Concentrated sets and the Hurewicz property

TL;DR

This paper investigates the structure of $$-concentrated sets of reals with the Hurewicz property, showing under certain set-theoretic assumptions that such sets are Hurewicz and productively so, with distinctions in different models.

Contribution

It provides ZFC results on $$-concentrated sets with the Hurewicz property and explores their behavior under the semifilter trichotomy and in the Laver model.

Findings

$$-concentrated sets with Hurewicz property are characterized using semifilters.

Under semifilter trichotomy, these sets are Hurewicz and productively Hurewicz.

The behavior of Hurewicz $$-concentrated sets varies in different set-theoretic models.

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.