# Logic Seminar

*,*University of Turin

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Let X and Y be two topological spaces. Given a subset A of X and a subset B of Y we say that A is Wadge-reducible (or sometimes continuously reducible) to B if there is a continuous function f from X to Y that satisfies f^{-1}(B)=A. Wadge reducibility is a particularly nice quasi-order on subsets of Polish zero-dimensional spaces: Wadge's Lemma guarantees indeed that its antichains are of size at most two, while Martin and Monk have proven that it is well-founded. This gives an ordinal ranking to every equivalence class for Wadge reducibility, thus generating various questions. I will talk about two of these questions.

First, given a Wadge equivalence class, can we build it using classes of lower ordinal rank, and how?

Second, given any Polish zero-dimensional space X, can we decide if there is an antichain of two classes or just one class of some specific ordinal rank of the Wadge quasi-order of X? For which ranks?