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Applications of Shrinkable Covers

J. C. Smith
Proceedings of the American Mathematical Society
Vol. 73, No. 3 (Mar., 1979), pp. 379-387
DOI: 10.2307/2042368
Stable URL: http://www.jstor.org/stable/2042368
Page Count: 9
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Applications of Shrinkable Covers
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Abstract

An open cover $\mathscr{G} = \{G_\alpha: \alpha \in A\}$ of a topological space $X$ is shrinkable if there exists a closed cover $\mathscr{F} = \{ F_\alpha: \alpha \in A\}$ 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 $\theta$-covers, weak $\overline \theta$-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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