Non-meager free sets and independent families

Medini, Andrea (author), Repovš, Dušan (author), Zdomskyy, Lyubomyr (author)

Our main result is that, given a collection ▫$\mathcal{R}$▫ of meager relations on a Polish space ▫$X$▫ such that ▫$\vert\mathcal{R} \vert \leq \omega $▫, there exists a dense Baire subspace ▫$F$▫ of ▫$X$▫ (equivalently, a nowhere meager subset ▫$F$▫ of ▫$X$▫) such that ▫$F$▫ is ▫$R$▫-free for every ▫$R \in \mathcal{R}$▫. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on ▫$\omega$▫, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "▫$\vert \mathcal{R} \vert \leq \omega$▫" with "▫$\vert \mathcal{R} \vert < \mathfrak{c}$▫" and "Baire" with "completely Baire".

• Language: English
• Keywords: free set, meager relation, completely Baire, hereditarily Baire, independent family
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2017
• Number of pages: Str. 4061-4073
• Numbering: Vol. 145, no. 9
• PID: 20.500.12556/RUL-110517
• UDC: 515.122:510.3
• ISSN on article: 0002-9939
• DOI: 10.1090/proc/13513
• COBISS.SI-ID: 18047577
• Publication date in RUL: 16.09.2019

Record is a part of a journal

• Title: Proceedings of the American Mathematical Society
• Shortened title: Proc. Am. Math. Soc.
• Publisher: American Mathematical Society
• ISSN: 0002-9939
• COBISS.SI-ID: 2335236