On countably perfectly meager sets

We study a strengthening of the notion of a perfectly meager set. We say that that a subset A of a perfect Polish space X is countably perfectly meager in X if for every sequence (P_n) of perfect subsets of X, there is an F_sigma set F in X containing A and such that the intersection of F with P_n is relatively meager in P_n for each n.

We give various characterizations and examples of countably perfectly meager sets. We prove that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński.

The results come from a joint paper with Roman Pol available at http://arxiv.org/abs/2010.08812.