Potential Wadge classes

Abstract

Let $\mathbf{\Gamma }$ be a Borel class, or a Wadge class of Borel sets, and $2\le d\le \omega$ a cardinal. We study the Borel subsets of ${\mathbb{R}}^d$ that can be made $\mathbf{\Gamma }$ by refining the Polish topology on the real line. These sets are called potentially $\mathbf{\Gamma }$. We give a test to recognize potentially $\mathbf{\Gamma }$ sets.