Topological Spaces Including a Treatment of

Excellent study of sets in topological spaces and topological vector spaces includes systematic development of the properties of multi-valued functions. Topics include families of sets, topological spaces, mappings of one set into another, ordered sets, more. Examples included from different domains. 1963 edition.

PREFACE

v

(2)

LIST OF SYMBOLS

vii

1 FAMILIES OF SETS

1

(19)

1. Sets: general notations

1

(3)

2. Elementary operations on sets

4

(1)

3. Families of sets

5

(2)

4. Operations in a family of sets

7

(1)

5. Partitions

8

(1)

6. Filter bases

9

(3)

7. Closure operations in a set

12

(3)

8. (*) Lattices of sets

15

(3)

9. Principal limits of a family of sets

18

(2)

II MAPPINGS OF ONE SET INTO ANOTHER

20

(8)

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

20

(2)

2. Operations on mappings

22

(2)

3. Upper and lower inverses of a mapping

24

(3)

4. Graphs

27

(1)

III ORDERED SETS

28

(17)

1. Order and equivalence

28

(2)

2. Countable infinite and continuum infinite sets

30

(2)

3. (*) Transfinite cardinal numbers

32

(4)

4. Ordered sets

36

(2)

5. (*) Transfinite ordinal numbers

38

(1)

6. (*) The different forms of the axiom of choice

39

(6)

IV TOPOLOGICAL SPACES

45

(37)

1. Metric spaces

45

(4)

2. (*) L(*)-and L(0)-spaces

49

(4)

3. Topological spaces

53

(5)

4. Sequences and filtered families

58

(5)

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

63

(3)

6. Compact sets

66

(5)

7. Connected sets

71

(3)

8. Numerical functions defined on a topological space

74

(3)

9. Products and sums of topological spaces

77

(5)

V TOPOLOGICAL PROPERTIES OF METRIC SPACES

82

(27)

1. Topology of a metric space

82

(3)

2. Sums and products of metric spaces

85

(2)

3. Sequences of elements

87

(3)

4. Totally bounded spaces and complete spaces

90

(3)

5. Separable sets

93

(1)

6. Compact sets

94

(2)

7. Connected sets

96

(3)

8. (*) Locally connected sets: curves

99

(4)

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

103

(7)

VI MAPPINGS FROM ONE TOPOLOGICAL SPACE INTO ANOTHER

109

(20)

1. Semi-continuous mappings

109

(4)

2. Properties of the two types of semi-continuity

113

(2)

3. Maximum theorem

115

(2)

4. Fixed points of a mapping of R into R

117

(1)

5. (*) Limits of a family of sets

118

(8)

6. (*) Hausdorff metrics

126

(3)

VII MAPPINGS OF ONE VECTOR SPACE INTO ANOTHER

129

(29)

1. Vector spaces

129

(4)

2. Linear mappings

133

(3)

3. Linear varieties, cones, convex sets

136

(8)

4. Dimension of a convex set

144

(4)

5. The gauge of a convex set

148

(6)

6. The Hahn-Banach theorem

154

(4)

VIII CONVEX SETS AND CONVEX FUNCTIONS IN THE SPACE R(n)

158

(73)

1. Topological properties of convex sets

158

(10)

2. Simplexes; Kakutani's Theorem

168

(8)

3. Matrices

176

(4)

4. Bistochastic matrices

180

(8)

5. Convex functions

188

(6)

6. Differentiable convex functions

194

(6)

7. The fundamental properties of convex functions

200

(7)

8. Quasi convex functions

207

(4)

9. The fundamental inequality of convexity

211

(4)

10. (*) Sub-XXX functions

215

(4)

11. S-convex functions

219

(7)

12. Extremal problems with convex and concave functions

226

(5)

IX TOPOLOGICAL VECTOR SPACES

231

(34)

1. Normed spaces

231

(5)

2. Topological vector spaces

236

(6)

3. General properties of convex sets

242

(3)

4. Separation by convex functions

245

(4)

5. Locally convex spaces

249

(3)

6. Banach spaces: strong convergence

252

(7)

7. Banach spaces: weak convergence

259

(6)

INDEX

265

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

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