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Proceedings of the American Mathematical Society

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



Properties of weak $ \bar \theta $-refinable spaces

Author: J. C. Smith
Journal: Proc. Amer. Math. Soc. 53 (1975), 511-517
MSC: Primary 54D20
MathSciNet review: 0380731
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Abstract: A space $ X$ is called weak $ \overline \theta $-refinable if every open cover of $ X$ has a refinement $ \bigcup\nolimits_{i = 1}^\infty {{\mathcal{G}_i}} $ satisfying (1) $ {\mathcal{G}_i} = \{ {G_\alpha }:\alpha \epsilon {A_i}\} $ is an open collection for each $ i$, (2) each $ x\epsilon X$ has finite positive order with respect to some $ {\mathcal{G}_i}$, (3) the open cover $ \{ {G_i} = \bigcup {[{G_\alpha }:\alpha \epsilon } {A_i}]\} _{i = 1}^\infty $ is point finite.

In this paper the author shows that the above property lies between the properties of $ \theta $-refinable and weak $ \theta $-refinable. The main result is the fact that if $ X$ is countably metacompact and satisfies property $ (\delta )$, every weak $ \overline \theta $-cover of $ X$ has a countable subcover. Results concerning paracompactness, metacompactness and the star-finite property are also obtained.

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Keywords: $ \theta $-refinable, weak $ \overline \theta $-refinable, countably compact, compact, paracompact, metacompact, property $ (\delta )$, discretely expandable, Lindelöf, starfinite property, quasi-developable
Article copyright: © Copyright 1975 American Mathematical Society