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Proceedings of the American Mathematical Society

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



Point-countability and compactness

Authors: H. H. Wicke and J. M. Worrell
Journal: Proc. Amer. Math. Soc. 55 (1976), 427-431
MathSciNet review: 0400166
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Abstract: We prove that if a countably compact space $ X$ has an open cover $ \mathcal{U} = \bigcup {\{ {\mathcal{V}_n}:n < \omega \} } $ such that each $ x \in X$ is in at least one but not more than countably many elements of some $ {\mathcal{V}_n}$, then some finite subcollection of $ \mathcal{U}$ covers $ X$. We apply the theorem in proving several metrization theorems for countably compact spaces and discuss consequences of weak $ \delta \theta $-refinability, a concept implicit in the statement of the theorem.

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Keywords: (Countably) compact, (weakly) $ \theta $-refinable, (weakly) $ \delta \theta $-refinable, $ p$-space, quasi-developable space, $ \theta $-base, $ {G_\delta }$-diagonal, primitive base, primitive structures, diagonal a primitive set of interior condensation
Article copyright: © Copyright 1976 American Mathematical Society