TY - JOUR

T1 - A note on weak θ-refinability

AU - Bennett, H. R.

AU - Lutzer, D. J.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1972/5

Y1 - 1972/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0013546453&partnerID=8YFLogxK

U2 - 10.1016/0016-660X(72)90035-9

DO - 10.1016/0016-660X(72)90035-9

M3 - Article

AN - SCOPUS:0013546453

VL - 2

SP - 49

EP - 54

JO - General Topology and its Applications

JF - General Topology and its Applications

SN - 0016-660X

IS - 1

ER -