour of uniform spaces would not occur. In order to remedy all the deﬁciencies of Top mentioned under (1)–(4), semiuniform convergence spaces are introduced. They have an easy deﬁnition, which is given soon, and all results on topological and uniform spaces remain important special cases, i.e., topological and uniform spaces are not all superﬂuous but they are studied in a better framework, namely in the framework of semiuniform convergence spaces. Thus, we obtain further nice results which cannot be obtained in topological and (or) uniform spaces in their original setting: (5) The localization of the concept of compactness leads to a cartesian closed category and ‘locally compact’ is equivalent to ‘compactly generated’ (the Hausdorﬀ axiom is not required!). (6) The localization of the concept of precompactness (= total boundedness) leads to a cartesian closed category in which quotients are hereditary, and ‘locally precompact’ is equivalent to ‘precompactly generated’. (7) The localization of connectedness leads to a cartesian closed category.

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(8) The (Hausdorﬀ ) completion of a separated uniform space can be easily constructed by means of function spaces. (9) Not only the structure of pointwise convergence and uniform convergence but also the structure of continuous convergence can be derived from the natural function space structure in the realm of semiuniform convergence spaces. In some sense the behaviour of subspaces in Top is strange: Though there is a diﬀerence of a topological nature between the removal of a point and the removal of a closed interval of length one from the usual topological space Rt of real numbers, the obtained topological spaces are homeomorphic. But if subspaces are formed in the realm of semiuniform convergence spaces and Rt is regarded as a semiuniform convergence space, then we obtain non-isomorphic semiuniform convergence spaces by means of the above procedure. Consequently, semiuniform convergence spaces behave well w.r.t. the formation of subspaces. The following enlargements of classical topological results can be obtained whenever subspaces of symmetric1 topological spaces are formed in the framework of semiuniform convergence spaces (where they need not be necessarily topological unless they are closed): (10) Subspaces of fully normal (normal) symmetric topological spaces are fully normal (normal). (11) Dense subspaces of connected (locally connected) symmetric topological spaces are connected (locally connected). (12) There is a dimension function for semiuniform convergence spaces such that the dimension of each subspace of a symmetric topological space X is less than or equal to the dimension of X, where for symmetric topological spaces this dimension function coincides with the (Lebesgue) covering dimension dim. By the way, even the formation of quotients in the category SUConv of semiuniform convergence spaces leads sometimes to better results than the formation of quotients in Top or Unif, e.g., (13) Quotients of ﬁrst countable (symmetric) topological spaces (regarded as semiuniform convergence spaces) are ﬁrst countable. (14) The quotient space of a uniform space X (regarded as a semiuniform convergence space) obtained by the decomposition of X into its uniform components is totally uniformly disconnected, i.e., the uniform component of each of its points is a singleton. As in the case of uniform spaces Cauchy ﬁlters are available in semiuniform convergence spaces. These ones can be axiomatized by means of ﬁlter spaces, which have been introduced by M. Katˇetov [48] in the realm of merotopic spaces where emphuniform covers are axiomatized. In this chapter they occur in the framework of semiuniform convergence spaces. But their relation to merotopic spaces make them a valuable tool for deﬁning normality, full normality (and paracompactness), and some kind of covering dimension in SUConv. Last but not least they form the bridge between symmetric convergence structures (such as symmetric topological structures and suitable generalizations of them) and uniform convergence structures (such as uniform structures and suitable generalizations of them). 1A topological space X is symmetric iﬀ for each pair (x, y) of points of X the following is satisﬁed: x belongs to the closure of singleton {y} implies that y belongs to closure of singleton {x}.

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2. Some important topological constructs and the relations between them 2.1. Deﬁnition of topological constructs. In order to handle problems of a topological nature topologists have created not only topological spaces but also uniform spaces, ﬁlter spaces, convergence spaces and so on. Since constructions in the corresponding concrete categories of these spaces have striking similarities it makes sense to deﬁne so-called ‘topological constructs’ where a construct means a category C whose objects are structured sets, i.e., pairs (X, ξ) where X is a set and ξ is a C-structure on X, whose morphisms f : (X, ξ) → (Y, η) are suitable maps between X and Y , and whose composition is the usual composition of maps. The class of all objects of a construct C is denoted by |C|. We often say C-object instead of object of C, and C-morphism instead of morphism of C. The class of all C-morphisms is denoted by Mor C. 2.1.1. Definition. A construct C is called topological iﬀ it satisﬁes the following conditions: (1) Existence of initial structures: For any set X, any family ((Xi , ξi ))i∈I of C-objects indexed by a class I, and any family (fi : X → Xi )i∈I indexed by I, there exists a unique C-structure ξ on X which is initial with respect to (X, fi , (Xi , ξi ), I) (shortly: w.r.t. (fi )), i.e., such that for each C-object (Y, η) a map g : (Y, η) → (X, ξ) is a C-morphism iﬀ for every i ∈ I the composite map fi ◦ g : (Y, η) → (Xi , ξi ) is a C-morphism. (2) For any set X, the class {(Y, η) ∈ |C| : X = Y } of all C-objects with underlying set X is a set. (3) For any set X with cardinality at most one, there exists exactly one Cobject with underlying set X (i.e., there exists exactly one C-structure on X). Let C be a topological construct, let X be a set, and let ξ, η be C-structures on X. The C-structure ξ is called ﬁner than η (or η coarser than ξ), denoted by ξ ≤ η, iﬀ 1X : (X, ξ) → (X, η) is a C-morphism. 2.1.2. Proposition (cf. [88, 1.1.5]). The initial structure ξ on a set X with respect to (X, fi , (Xi , ξi ), I) in a topological construct C is the coarsest C-structure on X such that fi is a C-morphism for each i ∈ I. In the deﬁnition of a topological construct (1) may be substituted by (1 ) For any set X, any family ((Xi , ξi ))i∈I of C-objects indexed by some class I, and any family (fi : Xi → X)i∈I of maps indexed by I, there exists a unique C-structure ξ on X which is ﬁnal with respect to ((Xi , ξi ), fi , X, I) (shortly: w.r.t. (fi )), i.e., such that for any C-object (Y, η) a map g : (X, ξ) → (Y, η) is a C-morphism iﬀ for every i ∈ I the composite map g ◦ fi : (Xi , ξi ) → (Y, η) is a C-morphism. (cf. [88, 1.2.1.1]) Analogously to Proposition 2.1.2, the ﬁnal C-structure w.r.t. (fi ) is the ﬁnest Cstructure such that each fi is a C-morphism. In topological constructs subspaces, quotient spaces, product spaces, and sum spaces can be constructed as follows: Let (X, ξ), (Y, η) be C-objects in a topological construct C. (Y, η) is called a subspace of (X, ξ) provided that there is an embedding f : (Y, η) → (X, ξ), i.e., an injective map f : Y → X such that η is the initial C-structure on

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Y w.r.t. f . If A ⊂ X and ξA is the initial C-structure on A w.r.t. the inclusion map i : A → X, then (A, ξA ) is called the subspace of (X, ξ) determined by A. (Y, η) is called a quotient space of (X, ξ) provided that there is a quotient map f : (X, ξ) → (Y, η), i.e., a surjective map f : X → Y such that η is the ﬁnal Cstructure w.r.t. f . If R is an equivalence relation on X then the ﬁnal C-structure on X/R w.r.t. the natural map ω : X → X/R is denoted by ξR , and (X/R, ξR ) is called the quotient space of (X, ξ) by R. Let ((Xi , ξi ))i∈I be a family of C-objects (in a topological construct C) indexed by some set I. The cartesian product i∈I Xi endowed with the initial C-structure w.r.t. the family (pi ) of projections pi : i∈I Xi → X i deﬁned by pi ((xi )) = xi is called the product space of ((Xi , ξi ))i∈I , denoted by i∈I (Xi , ξi )i∈I . The sum i∈I Xi × {i} endowed with the ﬁnal C-structure w.r.t. the family (ji ) of injections ji : Xi → i∈I Xi × {i} deﬁned by ji (xi ) = (xi , i), is called the sum space of ((Xi , ξi ))i∈I , denoted by i∈I (Xi , ξi ). Let M = {ξi : i ∈ I} be a set of C-structures on a set X and CX the set of all C-structures on X, where C is a topological construct. M has a greatest lower bound in the partially ordered set (CX , ≤), i.e., there is a coarsest C-structure ξ on X ﬁner than each ξi ∈ M, namely the initial C-structure on X w.r.t. the family (1iX : X → (X, ξi ))i∈I of identity maps. M has a least upper bound in the partially ordered set (CX , ≤), i.e., there is a ﬁnest C-structure ξ on X coarser than each ξi ∈ M, namely the ﬁnal C-structure on X w.r.t. the family (1iX : (X, ξi ) → X)i∈I of identity maps. 2.2. Examples of topological constructs. 2.2.1. Example. The construct Top of topological spaces (and continuous maps). 2.2.2. Example. The construct Unif of uniform spaces (and uniformly continuous maps). A uniform space is a pair (X, W), where X is a set and W a set of subsets of X × X such that the following are satisﬁed: (1) W1 , W2 ∈ W implies W1 ∩ W2 ∈ W. (2) V ∈ W and V ⊂ W ⊂ X × X imply W ∈ W. (3) W ∈ W implies ∆ = {(x, x) : x ∈ X} ⊂ W . (4) W ∈ W implies W −1 = {(x, y) : (y, x) ∈ W } ∈ W. (5) For each W ∈ W there is some W ∗ ∈ W such that W ∗2 = {(x, y) ∈ X×X : there is some z ∈ X with (x, z) ∈ W ∗ and (z, y) ∈ W ∗ } ⊂ W . If (X, W) is a uniform space, then W is called a uniformity on X, and the elements of W are called entourages. Further, a set B of subsets of X × X is called a base for the uniformity W provided that {W ⊂ X × X : B ⊂ W for some B ∈ B} = W. A map f : (X, W) → (X , W ) between uniform spaces is called uniformly continuous provided that (f ×f )−1 [W ] ∈ W for each W ∈ W , where f ×f : X ×X → X × X is deﬁned by (f × f )(x, y) = (f (x), f (y)) for each (x, y) ∈ X × X. Let X be a set, ((Xi , Wi ))i∈I a family of uniform spaces and (fi : X → Xi )i∈I a family of maps. Put gi = fi × fi for each i ∈ I. Then all ﬁnite intersections of elements of {gi−1 [Wi ] : Wi ∈ Wi , i ∈ I} form a base for a uniformity W on X which is initial w.r.t. (X, fi , (Xi , Wi ), I).

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2.2.3. Example. The construct GConv of generalized convergence spaces (and continuous maps). For each set X, let F (X) be the set of all ﬁlters on X. Then a generalized convergence space is a pair (X, q), where X is a set and q ⊂ F (X) × X such that the following are satisﬁed: (C1 )

(x, ˙ x) ∈ q for each x ∈ X, where x˙ = {A ⊂ X : x ∈ A}.

(C2 )

(G, x) ∈ q whenever (F, x) ∈ q and G ⊃ F. q

If (X, q) is a generalized convergence space, then we write F → x or shortly F → x instead of (F, x) ∈ q and say that F converges to x. A map f : (X, q) → (X , q ) between generalized convergence spaces is called continuous provided that (f (F), f (x)) ∈ q for each (F, x) ∈ q, where f (F) = {G ⊂ X : there is some F ∈ F with f [F ] = {f (x) : x ∈ F } ⊂ G}. Let X be a set, ((Xi , qi ))i∈I a family of generalized convergence spaces, and (fi : X → Xi )i∈I a family of maps. Then q = {(F, x) ∈ F (X) × X : (fi (F), fi (x)) ∈ qi for each i ∈ I} is a GConv-structure on X which is initial w.r.t. (fi ). 2.2.4. Example. The construct KConv of Kent convergence spaces (and continuous maps). A generalized convergence space (X, q) is called a Kent convergence space provided that the following is satisﬁed: (C3 )

(F ∩ x, ˙ x) ∈ q whenever (F, x) ∈ q.

The initial structures in KConv are formed as in GConv. 2.2.5. Example. The construct Lim of limit spaces (and continuous maps). A generalized convergence space (X, q) is called a limit space provided that the following is satisﬁed: (C4 )

(F ∩ G, x) ∈ q whenever (F, x) ∈ q and (G, x) ∈ q.

The initial structures in Lim are formed as in KConv (and thus as in GConv). 2.2.6. Example. The construct PsTop of pseudotopological spaces (and continuous maps). A generalized convergence space (X, q) is called a pseudotopological space (or a Choquet space) provided that the following is satisﬁed: (C5 )

(F, x) ∈ q whenever (U, x) ∈ q for every ultraﬁlter U ⊃ F.

The initial structures in PsTop are formed as in Lim. 2.2.7. Example. The construct PrTop of pretopological spaces (and continuous maps). A generalized convergence space (X, q) is called a pretopological space provided that the following is satisﬁed: (C6 ) (Uq (x), x) ∈ q for all x ∈ X where Uq (x) = {F ∈ F (X) : (F, x) ∈ q}. The initial structures in PrTop are formed as in PsTop. 2.2.8. Example. The construct Fil of ﬁlter spaces (and Cauchy continuous maps).

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A ﬁlter space is a pair (X, γ), where X is a set and γ a set of ﬁlters on X such that the following are satisﬁed: (1) x˙ ∈ γ for each x ∈ X. (2) G ∈ γ whenever F ∈ γ and F ⊂ G. If (X, γ) is a ﬁlter space, then the elements of γ are called Cauchy ﬁlters. A map f : (X, γ) → (X , γ ) between ﬁlter spaces is called Cauchy continuous provided that f (F) ∈ γ for each F ∈ γ. If X is a set, ((Xi , γi ))i∈I a family of ﬁlter spaces and (fi : X → Xi )i∈I a family of maps, then γ = {F ∈ F (X) : fi (F) ∈ γi for each i ∈ I} is the initial Fil-structure on X w.r.t. (fi ). If X is a set, ((Xi , γi ))i∈I a family of ﬁlter spaces, and (fi : Xi → X)i∈I a family of maps, then γ = {F ∈ F (X) : there is some i ∈ I and some Fi ∈ γi with fi (Fi ) ⊂ F} ∪ {x˙ : x ∈ X} is the ﬁnal 5 Fil-structure on X w.r.t. (fi ). If X = i∈I fi [Xi ], then γ = {F ∈ F (X) : there is some i ∈ I and some Fi ∈ γi with fi (Fi ) ⊂ F}. 2.2.9. Example. The construct SUConv of semiuniform convergence spaces (and uniformly continuous maps). A semiuniform convergence space is a pair (X, JX ), where X is a set and JX a set of ﬁlters on X × X such that the following are satisﬁed: (UC1 )

x˙ × x˙ = (x,˙ x) ∈ JX for each x ∈ X.

(UC2 )

G ∈ JX whenever F ∈ JX and F ⊂ G.

(UC3 )

F ∈ JX implies F −1 = {F −1 : F ∈ F} ∈ JX , where F −1 = {(y, x) : (x, y) ∈ F }.

If (X, JX ) is a semiuniform convergence space, then the elements of JX are called uniform ﬁlters. A map f : (X, JX ) → (Y, JY ) between semiuniform convergence spaces is called uniformly continuous provided that (f × f )(F) ∈ JY for each F ∈ JX , shortly: (f × f )(JX ) ⊂ Y . If X is a set, ((Xi , JXi ))i∈I a family of semiuniform convergence spaces, and (fi : X → Xi )i∈I a family of maps, then JX = {F ∈ F (X × X) : (fi × fi )(F) ∈ JXi for each i ∈ I} is the initial SUConv-structure on X w.r.t. (fi ). If X is a set, ((Xi , JXi ))i∈I a family of semiuniform convergence spaces, and (fi : Xi → X)i∈I a family of maps, then JX = {F ∈ F (X × X) : there is some i ∈ I and some Fi ∈ JXi with (fi × fi )(Fi ) ⊂ F} ∪ {x˙ × x˙ : x ∈ X} is the ﬁnal SUConv-structure on X w.r.t. (fi ). If X = i∈I fi [Xi ], then JX = {F ∈ F (X × X) : there is some i ∈ I and some Fi ∈ JXi with (fi × fi )(Fi ) ⊂ F}. 2.2.10. Example. The construct SULim of semiuniform limit spaces (and uniformly continuous maps). A semiuniform convergence space (X, JX ) is called a semiuniform limit space provided that the following is satisﬁed: (UC4 )

F ∈ JX and G ∈ JX imply F ∩ G ∈ JX .

