On hyperspaces of non-cut sets of continua
type: Item
instance of: scholarly article
title: On hyperspaces of non-cut sets of continua
zbMATH Open document ID: 1359.54010
author: Raúl Escobedo
author: Carolina Estrada-Obregón
author: Javier Sánchez-Martínez
published in: Topology and its Applications
publication date: 2017-01-09T00:00:00Z
review text: For a metric continuum \(X\), \(2^X\) denotes the hyperspace of nonempty closed subsets of \(X\), and \(F_{1}(X)\) is the hyperspace of singletons of \(X\), both hyperspaces are endowed with the Hausdorff metric. Given an element \(A \neq X\) in \(2^X\), we say that \(A\): (a) is a non-weak cut set of \(X\) (\(A \in NWC(X)\)) provided that for any two points \(x\) and \(y\) in \(X \setminus A\) there exists a subcontinuum \(B\) of \(X\) containing \(x\) and \(y\), and contained in \(X \setminus A\), (b) does not block the singletons of \(X\) (\(A \in NB(F_{1}(X))\)) provided that for each point \(x\) in \(X \setminus A\), the union of the subcontinua of \(X\) containing \(x\) and contained in \(X \setminus A\) is dense in \(X\), (c) does not block some point of \(X\) (\(A \in NB^*(F_{1}(X))\)) provided that there exists an \(x\) in \(X \setminus A\) such that the union of the subcontinua of \(X\) containing \(x\) and contained in \(X \setminus A\) is dense in \(X\), (d) is a shore set of \(X\) (\(A \in S(X)\)) if \(X\) is the limit (in the hyperspace \(2^X\)) of subcontinua contained in \(X \setminus A\), (e) is a non-cut set of \(X\) (\(A \in H(X)\)) provided that \(X \setminus A\) is connected. Define \(NC(X)\) as the set of elements \(A\) in \(H(X)\) such that \(\mathrm{int}(A) = \emptyset\). In this paper the authors show the following chain of inclusions: \(NWC(X) \subset NB(F_{1}(X) \subset NB^*(F_{1}(X)) \subset S(X) \subset NC(X)\). They show that all the inclusions are proper; and all these families coincide in locally connected continua. The authors offer several characterizations of the unit interval, the unit circle and indecomposable continua in terms of these concepts. They also include the following interesting question. Is there a non-locally connected continuum \(X\) such that \(NB(F_{1}(X))=F_{1}(X)\)
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6672886
zbMATH Keywords: arc
zbMATH Keywords: continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: indecomposable continua
zbMATH Keywords: non-block set
zbMATH Keywords: non-cut set
zbMATH Keywords: shore set
zbMATH Keywords: simple closed curve
MaRDI profile type: Publication
full work available at URL: j.topol.2016.12.004
OpenAlex ID: W2559889125
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DOI: 10.1016/J.TOPOL.2016.12.004
description: scientific article; zbMATH DE number 6672886