On 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 Xis countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology ton X, giving the original Borel structure of X, A is covered by an F-sigma-set Fin 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 Fin X with mu(F) = 0). We prove that if 2(N0) <= N-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)). (c) 2023 Elsevier B.V. All rights reserved.