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CONTEMPORARY 486

Beyond

Frédéric Mynard Elliott Pearl Editors

American Mathematical Society http://dx.doi.org/10.1090/conm/486

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 / Fr´ed´eric Mynard, Elliott Pearl, editors. p. cm. — (Contemporary mathematics ; v. 486) Includes bibliographical references and index. ISBN 978-0-8218-4279-9 (alk. paper) 1. Topological spaces. 2. Topology. I. Mynard, Fr´ed´eric, 1973– II. Pearl, Elliott.

QA611.3.B49 2009 514—dc22 2008050812

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational 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 acknowledg- ment 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 Math- ematical 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/ 10987654321 141312111009 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 and point-free Anna Di Concilio 89 An Initiation into Convergence Theory Szymon Dolecki 115 Closure Marcel Erne´ 163 An Introduction to Quasi-uniform Spaces Hans-Peter A. Kunzi¨ 239 Approach Theory Robert Lowen and Christophe Van Olmen 305 Semiuniform Convergence Spaces and 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 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 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 −1 topology induced by q|q−1(B) : q (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 ZX denotes the set of all functions from X to Z, the sets of functions are well-behaved in the sense that

 Y ZX×Y = ZX , where the equality represents the bijection f → tf where tf(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 tf : 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 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 eye- opener for the working mathematician, the day-to-day user of topology: there is sometimes much to gain in looking beyond topology. Any reference to theory has been carefully avoided so far, because the present volume is not a book on . 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 con- tributed to this book, and to my co-editor Elliott Pearl whose tremendous work and technical skills made it possible to finish this book.

Fr´ed´eric Mynard August 2008 Index

ACSp,174 PsTop, 2, 341 AFrm,319 PsTops,349 ATSp,174 P,71 ApVec, 313 QU,2,244 App,2 RegChy,68 Ap,307 RegNear,56 Ap0,321 RegNear0,58 C-Fil,348 RegTop,56 CAp, 325 Rng,3 CReg,307 SUConv,342 CSp,174 SULim,342 Cap,2 SepNear,57 Chy, 52, 355 SepNear0,59 Chy0,68 Seq,2 Cls,2 Set,2,141 CompUnif2,2 Sob,321 Comp2,2 T3Chy,68 Conv,2,141 TCSp,174 C,53 TNear,45 EpiAp,327 TopVec,313 EpiTop,25 Top, 2, 44, 174, 307, 340 Fil, 51, 341 Top0,2 Fil-D-SUConv,348 Top1,2 Fil-Mer,350 Tops,44,349 GConv,341 Tych,2 Grp,3 T-PrTop, 346 Haus,2 UAp,307 KConv,341 UG,2 KConvs,348 ULim, 79, 343 Lim,2,53,341 Unif, 2, 44, 340 Limr,54 Unif2,2 Lims, 53, 349 VectR,3 Mer, 41, 350 cChy0,70 Metc,2 cP,72 Near,2,43 cRegNear0,63 PUConv,368 lcApVec,313 Pos,2 lcTopVec,313 PrAp,325 pMetTopVec,313 PrTop, 2, 325, 341 pMetTop,313 PrTops, 349 pMetVec,313 PrULim, 347 pMet,2,307 Pretop,53 pqMet,307 Prost,2 sNormTopVec,313 PsAp,326 sNorm1,313

