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

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Beyond Topology


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 / Frédéric 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, Frédéric, 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


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. Frédéric 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

2

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


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.


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


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.


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.

8


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.

9


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

10


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

12


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

14


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].


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.

22


(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:


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


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


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


• 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.

References [1] J. Ad´ amek, Theory of mathematical structures, D. Reidel Publishing Co., Dordrecht, 1983. MR735079 (85c:18001) [2] J. Ad´ amek and H. Herrlich, Cartesian closed categories, quasitopoi and topological universes, Comment. Math. Univ. Carolin. 27 (1986), no. 2, 235–257. MR857544 (88c:18003) [3] J. Ad´ amek, H. Herrlich, and G. E. Strecker, Least and largest initial completions. I, II, Comment. Math. Univ. Carolin. 20 (1979), no. 1, 43–58, 59–77. MR526147 (80f:18009) , The structure of initial completions, Cahiers Topologie Géom. Différentielle 20 [4] (1979), no. 4, 333–352. MR558103 (81d:18009) , Abstract and concrete categories. The joy of cats, John Wiley & Sons, New York, [5] 1990, Revised online edition (2004), http://katmat.math.uni-bremen.de/acc. MR1051419 (91h:18001) [6] J. Ad´ amek, J. Reiterman, and F. Schwarz, On universally topological hulls and quasitopos hulls, Houston J. Math. 17 (1991), no. 3, 375–383. MR1126601 (93a:18002) [7] J. Ad´ amek, J. Reiterman, and G. E. Strecker, Realization of Cartesian closed topological hulls, Manuscripta Math. 53 (1985), no. 1–2, 1–33. MR804336 (86m:18005) [8] J. Ad´ amek and G. E. Strecker, Construction of Cartesian closed topological hulls, Comment. Math. Univ. Carolin. 22 (1981), no. 2, 235–254. MR620360 (83a:18017) [9] I. W. Alderton, F. Schwartz, and S. Weck-Schwarz, On the construction of the extensional topological hull, Cahiers Topologie Géom. Différentielle Catég. 35 (1994), no. 2, 165–174. MR1280988 (95h:18005) ´ [10] P. Antoine, Etude ´ el´ ementaire des cat´ egories d’ensembles structur´ es, Bull. Soc. Math. Belg. 18 (1966), 142–164. MR0200321 (34 #220)


[11] [12] [13] [14] [15] [16] [17] [18]

[19]

[20]

[21]

[22] [23] [24]

[25] [26] [27] [28] [29] [30]

[31] [32] [33] [34] [35] [36]

27

´ , Etude ´ el´ ementaire des cat´ egories d’ensembles structur´ es. II, Bull. Soc. Math. Belg. 18 (1966), 387–414. MR0213271 (35 #4135) , Extension minimale de la cat´ egorie des espaces topologiques, C. R. Acad. Sci. Paris S´ er. A 262 (1966), 1389–1392. MR0202788 (34 #2648) A. V. Arhangel ski˘ı and R. Wiegandt, Connectednesses and disconnectednesses in topology, General Topology and Appl. 5 (1975), 9–33. MR0367920 (51 #4162) C. E. Aull and R. Lowen (eds.), Handbook of the history of general topology, vol. 1, Kluwer Academic Publishers, Dordrecht, 1997. MR1619254 (99a:01001) C. E. Aull and R. Lowen (eds.), Handbook of the history of general topology, vol. 2, Kluwer Academic Publishers, Dordrecht, 1998. MR1795156 (2001e:54001) B. Banaschewski, Essential extensions of T0 -spaces, General Topology and Appl. 7 (1977), no. 3, 233–246. MR0458354 (56 #16557) R. Beattie and H.-P. Butzmann, Convergence structures and applications to functional analysis, Kluwer Academic Publishers, Dordrecht, 2002. MR2327514 (2008c:46001) H. L. Bentley and H. Herrlich, Extensions of topological spaces, Topology (Memphis, TN, 1975) (S. P. Franklin and B. V. Smith Thomas, eds.), Lecture Notes in Pure and Appl. Math., vol. 24, Marcel Dekker Inc., New York, 1976, pp. 129–184. MR0433392 (55 #6368) H. L. Bentley, H. Herrlich, and M. Huˇsek, The historical development of uniform, proximal, and nearness concepts in topology, Handbook of the history of general topology (C. E. Aull and R. Lowen, eds.), vol. 2, Kluwer Academic Publishers, Dordrecht, 1998, pp. 577–629. MR1795163 (2002i:54001) H. L. Bentley, H. Herrlich, and R. Lowen, Improving constructions in topology, Category theory at work (Bremen, 1990) (H. Herrlich and H.-E. Porst, eds.), Res. Exp. Math., vol. 18, Heldermann, Berlin, 1991, pp. 3–20. MR1147915 (93b:54009) H. L. Bentley, H. Herrlich, and E. Lowen-Colebunders, The category of Cauchy spaces is Cartesian closed, Topology Appl. 27 (1987), no. 2, 105–112, Proceedings of the 8th international conference on categorical topology (L’Aquila, 1986). MR911685 (89e:54062) , Convergence, J. Pure Appl. Algebra 68 (1990), no. 1–2, 27–45, Special issue in honor of B. Banaschewski. MR1082777 (91k:54052) H. L. Bentley, H. Herrlich, and W. A. Robertson, Convenient categories for topologists, Comment. Math. Univ. Carolin. 17 (1976), no. 2, 207–227. MR0425890 (54 #13840) H. L. Bentley and E. Lowen-Colebunders, The Cartesian closed topological hull of the category of completely regular filterspaces, Cahiers Topologie Géom. Différentielle Catég. 33 (1992), no. 4, 345–360. MR1197430 (94c:18006a) E. Binz and H. Herrlich (eds.), Categorical topology (Mannheim, 1975), Lecture Notes in Math., vol. 540, Springer, Berlin, 1976. MR0407787 (53 #11558) G. Birkhoff, Lattice theory, 3 ed., AMS Colloq. Publ., no. 25, American Mathematical Society, Providence, RI, 1967. MR0227053 (37 #2638) P. Booth and J. Tillotson, Monoidal closed, Cartesian closed and convenient categories of topological spaces, Pacific J. Math. 88 (1980), no. 1, 35–53. MR595812 (82a:55021) N. Bourbaki, Topologie g´ en´ erale. Ch. 9, Utilisation des nombres r´ eels en topologie g´ en´ erale, 2 ed., Actualits Sci. Indust., vol. 1045, Hermann, Paris, 1958. MR0173226 (30 #3439)) , Topologie g´ en´ erale. Ch. 3, Groupes topologiques. Ch. 4, Nombres r´ eels, 3 ed., Actualits Sci. Indust., vol. 1143, Hermann, Paris, 1960. MR0140603 (25 #4021) , Topologie g´ en´ erale. Ch. 5, Groupes a ` un param` etre. Ch. 6, Espaces num´ eriques et espaces projectifs. Ch. 7, les groupes additifs Rn , Ch. 8, Nombres complexes, 3 ed., Actualits Sci. Indust., vol. 1235, Hermann, Paris, 1963. , Topologie g´ en´ eral. Ch. 1, Structures topologiques. Ch. 2, Structures uniformes, 4 ed., Actualits Sci. Indust., vol. 1142, Hermann, Paris, 1965. MR0244924 (39 #6237) , Topologie g´ en´ erale. Ch. 10, Espaces fonctionnels, Actualits Sci. Indust., vol. 1084, Hermann, Paris, 1967. , Th´ eorie des ensembles, Hermann, Paris, 1970. MR0276101 (43 #1849) G. Bourdaud, Structures d’Antoine associ´ ees aux semi-topologies et aux topologies, C. R. Acad. Sci. Paris Sér. A 279 (1974), 591–594. MR0355930 (50 #8403) , Deux charact´ erizations des c-espaces, C. R. Acad. Sci. Paris Sér. A 281 (1975), 313–316. MR0380700 (52 #1597) , Espaces d’Antoine et semi-espaces d’Antoine, Cahiers Topologie Géom. Diff´ erentielle Catég. 16 (1975), no. 2, 107–134. MR0394529 (52 #15330)

