The foundations of analysis, - Topological ideas

Contents......Page 5
Introduction......Page 11
13.1 The space R^n......Page 13
13.3 Length and angle in R^n......Page 16
13.5 Some inequalities......Page 17
13.11 Distance......Page 19
13.14 Euclidean geometry and R^n......Page 20
13.17 Normed vector spaces......Page 26
13.18 Metric space......Page 27
13.19 Non-Euclidean geometry......Page 28
13.20 Distance between a poin and a se......Page 29
14.2 Boundary of a se......Page 33
14.3 Open balls......Page 34
14.7 Open and closed sets......Page 37
14.15 Open and closed sets in R^n......Page 41
15.1 Interior and closure......Page 43
15.4 Closure properties......Page 44
15.8 Interior properties......Page 45
15.11 Contiguous sets......Page 46
16.2 Continuous functions......Page 51
16.7 The continuity of algebraic operations......Page 53
16.13 Rational functions......Page 56
16.17 Complex-valued functions......Page 58
17.2 Connected sets......Page 59
17.6 Connected sets in R......Page 61
17.8 Continuity and connected sets......Page 62
17.15 Curves......Page 64
17.18 Pathwise connected sets......Page 65
17.21 Components......Page 68
17.25 Structure of open sets in R^n......Page 69
18.1 Cluster points......Page 72
18.4 Properties of cluster points......Page 74
18.9 The Cantor set......Page 75
19.1 Introduction......Page 78
19.2 Chinese boxes......Page 79
19.5 Compac sets and cluster points......Page 81
19.12 Compactness and continuity......Page 85
20.2 Open coverings......Page 89
20.7 Compactness in R"......Page 90
20.15 Completeness......Page 95
20.16 Compactness in general metric spaces......Page 96
20.20 A spherical cube......Page 98
21.1 Topological equivalence......Page 101
21.2 Maps......Page 102
21.3 Homeomorphisms between intervals......Page 103
21.4 Circles and spheres......Page 104
21.6 Continuous functions and open sets......Page 106
21.9 Relative topologies......Page 107
21.15 Introduction to topological spaces......Page 112
21.18 Produc topologies......Page 114
22.1 Introduction......Page 118
22.3 Limits......Page 121
22.4 Limits and continuity......Page 122
22.9 Limits and distance......Page 125
22.12 Righ and lef hand limits......Page 128
22.15 Some notation......Page 130
22.16 Monotone functions......Page 131
22.19 Inverse functions......Page 133
22.23 Roots......Page 135
22.25 Combining limits......Page 137
22.34 Complex functions......Page 140
23.1 Double limits......Page 142
23.5 Repeated limits......Page 144
23.11 Uniform convergence......Page 148
23.12 Distance between functions......Page 150
23.20 Uniform continuity......Page 157
24.1 Introduction......Page 161
24.2 One-poin compactification of the reals......Page 162
24.3 The Riemann sphere and the Gaussian plane......Page 165
24.4 Two-poin compactification of the reals......Page 166
24.5 Convergence and divergence......Page 169
24.8 Combination theorems......Page 175
24.12 Produc spaces......Page 177
25.1 Introduction......Page 181
25.2 Convergence of sequences......Page 182
25.7 Convergence of functions and sequences......Page 184
25.12 Sequences and closure......Page 187
25.18 Subsequences......Page 188
25.23 Sequences and compactness......Page 191
26.1 Divergence......Page 193
26.2 Limi points......Page 194
26.5 Oscillating functions......Page 196
26.11 Lim sup and lim inf......Page 198
27.2 Completeness......Page 202
27.8 Some complete spaces......Page 205
27.13 Incomplete spaces......Page 207
27.16 Completion of metric spaces......Page 209
27.18 Completeness and the continuum axiom......Page 211
28.1 Convergence of series......Page 213
28.8 Power series......Page 216
28.11 I Uniform convergence of series......Page 220
28.151 Series in function spaces......Page 221
28.19 Continuous operators......Page 224
28.26 Applications to power series......Page 227
29.1 Commutative and associative laws......Page 230
29.2 Infinite sums......Page 232
29.4 Infinite sums and series......Page 233
29.9 Complete spaces and the associative law......Page 235
29.17 Absolute sums......Page 238
29.23 Repeated series......Page 242
30.1 Introduction......Page 246
30.4 Separating hyperplanes......Page 247
30.8 Norms and topologies in R^n......Page 251
30.11 Curves and continua......Page 253
30.12 Simple curves......Page 254
30.14 Simply connected regions......Page 255
Notation......Page 257
Index......Page 259