The initial structures in SULim are formed as in SUConv.

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2.2.11. Example. The construct ULim of uniform limit spaces (and uniformly continuous maps). A semiuniform limit space (X, JX ) is called a uniform limit space provided that the following is satisﬁed: (UC5 )

F ∈ JX and G ∈ JX imply F ◦ G ∈ JX

(whenever F ◦ G exists, i.e., F ◦ G = {(x, y) : there is some z ∈ X with (x, z) ∈ G and (z, y) ∈ F } = ∅ for every F ∈ F, G ∈ G), where F ◦ G is the ﬁlter generated by the ﬁlter base {F ◦ G : F ∈ F, G ∈ G}. The initial structures in ULim are formed as in SULim (and thus as in SUConv). 2.3. Remarks on topological constructs. Many other examples of topological constructs have been studied by various authors, e.g., the constructs Prox of proximity spaces (and δ-maps) in [22] and [67], Near of nearness spaces (and uniformly continuous maps) in [38] and [41], Chy of Cauchy spaces (and Cauchy continuous maps) in [49] and [64], AP of approach spaces (and contractions) in [62] and [63], or QUnif in [24] and [66]. A generalization of nearness spaces has been introduced earlier under the name ‘merotopic spaces’ in [48]. The topological construct Mer of merotopic spaces is closely related to Fil (cf. 2.8). The relations of Chy to Fil and ULim are studied in [10, 49, 64, 75, 76, 77]. Furthermore, Chy is closely related to Prox (cf. [75]). The relations between semiuniform convergence spaces and merotopic spaces (including nearness spaces) are studied in [84, 86, 87]. Each subclass |A| of the object class |C| of a construct C deﬁnes a construct A, a so-called full subconstruct of C as follows: • |A| is the object class of A. • [A, B]A = [A, B]C for each (A, B) ∈ |A| × |A|. • The composition of morphisms in A coincides with the composition of these morphisms in C. • For each A ∈ |A| the identity 1A is the same in A and in C. Example: ULim is a full subconstruct of SULim. If C is a construct, then a C-morphism f : (X, ξ) → (Y, η) is called an isomorphism provided that there is a C-morphism g : (Y, η) → (X, ξ) such that g ◦ f = 1X and f ◦ g = 1Y , where 1X (resp. 1Y ) denotes the identity map on X (resp. Y ). A C-object (X, ξ) is called isomorphic to a C-object (Y, η) (abbreviated by (X, ξ) ∼ = (Y, η)) iﬀ there is an isomorphism f : (X, ξ) → (Y, η). If P is a property for the objects of a construct C, then P is called a C-invariant provided that the following is satisﬁed: whenever (X, ξ) ∈ |C| has P , then each (Y, η) ∈ |C| which is isomorphic to (X, ξ) has P too. A (full) subconstruct A of a construct C is called isomorphism-closed iﬀ each C-object being isomorphic to some A-object belongs to |A|. Example: Let P be a Cinvariant. Then the full subconstruct A of C deﬁned by |A| = {X ∈ |C| : X has P } is isomorphism-closed. 2.4. Bireﬂective and bicoreﬂective subconstructs. 2.4.1. Definition. A full subconstruct A of a topological construct C is called • bireﬂective in C iﬀ for each (X, ξ) ∈ |C|, there is an A-structure ξA on X coarser than ξ such that for each A-object (Y, η) and each Cmorphism f : (X, ξ) → (Y, η), f : (X, ξA ) → (Y, η) is an A-morphism (=

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C-morphism), i.e., the following diagram commutes. / (Y, η) u: u u uu 1X uu f u u (X, ξA ) (X, ξ)

f

If A is bireﬂective in C, then (X, ξA ) (resp. ξA ) is called the bireﬂective modiﬁcation of (X, ξ) (resp. ξ) w.r.t. A. • bicoreﬂective in C iﬀ for each (X, ξ) ∈ |C|, there is an A-structure ξA on X ﬁner than ξ such that for each A-object (Y, η) and each C-morphism f : (Y, η) → (X, ξ), f : (Y, η) → (X, ξA ) is an A-morphism (= C-morphism), i.e., the following diagram commutes. / (X, ξ) u: u u uu f uu f u u (Y, η)

(X, ξA ) O

1X

If A is bicoreﬂective in C, then (X, ξA ) (resp. ξA ) is called the bicoreﬂective modiﬁcation of (X, ξ) (resp. ξ) w.r.t. A. 2.4.2. Corollary. Every bireﬂective (or bicoreﬂective) subconstuct of a topological construct C is isomorphism-closed. Proof. If A is bireﬂective in C, let f : (X, ξ) → (Y, η) be an isomorphism from (X, ξ) ∈ |C| to (Y, η) ∈ |A| and (X, ξA ) the bireﬂective modiﬁcation of (X, ξ) f

f −1

w.r.t. A. Then the composition (X, ξA ) −→ (Y, η) −→ (X, ξ) is a C-morphism, i.e., 1X : (X, ξA ) → (X, ξ) is a C-morphism. Thus, since 1X : (X, ξ) → (X, ξA ) is a C-morphism by assumption, ξ = ξA (note: initial structures are unique!). In case A is bicoreﬂective in C the proof is similar. 2.4.3. Proposition. Let A be a full subconstruct of a topological construct C. Then the following hold: (1) A is bireﬂective in C iﬀ A is initially closed in C, i.e., whenever X is a set, ((Xi , ξi ))i∈I a family of A-objects, and (fi : X → Xi )i∈I a family of maps, then (X, ξ) ∈ |A|, where ξ is the initial C-structure on X w.r.t. (fi ). (2) A is bicoreﬂective in C iﬀ A is ﬁnally closed in C, i.e., whenever X is a set, ((Xi , ξi ))i∈I a family of A-objects, and (fi : Xi → X)i∈I a family of maps, then (X, ξ) ∈ |A|, where ξ is the ﬁnal C-structure on X w.r.t. (fi ). Proof. (1) ⇒. Let (fi : (X, ξ) → (Xi , ξi ))i∈I be a family of C-morphisms such that (Xi , ξi ) ∈ |A| for each i ∈ I, where ξ is the initial C-structure w.r.t (fi ). If (X, ξA ) is the bireﬂective modiﬁcation of (X, ξ) w.r.t. A, then 1X : (X, ξ) → (X, ξA ) is a C-morphism. In order to prove that 1X : (X, ξA ) → (X, ξ) is a C-morphism too, it suﬃces to prove that fi ◦ 1X = fi : (X, ξA ) → (Xi , ξi ) is a C-morphism for each i ∈ I. But this is obvious, since A is bireﬂective in C and fi : (X, ξ) → (Xi , ξi ) is a C-morphism for each i ∈ I. Thus ζ = ξA , i.e., (X, ξ) ∈ |A|.

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⇐. Let (X, ξ) ∈ |C|, and (fi : (X, ξ) → (Xi , ξi ))i∈I the family of all Cmorphisms from (X, ξ) into A-objects (Xi , ξi ). If ξA denotes the initial C-structure on X w.r.t. (fi ), then (X, ξA ) is the desired bireﬂective modiﬁcation of (X, ξ). The equivalence of (2) is proved analogously. A full and isomorphism-closed subconstruct A of a topological construct C is called • closed under the formation of subspaces (resp. quotient-spaces) in C iﬀ for each X ∈ |A|, each subspace (resp. quotient space) of X formed in C belongs to |A|, • closed under the formation of product spaces (resp. sum spaces) in C iﬀ whenever i∈I Xi (resp. i∈I Xi ) is a product space (resp. sum space) in C such that all Xi belong to |A|, then i∈I Xi (resp. i∈I Xi ) belongs to |A|. Furthermore, the initial C-structure ξi (resp. ﬁnal C-structure ξd ) on a set X with respect to the empty index class I is called indiscrete (resp. discrete), and (X, ξi ) (resp. (X, ξd )) an indiscrete (resp. a discrete) C-object. The following is valid: 2.4.4. Theorem (cf. [88, 2.2.11 and 2.2.4]). Let C be a topological construct and A a full and isomorphism-closed subconstruct. Then the following hold: (1) A is bireﬂective in C iﬀ A is closed under the formation of subspaces and product spaces in C, and all indiscrete C-objects belong to |A|. (2) A is bicoreﬂective in C iﬀ A is closed under the formation of quotient spaces and sum spaces in C, and all discrete C-objects belong to |A|. 2.4.5. Theorem. Let A be a full and isomorphism-closed subconstruct of a topological construct C. Then A is a topological construct provided that A is bireﬂective or bicoreﬂective in C. In particular, if A is bireﬂective (resp. bicoreﬂective) in C, then the initial structures (resp. ﬁnal structures) in A are formed as in C, whereas the ﬁnal structures (resp. initial structures) arise from the ﬁnal structures (resp. initial structures) in C by bireﬂective (resp. bicoreﬂective) modiﬁcation. Proof. The ﬁrst part of the theorem follows from Proposition 2.4.3, and the remaining part is easily checked (cf. e.g. [88, 2.2.12]). Sometimes it is useful to have an alternative description of the objects in a (topological) construct. This idea is captured by the following. 2.4.6. Definition. Two constructs C and D are called concretely isomorphic iﬀ there is a concrete functor F : C → D which is an isomorphism, i.e., there is a (concrete) functor G : D → C such that G ◦ F = 1C and F ◦ G = 1D , where 1C and 1D are the identity functors on C and D respectively. (A functor F : C → D is called concrete provided that K ◦ F = H, where H and C denote the forgetful functors from C and D in the construct Set of sets respectively.) If C and D are concretely isomorphic, we write C ∼ = D. 2.5. Applications to convergence structures. A pretopological space (X, q) is called topological (or a topologically pretopological space) provided the following is satisﬁed: For each x ∈ X and each U ∈ Uq (x) there is some V ∈ Uq (x) (C7 ) such that U ∈ Uq (y) for each y ∈ V .

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2.5.1. Proposition. Top can be embedded into PrTop as a bireﬂective subconstruct, i.e., Top is concretely isomorphic to a bireﬂective subconstruct of PrTop. Proof. The construct T-PrTop of topologically pretopological spaces (and continuous maps) is concretely isomorphic to Top: (1) If (X, q) is a generalized convergence space, then a topology Xq on X is deﬁned by O ∈ Xq ⇔ for each x ∈ O and each ﬁlter F on X with (F, x) ∈ q, O ∈ F. (2) If X is a topology on a set X, then a T-PrTop-structure qX on X is deﬁned by (F, x) ∈ qX ⇔ F ⊃ UX (x), where UX (x) denotes the neighborhood ﬁlter of x ∈ X in (X, X ), i.e., U ∈ UX (x) iﬀ there is some O ∈ X with x ∈ O ⊂ U. (3) It is easy to check that (a) XqX = X for each topology X , (b) qXq = q for each T-PrTop-structure q. [Concerning (b), note that a GConv-structure q on a set X is a T-PrTopstructure iﬀ there is a topology X on X such that q = qX , and use (a).] (4) If f : (X, q) → (X , q ) is a continuous map between generalized convergence spaces, then f : (X, Xq ) → (X , Xq ) is continuous. (5) If f : (X, X ) → (X , X ) is a continuous map between topological spaces deﬁned by means of open sets, then f : (X, qX ) → (X , qX ) is a T-PrTopmorphism. There are concrete functors F : Top → T-PrTop and G : T-PrTop → Top deﬁned by F((X, X )) = (X, qX ) for each (X, X ) ∈ |Top|, F(f ) = f for each f ∈ Mor Top, and G((X, q)) = (X, Xq ) for each (X, q) ∈ |T-PrTop|, G(f ) = f for each f ∈ Mor T-PrTop, respectively. Using (3), (4), (5), we obtain G ◦ F = 1Top and F ◦ G = 1T-PrTop , i.e., Top is concretely isomorphic to T-PrTop. T-PrTop (∼ = Top) is bireﬂective in PrTop: Let (X, q) ∈ |T-PrTop|. Then (X, qXq ) (resp. (X, Xq )) is the bireﬂective modiﬁcation of (X, q) w.r.t. T-PrTop (resp. Top). By Proposition 2.5.1 therefore there are two possibilities for describing topological spaces, namely by means of open set and by means of ﬁlter convergence. In particular, we need not distinguish between topologically pretopological spaces and topological spaces. 2.5.2. Proposition. PrTop is bireﬂective in PsTop. Proof. Note ﬁrst that PrTop is a full subconstruct of PsTop, i.e., every pretopological space (X, q) is pseudotopological. (Hint: For each ﬁlter F on a set X, {U : U is an ultraﬁlter on X with U ⊃ F} = F.) Let (X, q) ∈ |PsTop|. Then (X, q¯) is the bireﬂective modiﬁcation of (X, q) w.r.t. PrTop, where (F, x) ∈ q¯ iﬀ F ⊃ Uq (x) = {G ∈ F (X) : (G, x) ∈ q}. 2.5.3. Proposition. PsTop is bireﬂective in Lim.

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Proof. Note ﬁrst that PsTop is a full subconstruct of Lim, i.e., every pseudotopological space (X, q) is a limit space. Namely, if (C4 ) were not satisﬁed, there would be ﬁlters F and G on X such that (F, x) ∈ q and (G, x) ∈ q for some x ∈ X but (F ∩ G, x) ∈ q. Thus, there would exist some ultraﬁlter U containing F ∩ G such that (U, x) ∈ q. Especially, U ⊃ F and U ⊃ G, i.e., there would exist F ∈ F and G ∈ G with F ∈ U and G ∈ U. On the other hand we would have F ∪ G ∈ U. So it would follow that F ∈ U or G ∈ U since U is an ultraﬁlter. This is a contradiction. Let (X, q) ∈ |Lim|. Then (X, q) is the bireﬂective modiﬁcation of (X, q) w.r.t. PsTop, where (F, x) ∈ q iﬀ (U, x) ∈ q for each ultraﬁlter U ⊃ F. 2.5.4. Proposition. Lim is bireﬂective in KConv. Proof. If (X, q) ∈ |KConv|, then (X, q) is the bireﬂective modiﬁcation of (X, q) w.r.t. Lim, where (F, x) ∈ q iﬀ there exist ﬁnitely nmany F1 , . . . , Fn ∈ F (X) such that (Fi , x) ∈ q for each i ∈ {1, . . . , n} and F ⊃ i=1 Fi . 2.5.5. Proposition. KConv is bireﬂective and bicoreﬂective in GConv. Proof. Let (X, q) ∈ |GConv|. Deﬁne two KConv-structures qr and qc on X as follows: (F, x) ∈ qr ⇔ ∃(G, x) ∈ q such that G ∩ x˙ ⊂ F, (F, x) ∈ qc ⇔ (F ∩ x, ˙ x) ∈ q. Then (X, qr ) (resp. (X, qc )) is the bireﬂective (resp. bicoreﬂective) modiﬁcation of (X, q) w.r.t. KConv. 2.6. Applications to uniform convergence structures. 2.6.1. Proposition. SULim is bireﬂective in SUConv. Proof. Let (X, JX ) ∈ |SUConv|, and (JX )L = {F ∈ F (X × X) : there exist n ﬁnitely many F1 , . . . , Fn ∈ JX with F ⊃ i=1 Fi }. Then (X, (JX )L ) is the bireﬂective modiﬁcation of (X, JX ) w.r.t. SULim. 2.6.2. Proposition. ULim is bireﬂective in SULim. Proof. If (X, JX ) ∈ |SULim|, then (X, ((JX )Q )L ) is the bireﬂective modiﬁcation of (X, JX ) w.r.t. ULim, where (JX )Q = {F ∈ F (X × X) : there exist ﬁnitely many F1 , . . . , Fn ∈ JX with F ⊃ F1 ◦ · · · ◦ Fn } and L is deﬁned as under 2.6.1. 2.6.3. Proposition. Unif can be embedded into ULim as a bireﬂective subconstruct. Proof. A uniform limit space (X, JX ) is called a principal uniform limit space iﬀ there is a non-empty set F of subsets of X × X satisfying the conditions (1) F1 , F2 ∈ F implies F1 ∩ F2 ∈ F, and (2) F ∈ F whenever G ∈ F and G ⊂ F such that JX = [F], where [F] = {G ∈ F (X × X) : G ⊃ F}. The construct PrULim of principal uniform limit space (and uniformly continuous maps) is concretely isomorphic to Unif (note that a non-empty set F of subsets of X × X satisfying (1) and (2) is a uniformity on X iﬀ [F] is a PrULim-structure, and that a map f : (X, W) → (Y, V) between uniform spaces is uniformly continuous iﬀ f : (X, [W]) → (Y, [V]) is uniformly continuous). Let (X, JX ) be a uniform limit

