Abstract
We study structural properties of the collection of all σ-ideals in the σ-algebra of Borel subsets of the Cantor group \(2^{\mathbb{N}}\), especially those which satisfy the countable chain condition (ccc) and are translation invariant. We prove that the latter collection contains an uncountable family of pairwise orthogonal members and, as a consequence, a strictly decreasing sequence of length ω 1.
We also make some observations related to the σ-ideal I ccc on \(2^{\mathbb{N}}\), consisting of all Borel sets which belong to every translation invariant ccc σ-ideal on \(2^{\mathbb{N}}\). In particular, improving earlier results of Recław, Kraszewski and Komjáth, we show that:
-
every subset of \(2^{\mathbb{N}}\) of cardinality less than
can be covered by a set from I
ccc, -
there exists a set C∈I ccc such that every countable subset Y of \(2^{\mathbb{N}}\) is contained in a translate of C.
can be covered by a set from I
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Zakrzewski, P. On invariant ccc σ-ideals on \(2^{\mathbb{N}}\) . Acta Math. Hungar. 143, 367–377 (2014). https://doi.org/10.1007/s10474-013-0388-7
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DOI: https://doi.org/10.1007/s10474-013-0388-7