# Three topological reducibilities for discontinuous functions

## TL;DR

This paper introduces three related reducibility notions for functions on compact metric spaces, linking them to Baire classes and hierarchies, and characterizes the degree structures within Baire 1 functions.

## Contribution

It defines new reducibility notions for functions and characterizes their degree structures, connecting them to classical Baire class hierarchies.

## Key findings

- Most $ ext{equiv}_T$-classes align with proper Baire classes.
- Certain $eta$-jump functions are minimal in their Baire class under $ ext{reducibility}_m$.
- Complete characterization of degree structures for Baire 1 functions matching known hierarchies.

## Abstract

We define a family of three related reducibilities, $\leq_T$, $\leq_{tt}$ and $\leq_m$, for arbitrary functions $f,g:X\rightarrow\mathbb R$, where $X$ is a compact separable metric space. The $\equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $\alpha$-jump functions $j_\alpha:2^\omega\rightarrow \mathbb R$ are $\leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $\leq_{tt}$ and $\leq_m$, finding an exact match to the $\alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.