28

[37]

[38]

[39] [40] [41]

[42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58]

[59] [60] [61] [62] [63]

[64]


, Some Cartesian closed topological categories of convergence spaces, Categorical topology (Mannheim, 1975) (E. Binz and H. Herrlich, eds.), Lecture Notes in Math., vol. 540, Springer, Berlin, 1976, pp. 93–108. MR0493924 (58 #12880) H. Brandenburg and M. Huˇsek, A remark on Cartesian closedness, Category theory (Gummersbach, 1981) (K. H. Kamps, D. Pumpl¨ un, and W. Tholen, eds.), Lecture Notes in Math., vol. 962, Springer, Berlin, 1982, pp. 33–38. MR682941 (84a:18014) P. Brock and D. C. Kent, Approach spaces, limit tower spaces, and probabilistic convergence spaces, Appl. Categ. Structures 5 (1997), no. 2, 99–110. MR1456517 (98b:54017) , On convergence approach spaces, Appl. Categ. Structures 6 (1998), no. 1, 117–125. MR1614421 (98m:54007) G. C. L. Br¨ ummer, Topological functors and structure functors, Categorical topology (Mannheim, 1975) (E. Binz and H. Herrlich, eds.), Lecture Notes in Math., vol. 540, Springer, Berlin, 1976, pp. 109–135. MR0442052 (56 #440) ¨ H.-P. Butzmann, Uber die c-Reflexivit¨ at von Cc (X), Comment. Math. Helv. 47 (1972), 92–101. MR0308651 (46 #7765) F. Cagliari, Disconnectednesses cogenerated by Hausdorff spaces, Cahiers Topologie Géom. Diff´ erentielle Catég. 29 (1988), no. 1, 3–8. MR941648 (89f:54023) F. Cagliari and M. Cicchese, Disconnectednesses and closure operators, Rend. Circ. Mat. Palermo (2) Suppl. (1985), no. 11, 15–23. MR897967 (88j:54015) H. Cartan, Filtres et ultrafiltres, C. R. Acad. Sci. Paris 205 (1937), 777–779. , Th´ eorie des filtres, C. R. Acad. Sci. Paris 205 (1937), 595–598. ˇ E. Cech, Topological spaces, John Wiley & Sons, Prague, 1966. MR0211373 (35 #2254) G. Choquet, Convergences, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 23 (1948), 57–112. MR0025716 (10,53d) ˇ cura, Cartesian closed coreflective subcategories of the category of topological spaces, J. Cinˇ Topology Appl. 41 (1991), no. 3, 205–212. MR1135098 (92m:18013) M. M. Clementino, Hausdorff separation in categories, Appl. Categ. Structures 1 (1993), no. 3, 285–295. MR1254740 (94m:18005) , On categorical notions of compact objects, Appl. Categ. Structures 4 (1996), no. 1, 15–29. MR1393959 (97g:18002) M. M. Clementino, E. Giuli, and W. Tholen, Topology in a category: compactness, Portugal. Math. 53 (1996), no. 4, 397–433. MR1432147 (97k:54008) Maria Manuel Clementino and Walter Tholen, Separated and connected maps, Appl. Categ. Structures 6 (1998), no. 3, 373–401. MR1641863 (99h:18005) ¨ G. S. Cogosvili, Uber Konvergenzra¨ ume, Rec. Math. Moscou N.S. 9 (51) (1941), 377–381 (Russian), German summary. MR0004754 (3,56g) C. H. Cook and H. R. Fischer, On equicontinuity and continuous convergence, Math. Ann. 159 (1965), 94–104. MR0179752 (31 #3995) , Uniform convergence structures, Math. Ann. 173 (1967), 290–306. MR0217756 (36 #845) ´ Cs´ A. asz´ ar, Fondements de la topologie g´ en´ erale, Akad´ emiai Kiad´ o, Budapest, 1960. MR0113205 (22 #4043) , Erweiterung, Kompaktifizierung und Vervollst¨ andigung syntopogener R¨ aume, Contributions to extension theory of topological structures (Berlin, 1967) (J. Flacshmeyer, H. Poppe, and F. Terpe, eds.), Deutscher Verlag der Wissenschaften, Berlin, 1969, pp. 51–54. , Proximities, screens, merotopies, uniformities. I, Acta Math. Hungar. 49 (1987), no. 3–4, 459–479. MR891059 (88k:54002a) , Proximities, screens, merotopies, uniformities. II, Acta Math. Hungar. 50 (1987), no. 1–2, 97–109. MR893249 (88k:54002b) , Proximities, screens, merotopies, uniformities. III, Acta Math. Hungar. 51 (1988), no. 1–2, 23–33. MR934578 (89e:54019a) , Proximities, screens, merotopies, uniformities. IV, Acta Math. Hungar. 51 (1988), no. 1–2, 151–164. MR934592 (89e:54019b) B. J. Day, Limit spaces and closed span categories, Category Seminar (Sydney, 1972/1973) (G. M. Kelly, ed.), Lecture Notes in Math., vol. 420, Springer, Berlin, 1974, pp. 65–74. MR0374240 (51 #10440) B. J. Day and G. M. Kelly, On topologically quotient maps preserved by pullbacks or products, Proc. Cambridge Philos. Soc. 67 (1970), 553–558. MR0254817 (40 #8024)


