On 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 τ 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 τ (respectively, for every finite, non-atomic, Borel measure μ on X, A is covered by an $F_{\sigma }$-set F in X with $\mu \left(F\right)=0$).

We prove that if $2^{{\aleph }_0}\le {\aleph }_2$, then there exists a universally meager set in $2^N$ which is not countably perfectly meager in $2^N$ (respectively, a universally null set in $2^N$ which is not countably perfectly null in $2^N$).