Induced mappings on symmetric products, some answers
type: Item
instance of: scholarly article
title: Induced mappings on symmetric products, some answers
zbMATH Open document ID: 1395.54010
author: Alejandro Illanes
author: Jimmy A. Naranjo-Murillo
author: Jorge E. Vega
author: Yajaida N. Velázquez-Inzunza
published in: Topology and its Applications
publication date: 2018-06-07T00:00:00Z
review text: Let \(X\) be a nondegenerate metrizable continuum. Certain hyperspaces of \(X\) (with the Vietoris topology) are explored, in particular, \(F_n(X)\), the collection of nonempty subsets of \(X\) of cardinality \(\leq n\). The quotient space \(SF_n(X)=F_n(X)/F_1(X)\), \(n\geq 2\), comes into play in this study. In case \(f:X\to Y\) is a map of such continua, there are induced maps \(f_n:F_n(X)\to F_n(Y)\) and \(Sf_n:SF_n(X)\to SF_n(Y)\). The authors are interested in certain classes \(\mathcal{M}\) of maps and the relations among the statements \(f\in\mathcal{M}\), \(f_n\in\mathcal{M}\), and \(Sf_n\in\mathcal{M}\). Four classes \(\mathcal{M}\) of maps are considered: almost monotone, atriodic, freely decomposable, and joining. To give some flavor to the results, recall that a surjective map \(f:X\to Y\) of continua is called almost monotone if for every subcontinuum \(B\) of \(Y\) that has nonempty interior in \(Y\), \(f^{-1}(B)\) is connected. Consider the statements: \((1)\) \(f\) is almost monotone. \((2)\) \(f_n\) is almost monotone. \((3)\) \(Sf_n\) is almost monotone. The authors point out that \((2)\Rightarrow(1)\) and \((2)\Rightarrow(3)\) have previously been established. In Section 2, they show that \((1)\) does not imply \((3)\), but that in case \(n=2\), \((3)\Rightarrow(1)\). Let us just mention that the last three sections deal respectively with atriodic, freely decomposable, and joining maps.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 6882107
zbMATH Keywords: almost monotone map
zbMATH Keywords: atriodic map
zbMATH Keywords: continuum
zbMATH Keywords: freely decomposable map
zbMATH Keywords: joining map
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
full work available at URL: j.topol.2018.04.018
OpenAlex ID: W2800198365
cites work: Induced mappings on symmetric products of continua
cites work: Induced maps on \(n\)-fold symmetric product suspensions
cites work: More on induced maps on \(n\)-fold symmetric product suspensions
cites work: Local connectedness in hyperspaces
cites work: Q3064140
cites work: Q4230677
Wikidata QID: Q129816329
DOI: 10.1016/J.TOPOL.2018.04.018
description: scientific article; zbMATH DE number 6882107