A separation result for countable unions of Borel rectangles

TL;DR

This paper establishes a dichotomy for separating disjoint analytic binary relations using countable unions of Borel rectangles, advancing understanding of their descriptive set-theoretic complexity.

Contribution

It provides a clear characterization of when such relations can be separated by specific Borel classes, filling a gap in descriptive set theory.

Findings

Characterizes separation by countable unions of Borel rectangles

Identifies conditions for separation by specific Borel classes

Advances understanding of analytic binary relations in descriptive set theory

Abstract

We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bf\Sigma}^0_1 \!\times\! {\bf\Sigma}^0_\xi$ sets, or by a ${\bf\Pi}^0_1 \!\times\! {\bf\Pi}^0_\xi$ set.