Topologies on Closed and Closed Convex Sets

by Gerald Beer

Overview

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.

Product Details

ISBN-13:	9780792325314
Publisher:	Springer Netherlands
Publication date:	10/31/1993
Series:	Mathematics and Its Applications , #268
Edition description:	1993
Pages:	340
Product dimensions:	6.40(w) x 9.40(h) x 1.10(d)

Table of Contents

Preface. 1. Preliminaries. 2. Weak Topologies determined by Distance Functionals. 3. The Attouch—Wets and Hausdorff Metric Topologies. 4. Gap and Excess Functionals and Weak Topologies. 5. The Fell Topology and Kuratowski—Painlevé Convergence. 6. Multifunctions - the Rudiments. 7. The Attouch—Wets Topology for Convex Functions. 8. The Slice Topology for Convex Functions. Notes and References. Bibliography. Symbols and Notation. Subject Index.

Editorial Reviews

An introduction to the theory of topologies defined on the closed subsets of a metric apace, and on the closed convex subsets of 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: the Wijsman topology, the Attouch-Wets topology, and the slice topology. Over 350 exercises are included. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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