Topics in General Topology (eBook)

K. Morita, J.-I. Nagata (Herausgeber)

eBook Download: PDF

1989 | 1. Auflage
746 Seiten
Elsevier Science (Verlag)
9780080879888 (ISBN)

Infos

Being an advanced account of certain aspects of general topology, the primary purpose of this volume is to provide the reader with an overview of recent developments.

The papers cover basic fields such as metrization and extension of maps, as well as newly-developed fields like categorical topology and topological dynamics. Each chapter may be read independently of the others, with a few exceptions. It is assumed that the reader has some knowledge of set theory, algebra, analysis and basic general topology.

Inhalt

Front Cover 1
Topics in General Topology 4
Copyright Page 5
Contents 8
Preface 6
Chapter 1. Extensions of Mappings I 14
Introduction 14
1. Generalized uniform spaces and semi-uniform spaces 15
2. Completeness, completions and extensions of spaces 23
3. Extensions of continuous maps from dense subspaces 38
Appendix. A generalization of the Ascoli Theorem 48
References 51
Chapter 2. Extensions of Mappings II 54
Introduction 54
1. Preliminaries 55
2. C*-, C-, Pm- and P-embeddings 64
3. Unions of C*-embedded subsets 75
4. C*-embedding in product spaces 82
5. Homotopy extension property 88
References 91
Chapter 3. Normality of Product Spaces I 94
Introduction 94
1. Fundamental results 95
2. The first of Morita's three conjectures 105
3. The second and third of Morita's three conjectures 122
References 129
Chapter 4. Normality of Product Spaces II 134
1. Products of normal spaces with metric spaces 134
2. Products of normal spaces with generalized metric spaces 153
References 129
Chapter 5. Generalized Paracompactness 174
Introduction 174
1. Preliminaries 175
2. Characterizations of paracompactness 177
3. Characterizations of submetacompactness 188
4. Characterizations of metacompactness 191
5. Characterizations of subparacompactness 197
6. Shrinking properties 199
7. Examples 204
References 211
Chapter 6. The Tychonoff Functor and Related Topics 216
Introduction 216
1. The Tychonoff functor 217
2. Product spaces and the Tychonoff functor 220
3. w-Compact spaces 222
5. A space X such that t(X × = t( X ) × t( Y ) for any k-space Y 228
5. A space X such that t(X × ) = t( X ) × t( Y ) for any k-space Y 231
6. Products of w-compact spaces 234
7. w-Paracompact spaces and the Tychonoff functor 237
8. A generalization of Tamano's theorem 243
9. Rectangular products and w-paracompact spaces 247
References 254
Chapter 7. Metrization I 258
Introduction 258
1. Metrizability in terms of g-functions 258
2. Metrizability in terms of base of point-finite rank 264
3. Metrizability in terms of hereditary property 272
4. Some other aspects 278
References 284
Chapter 8. Metrization II 288
Iritroduction 288
1. Spaces which contain a copy of S. or S2 289
2. Spaces dominated by metric subsets 297
3. Spaces with s-hereditarily closure-preserving k-networks 300
4. Spaces with certain point-countable covers 304
5. Quotient s-images of locally separable metric spaces 314
6. .-Metrizable spaces and d-metrizable spaces 317
References 324
Chapter 9. Generalized Metric Spaces I 328
Introduction 328
1. Lašnev space and N-space 328
2. Developable space 339
3. M-space and related topics 350
4. Universal spaces 364
References 378
Chapter 10. Generalized Metric Spaces II 380
Introduction 380
1. Review of basic results 381
2. Closure-preserving collections and definitions of various stratifiable spaces 384
3. Expandability, extension property, and sup-characterization of stratifiable spaces 395
4. Classes of M1-spaces 401
5. Embeddings 409
6. Closed maps and perfect maps 413
7. Related topics 418
8. Problems 419
References 420
Chapter 11. Function Spaces 424
1. Introduction and notation 424
2. Some properties of Cn*(X) 429
3. Some properties of Ck(X) 430
4. Some properties of Cp(X) 440
5. Topological properties and linear topological properties 449
6. Topological properties and l-equivalence 460
References 470
Chapter 12. N-Compactness and Its Applications 472
Introduction 472
1. Conventions and notation 473
2. N-compactness and N-compactifications 474
3. N-compactness vs. realcompactness 480
4. N-compactness of knX 487
5. Rings and lattices of continuous functions 492
6. Applications to abelian groups 496
7. Applications to non-Archimedean Banach spaces 510
8. Problems 529
References 531
Chapter 13. Topological Games and Applications 536
Introduction 536
1. The games and winning strategies 537
2. K-scattered spaces 542
3. Closure-preserving covers by compact sets 549
4. Covering properties of product spaces 556
5. The games in product spaces 562
6. Applications to dimension theory 566
References 573
Chapter 14. Categorical Topology 576
Introduction 576
1. Categories and functors 578
2. Monomorphisms and epimorphisms 582
3. Diagrams and limits 588
4. Complete categories 592
5. Factorizations of morphisms 595
6. Reflective subcategories 601
7. Characterization theorem 605
8. (E, M)-categories 611
9. Epireflective vs. bireflective in Top 615
10. Separation axioms and connectedness 619
11. Simplicity of epireflective subcategories 624
12. Topological functors 627
13. Topological categories 631
References 635
Chapter 15. Topological Dynamics 638
Introduction 638
1. Orbit structures 640
2. Expansive behaviours 648
3. Expansivity and dimension 671
4. Pseudo-orbit-tracing property 680
5. Coordinate systems and topological stability 693
6. Representations of maps with hyperbolic coordinates 700
7. Chain components and decompositions 708
8. Markov partitions and subshifts 720
9. Topological entropy 740
References 749
Subject Index 754

Produkt-Details

• Erscheint lt. Verlag: 4.8.1989
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-13: 9780080879888 / 9780080879888