《TOPOLOGICAL SPACES》

ⅠFAMILIES OF SETS1

1．Sets:general notations1

2．Elementary operations on sets4

3．Families of sets5

4．Operations in a family of sets7

5．Partitions8

6．Filter bases9

7．Closure operations in a set12

8．Lattices of sets15

9．Principal limits of a family of sets18

ⅡMAPPINGS OF ONE SET INTO ANOTHER20

1．Single-valued,semi-single-valued and multi-valued mappings20

2．Operations on mappings22

3．Upper and lower inverses of a mapping24

4．Graphs27

ⅢORDERED SETS28

1．Order and equivalence28

2．Countable infinite and continuum infinite sets30

3．Transfinite cardinal numbers32

4．Ordered sets36

5．Transfinite ordinal numbers38

6．The different forms of the axiom of choice39

ⅣTOPOLOGICAL SPACES45

1．Metric spaces45

2．L＊-and L0-spaces49

3．Topological spaces53

4．Sequences and filtered families58

5．Separated,quasi-separated,regular and normal spaces63

6．Compact sets66

7．Connected sets71

8．Numerical functions defined on a topological space74

9．Products and sums of topological spaces77

ⅤTOPOLOGICAL PROPERTIES OF METRIC SPACES82

1．Topology of a metric spaces82

2．Sums and products of metric spaces85

3．Sequences of elements87

4．Totally bounded spaces and complets spaces90

5．Separable sets93

6．Compact sets94

7．Connected sets96

8．Locally connected sets:curves99

9．Single-valued mappings of one metric space into another103

ⅥMAPPINGS FROM ONE TOPOLOGICAL SPACE INTO ANOTHER109

1．Semi-continuous mappings109

2．Properties of the two types of semi-continuity113

3．Maximum theorem115

4．Fixed points of a mapping of R into R117

5．Limits of a family of sets118

6．Hausdorff metrics126

ⅦMAPPINGS OF ONE VECTOR SPACE INTO ANOTHER129

1．Vector spaces129

2．Linear mappings133

3．Linear varieties,cones,convex sets136

4．Dimension of a convex set144

5．The gauge of a convex set148

6．The Hahn-Banach theorem154

ⅧCONVEX SETS AND CONVEX FUNCTIONS IN THE SPACE Rn158

1．Topological properties of convex sets158

2．Simplexes;Kakutani＇s Theorem168

3．Matrices176

4．Bistochastic matrices180

5．Convex functions188

6．Differentiable convex functions194

7．The fundamental properties of convex functions200

8．Quasi convex functions207

9．The fundamental inequality of convexity211

10．Sub-Φ functions215

11．S-convex functions219

12．Extremal problems with convex and concave functions226

ⅨTOPOLOGICAL VECTOR SPACES231

1．Normed spaces231

2．Topological vector spaces236

3．Geaeral properties of convex sets242

4．Separation by convex functions245

5．Locally convex spaces249

6．Banach spaces:strong convergence252

7．Banach spaces:weak convergence259

INDEX265