A midpoint function and an end point function in continua
type: Item
instance of: scholarly article
title: A midpoint function and an end point function in continua
zbMATH Open document ID: 1386.54003
author: Maria de J. López
author: Iván Serapio
author: Patricia Pellicer-Covarrubias
published in: Topology and its Applications
publication date: 2018-02-09T00:00:00Z
review text: For a metric continuum \(X\), let \(2^X\) be the hyperspace of nonempty closed subsets of \(X\), considered with the Hausdorff metric. Let \(\mathcal{M}(X)\) be the hyperspace of arcs and singletons in \(X\) and \(F_2(X)\) the hyperspace of nonempty subsets of \(X\) with at most two points. In this paper the authors consider the natural function \(E:\mathcal{M}(X)\to F_2(X)\) given by \(E(A)=\{a,b\}\), where \(a,b\) are the end points of \(A\), if \(A\) is an arc, and \(a=b\), if \(A=\{a\}\). In this paper the continuity of the function \(E\) is studied. It is proved that if \(X\) is regular (this means that \(X\) has a basis of open sets with finite boundary), then \(E\) is continuous. With an example, it is shown that this result cannot be extended to hereditarily locally connected continua. A non-locally connected dendroid for which \(E\) is continuous is also shown. Some characterizations of dendrites are shown. For example, the authors prove that an arcwise connected continuum \(X\) is a dendrite if and only if \(\mathcal{M}(X)\) is compact and \(E\) is continuous. Using a Whitney mapping defined on \(2^X\), it is possible to define also a midpoint function \(m:\mathcal M (X)\to F_1(X)\). In a previous paper [Rev. Integr. 34, No. 1, 109--123 (2016; Zbl 1353.54015)] the authors showed that the continuity of \(E\) is equivalent to that of \(m\). In the paper under review, the authors investigate conditions under which \(m\) is one-to-one, a homeomorphism, atomic, closed, open, monotone, confluent, weakly confluent, atriodic and light. Finally, the authors pose some interesting questions. For example, they ask whether the continuity of \(E\) for a hereditarily locally connected continuum \(X\) implies that \(X\) is regular.
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 6836780
zbMATH Keywords: continuum
zbMATH Keywords: dendrite
zbMATH Keywords: end point function
zbMATH Keywords: hyperspace of arcs and singletons
zbMATH Keywords: midpoint function
MaRDI profile type: Publication
full work available at URL: j.topol.2017.11.036
OpenAlex ID: W2772343954
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DOI: 10.1016/J.TOPOL.2017.11.036
description: scientific article; zbMATH DE number 6836780