Proper hereditary shape equivalences preserve property C
type: Item
instance of: scholarly article
title: Proper hereditary shape equivalences preserve property C
zbMATH Open document ID: 0568.54017
DOI: 10.1016/0166-8641(85)90086-0
author: Fredric D. Ancel
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: A space X has property C provided each sequence \(\{\) \({\mathcal U}_ n: n\in N\}\) of open covers of X admits an open cover \({\mathcal V}=\cup \{{\mathcal V}_ n: n\in N\}\) of X such that for each n, \({\mathcal V}_ n\) refines \({\mathcal U}_ n\) and is a collection of pairwise disjoint sets. The author considers metric spaces and compact surjective maps \(f: X\to Y\), which are approximately invertible, i.e., whenever X is a closed subset of a metric space Z and \({\mathcal W}\) is a collection of open sets of Z, refined by the fibers \(f^{-1}(y)\), \(y\in Y\), then there is a map \(g: Y\to Z\) such that gf and the inclusion \(Y\to Z\) are \({\mathcal W}\)-near maps. The main theorem asserts that such maps f preserve property C, i.e., if X has property C, then so does Y. In particular, this applies to proper hereditary shape equivalences and generalizes a recent result of \textit{J. van Mill} and \textit{J. Mogilski} [Proc. Am. Math. Soc. 90, 118-120 (1984; Zbl 0548.57006)].
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55P55
zbMATH DE Number: 3907332
zbMATH Keywords: property C
zbMATH Keywords: proper hereditary shape equivalences
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90086-0
OpenAlex ID: W2068866968
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description: scientific article; zbMATH DE number 3907332