On minimal non-potentially closed subsets of the plane

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.

💡 Research Summary

The paper investigates a special class of Borel subsets of the Euclidean plane, namely those that cannot be turned into closed sets by any refinement of the Polish topology on the real line. Such sets are called “potentially closed”; a set is non‑potentially closed if no Polish topology on ℝ makes it closed. The authors introduce a reduction notion ≤_c based on products of continuous functions: for Borel sets A, B ⊆ ℝ², A ≤_c B iff there exist continuous f, g : ℝ → ℝ with (f × g)⁻¹