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.

Products of Menger spaces in the Miller model

“…There are many results of this kind where CH is relaxed to an equality between cardinal characteristics, see, e.g., [3,16,23,28]. Surprisingly, there are also inequalities between cardinal characteristics which imply that the Menger property is not productive even for sets of reals, see [27]. The following theorem, which is the main result of our paper, shows that an additional set-theoretic assumption in all these results was indeed necessary.…”

Products of general Menger spaces

“…In the realm of hereditarily Lindelöf spaces, if b = d, then every productively Menger real space is productively Hurewicz [19,Theorem 4.8]. It is unknown whether, when b = d, these classes provably coincide for real spaces [19,Problem 6.9]. Proposition 3.14.…”

A coanalytic Menger group that is not $\sigma$-compact

“…Theorem 2.1 and [27, Theorem 6.2] imply that there is a coanalytic productively Menger but nonproductively Lindelöf set of reals, but d-concentrated sets satisfy the Menger property [39] and then any unbounded tower (under V=L) is not productively Menger by Theorem 4.8 in [34].…”

“…A space X is called productively P if X × Y has the property P for each space Y satisfying P . Productively P properties have been studied by many authors (see, e.g., [3,27,34]). …”