How can we recognize potentially ${\mathbf{\Pi }}_{\xi }^0$ subsets of the plane?

Abstract

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