29

[65] J. De´ ak, A common topological supercategory for merotopies and syntopogenous structures, Topology. Theory and applications, II (Pécs, 1989), Colloq. Math. Soc. J´ anos Bolyai, vol. 55, North-Holland, Amsterdam, 1993, pp. 177–190. MR1244363 (94m:54028) [66] R. E. DeMarr, Order convergence and topological convergence, Proc. Amer. Math. Soc. 16 (1965), 588–590. MR0178449 (31 #2706) [67] D. Dikranjan and E. Giuli, Closure operators. I, Topology Appl. 27 (1987), no. 2, 129–143, Proceedings of the 8th international conference on categorical topology (L’Aquila, 1986). MR911687 (89e:18002) [68] 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) [69] S. Dolecki and G. H. Greco, Cyrtologies of convergences. I, Math. Nachr. 126 (1986), 327– 348. MR846584 (88b:54002a) [70] E. J. Dubuc, Concrete quasitopoi, Applications of sheaves (Durham, 1977), Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 239–254. MR555548 (81e:18010) [71] G. A. Edgar, A Cartesian closed category for topology, General Topology and Appl. 6 (1976), no. 1, 65–72. MR0400137 (53 #3972) [72] C. Ehresmann, Structures quotient, Comment. Math. Helv. 38 (1964), 219–283. MR0170919 (30 #1154) , Cat´ egories topologiques. I, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. [73] 28 (1966), 133–146. MR0215766 (35 #6601) [74] , Cat´ egories topologiques. I, II, III, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 133–146; 147–160; 161–175. MR0215766 (35 #6601) , Cat´ egories topologiques. II, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. [75] Math. 28 (1966), 147–160. MR0215766 (35 #6601) , Cat´ egories topologiques. III, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. [76] Math. 28 (1966), 161–175. MR0215766 (35 #6601) [77] , Cat´ egories structur´ ees g´ en´ eralis´ ees, Cahiers Topologie Géom. Différentielle 10 (1968), 139–167. MR0244337 (39 #5652) [78] T. H. Fay, G. C. L. Br¨ ummer, and K. A. Hardie, A characterization of topological functors, Math. Colloq. Univ. Cape Town 12 (1980) (1978/79), 85–93. MR565148 (81g:18002) [79] W. A. Feldman, Topological spaces and their associated convergence function algebras, Ph.d. thesis, Queen’s University, Kingston, 1971. [80] H. R. Fischer, Limesr¨ aume, Math. Ann. 137 (1959), 269–303. MR0109339 (22 #225) [81] A. Fr¨ olicher, Kompakt erzeugte R¨ aume und Limesr¨ aume, Math. Z. 129 (1972), 57–63. MR0341026 (49 #5776) , Cartesian closed categories and analysis of smooth maps, Categories in continuum [82] physics (Buffalo, NY, 1982) (F. W. Lawvere and S. H. Schanuel, eds.), Lecture Notes in Math., vol. 1174, Springer, Berlin, 1986, pp. 43–51. MR842916 (87k:58024) [83] A. Fr¨ olicher and W. Bucher, Calculus in vector spaces without norm, Lecture Notes in Math., vol. 30, Springer, Berlin, 1966. MR0213869 (35 #4723) [84] A. Fr¨ olicher and A. Kriegl, Linear spaces and differentiation theory, John Wiley & Sons, Chichester, 1988. MR961256 (90h:46076) [85] W. G¨ ahler, Grundstrukturen der Analysis. I, Lehrb¨ ucher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, vol. 58, Birkh¨ auser Verlag, Basel, 1977. MR519344 (80i:58001b) , Grundstrukturen der Analysis. II, Lehrb¨ ucher und Monographien aus dem Gebiete [86] der exakten Wissenschaften, Mathematische Reihe, vol. 61, Birkh¨ auser Verlag, Basel, 1978. MR519344 (80i:58001b) [87] , A topological approach to structure theory. I, Math. Nachr. 100 (1981), 93–144. MR632622 (83b:54009) , Convergence structures-historical remarks and the monadic approach, Categorical [88] methods in algebra and topology (H.-E. Porst, ed.), Mathematik Arbeitspapiere, vol. 48, Universit¨ at Bremen, 1997, pp. 171–193. [89] G. Greve, General construction of monoidal closed structures in topological, uniform and nearness spaces, Category theory (Gummersbach, 1981) (K. H. Kamps, D. Pumpl¨ un, and W. Tholen, eds.), Lecture Notes in Math., vol. 962, Springer, Berlin, 1982, pp. 100–114. MR682947 (84b:18005)

