Unbounded towers and products

We investigate products of sets of reals with combinatorial covering properties. A topological space satisfies S₁(Γ,Γ) if for each sequence of point-cofinite open covers of the space, one can pick one element from each cover and obtain a point-cofinite cover of the space. We prove that, if there is an unbounded tower, then there is a nontrivial set of reals satisfying S₁(Γ,Γ) in all finite powers. In contrast to earlier results, our proof does not require any additional set-theoretic assumptions. A topological space satisfies (ΩΓ) (also known as Gerlits–Nagy's property γ) if every open cover of the space such that each finite subset of the space is contained in a member of the cover, contains a point-cofinite cover of the space. We investigate products of sets satisfying (ΩΓ) and their relations to other classic combinatorial covering properties. We show that finite products of sets with a certain combinatorial structure satisfy (ΩΓ) and give necessary and sufficient conditions when these sets are productively (ΩΓ).