Products of Menger spaces: a combinatorial approach

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.

arXiv commentMinor revision

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Cited by in corpus (5)

• Products of Menger spaces in the Miller model
• On localization of the Menger property
• Null sets and combinatorial covering properties
• On the interplay between productively Menger and productively Hurewicz spaces in models of $\mathfrak{b}=\mathfrak{d}$
• Omission of Intervals: Deducing covering properties of subsets of the real line from their combinatorial structure