Countably perfectly meager and countably perfectly null sets

Abstract

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager (respectively, countably perfectly null) in $X$, if for every perfect Polish topology $\tau$ on $X$, giving the original Borel structure of $X$, $A$ is covered by an $F_\sigma$-set $F$ in $X$ with the original Polish topology such that $F$ is meager with respect to $\tau$ (respectively, for every finite, non-atomic, Borel measure $\mu$ on $X$, $A$ is covered by an $F_\sigma$-set $F$ in $X$ with $\mu(F)=0$). We prove that if $2^{\aleph_0}\leq \aleph_2$, then there exists a universally meager set in $2^{\mathbb N}$ which is not countably perfectly meager in $2^{\mathbb N}$ (respectively, a universally null set in $2^{\mathbb N}$ which is not countably perfectly null in $2^{\mathbb N}$).