On minimal non-potentially closed subsets of the plane

TL;DR

This paper investigates Borel subsets of the plane that can be made closed through topology refinement, establishing minimal non-potentially closed sets, and exploring implications for graphs and orders.

Contribution

It introduces a perfect antichain of minimal non-potentially closed sets and analyzes their properties in relation to graphs and orders.

Findings

Existence of a perfect antichain of minimal non-potentially closed sets

Non-injectivity in the Kechris-Solecki-Todorcevic dichotomy for analytic graphs

Comparison of Borel sets using products of continuous functions

Abstract

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic graphs.