Topologies on Closed and Closed Convex Sets

By: Gerald Beer

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Paperback | 15 December 2010

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356 Pages

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This monograph provides an introduction to the theory of topologies defined on the closed subsets of a metric space, and on the closed convex subsets of a normed linear space as well. A unifying theme is the relationship between topology and set convergence on the one hand, and set functionals on the other. The text includes for the first time anywhere an exposition of three topologies that over the past ten years have become fundamental tools in optimization, one-sided analysis, convex analysis, and the theory of multifunctions: the Wijsman topology, the Attouch--Wets topology, and the slice topology. Particular attention is given to topologies on lower semicontinuous functions, especially lower semicontinuous convex functions, as associated with their epigraphs. The interplay between convex duality and topology is carefully considered and a chapter on set-valued functions is included. The book contains over 350 exercises and is suitable as a graduate text.
This book is of interest to those working in general topology, set-valued analysis, geometric functional analysis, optimization, convex analysis and mathematical economics.

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Preface	p. ix
Preliminaries	p. 1
Notation and Background Material	p. 1
Weak Topologies	p. 8
Semicontinuous Functions	p. 13
Convex Sets and the Separation Theorem	p. 20
Gap and Excess	p. 28
Weak Topologies Determined by Distance Functionals	p. 34
The Wijsman Topology	p. 34
Hit-and-Miss Topologies and the Wijsman Topology	p. 43
UC Spaces	p. 54
The Slice Topology	p. 60
Complete Metrizability of the Wijsman and Slice Topologies	p. 69
The Attouch-Wets and Hausdorff Metric Topologies	p. 78
The Attouch-Wets Topology	p. 78
The Hausdorff Metric topology	p. 85
Varying the Metrics	p. 92
Set Convergence and Strong Convergence of Linear Functionals	p. 100
Gap and Excess Functionals and Weak Topologies	p. 106
Families of Gap and Excess Functionals	p. 106
Presentations of the Attouch-Wets and Hausdorff Metric Topologies	p. 113
The Scalar Topology and the Linear Topology for Convex Sets	p. 121
Weak Topologies determined by Infimal Value Functionals	p. 128
The Fell Topology and Kuratowski-Painleve Convergence	p. 138
The Fell Topology	p. 138
Kuratowski-Painleve Convergence	p. 145
Epi-convergencc	p. 155
Mosco Convergence and the Mosco Topology	p. 170
Mosco Convergence versus Wijsman Convergence	p. 178
Multifunctions: The Rudiments	p. 183
Multifunctions	p. 184
Lower and Upper Semicontinuity for Multifunctions	p. 192
Outer Semicontinuity versus Upper Semicontinuity	p. 199
KKM Maps and their Application	p. 208
Measurable Multifunctions	p. 216
Two Selection Theorems	p. 228
The Attouch-Wets Topology for Convex Functions	p. 235
Attouch-Wets Convergence of Epigraphs	p. 235
Continuity of Polarity and the Attouch-Wets Topology	p. 241
Regularization of Convex Functions and Attouch-Wets Convergence	p. 250
The Sum Theorem	p. 256
Convex Optimization and the Attouch-Wets Topology	p. 264
The Slice Topology for Convex Functions	p. 270
Slice and Dual Slice Convergence of Convex Functions	p. 270
Convex Duality and the Slice Topology	p. 276
Subdifferentials of Convex Functions and the Slice Topology	p. 287
Stability of the Geometric Ekeland Principle	p. 299
Notes and References	p. 306
Bibliography	p. 315
Symbols and Notation	p. 331
Subject Index	p. 335
Table of Contents provided by Ingram. All Rights Reserved.

Topologies on Closed and Closed Convex Sets

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