Perfectly meager sets in the transitive sense and the Hurewicz property

We work in the Cantor space with the usual group operation +. A set X  is perfectly meager in the transitive sense if for any perfect set P there is an F-sigma set F containing X such that for every point t the intersection of t+F and P is meager in the relative topology of P. A set X is Hurewicz if for any sequence of increasing open covers of X one can select one set from each cover such that the chosen sets formulate a gamma-cover of X, i.e., an infinite cover such that each point from X belongs to all but finitely many sets from the cover. Nowik proved that each Hurewicz set which cannot be mapped continuously onto the Cantor set is perfectly meager in the transitive sense. We answer a question of Nowik and Tsaban, whether of the same assertion holds for each Hurewicz set with no copy of the Cantor set inside. We solve this problem, under CH, in the negative. 

 

This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy. A preprint of a paper is available here: https://arxiv.org/pdf/2406.12609.

 

The research was funded by the National Science Centre, Poland  and the Austrian Science Found under the Weave-UNISONO call in the Weave programme, project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122