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space, and W the ﬁnest uniformity which is contained in each F ∈ JX . Then (X, [W]) is the bireﬂective modiﬁcation of (X, JX ) w.r.t. PrULim. 2.7. Fil as a link between convergence structures and uniform convergence structures. If (X, JX ) ∈ |SUConv|, then (X, γJX ) ∈ |Fil|, where γJX = {F ∈ F (X) : F × F ∈ JX } is the set of all JX -Cauchy ﬁlters on X and F × F denotes the product ﬁlter, i.e., F × F is generated by the ﬁlter base {F × F : F ∈ F}. Conversely, if (X, γ) ∈ |Fil|, then (X, Jγ ) ∈ |SUConv|, where Jγ = {F ∈ F (X ×X) : there is some G ∈ γ with G ×G ⊂ F}, and γJγ = γ, i.e., γ is the set of all Jγ -Cauchy ﬁlters on X. A semiuniform convergence space (X, JX ) is called Fil-determined iﬀ JX = JγJX (i.e., JX is ‘generated’ by all JX -Cauchy ﬁlters). Example: (X, Jγ ) is a Fil-determined semiuniform convergence space for each Fil-structure γ on a set X. 2.7.1. Proposition. Fil can be embedded into SUConv as a bireﬂective and bicoreﬂective subconstruct. Proof. The construct Fil-D-SUConv of all Fil-determined semiuniform convergence spaces (and uniformly continuous maps) is concretely isomorphic to Fil: (1) As above, γJγ = γ for each Fil-structure γ, and JX = JγJX for each Fil-D-SUConv-structure JX . (2) If f : (X, γ) → (X , γ ) is a Fil-morphism, then f : (X, Jγ ) → (X , Jγ ) is uniformly continuous (note: for each ﬁlter G on X, f × f (G × G) = f (G) × f (G)). (3) If f : (X, JX ) → (X , JX ) is a SUConv-morphism, then f : (X, γJX ) → (X , γJX ) is a Fil-morphism (shortly: every uniformly continuous map is Cauchy continuous). It follows from (1), (2), (3) that Fil-D-SUConv is concretely isomorphic to Fil. Fil-D-SUConv (∼ = Fil) is bicoreﬂective in SUConv: Let (X, JX ) ∈ |SUConv|. Then (X, JγJX ) is the bicoreﬂective modiﬁcation of (X, JX ) w.r.t. Fil-D-SUConv. (X, JγJX ) (or (X, γJX )) is called the underlying ﬁlter space of the semiuniform convergence space (X, JX ). Fil-D-SUConv is bireﬂective in SUConv since it is closed under formation of subspaces and product spaces (formed in SUConv) and contains all indiscrete SUConvobjects (cf. [88, 2.3.3.6] for the details). A Kent convergence spaces is called symmetric iﬀ (F, x) ∈ q and y ∈ F ∈F F imply (F, y) ∈ q. If (X, γ) is a ﬁlter space, then its underlying symmetric Kent convergence space (X, qγ ) is deﬁned by (F, x) ∈ qγ iﬀ F ∩ x˙ = {F ∪ {x} : F ∈ F} ∈ γ. A ﬁlter space (X, γ) is called complete iﬀ each F ∈ γ converges in (X, qγ ), i.e., there is some x ∈ X such that F ∩ x˙ ∈ γ. 2.7.2. Proposition. The construct KConvs of symmetric Kent convergence spaces (and continuous maps) can be embedded into Fil as a bicoreﬂective subconstruct. Proof. KConvs is concretely isomorphic to the construct C-Fil of complete ﬁlter spaces (and Cauchy continuous maps):

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(1) For each KConv-structure q on a set X a C-Fil-structure γq is deﬁned by F ∈ γq iﬀ there is some x ∈ X such that (F, x) ∈ q. Then qγq = q for each KConvs -structure q, and γqγ = γ for each C-Fil-structure γ. (2) If f : (X, γ) → (X , γ ) is a Cauchy continuous map between ﬁlter spaces, then f : (X, qγ ) → (X , qγ ) is continuous. (3) If f : (X, q) → (X , q ) is a continuous map between Kent convergence spaces, then f : (X, γq ) → (X , γq ) is Cauchy continuous. It follows from (1), (2), (3) that KConvs is concretely isomorphic to C-Fil. C-Fil is bicoreﬂective in Fil: Let (X, γ) ∈ |Fil|. Then (X, γc ) is the bicoreﬂective modiﬁcation of (X, γ) w.r.t. C-Fil, where γc = {F ∈ γ : there is some x ∈ X with F ∩ x˙ ∈ γ}. 2.7.3. Proposition. Let A be one of the topological constructs Top, PrTop, PsTop, Lim, or KConv. Then the full subconstruct As of symmetric A-objects is bireﬂective in A. Proof. Let X be a set, ((Xi , qi ))i∈I a family of As -objects and (fi : X → on X w.r.t. (fi ), Xi )i∈I a family of maps. If q denotes the initial A-structure then (X, q) is symmetric: If (F, x) ∈ q and y ∈ F ∈F F , then fi (y) ∈ fi [F ] for each i ∈ I and each F ∈ F, i.e., fi (y) ∈ H∈fi (F ) H for each i ∈ I, which implies that (fi (F), fi (y)) ∈ qi for each i ∈ I, since each (Xi , qi ) is symmetric and (fi (F), fi (x)) ∈ qi for each i ∈ I. Thus (F, y) ∈ q. By Proposition 2.4.3, As is bireﬂective in A. 2.7.4. Proposition. Each of the constructs in the following list is bireﬂective in the preceding ones: KConvs ⊃ Lims ⊃ PsTops ⊃ PrTops ⊃ Tops . Proof. Let (X, q) ∈ KConvs and (X, q) its bireﬂective modiﬁcation w.r.t. Lim (cf. 2.5.4). By 2.7.3, there is a bireﬂective modiﬁcation (X, ( q)s ) of (X, q) w.r.t. Lims . Thus, (X, ( q)s ) is the bireﬂective modiﬁcation of (X, q) w.r.t. Lims . The remaining cases are proved similarly. SUConv O iSSScS r S r SSS SULim Fil iSSSS O SScSS r KConv ULim O O s r

Unif

r

Lim O s r

PsTops O r

PrTops O r

Tops Figure 1. A summary of the relations between various subconstructs of SUConv, where r (resp. c) stands for bireﬂective (resp. bicoreﬂective) embedding

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2.8. Filter spaces and merotopic spaces. 2.8.1. Definition. A merotopic space is a pair (X, µ), where X is a set and µ is a non-empty set of non-empty covers of X such that the following are satisﬁed: • A ≺ B (i.e., for each A ∈ A, there is some B ∈ B such that A ⊂ B) and A ∈ µ imply B ∈ µ. • A ∈ µ and B ∈ µ imply A ∧ B = {A ∩ B : A ∈ A and B ∈ B} ∈ µ. If (X, µ) is a merotopic space, then the elements of µ are called uniform covers. Let (X, µ) be a merotopic space. Then: • A set C of subsets of X is called a Cauchy system in (X, µ) iﬀ for each A ∈ µ there are some A ∈ A and some C ∈ C such that C ⊂ A. • A ﬁlter on X is called a Cauchy ﬁlter in (X, µ) iﬀ it is a Cauchy system. • (X, µ) is called a ﬁlter-merotopic space iﬀ each Cauchy system C in (X, µ) is coreﬁned by some Cauchy ﬁlter F in (X, µ), i.e., for each F ∈ F there is some C ∈ C such that C ⊂ F . A map f : (X, µ) → (Y, η) between merotopic spaces is called uniformly continuous provided that f −1 A = {f −1 [A] : A ∈ A} ∈ µ for each A ∈ η. 2.8.2. Example. The construct Mer of merotopic spaces (and uniformly continuous maps) is topological, where initial and ﬁnal structures are formed as follows: If X is a set, ((Xi , µi ))i∈I a family of merotopic spaces and (fi : X → Xi )i∈I a family of maps, then µ = {A : A is a cover of X, and there are ﬁnitely many elements A1 , . . . , An of µ with A1 ∧ · · · ∧ An ≺ A} is the initial Mer-structure on X w.r.t. (fi ), where µ = {fi−1 Ai : Ai ∈ µi and i ∈ I}. If X is a set, ((Xi , µi ))i∈I a family of merotopic spaces and (fi : Xi → X)i∈I a family of maps, then µ = {A : A is a cover of X and fi−1 A ∈ µi for each i ∈ I} is the ﬁnal Mer-structure w.r.t. (fi ). 2.8.3. Proposition. The construct Fil-Mer of ﬁlter merotopic spaces (and uniformly continuous maps) is concretely isomorphic to Fil. Proof. If (X, γ) is a ﬁlter space, then (X, µγ ) is a ﬁlter-merotopic space, where µγ = {A : A is a cover of X and for each F ∈ γ there is some A ∈ A with A ∈ F}. If (X, µ) is a ﬁlter-merotopic space, then (X, γµ ) is a ﬁlter space, where γµ is the set of all Cauchy ﬁlters in (X, µ). The following are valid: • µγµ = µ for each Fil-Mer-structure µ on a set X, • γµγ = γ for each Fil-structure γ on a set X. If f : (X, γ) → (X , γ ) is a Cauchy continuous map between ﬁlter spaces, then f : (X, µγ ) → (X , µγ ) is uniformly continuous. If f : (X, µ) → (X , µ ) is a uniformly continuous map between merotopic spaces, then f : (X, γµ ) → (X , γµ ) is Cauchy continuous. 2.8.4. Proposition. Fil-Mer is bicoreﬂective in Mer. Proof. If (X, µ) ∈ |Mer|, then (X, µγµ) is the bicoreﬂective modiﬁcation of (X, µ).

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3. Convenient properties of topological constructs 3.1. Existence of natural function spaces. 3.1.1. Definition. A topological construct C is called cartesian closed iﬀ for any pair (A, B) ∈ |C| × |C|, the set [A, B]C of all C-morphisms from A to B can be endowed with the structure of a C-object denoted by B A (and called power object or natural function space) such that the following are satisﬁed: (a) The evaluation map eA,B : A × B A → B deﬁned by eA,B (a, g) = g(a) for each (a, g) ∈ A × B A is a C-morphism. (b) For each C-object C and each C-morphism f : A×C → B the map f¯: C → B A deﬁned by f¯(c)(a) = f (a, c), i.e., such that the following diagram commutes, is a C-morphism. eA,B /B A × B AeK y< KK y y KK yy KK yy f 1A ×f¯ KK y y A×C

3.1.2. Proposition. Let C be a cartesian closed topological construct. If A and B are C-objects and ([A, B]C , ξ) is the natural function space, then ξ is the coarsest C-structure for which the evaluation map eA,B : A × ([A, B]C , ξ) → B is a C-morphism. Proof. Let ξ be a C-structure on [A, B]C such that the evaluation map eA,B : A × ([A, B]C , ξ ) → B is a C-morphism. Since C is cartesian closed, it follows that eA,B : ([A, B]C , ξ ) → ([A, B]C , ξ) is a C-morphism such that the diagram commutes. eA,B / A × ([A, B]C , ξ) r8 B iRRR r r RRR rrr RRR rerA,B r r 1A ×eA,B RRRR rr A × ([A, B]C , ξ )

Obviously, eA,B = 1[A,B]C , i.e., ξ ≤ ξ.

3.1.3. Proposition. Let C be a cartesian closed topological construct, and A a bicoreﬂective subconstruct which is closed under formation of ﬁnite products in C. Then A is cartesian closed, and the natural function spaces in A arise from the natural function spaces in C by bicoreﬂective modiﬁcation. Proof. Let A, B be A-objects and ([A, B], ξ) the natural function space in C, where [A, B] = [A, B]A = [A, B]C . If eA,B : A × ([A, B], ξ) → B is the evaluation map, which is a C-morphism by assumption, and ([A, B], ξA ) the bicoreﬂective modiﬁcation of ([A, B], ξ) w.r.t. A, then e[A,B] ◦ (1A × 1[A,B] ) : A × ([A, B], ξA ) → B is easily seen to be the evaluation map fulﬁlling (a) and (b) of Deﬁnition 3.1.1 w.r.t. A, where the identity map 1[A,B] : ([A, B], ξA ) → ([A, B], ξ) is a C-morphism. 3.1.4. Remark. It is easily checked that a (full and isomorphism-closed) subconstruct A of a cartesian closed topological construct C which is closed under formation of ﬁnite products and natural function spaces in C is cartesian closed. In particular, if A is a bireﬂective subconstruct of a cartesian closed topological construct C which is closed under formation of natural function spaces in C, then

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A is a cartesian closed topological construct, and the natural function spaces in A are formed as in C. 3.2. Examples of natural function spaces. 3.2.1. Remark. Top and Unif are not cartesian closed. (cf. the introduction.) 3.2.2. Example. SUConv is cartesian closed, where the natural function spaces (= power objects) are formed as follows. Let X = (X, JX ) and Y = (Y, JY ) be semiuniform convergence spaces. Then the power object YX is the set [X, Y]SUConv endowed with the SUConv structure JX,Y = {Φ ∈ F ([X, Y]SUConv × [X, Y]SUConv : Φ(F) ∈ JY for each F ∈ JX }, where Φ(F) is the ﬁlter generated by the ﬁlter base {A(F ) : A ∈ Φ, F ∈ F} with A(F ) = {(f (a), g(b)) : (f, g) ∈ A, (a, b) ∈ F }. JX,Y is called the uniformly continuous SUConv-structure. JX,Y is a SUConv-structure: (UC1 ). f˙ × f˙ ∈ JX,Y for each f ∈ [X, Y]SUConv because f˙ × f˙(F) = f × f (F) ∈ JY for each F ∈ JX . (UC2 ) follows immediately from Φ(F) ⊂ Θ(F) for all Φ, Θ ∈ F ([X, Y]SUConv × [X, Y]SUConv ) with Φ ⊂ Θ and all F ∈ F (X × X). (UC3 ). Let Φ ∈ JX,Y . Then Φ−1 (F) = (Φ(F −1 ))−1 for each F ∈ JX [obviously, A−1 (F ) = (A(F −1 ))−1 for all A ∈ Φ and all F ∈ F]. Since F −1 ∈ JX , Φ(F −1 ) ∈ JY . Thus, (Φ(F −1 ))−1 ∈ JY because Y fulﬁlls (UC3 ). Consequently, Φ−1 ∈ JX,Y . The evaluation map eX,Y : X × ([X, Y]SUConv , JX,Y ) → Y is uniformly continuous because eX,Y × eX,Y (F × Φ) = Φ(F) (note that we do not distinguish between (X × [X, Y]) × (X × [X, Y]) and (X × X) × ([X, Y] × [X, Y]) and that F × Φ is generated by {F × A : F ∈ F, A ∈ Φ}). Let Z = (Z, JZ ) ∈ |SUConv| and f : X × Z → Y be a uniformly continuous map. Then f¯: Z → ([X, Y]SUConv , JX,Y ) deﬁned by f¯(z)(x) = f (x, z) for all x ∈ X and all z ∈ Z is uniformly continuous too, because by assumption (f¯ × f¯(G))(F) = f × f (F × G) ∈ JY for all G ∈ JZ and all F ∈ JX , i.e., f¯ × f¯(G) ∈ JX,Y for all G ∈ JZ . 3.2.3. Example. Fil is cartesian closed. Since Fil can be embedded into SUConv as a bicoreﬂective and bireﬂective subconstruct (cf. 2.7.1), it follows from Proposition 3.1.3 that Fil is cartesian closed, and the natural function spaces arise from the natural function spaces in SUConv by bicoreﬂective modiﬁcation, i.e., if X = (X, γ) and X = (X , γ ) are ﬁlter spaces, then the natural function space structure γ on [X, X ]Fil is given by Θ∈γ iﬀ eX,X (F × Θ) ∈ γ for each F ∈ γ. Thus, according to Proposition 3.1.2, it is the coarsest Fil-structure on [X, X ]Fil such that the evaluation map eX,X is Cauchy continuous, and it is called the Cauchy continuous Fil-structure. 3.2.4. Example. KConvs (∼ = C-Fil) is cartesian closed. KConvs is bicoreﬂectively embedded into Fil (cf. 2.7.2), and closed under formation of products in Fil. Let ((Xi , γi ))i∈I be a family of complete ﬁlter spaces, and (X, γ) their product in Fil. If F ∈ γ, then pi (F) ∈ γi for each i ∈ I, where

