Potential Wadge classes

Abstract

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