Three topological reducibilities for discontinuous functions

Abstract

We define a family of three related reducibilities, ${\le }_T$, ${\le }_{tt}$ and ${\le }_m$, for arbitrary functions $f,g:X\to \mathbb{R}$, 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 $j_{\alpha }:2^{\omega }\to \mathbb{R}$ are ${\le }_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to ${\le }_{tt}$ and ${\le }_m$, finding an exact match to the $\alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.

36 pages