Hurewicz-like tests for Borel subsets of the plane

TL;DR

This paper introduces a Hurewicz-like test to identify Borel subsets of the plane that can be transformed into a specific Borel class through topology refinement, advancing understanding of their structural properties.

Contribution

It provides a novel test for recognizing potentially Borel sets of a certain class in the plane, extending classical descriptive set theory techniques.

Findings

Developed a Hurewicz-like test for potentially $ormxi$ sets.

Characterized when Borel subsets can be refined to a specific Borel class.

Enhanced tools for analyzing the complexity of Borel sets in the plane.

Abstract

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.