Topologies on Closed and Closed Convex Sets

Description

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. Front Matter ....Pages i-xi Preliminaries (Gerald Beer)....Pages 1-33 Weak Topologies Determined by Distance Functionals (Gerald Beer)....Pages 34-77 The Attouch-Wets and Hausdorff Metric Topologies (Gerald Beer)....Pages 78-105 Gap and Excess Functionals and Weak Topologies (Gerald Beer)....Pages 106-137 The Fell Topology and Kuratowski-Painlevé Convergence (Gerald Beer)....Pages 138-182 Multifunctions: the Rudiments (Gerald Beer)....Pages 183-234 The Attouch-Wets Topology for Convex Functions (Gerald Beer)....Pages 235-269 The Slice Topology for Convex Functions (Gerald Beer)....Pages 270-305 Back Matter ....Pages 306-340