Abstract

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 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.