σ-fields of bad Borel sets

On Borel Mappings and Σ-ideals Generated by Closed Sets

We obtain some results about Borel maps with meager fibers on completely metrizable separable spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy. We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable s...

Borel homomorphisms of smooth σ-ideals

Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007,...

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...

On Borel Sets Belonging to Every Invariant Ccc Σ-ideal on 2n

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 dis...

A Course on Borel Sets