Unbounded towers and products

We investigate products of sets reals with combinatorial covering properties. A topological space satisfies S 1 ( ? , ) if for each sequence point-cofinite open covers the space, one can pick element from cover and obtain a space. prove that, there is an unbounded tower, then nontrivial set satisfying in all finite powers. In contrast to earlier results, our proof does not require any additional set-theoretic assumptions. ? (also known as Gerlits–Nagy's property ? every such that subset contained member cover, contains their relations other classic show certain structure satisfy give necessary sufficient conditions when these are productively .