Parametric Bing and Krasinkiewicz maps
type: Item
instance of: scholarly article
title: Parametric Bing and Krasinkiewicz maps
zbMATH Open document ID: 1147.54011
published in: Topology and its Applications
publication date: 2008-05-15T00:00:00Z
review text: All spaces in this paper are assumed to be metrizable and all maps continuous. A function space \(C(X,M)\) is endowed with the source limitation topology. A compact space \(X\) is called a Bing space if each of its subcontinua is hereditariy indecomposable. A map \(g\) is said to be a Bing map if all fibers of \(g\) are Bing spaces. The space \(M\) is a free space if for any compact space \(X\) the function space \(C(X,M)\) contains a dense subset consisting of Bing maps. A map \(g:X\to M\) between compact spaces is called a Krasinkiewicz map if every continuum in \(X\) is either contained in a fiber of \(g\) or contains a component of a fiber of \(g\). The class of free spaces contains all \(n\)-manifolds [\textit{J. Krasinkiewicz}, Bull. Pol. Acad. Sci., Math. 44, No. 2, 147--156 (1996; Zbl 0867.54020)], all locally finite polyhedra, all manifolds modeled on Menger cubes or Nöbeling spaces and 1-dimensional locally connected continua [\textit{J. Song} and \textit{E. D. Tymchatyn}, Fundam. Math. 163, No. 3, 229--239 (2000; Zbl 0971.54013)]. \textit{J. Krasinkiewicz} [Bull. Pol. Acad. Sci., Math. 44, No. 4, 431--440 (1996; Zbl 0885.54009)] proved that, for any compact space \(X\), the Krasinkiewicz maps form a dense subset \(C(X,[0,1])\). \textit{M. Levin} and \textit{W. Lewis} [Isr. J. Math. 133, 61--76 (2003; Zbl 1031.54019)] established that this set is also a \(G_\delta\)-subset of \(C(X,[0,1])\). In the paper under review, the author gives parametric versions of these results by proving the following theorems. Theorem 1.1. Let \(f:X\to Y\) be a perfect map with \(Y\) being a strongly countable-dimensional space. Then, for every complete \(ANR\) free space \(M\), the function space \(C(X,M)\) contains a dense \(G_\delta\)-set of maps \(g\) such that all restrictions \(g| f^{ - 1}(y), y\in Y\), are Bing maps. Moreover, if in addition \(M\) is a closed convex subset of a Banach space, then the same conclusion remains true provided that \(Y\) is a \(C\)-space. Theo…
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5275715
zbMATH Keywords: C-spaces
zbMATH Keywords: hereditarily indecomposable space
zbMATH Keywords: selections for set-valued maps
author: V. Valov
MaRDI profile type: Publication
full work available at URL: j.topol.2006.11.011
OpenAlex ID: W1997550272
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DOI: 10.1016/J.TOPOL.2006.11.011
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description: scientific article; zbMATH DE number 5275715