The infinite-dimensional topology of function spaces

J. van Mill9780444505576, 0444505571

In this book we study function spaces of low Borel complexity. Techniques from general topology, infinite-dimensional topology, functional analysis and descriptive set theory are primarily used for the study of these spaces. The mix of methods from several disciplines makes the subject particularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marciszewski-Mogilski Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented. In order to understand what is going on, a solid background in infinite-dimensional topology is needed. And for that a fair amount of knowledge of dimension theory as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensional topology, prerequisites and introduction’. A selection of what was done there can be found here as well, but completely revised and at many places expanded with recent results. A `scenic’ route has been chosen towards the Dobrowolski-Marciszewski-Mogilski Theorem, linking the results needed for its proof to interesting recent research developments in dimension theory and infinite-dimensional topology. The first five chapters of this book are intended as a text for graduate courses in topology. For a course in dimension theory, Chapters 2 and 3 and part of Chapter 1 should be covered. For a course in infinite-dimensional topology, Chapters 1, 4 and 5. In Chapter 6, which deals with function spaces, recent research results are discussed. It could also be used for a graduate course in topology but its flavor is more that of a research monograph than of a textbook; it is therefore more suitable as a text for a research seminar. The book consequently has the character of both textbook and a research monograph. In Chapters 1 through 5, unless stated otherwise, all spaces under discussion are separable and metrizable. In Chapter 6 results for more general classes of spaces are presented. In Appendix A for easy reference and some basic facts that are important in the book have been collected. The book is not intended as a basis for a course in topology; its purpose is to collect knowledge about general topology. The exercises in the book serve three purposes: 1) to test the reader’s understanding of the material 2) to supply proofs of statements that are used in the text, but are not proven there 3) to provide additional information not covered by the text. Solutions to selected exercises have been included in Appendix B. These exercises are important or difficult.

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Table of contents :
Contents ……Page 8
Introduction ……Page 12
1.1. Linear spaces ……Page 14
1.2. Extending continuous functions ……Page 34
1.3. Function spaces ……Page 42
1.4. The Borsuk homotopy extension theorem ……Page 50
1.5. Topological characterization of some familiar spaces ……Page 54
1.6. The inductive convergence criterion and applications ……Page 71
1.7. Bing’s shrinking criterion ……Page 79
1.8. Isotopies ……Page 83
1.9. Homogeneous zero-dimensional spaces ……Page 86
1.10. Inverse limits ……Page 93
1.11. Hyperspaces ……Page 108
2.1. Affine notions ……Page 124
2.2. Barycenters and subdivisions ……Page 138
2.3. The nerve of an open covering ……Page 145
2.4. Simplices in $R^n$ ……Page 151
2.5. The Lusternik-Schnirelman-Borsuk theorem ……Page 161
3.1. The covering dimension ……Page 164
3.2. Translation into open covers ……Page 170
3.3. The imbedding theorem ……Page 181
3.4. The inductive dimension functions ind and Ind ……Page 189
3.5. Dimensional properties of compactifications ……Page 196
3.6. Mappings into spheres ……Page 206
3.7. Dimension of subsets of $R^n$ and certain generalizations ……Page 217
3.8. Higher-dimensional hereditarily indecomposable continua ……Page 223
3.9. Totally disconnected spaces ……Page 229
3.10. The origins of dimension theory ……Page 234
3.11. The dimensional kernel of a space ……Page 240
3.12. Colorings of maps ……Page 250
3.13. Various kinds of infinite-dimensionality ……Page 264
3.14. The Brouwer fixed-point theorem revisited ……Page 270
4.1. Some properties of ANR’s ……Page 276
4.2. A characterization of ANR’s and AR’s ……Page 290
4.3. Open subspaces of ANR’s ……Page 314
5.1. Z-sets ……Page 320
5.2. Extending homeomorphisms in s ……Page 324
5.3. The estimated homeomorphism extension theorem ……Page 333
5.4. The compact absorption property ……Page 342
5.5. Absorbing systems ……Page 356
Chapter 6. Function spaces ……Page 380
6.1. Notation ……Page 381
6.2. The spaces $C_p(X)$: Introductory remarks ……Page 382
6.3. The Borel complexity of function spaces ……Page 385
6.4. The Baire property in function spaces ……Page 390
6.5. Filters and the Baire property in $C_p(N_F)$ ……Page 400
6.6. Extenders ……Page 406
6.7. The topological dual of $C_p(X)$ ……Page 412
6.8. The support function ……Page 417
6.9. Nonexistence of linear homeomorphisms ……Page 424
6.10. Bounded functions ……Page 429
6.11. Nonexistence of homeomorphisms ……Page 439
6.12. Topological equivalence of certain function spaces ……Page 447
6.13. Examples ……Page 458
A.l. Prerequisites and notation ……Page 470
A.2. Separable metrizable topological spaces ……Page 478
A.3. Limits of continuous functions ……Page 481
A.4. Normality type properties ……Page 482
A.5. Compactness type properties ……Page 486
A.6. Completeness type properties ……Page 492
A.7. A covering type property ……Page 498
A.8. Extension type properties ……Page 503
A.9. Wallman compactifications ……Page 507
A.10. Connectivity ……Page 513
A.11. The quotient topology ……Page 518
A.12. Homotopies ……Page 523
A.13. Borel and similar sets ……Page 530
Appendix B. Answers to selected exercises ……Page 540
Appendix C. Notes and comments ……Page 592
Bibliography ……Page 610
Special Symbols ……Page 626
Author Index ……Page 628
Subject Index ……Page 632