Selective Screenability and the Hurewicz Property

We characterize the Hurewicz covering property in metrizable spaces in terms of properties of the metrics of the space Theorem 1. Then we show that a weak version of selective screenability, when combined with the Hurewicz property, implies selective screenability Theorem 4. 1. Definitions and notation Let X be topological space. We shall use the notation O to denote the collection of all open covers of X. The symbol Ofin denotes the collection of all finite open covers of X. The relevant selection principle for this paper is as follows: Let S be an infinite set, and let A and B be collections of families of subsets of X. Then the selection principle Sc(A,B), introduced in [3], is defined as follows: For each sequence (An : n < ∞) of elements of the family A there exists a sequence (Bn : n < ∞) such that for each n Bn is a pairwise disjoint family refining An, and ⋃ n<∞Bn is a member of the family B. The instance Sc(O,O) of this selection property was introduced by Addis and Gresham in [1], where it was called property C. A topological space X has the Hurewicz property [8] if there is for each sequence (Un : n < ∞) of open covers of X a sequence (Vn : n < ∞) such that for each n, Vn is a finite subset of Un, and each element of X is in all but finitely many of the sets ∪Vn. The metrizable space X is said to be Haver [6] with respect to a metric d if there is for each sequence (ǫn : n < ∞) of positive reals a sequence (Vn : n < ∞) where each Vn is a pairwise disjoint family of open sets, each of d-diameter less than ǫn, such that ⋃ n<∞ Vn is a cover of X. A metric space X is totally bounded if there is for each ǫ > 0 a finite set F ⊂ X such that X ⊆ ⋃ f∈F B(f, ǫ). A metric space is σ-totally bounded if it is a union of countably many subsets, each totally bounded.