Topological SpacesPDF电子书下载

查看更多关于的内容

Ⅰ FAMILIES OF SETS 1

1．Sets:general notations 1

2．Elementary operations on sets 4

3．Families of sets 5

4．Operations in a family of sets 7

5．Partitions 8

6．Filter bases 9

7．Closure operations in a set 12

8．Lattices of sets 15

9．Principal limits of a family of sets 18

Ⅱ MAPPINGS OF ONE SET INTO ANOTHER 20

1．Single-valued,semi-single-valued and multi-valued mappings 20

2．Operations on mappings 22

3．Upper and lower inverses of a mapping 24

4．Graphs 27

Ⅲ ORDERED SETS 28

1．Order and equivalence 28

2．Countable infinite and continuum infinite sets 30

3．Transfinite cardinal numbers 32

4．Ordered sets 36

5．Transfinite ordinal numbers 38

6．The different forms of the axiom of choice 39

Ⅳ TOPOLOGICAL SPACES 45

1．Metric spaces 45

2．L＊-and L0-spaces 49

3．Topological spaces 53

4．Sequences and filtered families 58

5．Separated,quasi-separated,regular and normal spaces 63

6．Compact sets 66

7．Connected sets 71

8．Numerical functions defined on a topological space 74

9．Products and sums of topological spaces 77

Ⅴ TOPOLOGICAL PROPERTIES OF METRIC SPACES 82

1．Topology of a metric space 82

2．Sums and products of metric spaces 85

3．Sequences of elements 87

4．Totally bounded spaces and complete spaces 90

5．Separable sets 93

6．Compact sets 94

7．Connected sets 96

8．Locally connected sets:curves 99

9．Single-valued mappings of one metric space into another 103

Ⅵ MAPPINGS FROM ONE TOPOLOGICAL SPACE INTO ANOTHER 109

1．Semi-continuous mappings 109

2．Properties of the two types of semi-continuity 113

3．Maximum theorem 115

4．Fixed points of a mapping of R into R 117

5．Limits of a family of sets 118

6．Hausdorff metrics 126

Ⅶ MAPPINGS OF ONE VECTOR SPACE INTO ANOTHER 129

1．Vector spaces 129

2．Linear mappings 133

3．Linear varieties,cones,convex sets 136

4．Dimension of a convex set 144

5．The gauge of a convex set 148

6．The Hahn-Banach theorem 154

Ⅷ CONVEX SETS AND CONVEX FUNCTIONS IN THE SPACE Rn 158

1．Topological properties of convex sets 158

2．Simplexes;Kakutani＇s Theorem 168

3．Matrices 176

4．Bistochastic matrices 180

5．Convex functions 188

6．Differentiable convex functions 194

7．The fundamental properties of convex functions 200

8．Quasi convex functions 207

9．The fundamental inequality of convexity 211

10．Sub-Φ functions 215

11．S-convex functions 219

12．Extremal problems with convex and concave functions 226

Ⅸ TOPOLOGICAL VECTOR SPACES 231

1．Normed spaces 231

2．Topological vector spaces 236

3．General properties of convex sets 242

4．Separation by convex functions 245

5．Locally convex spaces 249

6．Banach spaces:strong convergence 252

7．Banach spaces:weak convergence 259

INDEX 265

查看更多关于的内容