375 376 INDEX

A-morphism, 3 cartesian closed hull, 327 ABC lattice, 189 cartesian closed topological construct, 20, 77 absolute subset selection, 208 cartesian closed topological hull, 22, 326 adherence, 127 categorical topology, 1 adherence of a set, 128 category, 3, 140 adherence operator, 310 category theory, 3 adjoint functor, 7 Cauchy colleciton, 47 adjoint pair of maps, 201 Cauchy continuous Cauchy structure, 78 adjoint situation, 217 Cauchy continuous map, 342 Alexandroff (discrete) space, 174 Cauchy filter, 47, 80, 270, 310, 342, 350 Alexandroff completion, 175, 183 Cauchy filter pair, 277 Alexandroff proximity, 94 , 52 Alexandroff space, 171 Cauchy system, 350 Alexandroff topology, 174 Cauchy-regular filter space, 68 (of a monad), 217 Cechˇ proximity, 94 algebraic closure spaces, 174 chain, 177 algebraic construct, 12 chaotic topology, 123 amnestic construct, 3 Choquet space, 341 Anti-Exchange Axiom, 169 close-to relation, 93 antisymmetric, 3 closed expansion, 316 Antoine space, 25, 328 , 173 approach frame, 319 closed sink, 26 approach prime, 320 closure, 166 approach space, 306 closure algebra, 223 approach vector space, 312 closure map, 190 associated topological convergence, 122 closure operation, 190 asymmetric norm, 298 closure operator, 49, 165, 173 atom, 188, 189 closure range, 192 atomistic lattice, 171 closure space, 165, 173 axiality, 207 closure system, 165, 173 cluster, 61, 104 base of a lattice, 170 co-adjoint functor, 7 base of a uniformity, 340 co-complete topological construct, 7 based lattice, 187 co-wellpowered category, 13 betweenness property, 98 coarser filter, 120 biBanach space, 298 coarser quasi-uniformity, 244 bicomplete quasi-uniformity, 270 coarser-than relation, 96, 339 bicompletion-true functorial quasi-uniformity, coarsest quasi-uniformity that a locally com- 284 pact topology admits, 265 bicontinuous map, 258 cocompact topology, 152, 265 bicoreflective modification, 344 cocushioned collection, 258 bicoreflective subcategory, 307 coequalizer, 9 bicoreflective subconstruct, 344 bidual convergence, 154 coessential family, 61 bilateral quasi-uniformity of a paratopolog- cofinite filter, 119 ical group, 286 coframe, 171 biquotient map, 145 colimit of a functor, 7 bireflective hull, 23 compact approach frame, 323 bireflective modification, 344 compact at a family, 130 bireflective subcategory, 14, 307 compact element, 188 bireflective subconstruct, 343 compact family, 130 bisequential topology, 144 compact frame, 323 bitopological space, 258 compact Hausdorff semiuniform convergence bounded set, 105 space, 359 boundedness, 105 compact semiuniform , 359 , 129 canonical completion, 63 compact-open topology, 107, 288 cartesian closed, 335, 351 compactoid, 130 cartesian closed category, 19, 117 compactoid space, 130 INDEX 377 comparison functor, 219 convergent filter, 47 compatible EF-proximity, 97 convex topology, 298 complemented approach space, 321 core-compact topology, 159, 269 complete approach space, 321 coreflective subcategory, 14, 142 complete closure algebra, 223 coreflector, 14, 142 complete cut space, 183 coseparator, 15 complete filter space, 348 costable filter, 275 complete join-semilattice, 191 counit, 217 complete kernel algebra, 223 countable compactness, 131 complete lattice, 216 countably based filter, 119, 140 complete nearness space, 62 countably core-compact topology, 159 complete semiuniform convergence space, 360 coupling convergence, 147 complete topological construct, 7 covariant functor, 140 complete uniformity, 270 cover, 41, 100, 128 completely bounded proximity space, 109 cover-compact convergence, 130 completely distributive lattice, 210 covering dimension, 362 completely regular semiuniform convergence covering uniformity, 100 space, 358 cushioned collection, 258 completely regular space, 94 cut, 183 completely regularly ordered space, 297 cut operator, 183 concentrated collection, 57 cut space, 183 concrete category, 3, 141 D-Cauchy filter, 277 concrete colimit, 7 decreasing set, 296 concrete functor, 17, 141, 345 Dedekind–MacNeille completion, 183 concrete limit, 7 density property, 95 concrete quasi-topos, 22 depending on countably many coordinates, concretely coreflective subcategory, 142 247 concretely coreflective subconstruct, 17 detachable closure space, 197 concretely isomorphic construct, 345 developable space, 252 concretely reflective subcategory, 43, 142 diagonal convergence, 137 concretely reflective subconstruct, 17 diagonal filter, 310 cone, 7 diagonal neighbourhood, 99 conjugate quasi-pseudometric, 242 diagonal semiuniform convergence space, 359 conjugate quasi-uniformity, 244 diagonal uniformity, 99 conjugate topology of a paratopological group, diagonalization property, 11 285 diagram functor, 7 connected semiuniform convergence space, , 174 359 directed subset, 177 construct, 3, 216, 339 discrete object, 3 construct invariant, 343 discrete proximity, 93 continuity for convergences, 125 discrete topology, 2 continuous convergence, 127, 334 discrete uniformity, 244 continuous lattice, 211 distance, 308 continuous map, 174 distance of S-convergence, 311 continuous morphism, 225 Doitchinov complete quasi-uniformity, 277 contour, 136 Doitchinov-Cauchy filter, 277 contraction, 307 Doitchinov-completion, 278 convenience property, 19 doubly residuated map, 223 convenient hull, 22 downset, 175 Convenient Topology, 357 downset operator, 183 convergence, 117, 122 dual category, 5 convergence approach space, 325 dual relation, 93 convergence complete quasi-uniformity, 271 dual strong inclusion relation, 95 convergence in proximity, 106 dually pseudocomplemented lattice, 198 convergence of countable character, 124, 140 convergence relation, 117 (E,M)-category, 11 convergence space, 117, 122, 360 (E,M)-factorization, 11, 315 convergence-approach limit, 324 E-reflective subcategory, 14 378 INDEX

