Borel-piecewise continuous reducibility for uniformization problems

TL;DR

This paper explores a detailed hierarchy of Borel-piecewise continuous functions, linking computability theory and topology to analyze uniformization problems and non-constructive principles.

Contribution

It introduces a fine hierarchy of Borel-piecewise continuous functions and connects priority arguments to $G_\delta$-piecewise continuity for separation results.

Findings

Established separation results for subclasses of $G_\delta$-piecewise continuous reductions.

Connected computability-theoretic priority arguments with topological continuity notions.

Applied methods to distinguish non-constructive principles in the Weihrauch lattice.

Abstract

We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and $G_\delta$-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of $G_\delta$-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of $G_\delta$-piecewise continuous reductions for uniformization problems on set-valued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.