Closed retracts and perfect retracts
The author studies conditions under which a retract \(E\) of a space \(X\) must be a perfect (respectively closed) retract of \(X\), that is, the image of \(X\) under a perfect (respectively closed) retraction. He first examines the case when \(E\) is compact and proves that in this case the following conditions are equivalent: (a) If \(E\) is a retract of a space \(X\) which is locally compact at every \(x\in E\), then \(E\) is a closed retract of \(X.\) (b) \(E\) \(\times\{0\}\) is a closed retract of \(E\times\mathbb{R}\). (c) \(E\) has a disjoint open cover \(\{E_{1},\dots,E_{n}\}\) with each \(E_{i}\) contractible. His next result is a partial generalization of the implication (c)\(\Rightarrow \)(a) in the previous result in which the local compactness requirement in (a) is slightly relaxed. He proves that if \(X\) is metrizable, \(E\subset X\) is a compact \(AR\) and \(X\) is locally compact at every point \(x\in E\) with one possible exception, then \(E\) is a closed retract of \(X.\) There are other results in the paper where the situation for \(E\) Lindelöf and locally compact or \(E\) Lindelöf and Čech complete is studied. Finally, the author proves the following theorem: Let \(Y\) be an abelian topological group. Suppose that \(E\) is a retract of a space \(X,\) and that there exists a perfect map \(h:X\rightarrow Y.\) Then there exists a perfect map \(g:X\rightarrow E\times Y\) such that \(g(x)=(x,0)\) for every \(x\in E.\)
scientific article; zbMATH DE number 1787538
scientific article