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

HOW CAN WE RECOGNIZE POTENTIALLY ${\bf\Pi}^{0}_{\xi}$ SUBSETS OF THE PLANE?