Hyperspaces of normed linear spaces with the Attouch-Wets topology
type: Item
instance of: scholarly article
title: Hyperspaces of normed linear spaces with the Attouch-Wets topology
zbMATH Open document ID: 1050.54009
DOI: 10.1023/A:1021981321137
author: Taras Banakh
author: Masayuki Kurihara
author: K. Sakai
published in: Set-Valued Analysis
publication date: 2003-04-27T00:00:00Z
review text: For a metric space \((X,d)\) let Cld\((X)\) denote the space of all non-empty closed subsets of \(X\). The space Cld\((X)\) can be seen as a subset of \(C(X)\), the space of all continuous real valued functions, by identifying each set \(A\in \text{Cld}(X)\) with the function \(a\in C(X)\) defined by \(a(x)=d(x,A)\) for \(x\in X\). The Attouch-Wets Topology of Cld\((X)\) is inherited from \(C(X)\) with the topology of uniform convergence on bounded sets. The authors investigate some topological properties in the hyperspace Cld\((X)\) equipped with the Attouch-Wets topology. The main results say that for any normed linear space \(X\), the hyperspace Cld\((X)\) and various subspaces, including the following: Bdd\((X)\) of all bounded closed sets Reg\((X)\) of all regular (i.e. \(A=\text{cl(int}A)\)) sets Conn\((X)\) of all closed sets with finite components Comp\((X)\) of all compacta Cont\((X)\) of all finite unions of continua Pol\((X)\) of all compact polyhedra Fin\((X)\) of all finite sets are ARs. Moreover if \(X\) is an infinite-dimensional Banach space with weight \(w(X)\), then Cld\((X)\) is homeomorphic to the Hilbert space with weight \(2^{w(X)}\). In particular, if \(X\) is a separable infinite dimensional Banach space, then Cld\((X)\) is homeomorphic to the space \(m=\ell_\infty\).
reviewed by: Nguyen To Nhu
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 1901529
zbMATH Keywords: hyperspace
zbMATH Keywords: Attouch-Wets topology
zbMATH Keywords: normed linear space
zbMATH Keywords: Banach space
zbMATH Keywords: Hilbert space
zbMATH Keywords: AR
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1901529