Hyperspaces through regular and meager subcontinua
type: Item
instance of: scholarly article
title: Hyperspaces through regular and meager subcontinua
zbMATH Open document ID: 1486.54045
author: Norberto Ordoñez
published in: Topology and its Applications
publication date: 2021-08-27T00:00:00Z
review text: A continuum is a nonempty compact connected metric space. By a hyperspace of a continuum \(X\) we mean a specified collection of subsets of \(X\) endowed with the Hausdorff metric. Given a continuum \(X\), the hyperspace of regular subcontinua \(D(X)\) is the collection of all regular subcontinua of \(X\) and the hyperspace of meager subcontinua \(M(X)\) is the collection of all subcontinua of \(X\) with empty interior. The article under review discusses the following problem: Problem. Characterize the families of continua for which any of the following equalities hold (a) \(F_1(X) = M(X)\); (b) \(D(X) = C(X) \setminus F_1(X)\); (c) \(D(X) = C(X) \setminus M(X)\); (d) \(C(X) = D(X) \cup M(X)\). \medskip Some results of the paper are the following: Theorem. Let \(X\) be a continuum. It is true that (a) If \(M(X) = F_1(X)\), then \(X\) is hereditarily locally connected. \medskip (b) If \(D(X) = C(X) \setminus F_1(X)\), then \(X\) is hereditarily locally connected. \medskip Theorem. Let \(X\) be a locally connected continuum. If \(D(X) = C(X) \setminus M(X)\), then \(M(X) = F_1(X)\). \medskip Theorem. If \(X\) is a hereditarily locally connected continuum, then the following conditions are equivalent: (i) \(X\) does not contain dendrites \(\mathcal{D}_1\); (ii) \(M(X) = F1(X)\); (iii) \(D(X) = C(X) \setminus F_1(X)\); (iv) \(D(X) = C(X) \setminus M(X)\); (v) \(C(X) = D(X) \cup M(X)\); Also, the author presents interesting problems in this line of research. This article contains some interesting results.
Mathematics Subject Classification ID: 54F16
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 7387398
zbMATH Keywords: continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: regular subcontinua
zbMATH Keywords: meager subcontinua
reviewed by: Alicia Santiago-Santos
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107760
OpenAlex ID: W3176972611
cites work: Q4230677
cites work: Q4004123
cites work: Q3869981
cites work: Q4995028
cites work: The hyperspace of regular subcontinua
DOI: 10.1016/J.TOPOL.2021.107760
description: scientific article; zbMATH DE number 7387398