Productively Scheepers spaces and their relatives

TL;DR

This paper explores the relationships between productively Scheepers and Hurewicz spaces under certain set-theoretic assumptions, using combinatorial methods and ultrafilter parametrizations.

Contribution

It establishes conditions under which productively Scheepers spaces are also productively Hurewicz, extending results to all topological spaces assuming =.

Findings

Under =, productively Scheepers spaces are productively Hurewicz in hereditarily Lindelf6f spaces.

The result extends to all topological spaces assuming =.

Scheepers property is equivalent to a Menger property parametrized by ultrafilters if near coherence of filters holds.

Abstract

We prove that assuming $\mathfrak{b}=\mathfrak{d}$, in the class of hereditarily Lindel\"of spaces, each productively Scheepers space is productively Hurewicz. The above statement remains true in the class of all general topological spaces assuming that $\mathfrak{d}=\aleph_1$. To this end we use combinatorial methods and the Menger covering property parametrized by ultrafilters. We also show that if near coherence of filters holds, then the Scheepers property is equivalent to a Menger property parametrized by any ultrafilter.