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pi : X → Xi denotes the i-th projection. Since (Xi , γi ) is complete, for each i ∈ I, there is some xi ∈ Xi with (pi (F), xi ) ∈ qγi . Thus, (F, x) ∈ qγ with x = (xi )i∈I . Therefore, by Proposition 3.1.3, KConvs is cartesian closed, and for X = (X, q), X = (X , q ) ∈ |KConvs |, the natural function space structure q ∗ on [X, X ]KConvs is given by (ψ, f ) ∈ q ∗ iﬀ eX,X (F × (ψ ∩ f˙)) converges in (X , q ) for each convergent ﬁlter F in (X, q). (Note: q ∗ = qγb with γ deﬁned as under 3.2.3.) 3.2.5. Example. Lim is cartesian closed. Let X = (X, q) and X = (X , q ) be limit spaces; then the natural function space X X is the set [X, X ]Lim endowed with the Lim-structure q deﬁned by (ψ, f ) ∈ q iﬀ (ψ(F), f (x)) ∈ q for each (F, x) ∈ q, where ψ(F) = eX,X (F × ψ), i.e., ψ(F) is generated by the ﬁlter base {A(F ) : A ∈ ψ, F ∈ F} with A(F ) = {f (z) : f ∈ A, z ∈ F }. q is called the Lim-structure of continuous convergence. q is a Lim-structure: (C1 ). (f˙, f ) ∈ q: Let (F, x) ∈ q. Since eX,X (F × f˙) = f (F), we obtain from the continuity of f that (eX,X (F × f˙), f (x)) ∈ q . (C2 ). Let (ψ, f ) ∈ q and ψ ⊃ ψ. Then for any (F, x) ∈ q, eX,X (F × ψ ) ⊃ eX,X (F × ψ). Since (eX,X (F × ψ), f (x)) ∈ q we have (eX,X (F × ψ ), f (x)) ∈ q . (C4 ). If (ψ, f ) ∈ q and (ψ , f ) ∈ q, i.e., (ψ(F), f (x)) ∈ q and (ψ (F), f (x)) ∈ q for each (F, x) ∈ q, it follows from ψ(F) ∩ ψ (F) = (ψ ∩ ψ )(F) [note: (A ∪ B)(F ) = A(F ) ∪ B(F ) for all A ∈ ψ, B ∈ ψ and F ∈ F] that ((ψ ∩ ψ )(F), f (x)) ∈ q for each (F, x) ∈ q, because (X , q ) fulﬁlls (C4 ). Thus, (ψ ∩ ψ , f ) ∈ q. It follows immediately from the deﬁnition of q that the evaluation map eX,X : X × ([X, X ]Lim , q) → X is continuous. Let X = (X , q ) be a limit space and f : X × X → X a continuous map. Then f¯: X → ([X, X ], q) deﬁned by f¯(x )(x) = f (x, x ) is continuous: Let ¯ )) ∈ q, we have to show that (G, x ) ∈ q . In order to prove that (f¯(G), f(x ¯ ¯ (eX,X (F × f (G)), f (x )(x)) ∈ q whenever (F, x) ∈ q. Since eX,X (F × f¯(G)) = f (F × G) and f¯(x )(x) = f (x, x ), this follows immediately from the continuity of f . 3.2.6. Example. Lims is cartesian closed. If X = (X, q) is a limit space and X = (X , q ) a symmetric limit space, then the Lim-structure q of continuous convergence on [X, X ]Lim is easily seen to be a Lims -structure. Hence, Lims is closed under formation of natural function spaces in Lim, and since Lim is cartesian closed (by 3.2.5) and contains Lims as a bireﬂective subconstruct (cf. 2.7.3), it follows from 3.1.4 that the natural function space structure in Lims is the structure of continuous convergence. 3.3. The role of natural function spaces in SUConv. 3.3.1. Proposition. Let X = (X, q) be a symmetric Kent convergence space and X = (X , q ) a symmetric limit space. Then the natural function space structure q ∗ on [X, X ]KConvs is the structure of continuous convergence. q

Proof. ⇒. Let (F, x) ∈ q. Then F = F ∩ x˙ →x. By assumption, there is some q that eX,X (F × ψ )→x where ψ = ψ ∩ f˙. Since (X , q ) is symmetric x ∈ X such q and f (x) ∈ {H : H ∈ eX,X (F × ψ )} it follows that eX,X (F × ψ )→f (x). q Because of eX,X (F × ψ ) ⊂ eX,X (F × ψ) we obtain eX,X (F × ψ)→f (x).

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⇐. (eX,X (F × ψ)) ∩ f (F) = eX,X (F × ψ ) which is easily veriﬁed. Let (F, x) ∈ q. Since f is continuous, it follows that (f (F), f (x)) ∈ q . By asumption, q eX,X (F × ψ)→f (x). Since (X , q ) is a limit space, we obtain (eX,X (F × ψ)) ∩ q q f (F)→f (x). Thus eX,X (F × ψ )→f (x). It follows from the above examples together with Proposition 3.3.1 that the Cauchy continuous Fil-structure as well as the structure of continuous convergence in Lims can be derived from the uniformly continuous SUConv-structure. Let (Y, V) be a uniform space and (Y, [V]) its corresponding principal uniform limit space. Furthermore, let (X, (∆X )) be a discrete uniform space, i.e., (∆X ) = {W ⊂ X × X : ∆X ⊂ W }, and (X, [(∆X )]) its corresponding principal uniform limit space. Then [(X, [(∆X )]), (Y, [V])]SUConv = Y X , i.e., the set of all maps from X into Y , and the uniformly continuous SUConv-structure JX,Y on Y X is equal to [U], where U is the usual uniformity of uniform convergence whose base is given by {W (V ) : V ∈ V} with W (V ) = {(f, g) ∈ Y X × Y X : f (x), g(x)) ∈ V for each x ∈ X}. In other words: (Y X , JX,Y ) is a uniform space (= principal uniform limit space). Let X be a set and JX = {x˙ × x˙ : x ∈ X}, i.e., JX is the discrete SUConvstructure on X. If (Y, JY ) is a semiuniform convergence space, then [(X, JX ), (Y, JY )]SUConv = Y X , and for the uniformly continuous SUConv-structure JX,Y on this set one obtains JX,Y = {Φ ∈ F (Y X × Y X ) : (px × px )(Φ) ∈ JY for each x ∈ X}, where px : Y X → Y denotes the x-th projection (note: Φ(x˙ × x) ˙ = px × px (Φ) for each Φ ∈ F (Y X × Y X )), i.e., JX,Y is the product structure (in SUConv) on Y X , which is called the structure of simple convergence. Thus, the structure of uniform convergence as well as the structure of simple convergence can also be derived from the uniformly continuous SUConv-structure. 3.3.2. Remark. Let C be a cartesian closed topological construct, and let A, B, C be C-objects. If ([A, B]C , ξ) denotes the natural function space B A , then the map − : [A × C, B]C → [C, ([A, B]C , ξ)]C deﬁned by −(f ) = f¯ is bijective (use Deﬁnition 3.1.1 (a) and (b)). That is, we need not distinguish between [A × C, B]C and [C, ([A, B]C , ξ)]C in a cartesian closed topological construct. This property is highly appreciated by mathematicians working e.g. in the ﬁeld of Algebraic Topology (Homotopy Theory) or Functional Analysis (Duality Theory). In this sense cartesian closedness is a convenient property for topological constructs. But there are also other convenient properties such as ‘extensionality’ and ‘productivity of quotients’. 3.4. Existence of one-point extensions. 3.4.1. Definition. In a topological construct C, a partial morphism from A to B is a C-morphism f : C → B whose domain is a subspace of A. A topological construct C is called extensional (or hereditary) provided that every C-object B has a one-point extension B ∗ ∈ |C|, i.e., every B ∈ |C| can be

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embedded via the addition of a single point ∞B into a C-object B ∗ such that, for every partial morphism f : C → B from A to B, the map f ∗ : A → B ∗ , deﬁned by f (a) if a ∈ C, f ∗ (a) = ∞B if a ∈ C, is a C-morphism. A topological construct C is called • strongly cartesian closed provided that it is cartesian closed and in C products of quotient maps are quotient maps. • a topological universe provided that it is cartesian closed and extensional. • a strong topological universe provided that it is strongly cartesian closed and extensional. 3.4.2. Remark. In every cartesian closed topological construct quotients are ﬁnitely productive (cf. e.g. [88, 3.1.3]), but there are cartesian closed topological constructs which are neither strong nor extensional, e.g., the construct Chy of Cauchy spaces (and Cauchy continuous maps) where a ﬁlter space (X, γ) is called a Cauchy space provided that the following is satisﬁed: (C) If F and G belong to γ such that every member of F meets every member of G, then F ∩ G belongs to γ. (cf. [9].) 3.4.3. Definition. A family (fi : (Xi , ξi ) → (X, ξ))i∈I of morphisms indexed by some class I in a topological construct C is called a ﬁnal sink in C provided that ξ is the ﬁnal C-structure w.r.t. (fi ). If additionally X = i∈I fi [X] then (fi : (Xi , ξi ) → (X, ξ))i∈I is called a ﬁnal epi-sink in C. A ﬁnal sink (fi : (Xi , ξi ) → (X, ξ))i∈I in C is called hereditary iﬀ the following is satisﬁed: If Y is a subset of X and (Y, η) the subspace of (X, ξ) determined by Y , (Yi , ηi ) the subspace of (Xi , ξi ) determined by Yi = fi−1 [Y ] and gi : (Yi , ηi ) → (Y, η) the corresponding restriction of fi , then (gi : (Yi , ηi ) → (Y, η))i∈I is a ﬁnal sink in C too. 3.4.4. Remark. If in a topological construct C ﬁnal sinks are hereditary, then • quotient maps in C are hereditary, i.e., for every quotient map f : (X, ξ) → (X, η) in C the ﬁnal epi-sink (f : (X, ξ) → (Y, η)) in C is hereditary, and • sums in C are hereditary, i.e., for every sum space i∈I (Xi , ξi ) in C the ﬁnal epi-sink (ji : (Xi , ξi ) → i∈I (Xi , ξi ))i∈I of injections is hereditary in C. 3.4.5. Theorem (cf. e.g. [88, 3.2.2 and 3.2.3]). For a topological construct C the following are equivalent: (1) (2) (3) (4)

C is extensional. Final sinks in C are hereditary. Final epi-sinks in C are hereditary. Quotient maps and sums in C are hereditary.

Let C be an extensional topological construct, and let (Y, η) ∈ |C|. If (Y ∗ , η ∗ ) denotes the one-point extension of (Y, η), then η ∗ is the coarsest C-structure on Y ∗ = Y ∪ {∞Y } such that (Y, η) is a subspace of (Y ∗ , η ∗ ). (If (Y, η) is a subspace

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of (Y ∗ , η ∗∗ ) ∈ |C|, then 1Y : (Y, η) → (Y, η) is a partial morphism from (Y ∗ , η ∗∗ ) to (Y, η). Hence (1Y )∗ = 1Y ∗ : (Y ∗ , η ∗∗ ) → (Y ∗ , η ∗ ) is a C-morphism, i.e., η ∗∗ ≤ η ∗ .) The following results are similar to the results 3.1.3 and 3.1.4 for cartesian closedness. 3.4.6. Proposition. Let A be a bicoreﬂective subconstruct of an extensional topological construct C which is closed under formation of subspaces in C. Then A is extensional, and the one-point extensions in A arise from the corresponding one-point extensions in C by bicoreﬂective modiﬁcation. (cf. e.g. [88, 3.2.5].) 3.4.7. Proposition. Let A be a bireﬂective subconstruct of an extensional topological construct C which is closed under formation of one-point extensions in C. Then A is extensional, and the one-point extensions in A are formed as in C. (cf. e.g. [88, 3.2.6].) 3.5. Examples of strong topological universes. 3.5.1. Remark. Top does not fulﬁll any convenient property: (1) Top is not cartesian closed (cf. (1) of the Introduction). (2) Top is not extensional, since in Top quotients are not hereditary (cf. (2) of the Introduction). (3) In Top quotients are not productive (cf. (3) of the Introduction). 3.5.2. Example. SUConv is a strong topological universe. (1) SUConv is cartesian closed (cf. 3.2.2). (2) SUConv is extensional. (3) In SUConv quotients are productive. Proof of 3.5.2 (2). Let (X, JX ) be a semiuniform convergence space. Put X ∗ = X ∪ {∞X } with ∞X ∈ X. For each M ⊂ X ∗ × X ∗ , let M ∗ = M ∪ (X ∗ × {∞X }) ∪ ({∞X } × X ∗ ). An SUConv-structure JX ∗ on X ∗ is deﬁned by JX ∗ = {F ∈ F (X ∗ × X ∗ ) : the trace of F on X × X exists and belongs to JX or {(∞X , ∞X )}∗ ∈ F}. Then (X ∗ , JX ∗ ) is the one-point extension of (X, JX ).

Proof of 3.5.2 (3). Let (fi : (Xi , JXi ) → (Yi , JYi ))i∈I be a non-empty family of quotient maps in SUConv indexed by some set I, and let Q

(X, JX )

fi

pi

pi

(Xi , JXi )

/ (Y, JY )

fi

/ (Yi , JY ) i

be thecorresponding product diagram in SUConv, where i∈I (Xi , JXi ) = (X, JX ) and ( i∈I (Yi , JYi ) = (Y, JY ). Since all fi are surjective, fi is surjective. For each i ∈ I, JYi = {F ∈ F (Yi × Yi ) : there exists Gi ∈ JXi with (fi × fi )(Gi ) ⊂ F}, because fi is a quotient map. Let JY = {K ∈ F (Y × Y ) : there exists G ∈ JX with ( fi × fi )(G) ⊂ K}. Then JY = {G ∈ F (Y × Y ) : pi × pi (G) ∈ JYi for each i ∈ I} is equal to JY , i.e., fi is a quotient map:

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If K ∈ JY , then there exists G ∈ JXwith ( fi × fi )(G) ⊂ K. Furthermore, (fi × fi )((pi × pi )(G)) = pi × pi (( fi × fi )(G)) ⊂ pi × pi (K) for each i ∈ I, which implies pi × pi (K) ∈ JYi for each i ∈ I, i.e., K ∈ JY . If G ∈ JY , then (pi × pi )(G) ∈ JYi for each i ∈ I. Thus, for each i ∈ I, there is some G ∈ J with (f × f )(G ) ⊂ (p × p )(G). Let product i X i i i i i i Gi be the Xi × Xi , where i∈I Xi × Xi isidentiﬁed with i∈I X × ﬁlter on i∈I i i∈I Xi . Then p × p ( G ) = G for each i ∈ I, i.e., G ∈ J . Since Y × Y i i i i i X i i and i∈I Y × Y can be identiﬁed, we obtain i i i∈I i∈I fi × fi Gi ⊂ (fi × fi )(Gi ) ⊂ (pi × pi )(G) ⊂ G. Hence, G ∈ JY

3.5.3. Example. Fil is a strong topological universe. (1) Fil is cartesian closed (cf. 3.2.3). (2) Fil is extensional. (3) In Fil quotient maps are productive. Proof of 3.5.3 (2). Since Fil can be embedded into SUConv as a bireﬂective and bicoreﬂective subconstruct (cf. 2.7.1) it follows from 3.4.6 that Fil is extensional, and the one-point extensions in Fil arise from the one-point extensions in SUConv by bicoreﬂective modiﬁcation, i.e., if (X, γ) is a ﬁlter-space, (X, Jγ ) its corresponding semiuniform convergence space, and (X ∗ , Jγ∗ ) the one-point extension of (X, Jγ ) in SUConv, then (X ∗ , γ ∗ ) is the one-point extension of (X, γ) in Fil, where ˙ X , i−1 (F) ∈ γ} ∪ {∞ ˙ X }. γ ∗ = γJγ∗ = {F ∈ F (X ∗ ) : F = ∞ (i : X → X ∗ denotes the inclusion map.)

