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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Applications of shrinkable covers


Author: J. C. Smith
Journal: Proc. Amer. Math. Soc. 73 (1979), 379-387
MSC: Primary 54D15
MathSciNet review: 518525
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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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DOI: https://doi.org/10.1090/S0002-9939-1979-0518525-6
Keywords: Shrinkable, normal, countably paracompact, $ \theta $-refinable, weakly $ \theta $-refinable, weakly $ \theta $-refinable, point countable cover, $ \delta \theta $-refinable, weak $ \overline {\delta \theta } $-cover, irreducible, closure-preserving, sequential, countable tightness
Article copyright: © Copyright 1979 American Mathematical Society