Applications of shrinkable covers

Smith, J. C.

doi:10.1090/S0002-9939-1979-0518525-6

Applications of shrinkable covers
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Proc. Amer. Math. Soc. 73 (1979), 379-387 Request permission

Abstract:

An open cover $\mathcal {G} = \{ {G_\alpha }:\alpha \in A\}$ of a topological space X is shrinkable if there exists a closed cover $\mathcal {F} = \{ {F_\alpha }:\alpha \in A\}$ of X such that ${F_\alpha } \subseteq {G_\alpha }$ for each $\alpha \in A$. In this paper the author determines conditions necessary for a variety of general covers to be shrinkable. In particular it is shown that the shrinkability of special types of covers provide characterizations for normal and countably paracompact, normal spaces. The types of covers investigated are, weak $\bar \theta$-covers, weak $\bar \theta$-covers, point countable covers, $\delta \theta$-covers and weak $\overline {\delta \theta }$-covers. Applications of these results are answers of unsolved problems and new results for irreducible spaces.

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