Hurewicz-like tests for Borel subsets of the plane

Let $\xi \geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made $\boldsymbol \Pi ^{0}_{\xi }$ by refining the Polish topology on the real line. These sets are called potentially $\boldsymbol \Pi ^{0}_{\xi }$. We give a Hurewicz-like test to recognize potentially $\boldsymbol \Pi ^{0}_{\xi }$ sets.