We show that if F is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on P(ωω) induced by F turns out to look like the Wadge hierarchy (which is the special case where F is the set of continuous functions).

An infinite binary word can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Σ 2 over the Cantor discontinuum. Note that the last coincides with the Borel complexit...

In partial answer to a question posed by Arnie Miller [5] and X. Caicedo [2] we obtain sufficient conditions for an Lω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: If Vaught’s conjecture holds, then every Lω1,ω theory in a countable language has an independent axiomatization; every intersection of a family of Borel sets can be formed as the inters...

We prove in this paper that there exists some infinitary rational relations which are Σ03-complete Borel sets and some others which are Π03-complete. These results give additional answers to questions of Simonnet [Sim92] and of Lescow and Thomas [Tho90] [LT94].

If A is a Σξ set and An (n < ω) are Borel sets then we call {An : n < ω} a presentation of A if A = S n<ω An and An (n < ω) have lower Borel class than A has. We show that for 2 ≤ ξ < ω1 it is not possible to assign a presentation to Σξ sets in a monotone way, i.e., it is not possible to define functions fn : Σ 0 ξ → Πξ (n < ω) such that for every A ∈ Σξ we have A = S n<ω fn(A) and A,A ′ ∈ Σξ, ...

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Froĺık for characterizing Čechcomplete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp(X); however the case of absolut...