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

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.