Products and Selection Principles

We study when the product of separable metric spaces has the selective screenability property, the Menger property, or the Rothberger property. Our results imply The product of a Lusin set and (1) a Sierpinski set always has the Menger property (Corollary 12); (2) a γ-set always has the Rothberger property (Corollary 13). All topological spaces considered in this paper are assumed to be separable metric spaces. Some cited theorems apply to more general spaces, as the reader could verify by consulting the appropriate references. Let A and B be given families of collections of subsets of some set S. The following selection principle was introduced in [2]: Sc(A,B): For each sequence (Om : m < ∞) of elements of A there is a sequence (Tm : m < ∞) with each Tm a pairwise disjoint family refining Om, and ∪{Tm : m < ∞} ∈ B. The special case of Sc(A,B) when A = B = O, the collection of open covers of a topological space, was introduced in [1] by Addis and Gresham. This property is related to the theory of covering dimension. One of the interesting questions about it, due to D. Rohm, asks when the product of two spaces with Sc(O,O) again has this property. The best known result regarding this question is due to [10] and [16]: Theorem 1 (Hattori-Yamada, Rohm). Let X and Y be topological spaces satisfying Sc(O,O). If X is σ-compact, then X×Y has the property Sc(O,O). The special case when A = T , the collection of two-element open covers of a space, and B = O, was introduced by Aleksandroff and is known as weak infinite dimensionality. This is not the original definition of weak infinite dimensionality, but Rohm has shown that Sc(T ,O) is equivalent to weak infinite dimensionality. It is unknown whether Sc(T ,O) is equivalent to Sc(O,O). Various alternatives of the hypothesis that one of the spaces be σ-compact have been investigated in attempts to generalize Theorem 1. To explain these we now recall two more selection principles from [19]: Sfin(A,B): For each sequence (Om : m < ∞) of elements of A there is a sequence (Tm : m < ∞) with each Tm a finite subset of Om, and ∪{Tm : m < ∞} ∈ B. The Menger property is Sfin(A,B) when A = B = O. Hurewicz introduced it in [11].