Topological Spaces (eBook)

Front Cover 1
Topological Spaces 4
Copyright Page 5
Table of Contents 8
PREFACE 6
Introduction 10
Chapter I. Elementary Principles 12
I. Fundamental Concepts of Set Theory 12
2. Ordered Sets and Lattices 20
3. Filters 30
Chapter II. Topological Spaces 42
4. Topology and Limits 42
5. Fundamental Concepts and Their Reformulations 47
6. The Order Topology and Metric Spaces 57
7. The Lattice of Topologies 64
8. Axioms of Separation 66
9. Subspaces 75
10. Cardinality Conditions 79
Chapter Ill. Compactness and Connectedness 88
11. Compactness 88
12. Locally Compact Spaces 97
13. Paracompact Spaces 100
14. Connectedness 110
15. Local Connectedness 116
Chapter IV. Mappings 124
16. Topological Maps, Homeomorphisms 124
17. Continuous Mappings 129
18. Completely Regular Spaces 136
19. Properties of Continuous Mappings 143
2o. Transference of Topologies 149
21. Quotient Spaces 152
22. Product Spaces 157
23. Spaces of Mappings 165
24. Sums of Spaces 168
Chapter V. Extensions and Characterization of Topological Spaces 174
25. Extensions of Topological Spaces 174
26. The Wallman Compactification 179
27. The Stone-Cech Compactification 183
28. Embeddings 186
29. Representation Theorems 191
Chapter VI. Metric and Uniform Spaces 200
30. Metric Spaces 200
31. Metrization 206
32. Extensions and Completions of Metric Spaces 211
33. Uniform Structures 219
34. Uniform Continuity and Uniform Convergence 226
35. Extension and Completion of Uniform Spaces 237
Chapter VII. Topological Groups. Applications 248
36. Topological Groups 248
37. Extensions and Completions of Topological Groups 257
38. Topological Rings and Fields 263
39. Topological Vector Spaces. 
269 
40. Inductive and Project!ve Limits 277
NOTATIONS EMPLOYED 288
BIBLIOGRAPHY 290
Subject Index 292