Potential Wadge Classes

Let $/bf/Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2/!/leq/! d/!/leq/!/omega$ be a cardinal. A Borel subset $B$ of ${/mathbb R}^d$ is potentially in $/bf/Gamma$ if there is a finer Polish topology on $/mathbb R$ such that $B$ is in $/bf/Gamma$ when ${/mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $/bf/Gamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.