Countably perfectly meager sets

TL;DR

The paper introduces and explores the concept of countably perfectly meager sets in perfect Polish spaces, providing characterizations, examples, and correcting previous assumptions about their relationship with universally meager sets.

Contribution

It defines countably perfectly meager sets, offers characterizations and examples, and clarifies their distinction from universally meager sets, correcting earlier misconceptions.

Findings

Not all universally meager sets are countably perfectly meager.

Provides new characterizations of countably perfectly meager sets.

Corrects previous results regarding the relationship between these classes.

Abstract

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 of perfect subsets $\{P_n: n \in {\mathbb N}\}$ of $X$, there exists an $F_\sigma$-set $F$ in $X$ such that $A \subseteq F$ and $F\cap P_n$ is 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 Bartoszy\'nski.