Local dendrites with unique hyperspace \(C(X)\)
type: Item
instance of: scholarly article
title: Local dendrites with unique hyperspace \(C(X)\)
zbMATH Open document ID: 1200.54018
author: David Herrera-Carrasco
author: Fernando Macías-Romero
author: Gerardo G. Acosta
published in: Topology and its Applications
publication date: 2010-08-27T00:00:00Z
review text: A continuum \(X\) is a nonempty, compact, connected metric space. The authors are concerned with the hyperspace \(C(X)\) of the subcontinua of \(X\). By definition, \(X\) has unique hyperspace \(C(X)\) if \(C(X)\cong C(Y)\) implies that \(X\cong Y\). Much work has been done on this subject, and the authors provide a list of references including many of these results. A dendrite is a locally connected continuum that contains no simple closed curve. Define \(\mathcal{D}\) to be the class of dendrites whose set of endpoints is closed. It is known, \textit{D. Herrera-Carrasco}, [Houston J. Math. 33, No.~3, 795--805 (2007; Zbl 1147.54005)], that if \(X\in\mathcal{D}\) and \(X\) is not an arc, then \(X\) has unique hyperspace \(C(X)\). Let \(\mathcal{L}\) be the class of local dendrites and \(\mathcal{L}\mathcal{D}\) the class of local dendrites each point of which has a neighborhood in \(\mathcal{D}\). The authors study the structures of \(\mathcal{D}\) and \(\mathcal{L}\mathcal{D}\). They prove, as an application, that if \(X \in\mathcal{L}\mathcal{D}\) and \(X\) is not an arc, then \(X\) has unique hyperspace \(C(X)\).
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 5775726
zbMATH Keywords: continuum
zbMATH Keywords: dendrite
zbMATH Keywords: dimension
zbMATH Keywords: hyperspace
zbMATH Keywords: local dendrite
zbMATH Keywords: unique hyperspace
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
full work available at URL: j.topol.2010.05.005
OpenAlex ID: W2019347528
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DOI: 10.1016/J.TOPOL.2010.05.005
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description: scientific article; zbMATH DE number 5775726