On Borel Sets Belonging to Every Invariant Ccc Σ-ideal on 2n
نویسنده
چکیده
Let Iccc be the σ-ideal of subsets of the Cantor group 2N generated by Borel sets which belong to every translation invariant σ-ideal on 2N satisfying the countable chain condition (ccc). We prove that Iccc strongly violates ccc. This generalizes a theorem of Balcerzak-Rosłanowski-Shelah stating the same for the σ-ideal on 2N generated by Borel sets B ⊆ 2N which have perfectly many pairwise disjoint translates. We show that the last condition does not follow from B ∈ Iccc even if B is assumed to be compact. Various other conditions which for a Borel B imply that B ∈ Iccc are also studied. As a consequence we prove in particular that: • If An are Borel sets, n ∈ N, and 2N = ⋃ n An, then there is n0 such that every perfect set P ⊆ 2N has a perfect subset Q a translate of which is contained in An0 . • CH is equivalent to the statement that 2N can be partitioned into א1 many disjoint translates of a closed set.
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