Products of Menger spaces: a combinatorial approach

TL;DR

This paper introduces a purely combinatorial method to construct Menger subsets of the real line with non-Menger products, explores their properties under set-theoretic hypotheses, and examines parameterized versions of Menger's property.

Contribution

It provides a new combinatorial construction of Menger sets with non-Menger products and analyzes their properties under various set-theoretic assumptions.

Findings

Constructed Menger sets with non-Menger products using combinatorial methods.

Under CH, every productively Menger set is productively Hurewicz.

Ultrafilter versions of Menger's property are intermediate between Menger and Hurewicz.

Abstract

We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.