Proof of 3.5.3 (3). This is obvious, since in Fil quotient maps and product spaces are formed as in SUConv, because Fil is bicoreﬂective and bireﬂective in SUConv, and 3.5.2 (3) is valid. 3.6. The aim of Convenient Topology. The so-called Convenient Topology consists in the study of strong topological universes in which convergence structures and uniform convergence structures are available. Furthermore, such a strong topological universe should not be too big. The strong topological universe SUConv of semiuniform convergence spaces fulﬁlls these criteria. Thus, in the realm of Convenient Topology we are mainly concerned with semiuniform convergence spaces or, more exactly, with the study of SUConv-invariants, i.e., properties of semiuniform convergence spaces which are preserved by isomorphisms in SUConv. This includes the study of full and isomorphism-closed subconstructs of SUConv (such as Unif and Tops as well as all the other subconstructs of SUConv according to Figure 1 of Section 2).

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4. The structural behaviour of SUConv-invariants 4.1. Deﬁnitions. A semiuniform convergence space (X, JX ) is called • a T1 -space iﬀ its underlying symmetric Kent convergence space (X, qγJX) ˙ y) ∈ qγJX implies is a T1 -space, i.e., iﬀ for each pair (x, y) ∈ X × X, (x, x = y. • a T2 -space (or separated ) iﬀ the underlying Kent convergence space (X, qγJX) is a T2 -space, i.e., each ﬁlter F on X converges to at most one element of X in (X, qγJX). • regular provided that for each F ∈ JX the subﬁlter F¯ generated by the ﬁlter base {F¯ : F ∈ F} belongs to JX , where F¯ denotes the closure of F in the underlying (symmetric) Kent convergence space of the product space (X, JX ) × (X, JX ). Note: For a Kent convergence space (X, q), the closure A¯ of a subset A of X is deﬁned by A¯ = {x ∈ X: there is some G ∈ F (X) with (G, x) ∈ q and A ∈ G}. • t-regular provided that for each F ∈ JX , the subﬁlter F¯ t , generated by the ﬁlter base of all closed elements of F, belongs to JX . Note: If (X, q) is a Kent convergence space, then a subset A of X is ¯ called closed iﬀ A = A. • completely regular iﬀ it is regular, and its underlying ﬁlter space (X, γJX ) is completely regular, i.e., (X, γJX ) is a subspace in Fil of some completely regular topological space (Z, Z) (regarded as a ﬁlter space, which means that the Cauchy ﬁlters on Z are exactly the convergent ﬁlters in (Z, Z)). • normal provided that it is regular and its underlying ﬁlter space (X, γJX ) is m-normal: – A ﬁlter space (X, γ) is called m-normal iﬀ its corresponding (ﬁlter-) merotopic space (X, µγ ) is normal. – A merotopic space (X, µ) is called normal iﬀ it is regular, and the merotopic space (X, µc ) is regular, where µc = {U ∈ µ : there is some ﬁnite V ∈ µ with V ≺ U}. – A merotopic space (X, µ) is called regular iﬀ for each U ∈ µ there is some (reﬁnement) V ∈ µ such that for each V ∈ V there exists some U ∈ U with {X \ V, U } ∈ µ. • fully normal provided that it is regular and its underlying ﬁlter space (X, γJX ) is m-uniform, where a ﬁlter space (X, γ) is called m-uniform iﬀ its corresponding (ﬁlter-) merotopic space (X, µγ ) is uniform, i.e., for each A ∈ µγ there is some B ∈ µγ such that for each B ∈ B there is some A ∈ A with St(B, B) ⊂ A, where St(B, B) = {B ∈ B : B ∩ B = ∅} (denoted by B ≺∗ A). • paracompact iﬀ it is a fully normal T1 -space. • topological iﬀ there is a symmetric topology X on X such that JX = {F ∈ F (X × X) : there is some x ∈ X with UX (x) × UX (x) ⊂ F}, where UX (x) denotes the neighborhood ﬁlter of x w.r.t. X , and UX (x) × UX (x) is the product ﬁlter. • subtopological iﬀ it is Fil-determined, and its corresponding ﬁlter space (X, γJX ) is a subspace (in Fil) of some symmetric topological space (regarded as a ﬁlter space).

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• sub-(compact Hausdorﬀ ) iﬀ it is Fil-determined, and its corresponding ﬁlter space (X, γJX ) is a subspace (in Fil) of some compact Hausdorﬀ space (regarded as a ﬁlter space). • diagonal provided that the ﬁlter (∆X ) generated by the diagonal ∆X = {(x, x) : x ∈ X} of X × X belongs to JX , if the ﬁlter (∆X ) exists. • uniform iﬀ there is a uniformity W on X such that JX = {F ∈ F (X ×X) : F ⊃ W}, i.e., iﬀ (X, JX ) is a principal uniform limit space. • compact iﬀ its underlying (symmetric) Kent convergence space (X, qγJX) is compact, i.e., each ultraﬁlter U on X converges in (X, qγJX). • locally compact iﬀ one of the following two equivalent conditions is satisﬁed: – Each F ∈ JX contains a compact subset of the product space (X, JX )× (X, JX ), where a subset A of a semiuniform convergence space (Y, JY ) is called compact provided that the subspace (A, JA ) of (Y, JY ) determined by A is compact. – (X, JX ) is compactly generated, i.e., JX is the ﬁnal SUConv-structure w.r.t. the family (ji : (Ki , JKi ) → (X, JX ))i∈I of the inclusions of all compact subspaces of (X, JX ). • precompact iﬀ each ultraﬁlter U on X is a JX -Cauchy ﬁlter. • locally precompact iﬀ one of the following two equivalent conditions is satisﬁed: – Each F ∈ JX contains a precompact subset of the product space (X, JX ) × (X, JX ), where a subset A of a semiuniform convergence space (Y, JY ) is called precompact provided that the subspace (A, JA ) of (Y, JY ) determined by A is precompact. – (X, JX ) is precompactly generated, i.e., JX is the ﬁnal SUConv-structure w.r.t. the family (ji : (Ci , JCi ) → (X, JX ))i∈I of the inclusions of all precompact subspaces of (X, JX ). ˙ • connected iﬀ each uniformly continuous map f : (X, JX ) → ({0, 1}, {0˙ × 0, ˙ from (X, JX ) into the two-point discrete semiuniform convergence 1˙ × 1}) space is constant. • uniformly connected iﬀ each uniformly continuous map f : (X, JX ) → D2∆ from (X, JX ) into the two-point discrete uniform space D2∆ (regarded as a principal uniform limit space) is constant. D2∆ = ({0, 1}, [(∆)]), where [(∆)] is the set of all ﬁlters on {0, 1} × {0, 1} containing the ﬁlter (∆) generated by the diagonal ∆ of {0, 1} × {0, 1}. • locally connected (resp. locally uniformly connected) iﬀ for each F ∈ JX there is some subﬁlter G ∈ JX of F together with a ﬁlter base B for G consisting of connected (resp. uniformly connected) subsets of (X, JX ) × (X, JX ), where a subset A of a semiuniform convergence space (Y, JY ) is called connected (resp. uniformly connected) provided that the subspace (A, JA ) of (Y, JY ) determined by A is connected (resp. uniformly connected). • totally disconnected (resp. totally uniformly disconnected ) iﬀ there are no connected (resp. uniformly connected) subsets containing more than one point.

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• ﬁrst countable (in other words: (X, JX ) fulﬁlls the ﬁrst axiom of countability) provided that for each F ∈ JX there is a subﬁlter G ∈ JX of F such that G has a countable base. • metrizable provided that there is a metric d on X such that JX = {F ∈ F (X × X) : F ⊃ Wd }, where Wd is the uniformity induced by d, i.e., Wd has the base {Vε : ε > 0} with Vε = {(x, y) ∈ X × X : d(x, y) < ε}. • topologically metrizable provided that there is a metric d on X such that JX = {F ∈ F (X × X) : there is some x ∈ X with UXd (x) × UXd (x) ⊂ F}, where Xd is the topology induced by d (cf. also the deﬁnition of a topological seminuniform convergence space). • a convergence space iﬀ JX = {F ∈ F (X ×X) : there is some (G, x) ∈ qγJX with G × G ⊂ F}. • complete (resp. weakly complete) iﬀ each JX -Cauchy ﬁlter (resp. JX Cauchy ultraﬁlter) converges (in (X, qγJX)). 4.2. Structural survey of SUConv invariants. An SUConv-invariant P is called hereditary, productive, summable, divisible, initially closed or ﬁnally closed iﬀ the full and isomorphism-closed subconstruct A of SUConv deﬁned by |A| = {(X, JX ) ∈ |SUConv| : (X, JX ) satisﬁes P } is closed under formation of subspaces, product spaces, sum spaces, quotient spaces, initially closed or ﬁnally closed respectively. The structural behaviour of the SUConv-invariants in Deﬁnition 4.1 is summarized in Figure 2. 4.3. Better behaviour of subspaces and quotients in SUConv. 4.3.1. Remark. The formation of subspaces in SUConv is better behaved than the formation of subspaces in Top (or Tops ). For example: 4.3.2. Proposition. Subspaces (in SUConv) of paracompact topological spaces (regarded as semiuniform convergence spaces) are paracompact. 4.3.3. Proposition. Dense subspaces (in SUConv) of connected symmetric topological spaces (regarded as semiuniform convergence spaces) are connected, where a subset A in a semiuniform convergence space (X, JX ) is dense iﬀ its closure A¯ formed in (X, qγJX) is equal to X. Concerning 4.3.2, it is well-known that subspaces (formed in Top or Tops ) of paracompact topological spaces are not paracompact unless they are closed. If 4.3.2 is true, it becomes clear that this result can be reobtained from 4.3.2, since closed subspaces in Tops are formed as in SUConv (note: Tops is bireﬂective in KConvs , and KConvs ∼ = C-Fil is closed under formation of closed subspaces in Fil and hence in SUConv). In order to prove 4.3.2, let (X, X ) be a paracompact topological space, i.e., (X, X ) is T1 and (X, µX ) is uniform (where µX is the set of all covers of X which are reﬁned some open cover of (X, X )), and (X, JX ) its corresponding paracompact semiuniform convergence space, i.e., JX = JγqX with µγqX = µX , U ⊂ X, and (U, JU ) the subspace (in SUConv) of (X, JX ) determined by U . The following facts are easily checked: (1) (U, JU ) is regular, since (X, JX ) is regular. (2) (U, JU ) is a T1 -space, since (X, JX ) is a T1 -space. (3) Subspaces in Fil (∼ = Fil-Mer) are formed as in Mer.

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SUConv-invariants T1 T2 regular or t-regular completely regular normal fully normal or paracompact topological subtopological sub-compact Hausdorﬀ diagonal uniform compact locally compact = compactly generated precompact locally precompact = precompactly generated connected or uniformly connected locally connected or locally uniformly connected totally disconnected or totally uniformly disconnected ﬁrst countable

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It remains to prove that (U, γJU ) is m-uniform, i.e., (U, µγJU) is uniform. By (3), (U, µγJU ) is a subspace in Mer of (X, µγJX). Since (X, µγJX) is uniform by assumption, and Unif is bireﬂective in Mer (where for each merotopic space (Y, η) the bireﬂective modiﬁcation w.r.t. Unif is given by (Y, ηU ) with ηU = {A ∈ η : there is a sequence (An )n∈N in η such that A1 = A and An+1 ≺∗ An for each n ∈ N}), (U, µγJU ) is uniform. (Here an alternative description of uniform spaces by means of uniform covers is used, which was originated by J. W. Tukey [104], i.e., a uniform space is a merotopic space (X, µ) such that for each A ∈ µ there is some B ∈ µ with B ≺∗ A.)

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Concerning 4.3.3, the following theorem is useful: 4.3.4. Theorem (cf. [88, 5.1.29]). Let (X, X ) be a symmetric topological space and (X, JX ) its corresponding topological semiuniform convergence space, i.e., JX = {F ∈ F (X × X) : there is some x ∈ X with UX (x) × UX (x) ⊂ F}. Then each uniformly continuous map f : (A, JA ) → (Y, JY ) from a dense subspace (A, JA ) of (X, JX ) (formed in SUConv) into a complete, regular and separated semiuniform convergence space (Y, JY ) has a unique uniformly continuous extension f¯: (X, JX ) → (Y, JY ). Obviously, a subspace (S, JS ) of a connected semiuniform convergence space (X, JX ) is connected iﬀ each uniformly continuous map f : (S, JS ) → D2 from (S, JS ) into the two-point discrete semiuniform convergence space D2 has a uniformly continuous extension f: (X, JX ) → D2 . Since D2 is complete, regular and separated, 4.3.3 follows from the above theorem. 4.3.5. Example. Consider the topological space Rt of real numbers, and the subsets A = R \ [0, 1] and B = R \ {0} of the real line R. The subspaces in Top (or Tops ) of Rt determined by A and B are homeomorphic, but the subspaces in SUConv of Rt (regarded as a topological semiuniform convergence space) determined by A and B are non-isomorphic. Proof. The subspace determined by B is connected by 4.3.3 and the subspace determined by A is not connected (otherwise the closure R \ [0, 1] would be connected too and as a closed subspace of Rt topological, i.e., it would be an interval, which is not true). Since the point 0 and the closed unit interval [0, 1] are nonhomeomorphic, there should be a diﬀerence in removing them from the real line. In the realm of topological spaces this diﬀerence cannot be expressed. 4.3.6. Remark. In topological dimension theory it would be desirable that the dimension of a subspace A of a topological space X is less than or equal to the dimension of X. Concerning Lebesgue’s covering dimension dim such a result is not true, unless A is a closed subspace of a symmetric topological space X. The dimension dim can be enlarged to semiuniform convergence spaces, and the resulting ‘ﬁlter dimension’ dimf has the property that for each subspace A of a semiuniform convergence space X, dimf A ≤ dimf X. Thus, if subspaces of symmetric topological spaces are formed in SUConv, we obtain better results even in dimension theory (cf. [84] or [88, 7.4.28]). 4.3.7. Remark. The formation of quotient spaces in SUConv is better behaved than the formation of quotient spaces in Top (or Tops ) and in Unif as the following two propositions show. 4.3.8. Proposition. Quotient spaces in SUConv of ﬁrst countable symmetric topological spaces are ﬁrst countable. 4.3.9. Proposition. Let (X, JX ) be a uniform space (= principal uniform limit space), and let R be the equivalence relation on X deﬁned by x R y iﬀ x and y belong to the same uniformly connected subset of X (the equivalence classes w.r.t. R are called uniform components of (X, JX )). Then the quotient space (in SUConv) of (X, JX ) by R is totally uniformly disconnected.

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R {N} ..

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Figure 3. The topological quotient space Rt /N Concerning 4.3.8, let (X, X ) be a ﬁrst countable symmetric topological space (regarded as a ﬁrst countable semiuniform convergence space (X, JX )) and let f : (X, JX ) → (Y, JY ) be a quotient map in SUConv. If F ∈ JY , then there is some G ∈ JX such that f ×f )(G) ⊂ F. By assumption, G contains H ∈ JX , where H has a countable base B. Hence {f ×f [B] : B ∈ B} is a countable base of f ×f (H) ∈ JX , and f × f (H) ⊂ F. Therefore, (Y, JY ) is ﬁrst countable. On the other hand, it is well-known that quotients in Top of ﬁrst countable topological spaces need not be ﬁrst countable, e.g., the usual topological space Rt of real numbers is ﬁrst countable, whereas the topological quotient space Rt /N, obtained by identifying the set N of positive integers to a point, is not ﬁrst countable, since the neighborhood ﬁlter of the marked point in Figure 3 does not have a countable base. Concerning 4.3.9, note ﬁrst that if f : (Y, JY ) → (Z, JZ ) is a quotient map in SUConv such that f −1 (z) is uniformly connected for each z ∈ Z, then (Y, JY ) is uniformly connected whenever (Z, JZ ) is uniformly connected (namely, if g : (Y, JY ) → D2∆ is uniformly continuous, then h : (Z, JZ ) → D2∆ deﬁned by h ◦ f = g is welldeﬁned and uniformly continuous, which implies that h is constant; therefore g is constant). In order to prove 4.3.9, suppose that C were a uniformly connected subset of X/R consisting of more than one point. If ω : X → X/R denotes the natural map, then its restriction (ω|ω−1 [C] ) : ω −1 [C] → C is a quotient map in SUConv (since quotient maps in SUConv are hereditary!) such that the inverse image of each point of C is uniformly connected, which implies that ω −1 [C] is uniformly connected, i.e., it is contained in an equivalence class w.r.t. R. This contradicts the fact that ω −1 [C] is the union of at least two equivalence classes w.r.t. R. Thus, the quotient space (in SUConv) of (X, JX ) by R is totally uniformly disconnected. If the quotient space of (X, JX ) by R is formed in Unif it need not be totally uniformly disconnected (cf. e.g., [88, 5.2.7]) for an example). 4.3.10. Remark. It is not hard to prove that the SUConv-invariants ‘locally compact’, ‘locally precompact’ and ‘locally connected’ (or ‘locally uniformly connected’) are ﬁnitely productive. First countability is even countably productive. Since all these properties are ﬁnally closed, they form cartesian closed topological constructs by Proposition 2.4.3, Example 3.2.2 and Proposition 3.1.3. Since ﬁrst countability and local precompactness are also hereditary, they form even topological universes by 3.4.6 and 3.5.2 (2).

