Potential Wadge Classes

Let $fGamma$ 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$fGamma$ if there is a finer Polish topology on $mathbb R$ such that $B$ is in $fGamma$ when ${mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $fGamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.