s-POINT FINITE REFINABLE SPACES

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) C ⊆ X, (i) for all nonempty V ∈ v, V ∩C ≠∅ and (ii) for all a ∈ C the set (v)_a = {V ∈ v : a ∈ V} 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 orderωif 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.