Induced mappings on hyperspaces \(C(p,X)\) and \(K(X)\)
type: Item
instance of: scholarly article
title: Induced mappings on hyperspaces \(C(p,X)\) and \(K(X)\)
zbMATH Open document ID: 1386.54015
author: D. Maya-Escudero
author: Norberto Ordoñez
author: Javier Sánchez-Martínez
published in: Topology and its Applications
publication date: 2017-11-09T00:00:00Z
review text: For a metric continuum \(X\) and a point \(p \in X\), consider the following hyperspaces: \(C(X)=\{A \subset X : A\) is a subcontinuum of \(X\}\), \(C_{p} (X) = \{A \in C(X) : p \in A\}\), and \(K(X)=\{C_{p} (X) \in C(C(X)) : p \in X\}\). Given a mapping between continua \(f:X \to Y\), the authors consider the following natural mappings: -- \(\overline{f} : C(X) \rightarrow C(Y)\), given by \(\overline{f} (A)=f(A)\) (the image of \(A\) under \(f\)), -- \(\tilde{f} : K(X) \rightarrow K(Y)\), given by \(\tilde{f} (C(p,X)) = C(f(p),Y)\), -- \(\Check{f} : K(X) \rightarrow C(C(Y))\) given by \(\Check{f} (C(p,X))=\overline{f}(C(p,X))\), and -- \(\overline{f}_{p} : C(p,X) \rightarrow C(f(p),Y)\) given by \(\overline{f}_{p}(A)=\overline{f}(A)\). Now let \(\mathcal{M}\) denote a class of mappings between continua. The main problem studied in this paper is to find all interrelations betweeen the following statements: (0) \(f \in \mathcal{M}\), (1) \(\overline{f} \in \mathcal{M}\), (2) \(\tilde{f} \in \mathcal{M}\), (3) \(\Check{f} \in \mathcal{M}\), and (4) \(\overline{f}_{p} \in \mathcal{M}\) for all \(p \in X\). There are a number of papers and authors that have studied the respective problem for many possible hyperspaces. The authors include a very complete list of references. In the paper under review the authors study this problem for the classes of monotone, confluent, weakly confluent, light and open mappings. They include many results on this topic and pose several interesting questions.
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 6804707
zbMATH Keywords: confluent
zbMATH Keywords: continuum
zbMATH Keywords: hyperspace
zbMATH Keywords: induced mapping
zbMATH Keywords: light
zbMATH Keywords: monotone
zbMATH Keywords: open
zbMATH Keywords: weakly confluent
MaRDI profile type: Publication
full work available at URL: j.topol.2017.10.008
OpenAlex ID: W2761826895
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DOI: 10.1016/J.TOPOL.2017.10.008
description: scientific article; zbMATH DE number 6804707