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

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- by J. C. Smith
- Proc. Amer. Math. Soc.
**53**(1975), 511-517 - DOI: https://doi.org/10.1090/S0002-9939-1975-0380731-8
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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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## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**53**(1975), 511-517 - MSC: Primary 54D20
- DOI: https://doi.org/10.1090/S0002-9939-1975-0380731-8
- MathSciNet review: 0380731