Various types of local connectedness in \(n\)-fold hyperspaces
type: Item
instance of: scholarly article
title: Various types of local connectedness in \(n\)-fold hyperspaces
zbMATH Open document ID: 1112.54006
author: Sam B. jun. Nadler
author: Sergio Macías
published in: Topology and its Applications
publication date: 2006-12-14T00:00:00Z
review text: For a metric continuum \(X\), with metric \(d\), the hyperspace of all closed nonempty subsets of \(X\) which have at most \(n\) components is denoted by \(C_n(X)\), and it is endowed with the Hausdorff metric \(H\). As usual, the hyperspace \(C_1(X)\) is denoted as \(C(X)\). Given a point \(p\in X\), the authors consider four notions related to local connectedness at \(p\). The continuum \(X\) is conected im kleinen (resp., locally connected, arcwise connected im kleinen, locally arcwise connected) provided that \(p\) has arbitrarily small connected (resp., open and connected, arcwise connected, arcwise connected and open) neighborhoods. The continuum \(X\) is said to have the property of Kelley, if for every \(\varepsilon>0\) there exists \(\delta>0\) such that for any \(x,y\in X\) with \(d(x,y)<\delta\) and each continuum \(A\subset X\) containing \(x\) there exists a continuum \(B\subset X\) containing \(y\) such that \(H(A,B)<\varepsilon\). The continuum \(X\) is \(C^*\)-smooth at \(A\) provided that, whenever a sequence of subcontinua \(A_n\) of \(X\) converges to a subcontinuum \(A\) of \(X\), then the sequence of hyperspaces \(C(A_n)\) converges to the hyperspace \(C(A)\) (in the space \(C(C(X))\)). Finally, the continuum \(X\) is said to be indecomposable, provided that \(X\) is not the union of two of its proper subcontinua. In this paper the authors claim: ``In fact, this is the first systematic study of points of local connectedness in \(C_n(X)\)''. Apparently, they are not aware that two years before they wrote this paper, there was written another paper on this subject by \textit{J. J. Charatonik} and the reviewer [Rocky Mountain J. Math. 36, 811--856 (2006; Zbl 1139.54008)], in which the four notions related to local connectedness were studied in an exhaustive and very complete way. In the paper under review, after obtaining some general results about local connectedness on \(C_n(X)\), the authors obtain characterizations of continua for which \(C_n(X…
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5080153
zbMATH Keywords: Connected im kleinen
zbMATH Keywords: continuum
zbMATH Keywords: C*-smooth
zbMATH Keywords: hyperspace
zbMATH Keywords: \(n\)-fold hyperspace
zbMATH Keywords: locally connected
zbMATH Keywords: property of Kelley.
MaRDI profile type: Publication
full work available at URL: j.topol.2006.03.015
OpenAlex ID: W1966542072
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DOI: 10.1016/J.TOPOL.2006.03.015
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description: scientific article; zbMATH DE number 5080153