On a σ-ideal of compact sets

On a σ-ideal of compact sets

We recall from [10] a Gδ σ-ideal of compact subsets of 2 ω and prove that it is not Tukey reducible to the ideal I1/n = {H ⊆ ω : ∑ h∈H 1/h <∞}. This result answers a question of S. Solecki and S. Todorčević in the negative.

The σ-ideal generated by H-sets∗

It is consistent with the axioms of set theory that the circle T can be covered by א1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it. In the proof of this theorem, the descriptive set theoretic phenomenon of overspill appears, and it is reformulated as a natural forcing preservation principle that persists through the operation of countable support ...

On Schmidt’s σ-ideal

A Gδ Ideal of Compact Sets Strictly above the Nowhere Dense Ideal in the Tukey Order

We prove that there is a Gδ σ-ideal of compact sets which is strictly above NWD in the Tukey order. Here NWD is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [4].

A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then f−1(y) is a Kσ set for all except countably many y ∈M , that M is also Luzin, and that the Borel classes of the sets f(F ), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the cla...