Metrics that generate the same hyperspace convergence
type: Item
instance of: scholarly article
title: Metrics that generate the same hyperspace convergence
zbMATH Open document ID: 0796.54016
DOI: 10.1007/BF01027689
author: Camillo Costantini
author: Sandro Levi
author: Jolanta Ziemińska
published in: Set-Valued Analysis
publication date: 1994-09-28T00:00:00Z
review text: Let \(c(X)\) be the family of all closed subsets of a metric space \((X,d)\). A net \((A_ j)\) in \(c(X)\) is Wijsman convergent (\(w_ d\)-convergent) to \(A\in c(X)\) if for any \(x\in X\) we have \(d(x,A)= \lim_ j d(x,A_ j)\). The convergence is topological and the authors give necessary and sufficient conditions for two equivalent metrics to generate the same Wijsman convergence. Some applications to normed linear spaces and to various classes of metrizable spaces are also included. One of typical results: Let \(\|\cdot\|_ 1\), \(\|\cdot\|_ 2\) be two equivalent non homothetic norms on a real linear space \(X\) and \(d_ 1\), \(d_ 2\) the metrics they respectively generate. Then \(w_{d_ 1}= w_{d_ 2}\) if and only if \(X\) is finite-dimensional.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 465282
zbMATH Keywords: hyperspace of a metric space
zbMATH Keywords: strict \(d\)-inclusion
zbMATH Keywords: Wijsman convergence
reviewed by: J. Chvalina
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 465282