On Borel sets belonging to every invariant ccc $\sigma$-ideal on $2^{\mathbb {N}}$
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Abstract:
Let $I_{ccc}$ be the $\sigma$-ideal of subsets of the Cantor group $2^{\mathbb {N}}$ generated by Borel sets which belong to every translation-invariant $\sigma$-ideal on $2^{\mathbb {N}}$ satisfying the countable chain condition (ccc). We prove that $I_{ccc}$ strongly violates ccc. This generalizes a theorem of Balcerzak-RosĆanowski-Shelah stating the same for the $\sigma$-ideal on $2^{\mathbb {N}}$ generated by Borel sets $B\subseteq 2^{\mathbb {N}}$ which have perfectly many pairwise disjoint translates. We show that the last condition does not follow from $B\in I_{ccc}$ even if $B$ is assumed to be compact. Various other conditions which for a Borel set $B$ imply that $B\in I_{ccc}$ are also studied. As a consequence we prove in particular that:
If $A_n$ are Borel sets, $n\in \mathbb N$, and $2^{\mathbb {N}}=\bigcup _n A_n$, then there is $n_0$ such that every perfect set $P\subseteq 2^{\mathbb {N}}$ has a perfect subset $Q$, a translate of which is contained in $A_{n_0}$.
CH is equivalent to the statement that $2^{\mathbb {N}}$ can be partitioned into $\aleph _1$ many disjoint translates of a closed set.
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Additional Information
- Piotr Zakrzewski
- Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
- MR Author ID: 239503
- Email: piotrzak@mimuw.edu.pl
- Received by editor(s): May 11, 2011
- Received by editor(s) in revised form: August 4, 2011
- Published electronically: August 3, 2012
- Additional Notes: This research was partially supported by MNiSW Grant Nr N N201 543638.
- Communicated by: Julia Knight
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1055-1065
- MSC (2000): Primary 03E15, 03E05; Secondary 54H05, 28C10
- DOI: https://doi.org/10.1090/S0002-9939-2012-11384-X
- MathSciNet review: 3003696