30


[90] D. W. Hajek and A. Mysior, On nonsimplicity of topological categories, Categorical topology (Berlin, 1978) (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 84–93. MR544634 (81j:18009) [91] J. M. Harvey, T0 -separation in topological categories, Quaestiones Math. 2 (1977/78), no. 1– 3, 177–190. MR0486050 (58 #5838) [92] Z. Hedrl´ın, A. Pultr, and V. Trnkov´ a, Concerning a categorical approach to topological and algebraic theories, General Topology and its Relations to Modern Analysis and Algebra, II (Prague, 1966), Academia, Prague, 1967, pp. 176–181. MR0230784 (37 #6344) [93] N. C. Heldermann, A remark on the embeddings of R0 Top into the category of nearness spaces, Topology, Vol. II (Budapest, 1978), Colloq. Math. Soc. J´ anos Bolyai, vol. 23, NorthHolland, Amsterdam, 1980, pp. 607–623. MR588809 (81k:54040) , Coreflections induced by fine functors in topological categories, Quaestiones Math. [94] 5 (1982/83), no. 2, 119–133. MR674752 (84j:54004) [95] H. Herrlich, E-kompakte R¨ aume, Math. Z. 96 (1967), 228–255. MR0205218 (34 #5051) , Topologische Reflexionen und Coreflexionen, Lecture Notes in Math., vol. 78, [96] Springer, Berlin, 1968. MR0256332 (41 #988) [97] , Categorical topology, General Topology and Appl. 1 (1971), no. 1, 1–15. MR0283744 (44 #974) , Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), [98] 1–16. MR0460414 (57 #408) , A concept of nearness, General Topology and Appl. 4 (1974), 191–212. MR0350701 [99] (50 #3193) , Topological functors, General Topology and Appl. 4 (1974), 125–142. MR0343226 [100] (49 #7970) [101] , Topological structures, Topological structures (Amsterdam, 1973), Mathematical Centre Tracts, vol. 52, Math. Centrum, Amsterdam, 1974, pp. 59–122. MR0358706 (50 #11165) , Initial completions, Math. Z. 150 (1976), no. 2, 101–110. MR0437614 (55 #10538) [102] , Initial and final completions, Categorical topology (Berlin, 1978) (H. Herrlich and [103] G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 137–149. MR544639 (80j:18015) , Are there convenient subcategories of Top?, Topology Appl. 15 (1983), no. 3, 263– [104] 271. MR694546 (84c:54015) , Categorical topology 1971–1981, General topology and its relations to modern anal[105] ysis and algebra, V (Prague, 1981), Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 279–383. MR698425 (84d:54016) , Universal topology, Categorical topology (Toledo, OH, 1983) (H. L. Bentley, [106] ed.), Sigma Ser. Pure Math., vol. 5, Heldermann, Berlin, 1984, pp. 223–281. MR785019 (86g:18004) , Topologie I: Topologische R¨ aume, Berliner Studienreihe zur Mathematik, vol. 2, [107] Heldermann, Berlin, 1986. MR880705 (88g:54001) [108] , Topological improvements of categories of structured sets, Topology Appl. 27 (1987), no. 2, 145–155, Proceedings of the 8th international conference on categorical topology (L’Aquila, 1986). MR911688 (88m:18006) , Hereditary topological constructs, General topology and its relations to modern [109] analysis and algebra, VI (Prague, 1986) (Z. Frolik, ed.), Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 249–262. MR952611 (90c:54008) , On the representability of partial morphisms in Top eand in related constructs, [110] Categorical algebra and its applications (Louvain-La-Neuve, 1987), Lecture Notes in Math., vol. 1348, Springer, Berlin, 1988, pp. 143–153. MR975967 (89m:18006) [111] H. Herrlich and M. Huˇsek, Categorical topology, Recent progress in general topology (Prague, 1991) (M. Huˇsek and J. van Mill, eds.), North-Holland, Amsterdam, 1992, pp. 369–403. MR1229132 , Some open categorical problems in Top, Appl. Categ. Structures 1 (1993), no. 1, [112] 1–19. MR1245790 (94g:54008) [113] H. Herrlich, E. Lowen-Colebunders, and F. Schwarz, Improving Top: PrTop and PsTop, Category theory at work (Bremen, 1990) (H. Herrlich and H.-E. Porst, eds.), Res. Exp. Math., vol. 18, Heldermann, Berlin, 1991, pp. 21–34. MR1147916 (93d:18007)


