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COUNTABLY PERFECTLY MEAGER SETS

Part of: Set theory Spaces with richer structures

Published online by Cambridge University Press:  07 June 2021

ROMAN POL  and
PIOTR ZAKRZEWSKI
ROMAN POL
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF WARSAW, UL. BANACHA 2 02-097 WARSAW, POLANDE-mail:pol@mimuw.edu.plE-mail:piotrzak@mimuw.edu.pl
PIOTR ZAKRZEWSKI
Affiliation:
INSTITUTE OF MATHEMATICS UNIVERSITY OF WARSAW, UL. BANACHA 2 02-097 WARSAW, POLANDE-mail:pol@mimuw.edu.plE-mail:piotrzak@mimuw.edu.pl
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Abstract

We study a strengthening of the notion of a perfectly meager set. We say 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 Bartoszyński.

Keywords

perfectly meager setuniversally meager setperfectly meager set in the transitive senseλ′-setσ-ideal

MSC classification

Primary: 03E20: Other classical set theory (including functions, relations, and set algebra)
Secondary: 03E15: Descriptive set theory 54E52: Baire category, Baire spaces
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1214 - 1227
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Copyright
© Association for Symbolic Logic 2021

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