Properties of weak 𝜃̄-refinable spaces

Abstract

A space X X is called weak θ ¯ \overline \theta -refinable if every open cover of X X has a refinement ⋃ i = 1 ∞ G i \bigcup \nolimits _{i = 1}^\infty {{\mathcal {G}_i}} satisfying (1) G i = { G α : α ϵ A i } {\mathcal {G}_i} = \{ {G_\alpha }:\alpha \epsilon {A_i}\} is an open collection for each i i , (2) each x ϵ X x\epsilon X has finite positive order with respect to some G i {\mathcal {G}_i} , (3) the open cover { G i = ⋃ [ G α : α ϵ A i ] } i = 1 ∞ \{ {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 θ \theta -refinable and weak θ \theta -refinable. The main result is the fact that if X X is countably metacompact and satisfies property ( δ ) (\delta ) , every weak θ ¯ \overline \theta -cover of X X has a countable subcover. Results concerning paracompactness, metacompactness and the star-finite property are also obtained.