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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)



On Borel sets belonging to every invariant ccc $ \sigma$-ideal on $ 2^{\mathbb{N}}$

Author: Piotr Zakrzewski
Journal: Proc. Amer. Math. Soc. 141 (2013), 1055-1065
MSC (2000): Primary 03E15, 03E05; Secondary 54H05, 28C10
Published electronically: August 3, 2012
MathSciNet review: 3003696
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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-Rosłanowski-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

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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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