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5. Completions 5.1. The simple completion. 5.1.1. Definition. If (X, γ) is a ﬁlter space, then an equivalence relation ∼ on γ is deﬁned by F ∼ G iﬀ there exist ﬁnitely many F0 , . . . , Fn ∈ γ with F0 = F, Fn = G, and such that sup{Fi−1 , Fi } exists for each i ∈ {1, . . . , n} (i.e., every member of Fi−1 meets every member of Fi ). The equivalence class of F ∈ γ w.r.t. ∼ is denoted by [F]. If (X, γ) is a Cauchy space, then F ∼ G iﬀ F ∩ G ∈ γ. 5.1.2. Proposition. Let (X, γ) be a Cauchy space and F ∈ γ. Then the following are equivalent: (1) F does not converge (in the underlying symmetric Kent convergence space). (2) [F] = [x] ˙ for each x ∈ X. (The proof is obvious.) 5.1.3. Theorem. Let (X, JX ) be a semiuniform convergence space. Put Y = X ∪ {[F] : F ∈ γJX does not converge}. If i : X → Y denotes the inclusion map, then an SUConv-structure JY on Y is deﬁned by F ∈ JY iﬀ there is some G ∈ BY with F ⊃ G, where BY = {(i × i)(H) : H ∈ JX } ∪ {(i(F) ∩ [F˙ ]) × (i(F) ∩ [F˙ ]) : F ∈ γJX does not converge}, and (Y, JY ) is a complete semiuniform convergence space containing ¯ = Y , where the closure X ¯ of X is formed in (X, JX ) as a dense subspace, i.e., X (Y, qγJY ). The set γJY of all JY -Cauchy ﬁlters is generated by {i(F) : F ∈ γJX converges} ∪ {i(F) ∩ [F˙ ] : F ∈ γJX does not converge}. Proof. It is easy to check that (Y, JY ) ∈ |SUConv|. Furthermore, (X, JX ) is a subspace of (Y, JY ): Since (i × i)(H) ∈ JY for each H ∈ JX , JX ⊂ (JY )X = {K ∈ F (X × X) : (i × i)(K) ∈ JY }. Conversely, if K ∈ (JY )X , then (1) (i × i)(K) ⊃ ˙ for some (i × i)(H) with H ∈ JX , or (2) (i × i)(H) ⊃ (i(F) ∩ [F˙ ]) × (i(F) ∩ [F]) non-convergent F ∈ γJX . In the ﬁrst case we obtain K = (i × i)−1 ((i × i)K)) ⊃ H = (i × i)−1 ((i × i)(H)) and thus K ∈ JX . In the second case we get K ⊃ ˙ = i−1 (i(F) ∩ [F˙ ]) × i−1 (i(F) ∩ [F˙ ]) = F × F. (i × i)−1 ((i(F) ∩ [F˙ ]) × (i(F) ∩ [F])) Since F × F ∈ JX , K ∈ JX . Next we prove γJY = γ with (∗)

γ = {K ∈ F (Y ) : there is some G ∈ B with K ⊃ G},

where B = {i(F) : F ∈ γJX converges} ∪ {i(F) ∩ [F˙ ] : F ∈ γJX does not converge}. If K ∈ γJY , then K × K ∈ JY . Thus, (1) K × K ⊃ (i × i)(H) with H ∈ JX , ˙ for some non-convergent F ∈ γJ . or (2) K × K ⊃ (i(F) ∩ [F˙ ]) × (i(F) ∩ [F]) X In the ﬁrst case X belongs to K (namely, X × X ∈ H ⊂ (i × i)(H) ⊂ K × K means X × X ⊃ K × K with K ∈ K, and consequently X ⊃ K, i.e., X ∈ K). Thus, i−1 (K) = F exists and, since K is a JY -Cauchy ﬁlter, it is a JX -Cauchy ﬁlter. Furthermore, i(F) = K (since X ∈ K), i.e., K ∈ γ . In the second case ˙ for some non-convergent F ∈ γJ , i.e., K ∈ γ . K ⊃ i(F) ∩ [F] X

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If K ∈ γ , then (1) K ⊃ i(F) with F ∈ γJX converges, or (2) K ⊃ i(F) ∩ [F˙ ] for some non-convergent F ∈ γJX . In the ﬁrst case K × K ⊃ i(F) × i(F) = (i × i)(F × F), i.e., K × K ∈ JY since F × F ∈ JX . Thus, K ∈ γJY . In the second case K × K ⊃ (i(F) ∩ [F˙ ]) × (i(F) ∩ [F˙ ]), i.e., K × K ∈ JY . Hence, K ∈ γJY . In order to prove that (Y, JY ) is complete, let H ∈ B. Then (1) H = i(F) where F converges in (X, qγJX), or (2) H = i(F) ∩ [F˙ ] where F ∈ γJX does not converge in (X, qγJX). Since we know already that (X, JX ) is a subspace of (Y, JY ), (X, qγJX) is a subspace of (Y, qγJY ), i.e., i : X → Y is continuous, and therefore H converges in (Y, qγJY ) in the ﬁrst case. In the second case H converges to [F] in (Y, qγJY ) because H ∩ [F˙ ] = H ∈ γJY by (∗). ¯ = {y ∈ Y : It remains to show that X is dense in Y : It suﬃces to prove Y ⊂ X there is some G ∈ F (Y ) with (G, y) ∈ qγJY and X ∈ G}. If y ∈ Y , then qγJ(1) y ∈ X, Y ˙ −→ i(y) = y, or (2) y = [F] where F does not converge in (X, qγJX). Since X ∈ i(y) qγJ Y ¯ in the ﬁrst case. In the second case we have X ∈ i(F) −→ we get y ∈ X [F] = y, ¯ since i(F) ∩ [F˙ ] ∈ γJY . Thus, y ∈ X. 5.1.4. Definition. Let (X, JX ) be a semiuniform convergence space and (Y, JY ) the complete semiuniform convergence space constructed in Theorem 5.1.3. Then the inclusion map i : (X, JX ) → (Y, JY ) is called the simple completion of (X, JX ). Occasionally, (Y, JY ) is already called the simple completion of (X, JX ). 5.1.5. Lemma (Extension Lemma). Let (X, JX ) be a semiuniform convergence space, i : (X, JX ) → (Y, JY ) its simple completion, (X , JX ) a separated complete semiuniform convergence space, and f : (X, JX ) → (X , JX ) a uniformly continuous map. Then there is a unique uniformly continuous map f¯: (Y, JY ) → (X , JX ) such that f¯ ◦ i = f . Proof. A map f¯: Y → X is deﬁned by f (y) if y ∈ X, ¯ f (y) = x if y ∈ Y \ X, where (f (F), x ) ∈ qγJX with y = [F]. x exists, since (X , JX ) is complete, and it is uniquely determined, since (X , JX ) is a T2 -space; furthermore, it is independent of the choice of the representative F. Obviously, f¯ ◦ i = f . In order to prove that f¯ is uniformly continuous, let K ∈ JY . Then (1) K ⊃ (i × i)(H) with H ∈ JX , or (2) K ⊃ (i(F) ∩ [F˙ ]) × (i(F) ∩ [F˙ ]) for some nonconvergent F ∈ γJX . In the ﬁrst case (f¯ × f¯)(K) ⊃ (f¯ × f¯)((i × i)(H)) = (f¯ ◦ i × f¯ ◦ i)(H) = (f × f )(H) ∈ JX , i.e., (f¯ × f¯)(K) ∈ JX . In the second case (f¯ × f¯)(K) ⊃ (f¯ × f¯)((i(F) ∩ [F˙ ]) × (i(F) ∩ [F˙ ])) = f¯(i(F) ∩ [F˙ ]) × f¯(i(F) ∩ ˙ = (f (F) ∩ x˙ ) × (f (F) ∩ x˙ ) ∈ JX , since [F˙ ]) = (f (F) ∩ f¯([F˙ ])) × (f (F) ∩ f¯([F])) f (F) ∩ x˙ ∈ γJX , i.e., (f (F), x ) ∈ qγJ . Thus (f¯ × f¯)(K) ∈ JX . Since now X f¯: Y → X is uniformly continuous, it is also continuous w.r.t. the underlying Kent convergence spaces. Hence, f¯ is uniquely determined by f¯ ◦ i = f , since X is dense in Y and X is a T2 space. 5.1.6. Proposition. If (X, JX ) is a uniform limit space, then (X, γJX) is a Cauchy space. Proof. Let F, G ∈ γJX such that sup{F, G} exists. Then (F × F) ◦ (G × G) = G ×F ∈ JX , and (G ×G)◦(F ×F) = F ×G ∈ JX . Consequently (F ∩G)×(F ∩G) =

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(F × F) ∩ (F × G) ∩ (G × F) ∩ (G × G) ∈ JX since F × F, G × G ∈ JX by assumption. Hence, F ∩ G ∈ γJX . 5.1.7. Proposition. Let (X, JX ) be a separated uniform limit space. Then j : X → {[x] ˙ : x ∈ X} deﬁned by j(x) = [x] ˙ is bijective. Proof. Obviously, j is surjective. In order to prove that j is injective, let [x] ˙ = [y]. ˙ Since (X, γJX ) is a Cauchy space by Proposition 5.1.6, x˙ ∩ y˙ ∈ γJX . By the deﬁnition of convergence in (X, qγJX), x˙ ∩ y˙ converges to x and y. Since (X, qγJX) is a T2 -space, x = y. 5.1.8. Remark. Let (X, JX ) be a separated uniform limit space, and let Y = ˙ : x ∈ X} ∪ {[F] : F ∈ γJX {[F] : F ∈ γJX }. Then by Proposition 5.1.2, Y = {[x] does not converge}. Since j : X → {[x] ˙ : x ∈ X} is bijective, X and {[x] ˙ : x ∈ X} can be identiﬁed, i.e., Y = Y , where (Y, JY ) is the simple completion of (X, JX ). 5.2. The Hausdorﬀ completion. For each separated uniform space (X, V) V) containing (X, V) (up to isomorthere is a separated complete uniform space (X, phism) as a dense subspace, where (X, V) can be constructed as follows (cf. [14]): be the set of all minimal Cauchy ﬁlters in (X, V), where a Cauchy ﬁlter in Let X (X, V) is deﬁned to be a Cauchy ﬁlter in (X, [V]), and a minimal Cauchy ﬁlter is a minimal element in the set γ[V] ordered by inclusion. For each V = V −1 ∈ V, let ×X : there is some M ∈ F ∩ G with M × M ⊂ V }. Further, V = {(F, G) ∈ X V) let V be generated by {V : V = V −1 ∈ V}. The embedding rX : (X, V) → (X, is deﬁned by rX (x) = U(x), where U(x) is the neighborhood ﬁlter of x w.r.t. the topology XW = {O ⊂ X : for each x ∈ O, there is some V ∈ V with V (x) = {y ∈ X : (x, y) ∈ V } ⊂ O}. This completion is unique up to isomorphism, V) (shortly: (X, V)) is called the Hausdorﬀ completion of where rX : (X, V) → (X, (X, V). It follows from Figure 1 of Section 2 that Unif is bireﬂective in SUConv. The bireﬂective modiﬁcation (X, U) of (X, JX ) ∈ |SUConv| w.r.t. Unif is called the underlying uniform space of (X, JX ). In particular, U is the ﬁnest uniformity which is contained in each F ∈ JX , i.e., U = {V ∈ V : there is a sequence (Vn )n∈N 2 with Vn ∈ V and Vn−1 ∈ V for each n ∈ N, and V1 = V , such that Vn+1 ⊂ Vn for each n ∈ N}, where V = F ∈JX F. 5.2.1. Theorem. Let (X, [V]) be a separated uniform space (regarded as a semiuniform convergence space). Then the underlying uniform space of the simple completion of (X, [V]) is the Hausdorﬀ completion of (X, V). Proof. Since (X, [V]) is separated, the underlying set of its simple completion may be identiﬁed with X ∗ = {[F] : F is a [V]-Cauchy ﬁlter}, and X with {[x] ˙ :x∈ X} (cf. 5.1.8). Each equivalence class [F] contains a minimal [V]-Cauchy ﬁlter V(F) which is generated by the ﬁlter base { x∈F V (x) : V ∈ V, F ∈ F} and independent of the choice of the representative, where V(x) ˙ is the neighborhood ﬁlter of x w.r.t. XV . If i : X → X ∗ denotes the inclusion map, and V ∗ is the uniformity on X ∗ generated by {V ∗ : V = V −1 ∈ V}, where V ∗ = {([F], [G]) ∈ X ∗ × X ∗ : there is some M ∈ V(F) ∩ V(G) with M × M ⊂ V } for each V = V −1 ∈ V, then i : (X, V) → (X ∗ , V ∗ ) is the Hausdorﬀ completion of (X, V). Let i : (X, [V]) → (Y, JY ) be the simple completion of (X, [V]), where Y = X ∗ , ˙ × (i(F) ∩ [F˙ ]) : F and let W = F ∈JX ∗ F. Then W = (i × i)(V) ∩ {(i(F) ∩ [F)])

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˙ ˙ is a [V]-Cauchy ﬁlter} = (i × i)(V) ∩ {(i(V(F)) ∩ [V(F)]) × (i(V(F)) ∩ [V(F)]) : F is a [V]-Cauchy ﬁlter}. We prove V ∗ ⊂ W. By the Extension Lemma 5.1.5, there is a unique uniformly continuous map h : (X ∗ , JX ∗ ) → (X ∗ , [V ∗ ]) such that the diagram commutes, / (X ∗ , [V ∗ ]) 8 ppp p p i ppp ppp h (X ∗ , JX ∗ ) (X, [V])

i

where for each [V]-Cauchy ﬁlter F, i(F) converges to h([F]) in (X ∗ , [V ∗ ]). On the other hand, i(F) converges to [F] in (X ∗ , V ∗ ), i.e., in (X ∗ , [V ∗ ]), and since (X ∗ , [V ∗ ]) is separated, h([F]) = [F] for each [V]-Cauchy ﬁlter F, i.e., h = 1X ∗ . Consequently, JX ∗ ⊂ [V ∗ ], which implies V ∗ ⊂ {F : F ∈ JX ∗ } = W. Furthermore, we prove V ∗ = W ◦ W ◦ W. Let W ∈ W. Then there is some V = V −1 ∈ V, and for each [F] = [V(F)] ∈ ∗ X , there is a set C[F ] ∈ V(F) such that (∗)

V ∪

{ C[F ] ∪ {[F]} × C[F ] ∪ {[F]} : [F(F, x) ∈ q] ∈ X ∗ } ⊂ W.

