Nonblockers in hyperspaces
type: Item
instance of: scholarly article
title: Nonblockers in hyperspaces
zbMATH Open document ID: 1256.54033
author: Raúl Escobedo
author: María de Jesús López
author: Hugo Villanueva
published in: Topology and its Applications
publication date: 2012-10-31T00:00:00Z
review text: For a continuum \(X\), let \(2^X\) denote its hyperspace of nonempty closed subsets. An element \(B \in 2^X\) does not block the singletons in \(X\) provided that, for each \(x \in X \backslash B\), the union of all the compact connected subsets of \(X \backslash B\) containing \(x\) is dense in \(X\). The set of all elements in \(2^X\) that do not block the singletons is denoted by \(\mathcal{NB}(F_{1}(X))\). The concept of blockers was introduced by \textit{A. Illanes} and \textit{P. Krupski} in [Topology Appl. 158, No. 5, 653--659 (2011; Zbl 1214.54025)] where it was used to study some properties of the hyperspaces. In the paper under review, the authors use the set \(\mathcal{NB}(F_{1}(X))\) to characterize some classes of locally connected continua such as: a simple closed curve, an arc, trees and dendrites. For example, they show that a simple closed curve is the only locally connected continuum for which the set \(\mathcal{NB}(F_{1}(X))\) is a continuum. They include the following interesting problem. Is a simple closed curve the only continuum \(X\) such that \(\mathcal{NB}(F_{1}(X))\) coincides with the set of singletons?
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6100636
zbMATH Keywords: arc
zbMATH Keywords: blockers
zbMATH Keywords: circle of pseudo-arcs
zbMATH Keywords: continuum
zbMATH Keywords: dendrite
zbMATH Keywords: hyperspace
zbMATH Keywords: simple closed curve
zbMATH Keywords: tree
MaRDI profile type: Publication
full work available at URL: j.topol.2012.09.002
OpenAlex ID: W1983694981
cites work: Another Homogeneous Plane Continuum
cites work: Blockers in hyperspaces
cites work: Q4230677
cites work: Q3869981
cites work: Q4004123
cites work: Q3878392
cites work: Q5837858
DOI: 10.1016/J.TOPOL.2012.09.002
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description: scientific article; zbMATH DE number 6100636