Proper hereditary shape equivalences preserve property C
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)].
scientific article; zbMATH DE number 3907332