Maps between continua with stable values
type: Item
instance of: scholarly article
title: Maps between continua with stable values
zbMATH Open document ID: 1020.54022
DOI: 10.1016/S0166-8641(02)00073-1
author: Sam B. jun. Nadler
published in: Topology and its Applications
publication date: 2003-04-27T00:00:00Z
review text: Let \(X\) and \(Y\) be metric continua. A stable value of a mapping \(f: X\to Y\) is a point \(p\in f(X)\) such that there exists \(\varepsilon> 0\) with the property that \(p\) is a value of all mappings within a distance of \(\varepsilon\) of \(f\) in the supremum metric. If \(f\) is onto, then \(f\) is called a pinch at \(p\in Y\) provided that there exist a continuum \(M\) and onto mappings \(g: X\to M\) and \(\pi: M\to Y\) such that \(f= \pi\circ g\) and there are points \(q_1,q_2\in M\) with \(\pi(q_1)= \pi(q_2)\) and \(\pi|_{(M\setminus\{q_1, q_2\})}\) is a homeomorphism onto \(Y\setminus\{p\}\). It is shown that if \(X\) and \(Y\) are Peano continua and a local separating point \(p\in Y\) is an instable value for some onto mapping \(f: X\to Y\), then \(f\) is a pinch at \(p\). Consequently, each local separating point is a stable value for any monotone mapping between Peano continua. The main result says that if \(X\) and \(Y\) are Peano continua, \(F\) is a finite separator of \(Y\) and \(f: X\to Y\) is onto, then some point of \(F\) is a stable value of \(f\). This implies that any mapping of a Peano continuum onto a regular curve has a stable value. In the last section continua are investigated that belong to Class(S), that is, such continua \(Y\) that every mapping of any continuum onto \(Y\) has a stable value. Among other results it is shown that every hereditarily decomposable arc-like continuum is in Class(S).
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 1900964
zbMATH Keywords: arc-like
zbMATH Keywords: class(S)
zbMATH Keywords: continuum
zbMATH Keywords: C-set
zbMATH Keywords: hereditarily decomposable continuum
zbMATH Keywords: lift
zbMATH Keywords: monotone map
zbMATH Keywords: Peano continuum
zbMATH Keywords: pinch at \(p\)
zbMATH Keywords: quotient space
zbMATH Keywords: regular curve
zbMATH Keywords: retraction
zbMATH Keywords: separating point
zbMATH Keywords: separator
zbMATH Keywords: stable value
zbMATH Keywords: usc decomposition
reviewed by: Janusz J. Charatonik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1900964