International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 2, Pages 367-375

s-point finite refinable spaces

Sheldon W. Davis, Elise M. Grabner, and Gray C. Grabner

229 Vincent Science Hall, Slippery Rock, PA 16057-1326, USA

Received 4 October 1996; Revised 28 October 1996

Copyright © 1999 Sheldon W. Davis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A space X is called s-point finite refinable (ds-point finite refinable) provided every open cover 𝒰 of X has an open refinement 𝒱 such that, for some (closed discrete) CX,

(i) for all nonempty V𝒱,VC and

(ii) for all aC the set (𝒱)a={V𝒱:aV} is finite.

In this paper we distinguish these spaces, study their basic properties and raise several interesting questions. If λ is an ordinal with cf(λ)=λ>ω and S is a stationary subset of λ then S is not s-point finite refinable. Countably compact ds-point finite refinable spaces are compact. A space X is irreducible of order ω if and only if it is ds-point finite refinable. If X is a strongly collectionwise Hausdorff ds-point finite refinable space without isolated points then X is irreducible.