Let $X$ be a set of reals and $\kappa$ be an uncountable cardinal number. The set $X$ is $\kappa$-concentrated, if $X$ has size at least $\kappa$ and contains a countable set $D$ such that each closed subset of $X$, disjoint with $D$, has size smaller than $\kappa$. Various forms of concentrated sets play an important role in the study of combinatorial covering properties such as Rothberger’s, Hurewicz’s, and Menger’s properties. We investigate the behavior of such sets in different models of set theory.

This is a joint work with Michał Pawlikowski and Lyubomyr Zdomskyy.

The research was funded by the Polish National Science Center and Austrian Science Fund; Grant: Weave-UNISONO, Project: Set-theoretic aspects of topological selections

2021/03/Y/ST1/00122.