31

[114] H. Herrlich and L. D. Nel, Cartesian closed topological hulls, Proc. Amer. Math. Soc. 62 (1977), no. 2, 215–222. MR0476831 (57 #16383) [115] H. Herrlich and G. Preuß (eds.), Categorical topology (Berlin, 1978), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979. MR544624 (80f:18002) [116] H. Herrlich and C. Prieto (eds.), Categorical topology. The complete work of Graciela Salicrup, Sociedad Matem´ atica Mexicana, M´ exico, 1988. MR1307636 (95g:18001) [117] H. Herrlich and M. Rajagopalan, The quasicategory of quasispaces is illegitimate, Arch. Math. (Basel) 40 (1983), no. 4, 364–366. MR708599 (85c:54013) [118] H. Herrlich, G. Salicrup, and R. V´ azquez, Light factorization structures, Quaestiones Math. 3 (1978/79), no. 3, 189–213. MR533531 (81e:18003) [119] H. Herrlich and G. E. Strecker, Coreflective subcategories, Trans. Amer. Math. Soc. 157 (1971), 205–226. MR0280561 (43 #6281) , Coreflective subcategories in general topology, Fund. Math. 73 (1971/72), no. 3, [120] 199–218. MR0302729 (46 #1872) , Category theory, Allyn and Bacon, Boston, 1973. MR0349791 (50 #2284) [121] [122] H. Herrlich and D. Zhang, Categorical properties of probabilistic convergence spaces, Appl. Categ. Structures 6 (1998), no. 4, 495–513. MR1657510 (99i:54007) [123] R. E. Hoffmann, Die kategorielle Auffassung der Initial- und Finaltopologie, Ph.d. thesis, Universit¨ at Bochum, 1972. , Topological functors and factorizations, Arch. Math. (Basel) 26 (1975), 1–7. [124] MR0428255 (55 #1280) [125] R.-E. Hoffmann, Note on universal topological completion, Cahiers Topologie Géom. Diff´ erentielle 20 (1979), no. 2, 199–216. MR548489 (81a:54011) [126] S. S. Hong, Limit-operators and reflective subcategories, TOPO 72—general topology and its applications (Pittsburgh, PA, 1972), Lecture Notes in Math., vol. 378, Springer, Berlin, 1974, pp. 219–227. MR0365456 (51 #1708) [127] Y. H. Hong, On initially structured functors, J. Korean Math. Soc. 14 (1977/78), no. 2, 159–165. MR0486038 (58 #5826) [128] W. N. Hunsaker and P. L. Sharma, Proximity spaces and topological functors, Proc. Amer. Math. Soc. 45 (1974), 419–425. MR0353265 (50 #5749) [129] M. Huˇsek, Generalized proximity and uniform spaces. I, Comment. Math. Univ. Carolin. 5 (1964), 247–266. MR0176441 (31 #713) , S-categories, Comment. Math. Univ. Carolin. 5 (1964), 37–46. MR0174027 (30 [130] #4234) , Generalized proximity and uniform spaces. II, Comment. Math. Univ. Carolin. 6 [131] (1965), 119–139. MR0177389 (31 #1652) , Remarks on reflections, Comment. Math. Univ. Carolin. 7 (1966), 249–259. [132] MR0202800 (34 #2659) [133] , Categorical methods in topology, General Topology and its Relations to Modern Analysis and Algebra, II (Prague, 1966), Academia, Prague, 1967, pp. 190–194. MR0235005 (38 #3317) , Construction of special functors and its applications, Comment. Math. Univ. Car[134] olin. 8 (1967), 555–566. MR0222136 (36 #5188) , Categorial connections between generalized proximity spaces and compactifications, [135] Contributions to extension theory of topological structures (Berlin, 1967), Deutscher Verlag der Wissenschaften, Berlin, 1969, pp. 127–132. MR0248730 (40 #1981) , Lattices of reflections and coreflections in continuous structures, Categorical topol[136] ogy (Mannheim, 1975) (E. Binz and H. Herrlich, eds.), Lecture Notes in Math., vol. 540, Springer, Berlin, 1976, pp. 404–424. MR0445458 (56 #3798) [137] , History and development of Hausdorff ’s work in extension in metric spaces, Recent developments of general topology and its applications (Berlin, 1992) (G. Preuß, G. G¨ ahler, and H. Herrlich, eds.), Mathematical Research, vol. 67, Akademie-Verlag, Berlin, 1992, pp. 160–169. MR1219777 (94c:01024) [138] M. Huˇsek and D. Pumpl¨ un, Disconnectedness, Quaestiones Math. 13 (1990), no. 3–4, 449– 459. MR1084754 (91k:54013) [139] M. Huˇsek and A. Tozzi, Generalized reflective cum coreflective classes in Top and Unif, Appl. Categ. Structures 4 (1996), no. 1, 57–68. MR1393962 (97c:54012)

