A space X is called s -point finite refinable ( d s -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 \omega$"> c f ( λ ) = λ > ω and S is a stationary subset of λ then S is not s -point finite refinable. Countably compact d s -point finite refinable spaces are compact. A space X is irreducible of order ω if and only if it is d s -point finite refinable. If X is a strongly collectionwise Hausdorff d s -point finite refinable space without isolated points then X is irreducible.