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

TL;DR

This paper investigates the classification of certain Borel subsets of the plane, called potentially ${f\Pi}^0_\xi$, by refining the topology, and provides a test to identify these sets.

Contribution

It introduces a Hurewicz-like test for recognizing potentially ${f\Pi}^0_\xi$ sets in the plane, advancing the understanding of their topological complexity.

Findings

Provides a criterion to identify potentially ${f\Pi}^0_\xi$ sets

Extends the theory of Borel hierarchy in the plane

Offers tools for topological refinement analysis

Abstract

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