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

by Berge, Claude

• ISBN13: 9780486696539
• ISBN10: 0486696537
• Format: Paperback
• Copyright: 2010-09-16
• Publisher: Dover Publications

Summary

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.

Table of Contents

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