English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let Iccc be the σ-ideal of subsets of the Cantor group 2 generated by Borel sets which belong to every translation-invariant σ-ideal on 2 satisfying the countable chain condition (ccc). We prove that Iccc strongly violates ccc. This generalizes a theorem of Balcerzak-Ros lanowski-Shelah stating the same for the σ-ideal on 2 generated by Borel sets B ⊆ 2 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 set B imply that B ∈ Iccc are also studied. As a consequence we prove in particular that: • If An are Borel sets, n ∈ N, and 2 = ⋃ n An, then there is n0 such that every perfect set P ⊆ 2 has a perfect subset Q, a translate of which is contained in An0 . • CH is equivalent to the statement that 2 can be partitioned into א1 many disjoint translates of a closed set.

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