Baire reductions and good Borel reducibilities

TL;DR

This paper studies the structure of Borel reducibilities, proving a dichotomy theorem for degrees induced by good Borel reducibilities, extending previous results to include a broader class of reductions like Baire class functions.

Contribution

It introduces a dichotomy theorem for degree-structures under good Borel reducibilities, expanding the scope to include more notions of reduction such as Baire class functions.

Findings

Established a dichotomy theorem for Borel reducibility degree-structures.

Extended previous results to include a larger class of reductions.

Improved understanding of the hierarchy induced by good Borel reducibilities.

Abstract

In reference [8] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class $\xi$ functions).