A note on weak θ-refinability

✍ Scribed by H.R. Bennett; D.J. Lutzer

• Publisher: Elsevier Science
• Year: 1972
• Weight: 880 KB
• Volume: 2
• Category: Article
• ISSN: 0016-660X
• DOI: 10.1016/0016-660x(72)90035-9

✦ Synopsis

A space is weakly b-refinable if every open cover u of X has an open refinement v = UCg/(n):n > 1) such that given XE X, one of the collections v(n) has finite, positive order at x. Several equ +!oct properkies of a space are given and ark used to prove that : (a) if X is weakly d-refinable and has closed sets G, then X is subparacompact ; (b) any quasi-developable space. (in rhe sense of Bennett) is weakly &refinable; (c) a space is quasi-devElopable if and only if it has a &base,'(d) a linearly ordered topological space is F aracompact if an Iy &refinable. Examples are given which show that weak &re:finabil.ity is strictly weaker than the notion of &-refinability introduced by Worrell and Wicke.