Whitney preserving maps on finite graphs
type: Item
instance of: scholarly article
title: Whitney preserving maps on finite graphs
zbMATH Open document ID: 1228.54031
author: Benjamin Espinoza
author: Alejandro Illanes
published in: Topology and its Applications
publication date: 2011-05-16T00:00:00Z
review text: For a metric continuum \(X\), let \(C(X)\) denote the hyperspace of subcontinua of \(X\). For a map \(f: X \rightarrow Y\) from a continuum \(X\) to a continuum \(Y\), let \(\hat{f}\) denote the induced map (from \(C(X)\) to \(C(Y)\) given by \(\hat{f}(A) = f(A)\) for all \(A \in C(X)\)). The map \(f: X \rightarrow Y\) is called Whitney preserving if there exist two Whitney maps \(\mu : C(X) \rightarrow \mathbb{R} \) and \(\nu : C(Y) \rightarrow \mathbb{R} \) such that for every \(t \in [0, \mu(X)]\) there exists \(s \in [0, \nu(Y)]\) such that \(\hat{f}(\mu^{-1}(t))=\nu^{-1}(s)\). The first author proved in [Topology Appl. 126, 351--358 (2002; Zbl 1019.54016)] that a Whitney preserving map from an arcwise connected continuum onto the unit interval is a homeomorphism. In this paper the result is extend as follows: 1) a Whitney preserving map from an arcwise connected continuum onto a finite graph, which is not a simple closed curve, is a homeomorphism; 2) if \(f: X \rightarrow S^{1}\) is a Whitney preserving map from a locally connected continuum \(X\) onto a simple closed curve \(S^{1}\), then \(X\) is homeomorphic to the unit interval or \(X\) is homeomorphic to \(S^{1}\). The authors also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C99
zbMATH DE Number: 5888206
zbMATH Keywords: arcwise connected
zbMATH Keywords: arcwise connected continuum
zbMATH Keywords: continuum
zbMATH Keywords: finite graph
zbMATH Keywords: hyperspace
zbMATH Keywords: locally connected continuum
zbMATH Keywords: weakly confluent map
zbMATH Keywords: Whitney levels
zbMATH Keywords: Whitney map
zbMATH Keywords: Whitney preserving map
MaRDI profile type: Publication
full work available at URL: j.topol.2011.02.010
OpenAlex ID: W1986035484
cites work: Hereditarily weakly confluent mappings onto \(S^1\)
cites work: Whitney preserving functions
cites work: Q4230677
cites work: Q3083071
cites work: Q3869981
cites work: Q4004123
DOI: 10.1016/J.TOPOL.2011.02.010
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description: scientific article; zbMATH DE number 5888206