Let ([F], [G]) ∈ V ∗ . Then V ∈ V(F) × V(G), which implies V ∩ (C[F ] × C[G] ) = ∅, i.e., there is some ([x], ˙ [y]) ˙ ∈ V ∩ (C[F ] × C[G] ). Hence it follows from (∗) that ([F], [x]) ˙ ∈ (C[F ] ∪ {[F]}) × (C[F ] ∪ {[F]}) ⊂ W, ([x], ˙ [y]) ˙ ∈ V ⊂ W, and ([y], ˙ [G]) ∈ (C[G] ∪ {[G]}) × (C[G] ∪ {[G]}) ⊂ W. Consequently, ([F], [G]) ∈ W 3 = W ◦ W ◦ W , i.e., V ∗ ⊂ W 3 . This proves W 3 = W ◦ W ◦ W ⊂ V ∗. Since V ∗ is a uniformity, V ∗ = V ∗ ◦ V ∗ . Because of V ∗ ⊂ W, V ∗ = V ∗ ◦ V ∗ = V ∗ ◦ V ∗ ◦ V ∗ ⊂ W ◦ W ◦ W. Now, let R be a uniformity on Y = X ∗ such that R ⊂ W. Then R = R ◦ R ◦ R ⊂ W ◦ W ◦ W = V ∗ , and since, V ∗ ⊂ W, V ∗ is the ﬁnest uniformity which is coarser than W, i.e., V ∗ is the ﬁnest uniformity which is contained in each F ∈ JX ∗ . Thus, (X ∗ , V ∗ ) is the underlying uniform space of (X ∗ , JX ∗ ). 5.2.2. Corollary. The Hausdorﬀ completion rX : (X, V) → (X ∗ , V ∗ ) of a separated uniform space (X, V) has the following universal property: For each complete separated uniform space (Y, R) and each uniformly continuous map f : (X, V) → (Y, R) there is a unique uniformly continuous map f¯: (X ∗ , V ∗ ) → (Y, R) such that f¯ ◦ rX = f . Proof. Identifying uniform spaces and principal uniform limit spaces, this follows immediately from the Extension Lemma and the fact that Unif is bireﬂective

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in SUConv, i.e., the following diagram commutes. / (Y, [R]) r8 A r rr r i r r rrr f¯ (X ∗ , JX ∗ ) ¯ f 1X ∗ " (X ∗ , [V ∗ ]) (X, [V])

rX

f

5.2.3. Remark. An alternative method to construct the Hausdorﬀ completion of a separated uniform space by means of function spaces has been found by R. J. Gazik, D. C. Kent and G. Richardson [32], who use the cartesian closedness of ULim as an intermediate step. The same procedure can be used to form function spaces in SUConv instead of ULim (cf. [88, 4.2.2]). The simple completion of a semiuniform convergence space has been constructed ﬁrst by G. Preuß [80]. 6. Preuniform convergence spaces Omitting the symmetry condition (UC3 ) in the deﬁnition of a semiuniform convergence space (Example 2.2.9), one obtains preuniform convergence spaces. The structure preserving maps between preuniform convergence spaces are deﬁned similarly to those between semiuniform convergence spaces and are also called uniformly continuous maps. The resulting construct PUConv of preuniform convergence spaces (and uniformly continuous maps) is a strong toplogical universe too (cf. [77, 3.1]). Non-symmetric spaces such as quasiuniform spaces, which are obtained by omitting the symmetry condition (4) in the deﬁnition of a uniform space (Example 2.2.2), topological spaces, and even generalized convergence spaces form subconstructs of PUConv (up to a concrete isomorphism), but subspaces of topological spaces formed in PUConv are not better behaved than subspaces of topological spaces formed in Top, since subspaces of generalized convergence spaces are formed as in PUConv (cf. [92, 2.11]), and Top is bireﬂective in GConv. Preuniform convergence spaces are mainly studied in the realm of non-symmetric convenient topology which improves non-symmetric topology and is closely related to convenient topology (cf. [92] for the details). References [1] J. Ad´ amek, H. Herrlich, and G. E. Strecker, Abstract and concrete categories. The joy of cats, John Wiley & Sons, New York, 1990, Revised online edition (2004), http://katmat.math.unibremen.de/acc. MR1051419 (91h:18001) [2] E. M. Alfsen and J. E. Fenstad, On the equivalence between proximity structures and totally bounded uniform structures, Math. Scand. 7 (1959), 353–360, Correction: ibid. 9 (1961), 258. MR0115156 (22 #5958) [3] R. F. Arens, A topology for spaces of transformations, Ann. of Math. (2) 47 (1946), 480–495. MR0017525 (8,165e) [4] A. V. Arhangel ski˘ı, Some types of factor mappings and the relations between classes of topological spaces, Doklady Akad. Nauk SSSR 153 (1963), 743–746 (Russian), English translation: Soviet Math. Dokl. 4 (1963) 1726–1729. MR0158362 (28 #1587) [5] R. Beattie and H.-P. Butzmann, Convergence structures and applications to functional analysis, Kluwer Academic Publishers, Dordrecht, 2002. MR2327514 (2008c:46001)

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Index

ACSp, 174 AFrm, 319 ATSp, 174 ApVec, 313 App, 2 Ap, 307 Ap0 , 321 C-Fil, 348 CAp, 325 CReg, 307 CSp, 174 Cap, 2 Chy, 52, 355 Chy0 , 68 Cls, 2 CompUnif 2 , 2 Comp2 , 2 Conv, 2, 141 C, 53 EpiAp, 327 EpiTop, 25 Fil, 51, 341 Fil-D-SUConv, 348 Fil-Mer, 350 GConv, 341 Grp, 3 Haus, 2 KConv, 341 KConvs , 348 Lim, 2, 53, 341 Limr , 54 Lims , 53, 349 Mer, 41, 350 Metc , 2 Near, 2, 43 PUConv, 368 Pos, 2 PrAp, 325 PrTop, 2, 325, 341 PrTops , 349 PrULim, 347 Pretop, 53 Prost, 2 PsAp, 326

PsTop, 2, 341 PsTops , 349 P, 71 QU, 2, 244 RegChy, 68 RegNear, 56 RegNear0 , 58 RegTop, 56 Rng, 3 SUConv, 342 SULim, 342 SepNear, 57 SepNear0 , 59 Seq, 2 Set, 2, 141 Sob, 321 T3 Chy, 68 TCSp, 174 TNear, 45 TopVec, 313 Top, 2, 44, 174, 307, 340 Top0 , 2 Top1 , 2 Tops , 44, 349 Tych, 2 T-PrTop, 346 UAp, 307 UG, 2 ULim, 79, 343 Unif, 2, 44, 340 Unif 2 , 2 VectR , 3 cChy0 , 70 cP, 72 cRegNear0 , 63 lcApVec, 313 lcTopVec, 313 pMetTopVec, 313 pMetTop, 313 pMetVec, 313 pMet, 2, 307 pqMet, 307 sNormTopVec, 313 sNorm1 , 313 375

376

INDEX

A-morphism, 3 ABC lattice, 189 absolute subset selection, 208 adherence, 127 adherence of a set, 128 adherence operator, 310 adjoint functor, 7 adjoint pair of maps, 201 adjoint situation, 217 Alexandroﬀ (discrete) space, 174 Alexandroﬀ completion, 175, 183 Alexandroﬀ proximity, 94 Alexandroﬀ space, 171 Alexandroﬀ topology, 174 algebra (of a monad), 217 algebraic closure spaces, 174 algebraic construct, 12 amnestic construct, 3 Anti-Exchange Axiom, 169 antisymmetric, 3 Antoine space, 25, 328 approach frame, 319 approach prime, 320 approach space, 306 approach vector space, 312 associated topological convergence, 122 asymmetric norm, 298 atom, 188, 189 atomistic lattice, 171 axiality, 207 base of a lattice, 170 base of a uniformity, 340 based lattice, 187 betweenness property, 98 biBanach space, 298 bicomplete quasi-uniformity, 270 bicompletion-true functorial quasi-uniformity, 284 bicontinuous map, 258 bicoreﬂective modiﬁcation, 344 bicoreﬂective subcategory, 307 bicoreﬂective subconstruct, 344 bidual convergence, 154 bilateral quasi-uniformity of a paratopological group, 286 biquotient map, 145 bireﬂective hull, 23 bireﬂective modiﬁcation, 344 bireﬂective subcategory, 14, 307 bireﬂective subconstruct, 343 bisequential topology, 144 bitopological space, 258 bounded set, 105 boundedness, 105 canonical completion, 63 cartesian closed, 335, 351 cartesian closed category, 19, 117

cartesian closed hull, 327 cartesian closed topological construct, 20, 77 cartesian closed topological hull, 22, 326 categorical topology, 1 category, 3, 140 category theory, 3 Cauchy colleciton, 47 Cauchy continuous Cauchy structure, 78 Cauchy continuous map, 342 Cauchy ﬁlter, 47, 80, 270, 310, 342, 350 Cauchy ﬁlter pair, 277 Cauchy space, 52 Cauchy system, 350 Cauchy-regular ﬁlter space, 68 ˇ Cech proximity, 94 chain, 177 chaotic topology, 123 Choquet space, 341 close-to relation, 93 closed expansion, 316 closed set, 173 closed sink, 26 closure, 166 closure algebra, 223 closure map, 190 closure operation, 190 closure operator, 49, 165, 173 closure range, 192 closure space, 165, 173 closure system, 165, 173 cluster, 61, 104 co-adjoint functor, 7 co-complete topological construct, 7 co-wellpowered category, 13 coarser ﬁlter, 120 coarser quasi-uniformity, 244 coarser-than relation, 96, 339 coarsest quasi-uniformity that a locally compact topology admits, 265 cocompact topology, 152, 265 cocushioned collection, 258 coequalizer, 9 coessential family, 61 coﬁnite ﬁlter, 119 coframe, 171 colimit of a functor, 7 compact approach frame, 323 compact at a family, 130 compact element, 188 compact family, 130 compact frame, 323 compact Hausdorﬀ semiuniform convergence space, 359 compact semiuniform convergence space, 359 compact space, 129 compact-open topology, 107, 288 compactoid, 130 compactoid space, 130

INDEX

comparison functor, 219 compatible EF-proximity, 97 complemented approach space, 321 complete approach space, 321 complete closure algebra, 223 complete cut space, 183 complete ﬁlter space, 348 complete join-semilattice, 191 complete kernel algebra, 223 complete lattice, 216 complete nearness space, 62 complete semiuniform convergence space, 360 complete topological construct, 7 complete uniformity, 270 completely bounded proximity space, 109 completely distributive lattice, 210 completely regular semiuniform convergence space, 358 completely regular space, 94 completely regularly ordered space, 297 concentrated collection, 57 concrete category, 3, 141 concrete colimit, 7 concrete functor, 17, 141, 345 concrete limit, 7 concrete quasi-topos, 22 concretely coreﬂective subcategory, 142 concretely coreﬂective subconstruct, 17 concretely isomorphic construct, 345 concretely reﬂective subcategory, 43, 142 concretely reﬂective subconstruct, 17 cone, 7 conjugate quasi-pseudometric, 242 conjugate quasi-uniformity, 244 conjugate topology of a paratopological group, 285 connected semiuniform convergence space, 359 construct, 3, 216, 339 construct invariant, 343 continuity for convergences, 125 continuous convergence, 127, 334 continuous lattice, 211 continuous map, 174 continuous morphism, 225 contour, 136 contraction, 307 convenience property, 19 convenient hull, 22 Convenient Topology, 357 convergence, 117, 122 convergence approach space, 325 convergence complete quasi-uniformity, 271 convergence in proximity, 106 convergence of countable character, 124, 140 convergence relation, 117 convergence space, 117, 122, 360 convergence-approach limit, 324

377

convergent ﬁlter, 47 convex topology, 298 core-compact topology, 159, 269 coreﬂective subcategory, 14, 142 coreﬂector, 14, 142 coseparator, 15 costable ﬁlter, 275 counit, 217 countable compactness, 131 countably based ﬁlter, 119, 140 countably core-compact topology, 159 coupling convergence, 147 covariant functor, 140 cover, 41, 100, 128 cover-compact convergence, 130 covering dimension, 362 covering uniformity, 100 cushioned collection, 258 cut, 183 cut operator, 183 cut space, 183 D-Cauchy ﬁlter, 277 decreasing set, 296 Dedekind–MacNeille completion, 183 density property, 95 depending on countably many coordinates, 247 detachable closure space, 197 developable space, 252 diagonal convergence, 137 diagonal ﬁlter, 310 diagonal neighbourhood, 99 diagonal semiuniform convergence space, 359 diagonal uniformity, 99 diagonalization property, 11 diagram functor, 7 directed set, 174 directed subset, 177 discrete object, 3 discrete proximity, 93 discrete topology, 2 discrete uniformity, 244 distance, 308 distance of S-convergence, 311 Doitchinov complete quasi-uniformity, 277 Doitchinov-Cauchy ﬁlter, 277 Doitchinov-completion, 278 doubly residuated map, 223 downset, 175 downset operator, 183 dual category, 5 dual relation, 93 dual strong inclusion relation, 95 dually pseudocomplemented lattice, 198 (E,M)-category, 11 (E,M)-factorization, 11, 315 E-reﬂective subcategory, 14

378

INDEX

EF-proximisable space, 97 EF-proximity, 93 EF-proximity space, 93 Efremoviˇ c Lemma, 102 Efremoviˇ c property, 93 Efremoviˇ c proximity, 93 Efremoviˇ c space, 93 elementary proximity, 94 embedding, 219 embedding morphism, 9 end ﬁlter, 103 entourage, 340 entourage of a quasi-uniformity, 241 epi-topological space, 25, 328 epimorphism, 9 epireﬂective subcategory, 14 episink morphism, 20 epitopologizer, 155 epitopology, 155 equalizer, 7 equentially based convergence, 123 equicontinuous family of maps, 291 essential embedding, 21 evaluation map, 127 Exchange Axiom, 169 exponenial laws, 19 exponential category, 117 exponential map, 147 exponential object, 19 extensional construct, 21, 354 extensional topological hull, 22, 325 extensive map, 173, 190 extremal epimorphism, 10 extremal monomorphism, 10 factorization structure, 11, 315 faithful functor, 3, 141 far-from relation, 93 far-miss set, 110 Fell compactiﬁcation, 280 ﬁber, 141 ﬁber of a set, 3, 42 ﬁber-small construct, 3 Fil-determined semiuniform convergence space, 348 ﬁlter, 41, 103, 118 ﬁlter decomposition, 120 ﬁlter space, 51, 342 ﬁlter-determined convergence, 139 ﬁlter-merotopic space, 350 ﬁnal convergence, 126 ﬁnal epi-sink, 355 ﬁnal lift, 4, 42 ﬁnal morphism, 4, 220 ﬁnal quasi-uniformity, 248 ﬁnal sink, 220, 355 ﬁnal source, 4 ﬁnally closed subconstruct, 17

ﬁnally dense, 143 ﬁnally dense subcategory, 18 ﬁnally tight extension topological construct, 23 ﬁne quasi-uniformity of a topology, 244 ﬁne transitive quasi-uniformity of a topology, 254 ﬁner convergence, 122 ﬁner ﬁlter, 120 ﬁner quasi-uniformity, 244 ﬁner-than relation, 96, 339 ﬁnest totally bounded quasi-uniformity coarser than, 261 ﬁnest transitive quasi-uniformity coarser than, 254 ﬁnitary closure, 176 ﬁnitary closure operator, 168 ﬁnitely compact, 131 ﬁrm, 321 ﬁrmly U-reﬂective subconstruct, 60 First coreﬂector, 143 First corelﬂector, 124 ﬁrst countable semiuniform convergence space, 359 Fletcher construction, 244 forgetful functor, 3 Fr´ echet space, 335 Fr´ echet topology, 144 frame, 171, 318 free algebra, 218 free cluster, 104 free ﬁlter, 118 Freudenthal compactiﬁcation, 104 Freudenthal proximity, 94 full concrete subconstruct, 14 full subconstruct, 343 fully normal semiuniform convergence space, 358 functionally indistinguishable proximity, 94 functor, 3, 140 functor-regular convergence, 142 functorial quasi-uniformity, 244 functorial subset selection, 177 Galois connection, 203 γ-space, 252 gauge, 306 gauge basis, 307 generalized convergence space, 341 global standard completion, 186 global standard extension, 186 globally saturated ideal, 306 grill, 41 group of self-isomorphisms of a quasi-uniformity, 290 handy relation, 267 Hausdorﬀ completion, 366 Hausdorﬀ convergence, 136

INDEX

Hausdorﬀ hyperuniformity, 108 Hausdorﬀ metric, 312 Hausdorﬀ quasi-pseudometric, 293 Hausdorﬀ quasi-uniformity, 292 hereditarily precompact quasi-uniformity, 260 hereditarily quotient map, 145 hereditary closure operator, 50 hereditary ﬁnal sink, 355 hit and far-miss topology, 110 hit and miss topology, 109 hit part, 109 hit set, 109 hypercompact convergence, 153 hyperspace, 108, 117 hypreproximity, 109 ideal, 129 ideal completion, 186 idempotent map, 173, 190 increasing map, 296 increasing set, 296 indiscrete object, 3 indiscrete proximity, 93 indiscrete topology, 2 indiscrete uniformity, 244 inductive closure, 176 inﬁmum proximity, 99 inherence, 128 initial convergence, 125, 126 initial hull, 23 initial lift, 4, 42 initial morphism, 4, 219 initial quasi-uniformity, 247 initial source, 4, 219 initial structure, 339 initially closed subconstruct, 17 initially dense, 143 initially dense object, 311 initially dense subcategory, 18 injective hull, 21 injective object, 21 interior cover, 44 interior operator, 42, 174 interior preserving open collection, 243 invariant, 343 invariant point selection, 188 inverse relation, 244 involution of a poset, 207 irreducible topology, 266 irregular, 137 isolated point, 118 isomorphic quasi-uniformities, 243 isomorphism-closed subconstruct, 343 isotone, 119 isotone map, 173, 175, 190 J-quotient convergence, 145 J-quotient map, 145 JE-convergence, 143