EF-proximisable space, 97 finally dense, 143 EF-proximity, 93 finally dense subcategory, 18 EF-proximity space, 93 finally tight extension topological construct, Efremoviˇc Lemma, 102 23 Efremoviˇc property, 93 fine quasi-uniformity of a topology, 244 Efremoviˇc proximity, 93 fine transitive quasi-uniformity of a topol- Efremoviˇc space, 93 ogy, 254 elementary proximity, 94 finer convergence, 122 embedding, 219 finer filter, 120 embedding morphism, 9 finer quasi-uniformity, 244 end filter, 103 finer-than relation, 96, 339 entourage, 340 finest totally bounded quasi-uniformity coarser entourage of a quasi-uniformity, 241 than, 261 epi-, 25, 328 finest transitive quasi-uniformity coarser than, epimorphism, 9 254 epireflective subcategory, 14 finitary closure, 176 episink morphism, 20 finitary closure operator, 168 epitopologizer, 155 finitely compact, 131 epitopology, 155 firm, 321 equalizer, 7 firmly U-reflective subconstruct, 60 equentially based convergence, 123 First coreflector, 143 equicontinuous family of maps, 291 First corelflector, 124 essential embedding, 21 first countable semiuniform convergence space, evaluation map, 127 359 Exchange Axiom, 169 Fletcher construction, 244 exponenial laws, 19 forgetful functor, 3 exponential category, 117 Fr´echet space, 335 exponential map, 147 Fr´echet topology, 144 , 19 frame, 171, 318 extensional construct, 21, 354 free algebra, 218 extensional topological hull, 22, 325 free cluster, 104 extensive map, 173, 190 free filter, 118 extremal epimorphism, 10 Freudenthal compactification, 104 extremal monomorphism, 10 Freudenthal proximity, 94 full concrete subconstruct, 14 factorization structure, 11, 315 full subconstruct, 343 faithful functor, 3, 141 fully normal semiuniform convergence space, far-from relation, 93 358 far-miss set, 110 functionally indistinguishable proximity, 94 Fell compactification, 280 functor, 3, 140 fiber, 141 functor-regular convergence, 142 fiber of a set, 3, 42 functorial quasi-uniformity, 244 fiber-small construct, 3 functorial subset selection, 177 Fil-determined semiuniform convergence space, Galois connection, 203 348 γ-space, 252 filter, 41, 103, 118 gauge, 306 filter decomposition, 120 gauge basis, 307 filter space, 51, 342 generalized convergence space, 341 filter-determined convergence, 139 global standard completion, 186 filter-merotopic space, 350 global standard extension, 186 final convergence, 126 globally saturated ideal, 306 final epi-sink, 355 grill, 41 final lift, 4, 42 group of self-isomorphisms of a quasi-uniformity, final morphism, 4, 220 290 final quasi-uniformity, 248 final sink, 220, 355 handy relation, 267 final source, 4 Hausdorff completion, 366 finally closed subconstruct, 17 Hausdorff convergence, 136 INDEX 379

