A note on weak θ-refinability

H. R. Bennett, D. J. Lutzer

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Abstract

A space is weakly θ-refinable if every open cover U of X has an open refinement V = ∪{V(n): n≥ 1} such that given xε{lunate}X, one of the collections V(n) has finite, positive order at x. Several equivalent properties of a space are given and are used to prove that: (a) if X is weakly θ-refinable and has closed sets Gδ then X is subparacompact; (b) any quasi-developable space (in the sense of Bennett) is weekly θ-refinable; (c) a space is quasi-developable if and only if it has a θ-base, (d) a linearly ordered topological space is paracompact if and only if it is weakly θ-refinable. Examples are given which show that weak θ-refinability is strictly weaker than the notion of θ-refinability introduced by Worrell and Wicke.

Original languageEnglish
Pages (from-to)49-54
Number of pages6
JournalGeneral Topology and its Applications
Volume2
Issue number1
DOIs
StatePublished - May 1972

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