Hurewicz-like tests for Borel subsets of the plane

Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\text{bormxi}$ by refining the Polish topology on the real line. These sets are called potentially $\text{bormxi}$. We give a Hurewicz-like test to recognize potentially $\text{bormxi}$ sets.