32


[140] J. R. Isbell, Some remarks concerning categories and subspaces, Canad. J. Math. 9 (1957), 563–577. MR0094405 (20 #923) , Uniform spaces, Mathematical Surveys, vol. 12, American Mathematical Society, [141] Providence, RI, 1964. MR0170323 (30 #561) [142] V. Kannan, Reflexive cum coreflexive subcategories in topology, Math. Ann. 195 (1972), 168–174. MR0291048 (45 #142) [143] M. Katˇetov, Convergence structures, General Topology and its Relations to Modern Analysis and Algebra, II (Prague, 1966), Academia, Prague, 1967, pp. 207–216. MR0232331 (38 #656) [144] H. H. Keller, R¨ aume stetiger multilinearer Abbildungen als Limesr¨ aume, Math. Ann. 159 (1965), 259–270. MR0193475 (33 #1695) , Die Limes-Uniformisierbarkeit der Limesr¨ aume, Math. Ann. 176 (1968), 334–341. [145] MR0225280 (37 #874) [146] J. F. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303–315. MR0174611 (30 #4812) [147] D. C. Kent, Convergence quotient maps, Fund. Math. 65 (1969), 197–205. MR0250258 (40 #3497) [148] D. C. Kent and G. D. Richardson, Compactifications of convergence spaces, Internat. J. Math. Math. Sci. 2 (1979), no. 3, 345–368. MR542954 (80i:54004) [149] H.-J. Kowalsky, Limesr¨ aume und Komplettierung, Math. Nachr. 12 (1954), 301–340. MR0073147 (17,390b) , Topologische R¨ aume, Lehrb¨ ucher und Monographien aus dem Gebiete der exakten [150] Wissenschaften, Mathematische Reihe, vol. 26, Birkh¨ auser Verlag, Basel, 1961. MR0121768 (22 #12502) [151] A. Kriegl, Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektorr¨ aumen, Monatsh. Math. 95 (1983), no. 4, 287–309. MR718065 (85e:58018) , A Cartesian closed extension of the category of smooth Banach manifolds, Cate[152] gorical topology (Toledo, OH, 1983) (H. L. Bentley, ed.), Sigma Ser. Pure Math., vol. 5, Heldermann, Berlin, 1984, pp. 323–336. MR785022 (86i:58005) [153] C. Kuratowski, Topologie, Monografie Matematyczne, vol. 3, PWN—Polish Scientific Publishers, Warsaw–Lw´ ow, 1933. [154] R. S. Lee, The category of uniform convergence spaces is Cartesian closed, Bull. Austral. Math. Soc. 15 (1976), no. 3, 461–465. MR0431081 (55 #4083) [155] S. J. Lee and L. D. Nel, Foundational isomorphisms in continuous differentiation theory, Appl. Categ. Structures 6 (1998), no. 1, 127–135. MR1614422 (99d:58027) [156] H. Lord, Factorization, diagonal separation, and disconnectedness, Topology Appl. 47 (1992), no. 2, 83–96. MR1193192 (94h:18002) , Factorizations and disconnectedness, Recent developments of general topology and [157] its applications (Berlin, 1992) (G. Preuß, G. G¨ ahler, and H. Herrlich, eds.), Mathematical Research, vol. 67, Akademie-Verlag, Berlin, 1992, pp. 197–202. MR1219782 (94d:54036) [158] E. Lowen and R. Lowen, A quasitopos containing CONV and MET as full subcategories, Internat. J. Math. Math. Sci. 11 (1988), no. 3, 417–438. MR947271 (89g:54024) , Topological quasitopos hulls of categories containing topological and metric ob[159] jects, Cahiers Topologie Géom. Différentielle Catég. 30 (1989), no. 3, 213–228. MR1029625 (91a:54015) [160] E. Lowen, R. Lowen, and C. Verbeeck, Exponential objects in the construct PRAP, Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 4, 259–276. MR1487961 (99e:54007) [161] R. Lowen, Kuratowski’s measure of noncompactness revisited, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 154, 235–254. MR947504 (89i:54010) , Approach spaces: a common supercategory of TOP and MET, Math. Nachr. 141 [162] (1989), 183–226. MR1014427 (90i:54025) [163] , Cantor-connectedness revisited, Comment. Math. Univ. Carolin. 33 (1992), no. 3, 525–532. MR1209293 (94b:54064) , Approach spaces: The missing link in the topology-uniformity-metric triad, Oxford [164] Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1997. MR1472024 (98f:54002)


