Selective covering properties of product spaces

TL;DR

This paper investigates how selective covering properties like Menger, Hurewicz, and Rothberger are preserved under products with certain sets of reals, introducing new methods and extending previous results in set-theoretic topology.

Contribution

The authors develop new techniques, including the projection method, to analyze the preservation of covering properties under products with concentrated sets, and establish several consistency results.

Findings

Product of concentrated space with Hurewicz space satisfies Hurewicz property.

Under Semifilter Trichotomy, concentrated spaces are productively Menger and Rothberger.

Scale sets are productively Hurewicz, Menger, Scheepers, and Gerlits–Nagy.

Abstract

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpi\'nski set $S$ there is a Luzin set $L$ such that $L\x S$ can be mapped onto the real line by a Borel function. \item Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. \item Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits--Nagy. \item Assuming $\fd=\aleph_1$, every productively Lindel\"of space is productively Hurewicz, productively Menger, and productively Scheepers. \ee A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than $\add(\cN)$, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy with $\add(\cN)<\cov(\cM)$. Our results improve upon and unify a number of results, established earlier by many authors.