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 $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.

💡 Research Summary

The paper investigates a refined notion of Borel complexity for subsets of the plane, introducing the class of “potentially (\Pi^0_\xi)” sets. For a non‑zero countable ordinal (\xi), a set (A\subseteq\mathbb R^2) is called potentially (\Pi^0_\xi) if there exists a Polish topology (\tau) on the real line such that, when (\mathbb R^2) is equipped with the product topology (\tau\times\tau), the set (A) becomes a (\Pi^0_\xi) set in the Borel hierarchy. This definition captures the idea that by suitably refining the underlying topology one can lower the descriptive‑set‑theoretic complexity of a set without changing its underlying point‑set.

The central contribution is a Hurewicz‑like test that characterizes exactly when a Borel subset of the plane fails to be potentially (\Pi^0_\xi). The test is a higher‑level analogue of the classical Hurewicz theorem, which characterizes when an analytic set can be made (G_\delta) by a continuous reduction. Here the authors construct, for each countable (\xi>0), a universal (\Sigma^1_1) set (U_\xi\subseteq\mathbb R^2) that is complete for the class of sets strictly more complex than (\Pi^0_\xi). They then prove:

*If there exists a continuous map (f:\mathbb R^2\to\mathbb R^2) such that (f^{-1}