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 continuously reducible) to $B$ if there is a continuous function $f: X \to Y$ 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?