Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E15, 03E05, 54H05, 28C10

Retrieve articles in all journals with MSC (2000): 03E15, 03E05, 54H05, 28C10


Additional Information

Piotr Zakrzewski
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Email: piotrzak@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11384-X
PII: S 0002-9939(2012)11384-X
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.