Products of general Menger spaces

TL;DR

This paper investigates how Menger's covering property behaves under products of topological spaces, extending existing methods and establishing implications under the Continuum Hypothesis.

Contribution

It extends the projection method to the Michael topology and proves new implications between productively Lindelöf, Menger, and Hurewicz spaces under CH.

Findings

Under CH, every productively Lindelöf space is productively Menger.

Under CH, every productively Menger space is productively Hurewicz.

Implications between these properties are not reversible.

Abstract

We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindel\"of space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.