379

join, 191 join-closed subset of a lattice, 194 join-complete subset of a lattice, 194 join-completion, 185 join-dense subset of a lattice, 170 join-preserving map, 184 join-semilattice, 191 joincompact bitopology, 273 K coreﬂector, 143 k -topology, 144 k-topology, 144 Kent convergence space, 341 kernel algebra, 223 kernel map, 191 kernel operation, 191 kernel operator, 174 kernel system, 173 Kofner plane, 255 Kuratowski closure, 167 Kuratowski convergence, 153 λ-base of a topology, 276 largest ﬁnally tight extension construct, 24 Leader convergence, 106 Lebesgue quasi-uniformity, 245 left K-Cauchy ﬁlter, 272 left K-Cauchy net, 272 left K-complete, 272 left adjoint functor, 217 left invariant quasi-pseudometric, 286 left quasi-uniformity of paratopological group, 286 left shift, 225 limit of a ﬁlter, 25, 53, 122, 328, 341 limit of a functor, 7 limit operator, 310 limit space, 53, 341 local proximity, 105 local quasi-uniformity, 251 local uniformity, 252 locale, 171, 200 locally compact convergence, 140 locally compact ﬁlter, 140 locally compact semiuniform convergence space, 359 locally compact space, 94 locally compact topology, 265 locally connected semiuniform convergence space, 359 locally convex approach vector space, 312 locally precompact semiuniform convergence space, 359 locally saturated ideal, 306 locally symmetric quasi-uniformity, 245 lower adjoint, 201 lower Kuratowski convergence, 153 lower quasi-uniformity, 292 lower semimodular lattice, 197

380

INDEX

lower topology, 296 lower Vietoris topology, 295 m-directed set, 179 m-frame, 210 m-normal ﬁlter space, 358 M -quotient, 146 m-uniform ﬁlter space, 358 Mackey structure, 314 MacNeille completion, 25 measure of compactness, 314 V meet-closed ( -closed) subset of a lattice, 194 meet-subcomplete subset of a lattice, 194 merotopic space, 41, 350 merotopy, 41 mesh, 119 mesh relation, 40 metacompact, 256 metric proximity, 94 metric space, 94 metrizable semiuniform convergence space, 360 micromeric collection, 47 Minkowski system, 313 miss part, 109 miss set, 109 modiﬁer of countable character, 124 monad, 218 monadic construct, 219 monadic functor, 219 monomorphism, 8 monoreﬂectives subcategory, 14 monotopological construct, 14 morphism, 3, 140 natural function space, 351 natural source, 7 natural topology underlying a proximity, 97 near collection, 48 near-to relation, 93 nearness space, 43 neighborhood ﬁlter, 118 neighbornet, 252 network of a topology, 107, 125 non-archimedean quasi-pseudometric, 242 non-archimedeanly quasi-pseudometrizable quasiuniformity, 250 normal approach frame, 324 normal frame, 324 normal merotopic space, 358 normal semiuniform convergence space, 358 normal space, 94 nucleus, 200 object, 3, 140 ω-bounded space, 271 one-point extension, 354 open ﬁlter, 271

open morphism, 225 open set, 173 opposite category, 5 order-embedding, 185 orthocomplementation of a poset, 207 p-cover, 101 p-function, 96 p-isomorphism, 96 p-metric, 306 p-neighbourhood, 94 p-neighbourhood relation, 95 pairbase of a topology, 258 pairwise completely regular bitopology, 260 pairwise-local quasi-uniformity, 252 parabitopological group, 285 paracompact semiuniform convergence space, 358 paratopological group, 285 paratopologizer, 144 paratopology, 133 partial morphism, 354 partial order, 165, 175 Pervin quasi-uniformity, 243 Pervin–Sieber–Cauchy ﬁlter, 271 Pervin–Sieber-complete quasi-uniformity, 271 pesudotopology, 131 ϕ-tower, 200 point-cluster, 104 point-free geometry, 111 point-operator, 328 point-symmetric quasi-uniformity, 245 pointwise convergence, 334 power convergence, 127 power set monad, 218 power space monad, 230 pq-metric, 306 pre-approach space, 325 pre-distance, 325 pre-regular approach frame, 322 pre-regular approach space, 323 precompact quasi-uniformity, 260 precompact semiuniform convergence space, 359 prel-limit operator, 325 prenorm, 312 preorder, 242 pretopological space, 53, 341 pretopologically diagonal convergence, 137 pretopologizer, 125, 144 pretopology, 124 preuniform convergence space, 368 prime topology, 118 primitive ﬁlter, 267 principal ﬁlter, 119, 175, 310 principal ideal, 175 principal quasi-order, 193 principal uniform limit space, 347

INDEX

product convergence, 126 product proximity, 98 product quasi-uniformity, 247 projection algebra, 222 projection operator, 191 promorphism, 222 proper contraction, 316 prototopology, 132, 156 proximal neighbourhood, 94 proximal set-open topology, 107 proximally continuous map, 263 proximally symmetric quasi-uniformity, 263 proximally transitive quasi-uniformity, 264 proximity, 262 proximity boolean algebra, 111 proximity function, 96 proximity invariant, 96 proximity isomorphism, 96 proximity lattice, 110 proximity naturally associated with a uniformity, 100 pseudo-approach limit, 326 pseudo-limit operator, 326 pseudo-metric, 306 pseudo-open map, 335 pseudo-quasi-metric, 306 pseudocomplement, 206 pseudocomplemented semilattice, 206 pseudometric, 242 pseudotopological space, 341 pseudotopologization, 132 pseudotopologizer, 132, 144 PSOT, 107 QH-equivalent quasi-uniformity, 296 quantale, 200 quasi-category, 24 quasi-construct, 23 quasi-metric, 243 quasi-order, 175 quasi-proximity, 262 quasi-proximity class of a quasi-proximity, 263 quasi-proximity induced by a quasi-uniformity, 262 quasi-pseudometric, 242 quasi-pseudometric bicompletion, 279 quasi-pseudometric quasi-uniformity, 242 quasi-pseudometrizable bitopology, 258 quasi-pseudometrizable quasi-uniformity, 251 quasi-sober topology, 266 quasi-topology, 139 quasi-topos hull, 327 quasi-uniform bicompletion, 279 quasi-uniform space, 241 quasi-uniformity, 241 quasi-uniformity compatible with a quasiproximity, 263

381

quasi-uniformity of compact convergence, 288 quasi-uniformity of pointwise convergence, 288 quasi-uniformity of uniform convergence, 288 quasi-uniformity of uniform convergence of A, 288 quasi-uniformizable topology, 243 quiet quasi-uniformity, 277 quotient convergence, 126 quotient map, 145, 335 quotient morphism, 9, 220 quotient object, 12 quotient proximity, 98 quotient quasi-uniformity, 248 quotient reﬂective subcategory, 58 rather below relation, 322 reciprocal limit space, 54 reciprocal space, 55 reﬁnement of a cover, 41, 100 reﬂective subcategory, 14, 142 reﬂector, 14, 142 reﬂector-quotient convergence, 145 reﬂector-quotient map, 145 regular approach frame, 322 regular approach space, 323 regular cardinal, 179 regular Cauchy space, 68 regular closed set, 98 regular convergence, 137 regular epimorphism, 8 regular ﬁlter space, 68 regular frame, 322 regular function frame, 318 regular limit space, 68 regular merotopic space, 56, 358 regular monomorphism, 8 regular nearness space, 56 regular reﬁnement of a cover, 56 regular semiuniform convergence space, 358 regular space, 56 regularity sequence, 69 regularization, 137 relation, 93 relative proximity, 98 relatively compact, 269 relatively compact family, 130 relatively compact space, 129 relativization, 170 remote-from relation, 93 residual map, 202 residuated map, 202 right K-Cauchy ﬁlter, 272 right K-Cauchy net, 272 right K-complete, 272 right adjoint functor, 20, 217 right quasi-uniformity of a paratopological group, 286

382

INDEX

right shift, 225 rim-compact space, 94 round ﬁlter, 103 Samuel compactiﬁcation, 101 Samuel uniformity, 101 Scott completion, 186 Scott topology, 186 sec operator, 40 semiuniform convergence space, 342 semiuniform limit space, 342 separated local proximity space, 106 separated merotopic space, 57 separated nearness space, 57 separated proximity, 93 separated relation, 95 separated semiuniform convergence space, 358 separated uniform space, 99 separator, 15 sequential ﬁlter, 119, 139 sequential modiﬁer, 124 sequential topology, 123, 144 sequentially based ﬁlter, 139 sequentially closed set, 144 sequentially compact convergence, 131 σ-interior preserving base of a topology, 250 Σ-operator, 71 Σ-based lattice, 188 Σ-complete closure space, 188 simple completion, 66, 365 sink, 4 Skula topology of a topology, 282 small collection, 47 small-set symmetric quasi-uniformity, 245 Smirnov compactiﬁcation, 103 Smirnov Compactiﬁcation Theorem, 103 Smyth-completable quasi-uniformity, 276 Smyth-complete quasi-uniformity, 276 sober approach space, 320 sober topology, 266 Sorgenfrey quasi-metric, 286 source, 4, 219 Souslin property, 271 spatial approach frame, 320 specialization order, 175 spectrum, 319 stable family, 152 stable ﬁlter, 262 stable quasi-uniformity, 278 stack, 40, 310 standard T0 -quasi-pseudometric on R, 285 standard completion, 183 standard extension of a poset, 183 star of a set w.r.t. a cover, 41, 100 star reﬁnement of a cover, 41, 100 Stone space, 121 Stone transform, 121 stratiﬁable space, 259

strictly dense map, 184 strong hyperproximity, 109 strong inclusion of a quasi-proximity, 263 strong inclusion relation, 95 strong topological universe, 355 W strongly Z- -closed, 208 strongly Z-continuous, 212 strongly k -topology, 144 strongly (quasi-)sober topology, 280 strongly cartesian closed construct, 355 strongly contained, 95 strongly Fr´echet topology, 144 structured sink, 4 structured source, 4 subcategory, 142 subconvergence, 125 subﬁrmly U-reﬂective subconstruct, 60 subobject, 12 subset selection, 177 subspace quasi-uniformity, 247 subtopological nearness space, 75 subtopological semiuniform convergence space, 358 supercomplete uniformity, 275 supercontinuous lattice, 211 superquasi-sober topology, 269 supersober topology, 269 supertight map, 321 supremum, 191 supremum proximity, 99 symmetric Kent convergence space, 348 symmetric limit space, 53 symmetric space, 44, 338 symmetrization of a quasi-pseudometric, 242 symmetrization of a quasi-uniformity, 244 system (of sets), 172 T0 -closure space, 184 T0 -object, 58 T0 -quasi-pseudometric, 242 T0 -quotient of a quasi-uniformity, 248 T0 -system, 175 T1 semiuniform convergence space, 358 T2 -Cauchy ﬁlter space, 68 T2 -ordered compactiﬁcation, 297 T2 -ordered space, 296 T2 semiuniform convergence space, 358 T3 -Cauchy ﬁlter space, 68 t-regular semiuniform convergence space, 358 θ-rather below relation, 322 θ-regular convergence, 137 toally disconnected semiuniform convergence space, 359 topological category, 220 topological closure space, 174 topological construct, 4, 141, 339 topological convergence, 117 Topological Duality Theorem, 5, 42

INDEX

topological functor, 4 topological group, 285 topological hull, 25 topological nearness space, 45 topological preordered space, 296 topological preordered space determined by a quasi-uniformity, 296 topological semiuniform convergence space, 358 topological universe, 22, 355 topological universe hull, 22, 327 topologically Hausdorﬀ convergence, 136 topologically metrizable semiuniform convergence space, 360 topologically pretopological space, 345 topologically regular convergence, 137 topologization of a convergence, 123 topologizer, 144 topology compatible with a quasi-uniformity, 243 topology induced by a quasi-proximity, 263 topology induced by a quasi-uniformity, 242 topology of uniform convergence, 106 totally bounded quasi-uniformity, 260 totally bounded uniformity, 99 totally uniformly disconnected semiuniform convergence space, 359 trace, 118 trace of ﬁlter, 118 transitive base of a quasi-uniformity, 250 transitive quasi-uniformity, 250 transitive topology, 254 trivial proximity, 93 U-completion, 60 U-reﬂective subconstruct, 60 U-remote relation, 100 U -structured source, 4 ultraﬁlter, 120 underlying set functor, 3 underlying set of a convergence, 122 uniform approach space, 307 uniform convergence, 106 uniform cover, 41, 100, 350 uniform gauge, 306 uniform gauge space, 306 uniform limit space, 79, 343 uniform limit structure, 79 uniform merotopic space, 358 uniform semiuniform convergence space, 359 uniform space, 44, 99, 340 uniform topology, 106 uniformity, 99, 244, 340 uniformity of uniform convergence, 106 uniformizable topology, 251 uniformly connected semiuniform convergence space, 359 uniformly continuous map, 243, 340

383

uniformly equicontinuous family of maps, 291 uniformly open map, 246 uniformly regular quasi-uniformity, 278 uniformly saturated ideal, 306 uniformly Urysohn family, 110 union complete subset selection, 177 unit, 217 universal topological construct, 59 upper adjoint, 201 upper Kuratowski convergence, 151 upper quasi-uniformity, 292 upper topology, 186, 296 upper Vietoris topology, 295 upset, 175 Urysohn function, 96 W -irreducible element, 188 ∨-irreducible element, 188 W -prime element, 188 ∨-prime element, 188 vicinity, 124 vicinity ﬁlter, 124 Vietoris topology, 109, 295 weak approach structure, 311 weak base of a topology, 125 weak hyperproximity, 109 weak star reﬁnement of a cover, 41 weak topology, 311 well-ﬁbered topological construct, 6 well-monotone quasi-uniformity, 244 well-quasi-order, 261 wellpowered category, 12 Wijsman distance, 312 Wijsman topology, 312 Wyler completion, 70 W Z- -closed set, 208 V Z- -closed set, 208 Z-below relation, 212 Z-below set, 211, 212 Z-closure space, 178 Z-closure system, 178 Z-complete poset, 186, 212 Z-continuous poset, 212 Z-cut completion, 186 Z-distributive lattice, 211 Z-downsets, 177 Z-join ideal, 186, 212 Z-join ideal completion, 186 Z-prime element, 189 Z-subalgebra, 208 Z-union closed, 178 Z-union preserving, 178 0-compact approach space, 314 zero-ideal, 323

Titles in This Series 489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic Forms and L-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and L-functions I. Global aspects, 2009 486 Fr´ ed´ eric Mynard and Elliott Pearl, Editors, Beyond topology, 2009 485 Idris Assani, Editor, Ergodic theory, 2009 484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and number theory, 2009 483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Eﬁm Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘ glu, and Dietmar Vogt, Editors, Functional analysis and complex analysis, 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and mathematical physics: Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, Jos´ e Luis Cisneros-Molina, David Massey, Jos´ e Seade, and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, Jos´ e Luis Cisneros-Molina, David Massey, Jos´ e Seade, and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. No¨ el, Donald R. King, Gaston M. N’Gu´ er´ ekata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical sciences: Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in ﬂuid dynamics, 2008 465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 ´ 464 Gestur Olafsson, Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmetic geometry, 2008

The purpose of this collection is to guide the non-specialist through the basic theory of various generalizations of topology, starting with clear motivations for their introduction. Structures considered include closure spaces, convergence spaces, proximity spaces, quasi-uniform spaces, merotopic spaces, nearness and filter spaces, semi-uniform convergence spaces, and approach spaces. Each chapter is self-contained and accessible to the graduate student, and focuses on motivations to introduce the generalization of topologies considered, presenting examples where desirable properties are not present in the realm of topologies and the problem is remedied in the more general context. Then, enough material will be covered to prepare the reader for more advanced papers on the topic. While category theory is not the focus of the book, it is a convenient language to study these structures and, while kept as a tool rather than an object of study, will be used throughout the book. For this reason, the book contains an introductory chapter on categorical topology.

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