Hausdorff hyperuniformity, 108 join, 191 Hausdorff , 312 join-closed subset of a lattice, 194 Hausdorff quasi-pseudometric, 293 join-complete subset of a lattice, 194 Hausdorff quasi-uniformity, 292 join-completion, 185 hereditarily precompact quasi-uniformity, 260 join-dense subset of a lattice, 170 hereditarily quotient map, 145 join-preserving map, 184 hereditary closure operator, 50 join-semilattice, 191 hereditary final sink, 355 joincompact bitopology, 273 hit and far-miss topology, 110 K hit and miss topology, 109 coreflector, 143 k hit part, 109 -topology, 144 k hit set, 109 -topology, 144 hypercompact convergence, 153 Kent convergence space, 341 hyperspace, 108, 117 kernel algebra, 223 hypreproximity, 109 kernel map, 191 kernel operation, 191 ideal, 129 kernel operator, 174 ideal completion, 186 kernel system, 173 idempotent map, 173, 190 Kofner plane, 255 increasing map, 296 Kuratowski closure, 167 increasing set, 296 Kuratowski convergence, 153 indiscrete object, 3 λ indiscrete proximity, 93 -base of a topology, 276 indiscrete topology, 2 largest finally tight extension construct, 24 indiscrete uniformity, 244 Leader convergence, 106 inductive closure, 176 Lebesgue quasi-uniformity, 245 K infimum proximity, 99 left -Cauchy filter, 272 K inherence, 128 left -Cauchy , 272 K initial convergence, 125, 126 left -complete, 272 initial hull, 23 left adjoint functor, 217 initial lift, 4, 42 left invariant quasi-pseudometric, 286 initial morphism, 4, 219 left quasi-uniformity of paratopological group, initial quasi-uniformity, 247 286 initial source, 4, 219 left shift, 225 initial structure, 339 limit of a filter, 25, 53, 122, 328, 341 initially closed subconstruct, 17 limit of a functor, 7 initially dense, 143 limit operator, 310 initially dense object, 311 limit space, 53, 341 initially dense subcategory, 18 local proximity, 105 injective hull, 21 local quasi-uniformity, 251 injective object, 21 local uniformity, 252 interior cover, 44 locale, 171, 200 interior operator, 42, 174 locally compact convergence, 140 interior preserving open collection, 243 locally compact filter, 140 invariant, 343 locally compact semiuniform convergence space, invariant point selection, 188 359 inverse relation, 244 , 94 involution of a poset, 207 locally compact topology, 265 irreducible topology, 266 locally connected semiuniform convergence irregular, 137 space, 359 isolated point, 118 locally convex approach vector space, 312 isomorphic quasi-uniformities, 243 locally precompact semiuniform convergence isomorphism-closed subconstruct, 343 space, 359 isotone, 119 locally saturated ideal, 306 isotone map, 173, 175, 190 locally symmetric quasi-uniformity, 245 lower adjoint, 201 J-quotient convergence, 145 lower Kuratowski convergence, 153 J-quotient map, 145 lower quasi-uniformity, 292 JE-convergence, 143 lower semimodular lattice, 197 380 INDEX lower topology, 296 open morphism, 225 lower Vietoris topology, 295 , 173 opposite category, 5 m -directed set, 179 order-embedding, 185 m -frame, 210 orthocomplementation of a poset, 207 m-normal filter space, 358 M-quotient, 146 p-cover, 101 m-uniform filter space, 358 p-function, 96 Mackey structure, 314 p-isomorphism, 96 MacNeille completion, 25 p-metric, 306 measure of compactness,V 314 p-neighbourhood, 94 meet-closed ( -closed) subset of a lattice, p-neighbourhood relation, 95 194 pairbase of a topology, 258 meet-subcomplete subset of a lattice, 194 pairwise completely regular bitopology, 260 merotopic space, 41, 350 pairwise-local quasi-uniformity, 252 merotopy, 41 parabitopological group, 285 mesh, 119 paracompact semiuniform convergence space, mesh relation, 40 358 metacompact, 256 paratopological group, 285 metric proximity, 94 paratopologizer, 144 , 94 paratopology, 133 metrizable semiuniform convergence space, partial morphism, 354 360 partial order, 165, 175 micromeric collection, 47 Pervin quasi-uniformity, 243 Minkowski system, 313 Pervin–Sieber–Cauchy filter, 271 miss part, 109 Pervin–Sieber-complete quasi-uniformity, 271 miss set, 109 pesudotopology, 131 modifier of countable character, 124 ϕ-tower, 200 monad, 218 point-cluster, 104 monadic construct, 219 point-free geometry, 111 monadic functor, 219 point-operator, 328 monomorphism, 8 point-symmetric quasi-uniformity, 245 monoreflectives subcategory, 14 , 334 monotopological construct, 14 power convergence, 127 morphism, 3, 140 monad, 218 natural function space, 351 power space monad, 230 natural source, 7 pq-metric, 306 natural topology underlying a proximity, 97 pre-approach space, 325 near collection, 48 pre-distance, 325 near-to relation, 93 pre-regular approach frame, 322 nearness space, 43 pre-regular approach space, 323 neighborhood filter, 118 precompact quasi-uniformity, 260 neighbornet, 252 precompact semiuniform convergence space, network of a topology, 107, 125 359 non-archimedean quasi-pseudometric, 