Introduction to metric and topological spaces [2nd ed., repr] 9780199563074, 0199563071, 9780199563081, 019956308X

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English Pages XI, [1], 206 s. : il.; 24 cm [216] Year 2009

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Introduction to metric and topological spaces [2nd ed., repr]
 9780199563074, 0199563071, 9780199563081, 019956308X

Table of contents :
Title page......Page 1
Preface......Page 3
Preface to the first edition......Page 4
Preface to reprinted edition......Page 5
Contents......Page 7
1. Introduction......Page 11
2. Notation and terminology......Page 15
3. More on sets and functions......Page 19
Direct and inverse images......Page 19
Inverse functions......Page 23
4. Review of some real analysis......Page 27
Real numbers......Page 27
Real sequences......Page 30
Limits of functions......Page 35
Continuity......Page 37
Examples of continuous functions......Page 40
5. Metric spaces......Page 47
Motivation and definition......Page 47
Examples of metric spaces......Page 50
Results about continuous functions on metric spaces......Page 58
Bounded sets in metric spaces......Page 60
Open balls in metric spaces......Page 61
Open sets in metric spaces......Page 63
6. More concepts in metric spaces......Page 71
Closed sets......Page 71
Closure......Page 72
Limit points......Page 74
Interior......Page 75
Boundary......Page 77
Convergence in metric spaces......Page 78
Equivalent metrics......Page 79
Review......Page 82
7. Topological spaces......Page 87
Definition......Page 87
Examples......Page 88
8. Continuity in topological spaces; bases......Page 93
Definition......Page 93
Homeomorphisms......Page 94
Bases......Page 95
9. Some concepts in topological spaces......Page 99
10. Subspaces and product spaces......Page 107
Subspaces......Page 107
Products......Page 109
Graphs......Page 114
Postscript on products......Page 115
11. The HausdorfF condition......Page 119
Motivation......Page 119
Separation conditions......Page 120
12. Connected spaces......Page 123
Motivation......Page 123
Connectedness......Page 123
Path-connectedness......Page 129
Comparison of definitions......Page 130
Connectedness and homeomorphisms......Page 132
13. Compact spaces......Page 135
Motivation......Page 135
Definition of compactness......Page 137
Compactness of closed bounded intervals......Page 139
Properties of compact spaces......Page 139
Continuous maps on compact spaces......Page 141
Compactness of subspaces and products......Page 142
Compact subsets of Euclidean spaces......Page 144
Compactness and uniform continuity......Page 145
An inverse function theorem......Page 145
14. Sequential compactness......Page 151
Sequential compactness for real numbers......Page 151
Sequential compactness for metric spaces......Page 152
15. Quotient spaces and surfaces......Page 161
Motivation......Page 161
A formal approach......Page 163
The quotient topology......Page 165
Main property of quotients......Page 167
The circle......Page 168
The torus......Page 169
The real projective plane and the Klein bottle......Page 170
Cutting and pasting......Page 177
The shape of things to come......Page 178
16. Uniform convergence......Page 183
Motivation......Page 183
Definition and examples......Page 183
Cauchy's criterion......Page 187
Uniform limits of sequences......Page 188
Generalizations......Page 190
17. Complete metric spaces......Page 193
Definition and examples......Page 194
Banach's fixed point theorem......Page 200
Contraction mappings......Page 202
Applications of Banach's fixed point theorem......Page 203
Bibliography......Page 211
Index......Page 213

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