Countably perfectly meager and countably perfectly null sets

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^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). The results come from a joint paper with Piotr Zakrzewski (T. Weiss, P. Zakrzewski, "Countably perfectly meager and countably perfectly null sets", submitted).