Noncut subsets of the hyperspace of subcontinua
type: Item
instance of: scholarly article
title: Noncut subsets of the hyperspace of subcontinua
zbMATH Open document ID: 1480.54023
author: Javier Camargo
author: D. Maya-Escudero
author: Patricia Pellicer-Covarrubias
published in: Topology and its Applications
publication date: 2021-11-26T00:00:00Z
review text: A continuum is a compact, connected, nonempty, metric space. The hyperspace of all nonempty closed subsets of a continuum \(X\) is denoted by \(2^X\). The symbol \(C(X)\) represents the hyperspace of all subcontinua of \(X\), whereas \(F_1(X)\) denotes the hyperspace of all singletons of \(X\). Moreover, if \(p\in X\), \(C_p(X) = \{B\in C(X) : p\in B\}\). These hyperspaces are considered with the Hausdorff metric. Given a continuum \(X\), a Whitney map \(\mu\colon C(X) \to [0,1]\) and \(p\in X\), the set \(\mu^{-1}(t) \cap C_p(X)\) is denoted by \(\mu^{-1}_p(t)\). In continuum theory there are different concepts related to the idea of being on the ``edge'' of a continuum or being a sort of noncut subset of a continuum. In this regard, for a nonempty closed subset \(B\) of a continuum \(X\) such that \(\text{int}_X(B) = \emptyset\), the following properties are defined: \(B\) is a subset of colocal connectedness, strong noncut subset, weak nonblock subset, shore subset, not strong center and noncut subset of \(X\). Furthermore, with these notions, the so-called hyperspaces of noncut sets of \(X\) are defined: \(CLC(X)\), \(SNC(X)\), \(NB(X)\), \(NB^\ast(X)\), \(S(X)\), \(NSC(X)\), \(NC(X)\). A study of these concepts, in the context of hyperspaces of continua, has been made by \textit{V. Martínez-de-la-Vega} and \textit{J. M. Martínez-Montejano} [Colloq. Math. 160, No. 2, 297--307 (2020; Zbl 1462.54029)]. They proved that \(F_1(X)\) belongs to some hyperspace of noncut sets of the hyperspaces \(2^X\), \(C_n(X)\) and \(F_n(X)\), where \(C_n(X)\) consists of those subsets of \(X\) with at most \(n\) components and \(F_n(X)\) of those subsets of \(X\) with at most \(n\) elements. In the paper under review, the authors present results on the study of all possible relationships among the conditions (a) and (b) and between (a) and (c) for: \begin{itemize} \item[(a)] \(\{p\}\) is an element of a certain hyperspace of noncut sets \(\mathcal{H}(X)\) of a continuum \(X\); \ite…
reviewed by: Jesús F. Tenorio
Mathematics Subject Classification ID: 54F16
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 7434144
zbMATH Keywords: continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: nonblock set
zbMATH Keywords: noncut set
zbMATH Keywords: order arc
zbMATH Keywords: shore set
zbMATH Keywords: strong center set
zbMATH Keywords: strong noncut set
zbMATH Keywords: weak nonblock set
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107867
OpenAlex ID: W3203815760
cites work: Union of shore sets in a dendroid
cites work: Non-cut, shore and non-block points in continua
cites work: On blockers in continua
cites work: Continua whose hyperspace of nonblockers of \(\mathcal{F}_1(X)\) is a continuum
cites work: The hyperspace of nonblockers of \(\mathcal{F}_1(X)\)
cites work: Q3311300
cites work: Application of a Selection Theorem to Hyperspace Contractibility
cites work: On hyperspaces of non-cut sets of continua
cites work: More on Connectedness im Kleinen and Local Connectedness in C(X)
cites work: Q3906104
cites work: A characterization of solenoids
cites work: Centers of a dendroid
cites work: Induced mappings on hyperspaces
cites work: The cone = hyperspace property, a characterization
cites work: Finite unions of shore sets
cites work: Blockers in hyperspaces
cites work: Q4230677
cites work: Q3304890
cites work: Shore points of a continuum
cites work: Q5109949
cites work: Concerning when $F_1(X)$ is a continuum of colocal connectedness in hyperspaces and symmetric products
cites work: Q3869981
cites work: Q4004123
cites work: Centers and shore points of a dendroid
cites work: Shore Points and Dendrites
cites work: The hyperspaces \(K(X)\)
cites work: Nonblockers in homogeneous continua
cites work: Embedding the Hyperspaces of Circle-Like Plane Continua
cites work: Whitney continua in the hyperspace C(X)
cites work: Q5837858
DOI: 10.1016/J.TOPOL.2021.107867
description: scientific article; zbMATH DE number 7434144