Point Set Topology (eBook)

Front Cover 1
Point Set Topology 4
Copyright Page 5
Contents 10
Preface 6
Notation 8
Introduction to Set Theory 16
1. Elementary Operations on Sets 16
2. Set Theoretical Equivalence and Denumerability 22
3. The Axiom of Choice and Its Equivalents 28
References 34
Chapter I. Topological Spaces 36
1. Open Sets and Closed Sets 36
2. Interior, Exterior, Boundary, and Closure 39
3. Closure Operators 44
4. Bases and Subbases 48
5. Topologies on Linearly Ordered Sets 50
6. Metric Spaces 53
7. Neighborhood Filters 57
8. Uniform Structures 60
9. Simple Results on Uniform Structures and Uniform Spaces 67
10. Subspaces 70
11. Product Spaces 73
12. Products of Uniformizable Spaces 79
13. Inverse and Direct Images of Topologies 83
14. Quotient Spaces 86
Notes 90
References 91
Chapter II. Separation Properties 92
1. (T0) and (T1) Axioms, Hausdorff Spaces 92
2. (T3) Spaces, Regular and Semiregular Spaces 95
3. (T,) Spaces and Normal Spaces 100
4. Point-Finite and Star-Finite Open Coverings 103
5. (T5) Spaces and Completely Normal Spaces 106
6. Separated Sets 111
7. Connected Spaces and Sets 114
8. Maximal Connected Subsets 120
9. (T) Axiom and Complete Regularity 123
10. Uniformization and Axiom (T) 127
11. Axioms of Separation in Product Spaces 131
12. Separable Spaces and Countability Axioms 134
Notes 139
References 140
Chapter III. Compactness and Uniformization 142
1. Compactness 142
2. Compact Metric Spaces 147
3. Subspaces and Separation Properties of Compact Spaces 152
4. The Product of Compact Topological Spaces 159
5. Locally Compact Spaces 163
6. Paracompactness and Full-Normality 168
7. The Equivalence of Paracompactness and Full-Normality 174
8. Metrizable Uniform Structures and Structure Gages 179
9. Metrizability Conditions 182
Notes 185
References 187
Chapter IV. Continuity 190
1. Functional Relations and Functions 190
2. Local Continuity 192
3. Continuous Functions 197
4. Homeomorphisms, Open and Closed Maps 200
5. Real-Valued Functions 206
6. Continuity and Axioms of Separation 213
7. Continuity and Compactness 221
8. Continuity and connectedness 228
9. Continuity in Product Spaces 233
10. Uniform Continuity and Equicontinuity 239
11. The Topology of Uniform Convergence 247
12. The Algebra of Continuous Functions 256
Notes 262
References 264
Chapter V. Theory of Convergence 268
1. Filters and Nets 269
2. Convergence of Filters, Nets, and Sequences 273
3. Ultrafllters and Universal Nets 279
4. Bounds, Traces, and Products of Filters 282
5. Applications of Filters and Nets to Compactness 284
6. Cauchy Filters and Complete Spaces 289
7. Completion of Metric Structures 296
8. Baire’s Category Theorem, the Principles of Uniform Boundedness and of the Condensation of Singularities 302
9. Completions and Compactifications 311
Notes 321
References 322
Author Index 324
Subject Index 326