242 prel-limit operator, 325 non-archimedeanly quasi-pseudometrizable quasi- prenorm, 312 uniformity, 250 , 242 normal approach frame, 324 pretopological space, 53, 341 normal frame, 324 pretopologically diagonal convergence, 137 normal merotopic space, 358 pretopologizer, 125, 144 normal semiuniform convergence space, 358 pretopology, 124 normal space, 94 preuniform convergence space, 368 nucleus, 200 prime topology, 118 primitive filter, 267 object, 3, 140 principal filter, 119, 175, 310 ω-bounded space, 271 principal ideal, 175 one-point extension, 354 principal quasi-order, 193 open filter, 271 principal uniform limit space, 347 INDEX 381 product convergence, 126 quasi-uniformity of compact convergence, 288 product proximity, 98 quasi-uniformity of pointwise convergence, product quasi-uniformity, 247 288 projection algebra, 222 quasi-uniformity of , 288 projection operator, 191 quasi-uniformity of uniform convergence of promorphism, 222 A,288 proper contraction, 316 quasi-uniformizable topology, 243 prototopology, 132, 156 quiet quasi-uniformity, 277 proximal neighbourhood, 94 quotient convergence, 126 proximal set-open topology, 107 quotient map, 145, 335 proximally continuous map, 263 quotient morphism, 9, 220 proximally symmetric quasi-uniformity, 263 quotient object, 12 proximally transitive quasi-uniformity, 264 quotient proximity, 98 proximity, 262 quotient quasi-uniformity, 248 proximity boolean algebra, 111 quotient reflective subcategory, 58 proximity function, 96 proximity invariant, 96 rather below relation, 322 proximity isomorphism, 96 reciprocal limit space, 54 proximity lattice, 110 reciprocal space, 55 proximity naturally associated with a uni- refinement of a cover, 41, 100 formity, 100 reflective subcategory, 14, 142 pseudo-approach limit, 326 reflector, 14, 142 pseudo-limit operator, 326 reflector-quotient convergence, 145 pseudo-metric, 306 reflector-quotient map, 145 pseudo-open map, 335 regular approach frame, 322 pseudo-quasi-metric, 306 regular approach space, 323 pseudocomplement, 206 regular cardinal, 179 pseudocomplemented semilattice, 206 regular Cauchy space, 68 pseudometric, 242 regular closed set, 98 pseudotopological space, 341 regular convergence, 137 pseudotopologization, 132 regular epimorphism, 8 pseudotopologizer, 132, 144 regular filter space, 68 PSOT, 107 regular frame, 322 regular function frame, 318 QH-equivalent quasi-uniformity, 296 regular limit space, 68 quantale, 200 regular merotopic space, 56, 358 quasi-category, 24 regular monomorphism, 8 quasi-construct, 23 regular nearness space, 56 quasi-metric, 243 regular refinement of a cover, 56 quasi-order, 175 regular semiuniform convergence space, 358 quasi-proximity, 262 regular space, 56 quasi-proximity class of a quasi-proximity, regularity sequence, 69 263 regularization, 137 quasi-proximity induced by a quasi-uniformity, relation, 93 262 relative proximity, 98 quasi-pseudometric, 242 relatively compact, 269 quasi-pseudometric bicompletion, 279 relatively compact family, 130 quasi-pseudometric quasi-uniformity, 242 relatively compact space, 129 quasi-pseudometrizable bitopology, 258 relativization, 170 quasi-pseudometrizable quasi-uniformity, 251 remote-from relation, 93 quasi-sober topology, 266 residual map, 202 quasi-topology, 139 residuated map, 202 quasi-topos hull, 327 right K-Cauchy filter, 272 quasi-uniform bicompletion, 279 right K-Cauchy net, 272 quasi-, 241 right K-complete, 272 quasi-uniformity, 241 right adjoint functor, 20, 217 quasi-uniformity compatible with a quasi- right quasi-uniformity of a paratopological proximity, 263 group, 286 382 INDEX right shift, 225 strictly dense map, 184 rim-compact space, 94 strong hyperproximity, 109 round filter, 103 strong inclusion of a quasi-proximity, 263 strong inclusion relation, 95 Samuel compactification, 101 strong topologicalW universe, 355 Samuel uniformity, 101 strongly Z- -closed, 208 Scott completion, 186 strongly Z-continuous, 212 Scott topology, 186 strongly k-topology, 144 sec operator, 40 strongly (quasi-)sober topology, 280 semiuniform convergence space, 342 strongly cartesian closed construct, 355 semiuniform limit space, 342 strongly contained, 95 separated local proximity space, 106 strongly Fr´echet topology, 144 separated merotopic space, 57 structured sink, 4 separated nearness space, 57 structured source, 4 separated proximity, 93 subcategory, 142 separated relation, 95 subconvergence, 125 separated semiuniform convergence space, 358 subfirmly U-reflective subconstruct, 60 separated uniform space, 99 subobject, 12 separator, 15 subset selection, 177 sequential filter, 119, 139 subspace quasi-uniformity, 247 sequential modifier, 124 subtopological nearness space, 75 sequential topology, 123, 144 subtopological semiuniform convergence space, sequentially based filter, 139 358 sequentially closed set, 144 supercomplete uniformity, 