Homogeneous continua for which the set function \(\mathcal T\) is continuous
type: Item
instance of: scholarly article
title: Homogeneous continua for which the set function \(\mathcal T\) is continuous
zbMATH Open document ID: 1116.54011
published in: Topology and its Applications
publication date: 2007-01-09T00:00:00Z
review text: \textit{F. Burton Jones} [Am. J. Math. 70, 403--413 (1948; Zbl 0035.10904)] defined the set function \({\mathcal T}\) and \textit{D. Bellamy} [Trans. Am. Math. Soc. 151, 581--587 (1970; Zbl 0207.53001)] gave conditions on a continuum for which \({\mathcal T}\) is continuous. The author of this paper in [Houston J. Math 32, No. 1, 161--165 (2006; Zbl 1102.54021)] gave a class of decomposable nonlocally connected one-dimensional continua for which \({\mathcal T}\) is continuous. In 2004 David Bellamy and Sam Nadler conjectured that if \(X\) is a homogeneous one-dimensional continuum, then \({\mathcal T}\) is continuous for \(X\). In this paper the author gives a negative answer to this question and establishes that \({\mathcal T}\) is continuous for a homogeneous continuum \(X\) if and only if one of the following conditions holds: 1. \(X\) is locally connected; 2. \(X\) is indecomposable; 3. \(X\) is not aposyndetic and is homeomorphic to the simple closed curve \(S^1\) or to the Menger universal curve \({\mathcal M}\), where \({\mathcal G}= \{{\mathcal T}(\{x\}): x\in X\}\).
reviewed by: Donald E. Bennett
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5083208
zbMATH Keywords: continuum
zbMATH Keywords: homogeneous
zbMATH Keywords: aposyndetic
zbMATH Keywords: indecomposable
zbMATH Keywords: locally connected
zbMATH Keywords: continuous
author: Sergio Macías
MaRDI profile type: Publication
full work available at URL: j.topol.2006.02.003
OpenAlex ID: W2074814365
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DOI: 10.1016/J.TOPOL.2006.02.003
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description: scientific article; zbMATH DE number 5083208