Schaum's outline of general topology

SCHAUM\'S OUTLINES OF GENERAL TOPOLOGY......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
Contents......Page 5
Sets, Elements......Page 7
Classes, Collections, Families and Spaces......Page 8
Set Operations......Page 9
Product Sets......Page 10
Relations......Page 11
Equivalence Relations......Page 12
Composition of Relations......Page 13
Solved Problems......Page 14
Supplementary Problems......Page 20
Answers to Supplementary Problems......Page 21
Functions......Page 23
One-One, Onto, Inverse and Identity Functions......Page 24
Generalized Operations......Page 25
Associated Set Functions......Page 26
Algebra of Real-Valued Functions......Page 27
Solved Problems......Page 29
Supplementary Problems......Page 34
Answers to Supplementary Problems......Page 37
Denumerable and Countable Sets......Page 38
Schroeder–Bernstein Theorem......Page 39
Partially Ordered Sets......Page 40
First and Last Elements......Page 41
Upper and Lower Bounds......Page 42
Solved Problems......Page 43
Supplementary Problems......Page 50
Answers to Supplementary Problems......Page 52
Open Sets in R......Page 53
Closed Sets......Page 54
Sequences......Page 55
Convergent Sequences......Page 56
Completeness......Page 57
Continuous Functions......Page 58
Topology of the Plane......Page 59
Solved Problems......Page 60
Supplementary Problems......Page 70
Answers to Supplementary Problems......Page 71
Topological Spaces......Page 72
Accumulation Points......Page 73
Closure of a Set......Page 74
Interior, Exterior, Boundary......Page 75
Neighborhoods and Neighborhood Systems......Page 76
Coarser and Finer Topologies......Page 77
Equivalent Definitions of Topologies......Page 78
Solved Problems......Page 79
Supplementary Problems......Page 90
Answers to Supplementary Problems......Page 92
Base for a Topology......Page 93
Topologies Generated by Classes of Sets......Page 94
Local Bases......Page 95
Solved Problems......Page 96
Supplementary Problems......Page 101
Answers to Supplementary Problems......Page 102
Continuous Functions......Page 103
Continuous Functions and Arbitrary Closeness......Page 104
Open and Closed Functions......Page 105
Topological Properties......Page 106
Topologies Induced by Functions......Page 107
Solved Problems......Page 108
Supplementary Problems......Page 115
Answers to Supplementary Problems......Page 116
Metrics......Page 117
Distance between Sets, Diameters......Page 118
Open Spheres......Page 119
Properties of Metric Topologies......Page 120
Equivalent Metrics......Page 121
Isometric Metric Spaces......Page 122
Hilbert Space......Page 123
Normed Spaces......Page 124
Solved Problems......Page 126
Supplementary Problems......Page 133
Answers to Supplementary Problems......Page 136
Second Countable Spaces......Page 137
Separable Spaces......Page 138
Hereditary Properties......Page 139
Solved Problems......Page 140
Supplementary Problems......Page 143
Hausdorff Spaces......Page 145
Regular Spaces......Page 146
Normal Spaces......Page 147
Completely Regular Spaces......Page 148
Solved Problems......Page 149
Supplementary Problems......Page 155
Answers to Supplementary Problems......Page 156
Compact Sets......Page 157
Subsets of Compact Spaces......Page 158
Compactness and Hausdorff Spaces......Page 159
Sequentially Compact Sets......Page 160
Locally Compact Spaces......Page 161
Compactification......Page 162
Totally Bounded Sets......Page 163
Solved Problems......Page 164
Supplementary Problems......Page 171
Product Topology......Page 173
Defining Subbase and Defining Base for the Product Topology......Page 174
Example of a Product Space......Page 175
Tychonoff\'s Product Theorem......Page 176
Cantor Set......Page 177
Properties of the Cantor Set......Page 178
Solved Problems......Page 179
Supplementary Problems......Page 183
Connected Sets......Page 186
Connected Spaces......Page 187
Connectedness on the Real Line......Page 188
Locally Connected Spaces......Page 189
Arcwise Connected Sets......Page 190
Homotopic Paths......Page 191
Solved Problems......Page 192
Supplementary Problems......Page 199
Cauchy Sequences......Page 201
Completeness and Contracting Mappings......Page 202
Completions......Page 203
Construction of the Real Numbers......Page 204
Solved Problems......Page 205
Supplementary Problems......Page 212
Point Open Topology......Page 213
Pointwise Convergence......Page 214
Uniform Convergence......Page 215
The Function Space C[0, 1]......Page 216
Compact Open Topology......Page 217
Topology of Compact Convergence......Page 218
Functionals on Normed Spaces......Page 219
Solved Problems......Page 220
Supplementary Problems......Page 228
Field Axioms......Page 231
Positive Numbers......Page 232
Absolute Value......Page 233
Least Upper Bound Axiom......Page 234
Solved Problems......Page 235
Supplementary Problems......Page 239
Answers to Supplementary Problems......Page 240
Index......Page 241
Index of Symbols......Page 245