Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity

Instead of the cover......Page 1
Title page......Page 3
Copyright page......Page 4
PREFACE......Page 5
LIST OF SYMBOLS......Page 7
CONTENTS......Page 11
1. Sets: general notations......Page 15
2. Elementary operations on sets......Page 18
3. Families of sets......Page 19
4. Operations in a family of sets......Page 21
5. Partitions......Page 22
6. Filter bases......Page 23
7. Closure operations in a set......Page 26
8.* Lattices of sets......Page 29
9. Principal limits of a family of sets......Page 32
1. Single-valued, semi-single-valued and multi-valued mappings......Page 34
2. Operations on mappings......Page 36
3. Upper and lower inverses of a mapping......Page 38
4. Graphs......Page 41
1. Order and equivalence......Page 42
2. Countable infinite and continuum infinite sets......Page 44
3.* Transfinite cardinal numbers......Page 46
4. Ordered sets......Page 50
5.* Transfinite ordinal numbers......Page 52
6.* The different forms of the axiom of choice......Page 53
1. Metric spaces......Page 59
2.* $L^\ast$- and $L^0$-spaces......Page 63
3. Topological spaces......Page 67
4. Sequences and filtered families......Page 72
5. Separated, quasi-separated, regular and normal spaces......Page 77
6. Compact sets......Page 80
7. Connected sets......Page 85
8. Numerical functions defined on a topological space......Page 88
9. Products and sums of topological spaces......Page 91
1. Topology of a metric space......Page 96
2. Sums and products of metric spaces......Page 99
3. Sequences of elements......Page 101
4. Totally bounded spaces and complete spaces......Page 104
5. Separable sets......Page 107
6. Compact sets......Page 108
7. Connected sets......Page 110
8.* Locally connected sets: curves......Page 113
9. Single-valued mappings of one metric space into another......Page 117
1. Semi-continuous mappings......Page 123
2. Properties of the two types of semi-continuity......Page 127
3. Maximum theorem......Page 129
4. Fixed points of a mapping of $\mathbb{R}$ into $\mathbb{R}$......Page 131
5.* Limits of a family of sets......Page 132
6.* Hausdorff metrics......Page 140
1. Vector spaces......Page 143
2. Linear mappings......Page 147
3. Linear varieties, cones, convex sets......Page 150
4. Dimension of a convex set......Page 158
5. The gauge of a convex set......Page 162
6. The Hahn-Banach theorem......Page 168
1. Topological properties of convex sets......Page 172
2. Simplexes; Kakutani's Theorem......Page 182
3. Matrices......Page 190
4. Bistochastic matrices......Page 194
5. Convex functions......Page 202
6. Differentiable convex functions......Page 208
7. The fundamental properties of convex functions......Page 214
8. Quasi convex functions......Page 221
9. The fundamental inequality of convexity......Page 225
10.* Sub-$\Phi$ functions......Page 229
11. S-convex functions......Page 233
12. Extremal problems with convex and concave functions......Page 240
1. Nonned spaces......Page 245
2. Topological vector spaces......Page 250
3. General properties of convex sets......Page 256
4. Separation by convex functions......Page 259
5. Locally convex spaces......Page 263
6. Banach spaces: strong convergence......Page 266
7. Banach spaces: weak convergence......Page 273
INDEX......Page 279