Hyperspaces of separable Banach spaces with the Wijsman topology
type: Item
instance of: scholarly article
title: Hyperspaces of separable Banach spaces with the Wijsman topology
zbMATH Open document ID: 1068.54011
author: Wiesław Kubiś
author: K. Sakai
author: Masato Yaguchi
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: Let \(\text{Cld}(X)\) be the set of all non-empty closed sets in a topological space \(X\). Let \(X= (X,d)\) be a metric space, for each \(x\in X\) and \(r> 0\), let \[ U^-(x,r)= \{A\in \text{Cld}(X): d(x,A)< r\};\;U^+(x,r)= \{A\in\text{Cld}(X): d(x,A)> r\}. \] The Wijsman topology on \(\text{Cld}(X)\) is the topology induced by the family \(\{U^-(x,r), U^+(x,r): x\in X, r> 0\}\) as an open subbasis and the space \(\text{Cld}(X)\) with the Wijsman topology is denoted by \(\text{Cld}_W(X)\). The authors discuss \(\text{Cld}_W(X)\) for separable metric spaces relating to Banach space and retract theory, and proved the following theorems. Theorem I. If \(X\) is an infinite-dimensional separable Banach space, then \(\text{Cld}_W(X)\) is homeomorphic to the separable Hilbert space \(\ell_2\). Theorem II. If \(x\) is an infinite-dimensional separable Banach space, then \(\text{Fin}_W(X)\!\approx\! \text{Bdd}_W(X)\!\approx \ell_2\times \ell^f_2\), where \(\text{Fin}_W(X)\) and \(\text{Bdd}_W(X)\) are the subspaces of \(\text{Cld}_W(X)\) consisting of all non-empty finite sets and consisting of all non-empty bounded closed sets, respectively, and \[ \ell^f_2= \{(x_i)_{i\in\mathbb{N}}\in \ell_2:x_i= 0\text{ except for finitely many }i\in\mathbb{N}\}. \] Theorem III. Let \(X\) be a separable metric space. If the component of any finite union of open balls in \(X\) has only finitely many path-components and all of them are unbounded closed (respectively each of them is compact or unbounded closed), then \(\text{Cld}_W(X)\) is an AR (respectively an ANR) and \(\text{Fin}_W(X)\) is homotopy dense in \(\text{Cld}_W(X)\). In particular, \({\mathfrak h}\) is an AR (respectively an ANR) if \(\text{Fin}_W(X)\subset{\mathfrak h}\subset \text{Cld}_W(X)\). It is said that \(Y\subset X\) is homotopy dense in \(X\) if there is a homotopy \(h: X\times 1\to X\) such that \(h_0= \text{id}\) and \(h_t(X)\in Y\) for every \(t> 0\). Theorem IV. Let \(X\) be a separable metric space. If the comp…
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 2142277
zbMATH Keywords: hyperspace
zbMATH Keywords: Wijsman topology
zbMATH Keywords: infinite-dimensional separable Banach space
zbMATH Keywords: separable metric space
zbMATH Keywords: separable Hilbert space
zbMATH Keywords: AR
zbMATH Keywords: ANR
zbMATH Keywords: homotopy dense
MaRDI profile type: Publication
full work available at URL: j.topol.2004.07.009
OpenAlex ID: W2095104572
cites work: A Fake Topological Hilbert Space
cites work: Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold
cites work: Hyperspaces of normed linear spaces with the Attouch-Wets topology
cites work: Q4938163
cites work: Q4277838
cites work: Q4060144
cites work: Characterizing certain incomplete infinite-dimensional absolute retracts
cites work: Boundary sets in the Hilbert cube
cites work: Some Applications of The Topological Characterizations of the Sigma-Compact Spaces l 2 f and &#8721
cites work: Hyperspaces of finite subsets which are homeomorphic to \(\aleph _ 0\)- dimensional linear metric spaces
cites work: Hyperspaces of Peano continua are Hubert cubes
cites work: Q5522732
cites work: Q4323294
cites work: Complete metric absolute neighborhood retracts
cites work: Topological Semilattices with Small Semilattices
cites work: Q5676645
cites work: The Wijsman and Attouch-Wets topologies on hyperspaces revisited
cites work: Q3812929
cites work: The completions of metric ANR's and homotopy dense subsets
cites work: Hyperspaces of Banach spaces with the Attouch-Wets topology
cites work: Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube.
cites work: Characterizing Hilbert space topology
cites work: A correction of two papers concerning Hilbert manifolds
DOI: 10.1016/J.TOPOL.2004.07.009
description: scientific article; zbMATH DE number 2142277