275 sequentially compact convergence, 131 supercontinuous lattice, 211 σ-interior preserving base of a topology, 250 superquasi-sober topology, 269 Σ-operator, 71 supersober topology, 269 Σ-based lattice, 188 supertight map, 321 Σ-complete closure space, 188 supremum, 191 simple completion, 66, 365 supremum proximity, 99 sink, 4 symmetric Kent convergence space, 348 Skula topology of a topology, 282 symmetric limit space, 53 small collection, 47 symmetric space, 44, 338 small-set symmetric quasi-uniformity, 245 symmetrization of a quasi-pseudometric, 242 Smirnov compactification, 103 symmetrization of a quasi-uniformity, 244 Smirnov Compactification Theorem, 103 system (of sets), 172 Smyth-completable quasi-uniformity, 276 Smyth-complete quasi-uniformity, 276 T0-closure space, 184 sober approach space, 320 T0-object, 58 sober topology, 266 T0-quasi-pseudometric, 242 Sorgenfrey quasi-metric, 286 T0-quotient of a quasi-uniformity, 248 source, 4, 219 T0-system, 175 Souslin property, 271 T1 semiuniform convergence space, 358 spatial approach frame, 320 T2-Cauchy filter space, 68 specialization order, 175 T2-ordered compactification, 297 spectrum, 319 T2-ordered space, 296 stable family, 152 T2 semiuniform convergence space, 358 stable filter, 262 T3-Cauchy filter space, 68 stable quasi-uniformity, 278 t-regular semiuniform convergence space, 358 stack, 40, 310 θ-rather below relation, 322 standard T0-quasi-pseudometric on R, 285 θ-regular convergence, 137 standard completion, 183 toally disconnected semiuniform convergence standard extension of a poset, 183 space, 359 star of a set w.r.t. a cover, 41, 100 topological category, 220 star refinement of a cover, 41, 100 topological closure space, 174 Stone space, 121 topological construct, 4, 141, 339 Stone transform, 121 topological convergence, 117 stratifiable space, 259 Topological Duality Theorem, 5, 42 INDEX 383 topological functor, 4 uniformly equicontinuous family of maps, 291 , 285 uniformly open map, 246 topological hull, 25 uniformly regular quasi-uniformity, 278 topological nearness space, 45 uniformly saturated ideal, 306 topological preordered space, 296 uniformly Urysohn family, 110 topological preordered space determined by union complete subset selection, 177 a quasi-uniformity, 296 unit, 217 topological semiuniform convergence space, universal topological construct, 59 358 upper adjoint, 201 topological universe, 22, 355 upper Kuratowski convergence, 151 topological universe hull, 22, 327 upper quasi-uniformity, 292 topologically Hausdorff convergence, 136 upper topology, 186, 296 topologically metrizable semiuniform conver- upper Vietoris topology, 295 gence space, 360 upset, 175 topologically pretopological space, 345 Urysohn function, 96 topologically regular convergence, 137 W -irreducible element, 188 topologization of a convergence, 123 ∨-irreducible element, 188 topologizer, 144 W -prime element, 188 topology compatible with a quasi-uniformity, ∨-prime element, 188 243 vicinity, 124 topology induced by a quasi-proximity, 263 vicinity filter, 124 topology induced by a quasi-uniformity, 242 Vietoris topology, 109, 295 topology of uniform convergence, 106 totally bounded quasi-uniformity, 260 weak approach structure, 311 totally bounded uniformity, 99 weak base of a topology, 125 totally uniformly disconnected semiuniform weak hyperproximity, 109 convergence space, 359 weak star refinement of a cover, 41 trace, 118 weak topology, 311 trace of filter, 118 well-fibered topological construct, 6 transitive base of a quasi-uniformity, 250 well-monotone quasi-uniformity, 244 transitive quasi-uniformity, 250 well-quasi-order, 261 transitive topology, 254 wellpowered category, 12 trivial proximity, 93 Wijsman distance, 312 Wijsman topology, 312 U-completion, 60 Wyler completion, 70 U-reflective subconstruct, 60 W Z U-remote relation, 100 -V-closed set, 208 U-structured source, 4 Z- -closed set, 208 ultrafilter, 120 Z-below relation, 212 underlying set functor, 3 Z-below set, 211, 212 underlying set of a convergence, 122 Z-closure space, 178 uniform approach space, 307 Z-closure system, 178 uniform convergence, 106 Z-complete poset, 186, 212 uniform cover, 41, 100, 350 Z-continuous poset, 212 uniform gauge, 306 Z-cut completion, 186 uniform gauge space, 306 Z-distributive lattice, 211 uniform limit space, 79, 343 Z-downsets, 177 uniform limit structure, 79 Z-join ideal, 186, 212 uniform merotopic space, 358 Z-join ideal completion, 186 uniform semiuniform convergence space, 359 Z-prime element, 189 uniform space, 44, 99, 340 Z-subalgebra, 208 uniform topology, 106 Z-union closed, 178 Z uniformity, 99, 244, 340 -union preserving, 178 uniformity of uniform convergence, 106 0-compact approach space, 314 uniformizable topology, 251 zero-ideal, 323 uniformly connected semiuniform convergence space, 359 uniformly continuous map, 243, 340 Titles in This

