We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin’s ultrafilter on the Turing degrees.

We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ERs of a family which includes Polish actions of S∞, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T2, an equivalence relation called “the...

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...

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly ...

Wadge reducibility provides a rich and nice analysis of Borel sets in Polish zero dimensional spaces. However, outside this framework, reducibility by continuous functions was shown to be ill behaved in many important cases. We define a notion of reducibility for subsets of a second countable T0 topological space based on the notions of admissible representations and relatively continuous relat...

Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure μ on G such that μ(B+g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G...

Metric spaces, Borel and analytic sets, Baire property and measurability, dichotomies, equivalence relations. Possible topics: determinacy, group actions, rigidity, turbulence.

We show that Monadic Second Order Logic on ω-words extended with the unbounding quantifier (MSO+U) can define non-Borel sets. We conclude that there is no model of nondeterministic automata with a Borel acceptance condition which captures all of MSO+U. We also give an exact topological complexity of the classes of languages recognized by nondeterministic ωB-, ωSand ωBS-automata studied by Bojań...