Topological reducibilities for discontinuous functions and their structures

TL;DR

This paper thoroughly describes the topological many-one degree structure of real-valued functions, extends the Bourgain rank characterization to noncompact domains, and explores the relationship between the Martin conjecture and topological reducibility under determinacy.

Contribution

It provides a complete description of the topological degree structure, extends Bourgain rank results, and clarifies the connection between the Martin conjecture and topological Weihrauch reducibility.

Findings

The topological many-one degree structure of real-valued functions is fully characterized.

The Bourgain rank characterization extends to noncompact Polish domains.

Under the axiom of determinacy, continuous Weihrauch degrees are well-ordered, and ranks increase with Turing jumps.

Abstract

In this article, we give a full description of a topological many-one degree structure of real-valued functions, recently introduced by Day-Downey-Westrick. We also point out that their characterization of the Bourgain rank of a Baire-one function of compact Polish domain can be extended to noncompact Polish domain. Finally, we clarify the relationship between the Martin conjecture and Day-Downey-Westrick's topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if $f$ is has continuous Weihrauch rank $\alpha$, then $f'$ has continuous Weihrauch rank $\alpha+1$, where $f'(x)$ is defined as the Turing jump of $f(x)$.