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Non-meager free sets and independent families
ID
Medini, Andrea
(
Author
),
ID
Repovš, Dušan
(
Author
),
ID
Zdomskyy, Lyubomyr
(
Author
)
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Abstract
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
Views:
1006
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432
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Title:
Proceedings of the American Mathematical Society
Shortened title:
Proc. Am. Math. Soc.
Publisher:
American Mathematical Society
ISSN:
0002-9939
COBISS.SI-ID:
2335236
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