A characterization of absolute neighborhood retracts in general spaces
Let Q be a class of topological spaces satisfying the following conditions: (1) every \(X\in Q\) is paracompact normal, (2) if A is a closed (resp. open) subspace of \(X\in Q\), then \(A\in Q\), (3) for every \(X\in Q\), \(X^ 2\in Q\), (4) a space \(X\in Q\) is an absolute neighborhood retract for the class Q (denote \(X\in ANR(Q))\) iff X is an absolute neighborhood extensor for Q (denote \(X\in ANE(Q)).\) In the paper under review the author in particular gives the following characterization of ANR(Q) spaces. Theorem. A necessary and sufficient condition for a space \(Y\in Q\) to be an ANR(Q) is the existence of an open covering W of Y such that, for any two W-near maps f,g: \(X\to Y\) defined on a space \(X\in Q\) and any W-homotopy \(F_ t: A\to Y\) (0\(\leq t\leq 1)\), defined on a closed subspace A of X with \(F_ 0=f| A\) and \(F_ 1=g| A\), there exists a homotopy \(H_ t: X\to Y\), (0\(\leq t\leq 1)\) with \(H_ 0=f\), \(H_ 1=g\) and \(H_ t| A=F_ t\) for every \(0\leq t\leq 1.\) Let us observe that this characterization is an extension of the well- known theorem first proved by \textit{O. Hanner} for separable metric spaces [Ark. Mat. 1, Nr. 30, 389-408 (1951; Zbl 0042.411)]. Hanner's characterization was extended to the metric spaces by \textit{S. T. Hu} [see Theory of retracts (1965; Zbl 0145.430), Chapter 4].
scientific article; zbMATH DE number 3944838