Spaces of densely continuous forms, USCO and minimal USCO maps
type: Item
instance of: scholarly article
title: Spaces of densely continuous forms, USCO and minimal USCO maps
zbMATH Open document ID: 1022.54012
DOI: 10.1023/A:1022982924962
published in: Set-Valued Analysis
publication date: 2003-06-09T00:00:00Z
review text: The present paper is a study on topologies and corresponding convergences of set-valued maps, putting an emphasis on the topology of uniform convergence on compact sets on the space of densely continuous forms, upper semicontinuous (usco) and minimal upper semicontinuous maps. The author generalizes, completes and improves results by Hammer and McCoy which concern the space \(D(X,Y)\) of densely continuous forms from \(X\) to \(Y\). Some of the main results are the following: For a Hausdorff topological space \(X\) and a metric space \((Y,d)\), the topology \(D_k(X,Y)\) of uniform convergence on compact sets on the space \(D(X,Y)\) is metrizable iff \(D_k(X,Y)\) is first countable iff \(X\) is hemicompact. Moreover, if \(X\) is a Hausdorff locally compact hemicompact space and \((Y,d)\) is a locally compact complete metric space, then \(D_k(X,Y)\) is completely metrizable. Finally, the author studies a question posed by Hammer and McCoy (1998) whether two compatible metrics on \(Y\) generate the same topologies of uniform convergence on compact sets on \(D(X,Y)\) and discusses the completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps.
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1924924
zbMATH Keywords: densely continuous form
zbMATH Keywords: locally bounded densely continuous form
zbMATH Keywords: Hausdorff metric topology
zbMATH Keywords: hemicompact
zbMATH Keywords: locally compact
zbMATH Keywords: \(k\)-space
zbMATH Keywords: multifunction with closed graph
zbMATH Keywords: usco map
zbMATH Keywords: minimal usco map
author: Ľubica Holá
reviewed by: Chariklia Konstadilaki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1924924