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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.

##### MSC:
 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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