489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic Forms and L-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and L-functions I. Global aspects, 2009 486 Fr´ed´eric Mynard and Elliott Pearl, Editors, Beyond topology, 2009 485 Idris Assani, Editor, Ergodic theory, 2009 484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and , 2009 483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Efim Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘glu, and Dietmar Vogt, Editors, and , 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and : Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. No¨el, Donald R. King, Gaston M. N’Gu´er´ekata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical sciences: Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in fluid dynamics, 2008 465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 464 Gestur Olafsson,´ Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational geometry, 2008 The purpose of this collection is to guide the non-specialist through the basic theory of various generalizations of topology, starting with clear motivations for their intro- duction. Structures considered include closure spaces, convergence spaces, proximity spaces, quasi-uniform spaces, merotopic spaces, nearness and filter spaces, semi-uniform convergence spaces, and approach spaces. Each chapter is self-contained and accessible to the graduate student, and focuses on motivations to introduce the generalization of topologies considered, presenting examples where desirable properties are not present in the realm of topologies and the problem is remedied in the more general context. Then, enough material will be covered to prepare the reader for more advanced papers on the topic. While category theory is not the focus of the book, it is a convenient language to study these structures and, while kept as a tool rather than an object of study, will be used throughout the book. For this reason, the book contains an introductory chapter on categorical topology.

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