33

[165] R. Lowen and K. Robeys, Compactifications of products of metric spaces and their relations ˇ to Cech–Stone and Smirnov compactifications, Topology Appl. 55 (1994), no. 2, 163–183. MR1256218 (94m:54063) [166] R. Lowen and M. Sioen, A small note on separation in AP, To appear. [167] E. Lowen-Colebunders, Uniformizable Cauchy spaces, Internat. J. Math. Math. Sci. 5 (1982), no. 3, 513–527. MR669105 (83k:54031) , Function classes of Cauchy continuous maps, Monographs and Textbooks in Pure [168] and Applied Mathematics, vol. 123, Marcel Dekker Inc., New York, 1989. MR1001563 (91c:54002) [169] E. Lowen-Colebunders, R. Lowen, and M. Nauwelaerts, The Cartesian closed hull of the category of approach spaces, Cahiers Topologie Géom. Différentielle Catég. 42 (2001), no. 4, 242–260. MR1876866 (2002k:54012) [170] E. Lowen-Colebunders and G. Sonck, Exponential objects and Cartesian closedness in the construct Prtop, Appl. Categ. Structures 1 (1993), no. 4, 345–360. MR1268508 (94m:54029) , On the largest coreflective Cartesian closed subconstruct of Prtop, Appl. Categ. [171] Structures 4 (1996), no. 1, 69–79. MR1393963 (97c:54013) [172] A. Machado, Espaces d’Antoine et pseudo-topologies, Cahiers Topologie Géom. Différentielle 14 (1973), 309–327. MR0345054 (49 #9793) [173] T. Marny, Rechts-Bikategoriestrukturen in toplogischen Kategorien, Ph.d. thesis, Free University Berlin, 1973. , Limit-metrizability of limit spaces and uniform limit spaces, Categorical topology [174] (Berlin, 1978) (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 234–242. MR544649 (81b:54030) [175] Th. Marny, On epireflective subcategories of topological categories, General Topology and Appl. 10 (1979), no. 2, 175–181. MR527843 (80g:18004) [176] E. H. Moore and H. L. Smith, A general theory of limits, Amer. J. Math. 44 (1922), no. 2, 102–121. MR1506463 [177] K. Morita and J. Nagata (eds.), Topics in general topology, North-Holland Mathematical Library, vol. 41, North-Holland, Amsterdam, 1989. MR1053191 (91a:54001) [178] S. Mr´ owka, On the convergence of nets of sets, Fund. Math. 45 (1958), 237–246. MR0098359 (20 #4820) [179] S. G. Mr´ owka and W. J. Pervin, On uniform connectedness, Proc. Amer. Math. Soc. 15 (1964), 446–449. MR0161307 (28 #4515) [180] S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 59, Cambridge University Press, London, 1970. MR0278261 (43 #3992) [181] R. Nakagawa, Natural relations, separations and connections, Questions Answers Gen. Topology 3 (1985), no. 1, 28–46. MR798584 (87a:18004) [182] L. D. Nel, Initially structured categories and Cartesian closedness, Canad. J. Math. 27 (1975), no. 6, 1361–1377. MR0393183 (52 #13993) , Cartesian closed topological categories, Categorical topology (Mannheim, 1975) [183] (E. Binz and H. Herrlich, eds.), Lecture Notes in Math., vol. 540, Springer, Berlin, 1976, pp. 439–451. MR0447369 (56 #5682) , Cartesian closed coreflective hulls, Quaestiones Math. 2 (1977/78), no. 1–3, 269– [184] 283. MR0480687 (58 #843) [185] , Topological universes and smooth Gel fand–Na˘ımark duality, Mathematical applications of category theory (Denver, CO, 1983), Contemporary Mathematics, vol. 30, American Mathematical Society, Providence, RI, 1984, pp. 244–276. MR749775 (86b:18007) , Infinite-dimensional calculus allowing nonconvex domains with empty interior, [186] Monatsh. Math. 110 (1990), no. 2, 145–166. MR1076329 (91i:58015) , Introduction to categorical methods. I and II, Carleton-Ottawa Mathematical Lec[187] ture Notes Series, vol. 11, Carleton University, Ottawa, 1991. MR1153792 (92j:46132a) [188] , Introduction to categorical methods. III, Carleton-Ottawa Mathematical Lecture Notes Series, vol. 12, Carleton University, Ottawa, 1991. MR1153792 (92j:46132a) , Differential calculus founded on an isomorphism, Appl. Categ. Structures 1 (1993), [189] no. 1, 51–57. MR1245792 (94h:58023) [190] E. T. Ordman, Classroom notes: convergence almost everywhere is not topological, Amer. Math. Monthly 73 (1966), no. 2, 182–183. MR1533643

34


[191] J. Penon, Quasi-topos, C. R. Acad. Sci. Paris Sér. A 276 (1973), 237–240 (French). MR0332921 (48 #11246) , Sur les quasi-topos, Cahiers Topologie Géom. Différentielle 18 (1977), no. 2, 181– [192] 218. MR0480693 (58 #848) [193] G. Preuß, Trennung und Zusammenhang, Monatsh. Math. 74 (1970), 70–87. MR0261543 (41 #6156) , Allgemeine Topologie, Hochschultext, Springer, Berlin, 1972. MR0370455 (51 [194] #6682) , Relative connectednesses and disconnectednesses in topological categories, Quaes[195] tiones Math. 2 (1977/78), no. 1–3, 297–306. MR0500841 (58 #18355) , Connection properties in topological categories and related topics, Categorical topol[196] ogy (Berlin, 1978) (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 293–307. MR544654 (80j:18009) , Connectednesses and disconnectednesses in S-Near, Categorical aspects of topology [197] and analysis (Ottawa, 1980), Lecture Notes in Math., vol. 915, Springer, Berlin, 1982, pp. 275–292. MR659898 (83h:54036) [198] , Theory of topological structures. An approach to categorical topology, Mathematics and its Applications, vol. 39, D. Reidel Publishing Co., Dordrecht, 1988. MR937052 (89m:54014) , Point separation axioms, monotopological categories and MacNeille completions, [199] Category theory at work (Bremen, 1990) (H. Herrlich and H.-E. Porst, eds.), Res. Exp. Math., vol. 18, Heldermann, Berlin, 1991, pp. 47–55. MR1147918 (93d:18010) , Semiuniform limit spaces and semi-Cauchy spaces, Recent developments of general [200] topology and its applications (Berlin, 1992) (G. Preuß, G. G¨ ahler, and H. Herrlich, eds.), Mathematical Research, vol. 67, Akademie-Verlag, Berlin, 1992, pp. 268–278. MR1219792 (94m:54005) [201] A. Pultr, On full embeddings of concrete categories with respect to forgetful functors, Comment. Math. Univ. Carolin. 9 (1968), 281–305. MR0240166 (39 #1520) [202] A. Pultr and A. Tozzi, Separation axioms and frame representation of some topological facts, Appl. Categ. Structures 2 (1994), no. 1, 107–118. MR1283218 (95c:54034) [203] J. F. Ramaley and O. Wyler, Cauchy spaces. I. Structure and uniformization theorems, Math. Ann. 187 (1970), 175–186. MR0266141 (42 #1050a) , Cauchy spaces. II. Regular completions and compactifications, Math. Ann. 187 [204] (1970), 187–199. MR0266142 (42 #1050b) [205] B. C. Rennie, Lattices, Proc. London Math. Soc. (2) 52 (1951), 386–400. MR690634 (13,7c) ˇ [206] G. D. Richardson, A Stone–Cech compactification for limit spaces, Proc. Amer. Math. Soc. 25 (1970), 403–404. MR0256336 (41 #992) [207] G. D. Richardson and D. C. Kent, Regular compactifications of convergence spaces, Proc. Amer. Math. Soc. 31 (1972), 571–573. MR0286074 (44 #3290) [208] G. Richter, More on exponential objects in categories of pretopological spaces, Appl. Categ. Structures 5 (1997), no. 4, 309–319. MR1478523 (99g:54006) [209] G. Salicrup and R. V´ azquez, Connection and disconnection, Categorical topology (Berlin, 1978) (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 326–344. MR544657 (80m:54015) [210] F. Schwarz, Exponential objects in categories of (pre)topological spaces and their natural function spaces, C. R. Math. Rep. Acad. Sci. Canada 4 (1982), no. 6, 321–326. MR681186 (84g:18016) , Cartesian closedness, exponentiality, and final hulls in pseudotopological spaces, [211] Quaestiones Math. 5 (1982/83), no. 3, 289–304. MR690033 (84d:18005) [212] , Powers and exponential objects in initially structured categories and applications to categories of limit spaces, Quaestiones Math. 6 (1983), no. 1–3, 227–254. MR700250 (85c:54023) , Hereditary topological categories and topological universes, Quaestiones Math. 10 [213] (1986), no. 2, 197–216. MR871020 (88a:18011) , Description of the topological universe hull, Categorical methods in computer sci[214] ence (Berlin, 1988), Lecture Notes in Comput. Sci., vol. 393, Springer, Berlin, 1989, pp. 325– 332. MR1048372 (91j:18006)


