On Sierpiński sets, Hurewicz spaces and Hilgers functions
Witold Marciszewski, Roman Pol, Piotr Zakrzewski
TL;DR
The paper addresses how the Hurewicz property interacts with Sierpiński sets and product spaces, solving key open questions by constructing counterexamples and delineating separation properties. It develops and employs Hilgers functions and the $\mathfrak b$-scale set $\mathfrak B$ to build $\lambda'$-sets $H$ whose product with certain Sierpiński sets $S$ fails to be Hurewicz (and sometimes fails to be Menger) when $|S|\ge \mathfrak b$. It also introduces a strengthened Hurewicz notion via $\mathscr E$-separation, producing Gr$(f)$ with strong separation properties that contradict PMT and reveal limits of separation under $V=L$ and CH. Finally, it analyzes $C$-space products, showing CH/CH-like assumptions yield negative results for $S\times H$ being a $C$-space, while structured constructions yield $E$ with high powers that remain Hurewicz, yet fail to preserve $C$-space status in products. These results deepen understanding of how classical selection principles behave under products with pathological sets and highlight intricate interactions among Sierpiński sets, $\lambda'$-sets, and $C$-spaces.
Abstract
The Hurewicz property is a classical generalization of $σ$-compactness and Sierpiński sets (whose existence follows from CH) are standard examples of non-$σ$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban, that for each Sierpiński set S of cardinality at least $\mathfrak b$ there is a Hurewicz space H with $S\times H$ not Hurewicz. Some other questions in the literature concerning this topic are also answered.