Concentrated sets and the Hurewicz property
Valentin Haberl, Piotr Szewczak, Lyubomyr Zdomskyy
TL;DR
This work investigates how the Hurewicz, Menger, and Rothberger covering properties interact with the structure of ${\mathfrak{b}}$-concentrated sets of reals, using semifilter methods to unify ZFC results with model-dependent phenomena. It introduces and exploits semifilters, ${S}$-scales, and notions like meager-unboundedness and bi-${\mathfrak{d}}$-unboundedness to characterize when products preserve Hurewicz and Menger properties. Under the semifilter trichotomy, every ${\mathfrak{b}}$-concentrated set becomes Hurewicz and productively Hurewicz, while the Laver model yields a distinct combinatorial picture, with Hurewicz equating to a $(\dagger)$-type property. The paper develops a broad, filter- and scale-based framework for productive versions of Menger-type properties, proving both positive and negative productivity results and posing extensive open questions for further exploration in set-theoretic topology. The results illuminate how set-theoretic hypotheses shape the landscape of productiveness for Hurewicz, Menger, and related properties in real-analytic spaces.
Abstract
A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.