35

[215] F. Schwarz and S. Weck-Schwarz, Internal description of hulls: a unifying approach, Category theory at work (Bremen, 1990) (H. Herrlich and H.-E. Porst, eds.), Res. Exp. Math., vol. 18, Heldermann, Berlin, 1991, pp. 35–45. MR1147917 (93b:18014) , On hereditary and product-stable quotient maps, Comment. Math. Univ. Carolin. [216] 33 (1992), no. 2, 345–352. MR1189666 (93h:18002) [217] Yu. Smirnov, On proximity spaces in the sense of V. A. Efremoviˇ c, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 895–898 (Russian), English translation: Amer. Math. Soc. Transl. Ser. 2 38 (1964) 1–4. MR0055660 (14,1107a) , On completeness of uniform spaces and proximity spaces, Doklady Akad. Nauk [218] SSSR (N.S.) 91 (1953), 1281–1284 (Russian). MR0063014 (16,58e) [219] E. Spanier, Quasi-topologies, Duke Math. J. 30 (1963), 1–14. MR0144300 (26 #1847) [220] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR0210075 (35 #970) [221] G. E. Strecker, On Cartesian closed topological hulls, Categorical topology (Toledo, OH, 1983) (H. L. Bentley, ed.), Sigma Ser. Pure Math., vol. 5, Heldermann, Berlin, 1984, pp. 523– 539. MR785033 (87f:18006) [222] W. Tholen, Reflective subcategories, Topology Appl. 27 (1987), no. 2, 201–212, Proceedings of the 8th international conference on categorical topology (L’Aquila, 1986). MR911692 (89b:18006) [223] A. Tozzi and O. Wyler, On categories of supertopological spaces, Acta Univ. Carolin. Math. Phys. 28 (1987), no. 2, 137–149. MR932750 (89b:54014) [224] A. J. Ward, On relations between certain intrinsic topologies in partially ordered sets, Proc. Cambridge Philos. Soc. 51 (1955), 254–261. MR0070995 (17,67b) [225] S. Weck-Schwarz, Cartesian closed topological and monotopological hulls: a comparison, Topology Appl. 38 (1991), no. 3, 263–274. MR1098906 (92c:18010) [226] A. Weil, Les recouvrements des espaces topologiques: espaces complets, espaces bicompacts, C. R. Acad. Sci. Paris 202 (1936), 1002–1005. , Sur les espaces a ` structure uniforme et sur la topologie g´ en´ erale, Hermann, Paris, [227] 1937. ˇ [228] O. Wyler, The Stone–Cech compactification for limit spaces, Notices Amer. Math. Soc. 15 (1968), 169. , On the categories of general topology and topological algebra, Arch. Math. (Basel) [229] 22 (1971), 7–17. MR0287563 (44 #4767) , Top categories and categorical topology, General Topology and Appl. 1 (1971), [230] no. 1, 17–28. MR0282324 (43 #8036) [231] , Function spaces in topological categories, Categorical topology (Berlin, 1978) (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math., vol. 719, Springer, Berlin, 1979, pp. 411–420. MR544664 (82m:54004) , Lecture notes on topoi and quasitopoi, World Scientific Publishing Co. Inc., Teaneck, [232] NJ, 1991. MR1094373 (92c:18004) University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, 2020 Antwerp, Belgium E-mail address: [email protected] Department of Mathematics, Free University of Brussels, Pleinlaan 2, 1000 Brussels, Belgium E-mail address: [email protected] University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, 2020 Antwerp, Belgium E-mail address: [email protected]


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

A CONVENIENT SETTING FOR COMPLETIONS AND FUNCTION SPACES

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

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 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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S. DOLECKI

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).


135

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 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

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