Potential Wadge classes

TL;DR

This paper investigates the classification of Borel subsets of Euclidean spaces that can be transformed into a specific Borel class through topology refinement, providing a test to identify such sets.

Contribution

It introduces a method to determine when Borel subsets of Euclidean spaces are potentially within a given Borel class after topology refinement.

Findings

Developed a test for recognizing potentially mma sets.

Characterized the conditions under which Borel sets can be refined into a specific class.

Extended the understanding of Borel set classifications in Euclidean spaces.

Abstract

Let $\bf\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leq\omega$ a cardinal. We study the Borel subsets of ${\mathbb R}^d$ that can be made $\bf\Gamma$ by refining the Polish topology on the real line. These sets are called potentially $\bf\Gamma$. We give a test to recognize potentially $\bf\Gamma$ sets.