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.

Automated summary

This article studies a strengthening of the concepts of universally meager and universally null sets, introducing the notions of countably perfectly meager and countably perfectly null sets. The article proves the existence of universally meager and universally null sets that do not satisfy the properties of being countably perfectly meager and countably perfectly null, respectively, under certain conditions.