Local connectedness and retractions in hyperspaces
type: Item
instance of: scholarly article
title: Local connectedness and retractions in hyperspaces
zbMATH Open document ID: 1211.54024
DOI: 10.1216/RMJ-2008-38-6-1927
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-04-07T00:00:00Z
review text: The results obtained in this article are related to certain ones obtained earlier by the second author in [Topology Appl. 135, No. 1--3, 277--291 (2004; Zbl 1036.54002)]. Given a metric continuum \(X\), let \(2^{X}\) denote the hyperspace of all nonempty closed subsets of \(X\), \(C(X)\) the hyperspace of subcontinua of \(X\) and \(C_{2}(X)\) the hyperspace composed of all members of \(2^{X}\) which have at most two components. For a continuum \(X\) and a point \(p \in X\), define \(\psi_{p} : 2^X \rightarrow 2^X\) by \(\psi_{p}(A) = A \cup \{p\}\) for each \(A \in 2^X\); similarly, \(\varphi_{p} : C(X) \rightarrow C_{2}(X)\) by \(\varphi_{p}(A) = A \cup \{p\}\) for each \(A \in C(X)\). Among other results it is shown in this paper that the following two conditions are equivalent: (1) \(\psi_{p}\) is a strong deformation retraction, (2) \(\varphi_{p}\) is a strong deformation retraction in \(C_{2}(X)\). Furthermore, it is shown that if \(\psi_{p}\) is a strong deformation retraction, then \(X\) is connected im kleinen at \(p\).
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C15
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5541304
zbMATH Keywords: connected im kleinen
zbMATH Keywords: continuum
zbMATH Keywords: contractible
zbMATH Keywords: deformation retraction
zbMATH Keywords: hyperspace
zbMATH Keywords: local basis
zbMATH Keywords: locally connected
zbMATH Keywords: retraction
zbMATH Keywords: strong deformation retraction
author: Janusz J. Charatonik
author: Patricia Pellicer-Covarrubias
MaRDI profile type: Publication
cites work: On smooth dendroids
cites work: On the hyperspaces \({\mathcal C}_n(X)\) of a continuum \(X\)
cites work: Q4252510
cites work: Retractions in hyperspaces.
full work available at URL: rmj-2008-38-6-1927
OpenAlex ID: W2076160666
description: scientific article; zbMATH DE number 5541304