Relative collectionwise normality. (English) Zbl 1066.54025

The authors study relative collectionwise normality and prove relative versions of Bing’s theorem asserting that every paracompact space is collectionwise normal and of Michael-Nagami’s theorem asserting that every metacompact collectionwise normal space is paracompact. Let \(Y\) be a subspace of a topological space \(X\). They define \(Y\) to be (strongly) collectionwise normal in \(X\) if for every discrete collection \({\mathcal F}\) of closed sets in \(X\), there exists a collection \(\{U(F):F\in{\mathcal F}\}\) of open sets in \(X\) which is discrete with respect to \(Y\) and such that \(F\cap Y\subseteq U(F)\subseteq X\setminus\bigcup({\mathcal F}\setminus\{F\})\) (\(F\cap Y\subseteq U(F)\subseteq X\setminus\overline{\bigcup({\mathcal F}\setminus\{F\})}\)) for each \(F\in{\mathcal F}\). It is proved that (1) the subspace \(Y\) is collectionwise normal in \(X\) provided that \(Y\) is strongly regular in \(X\) and paracompact in \(X\); and (2) \(Y\) is paracompact in \(X\) provided that \(Y\) is strongly regular in \(X\), metacompact in \(X\) and strongly collectionwise normal in \(X\). Several related results and examples which show the limits of the results are also given.


54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B05 Subspaces in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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