Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

TL;DR

This paper extends classical theorems on Polish group actions to more general groupoid actions, providing a unified topological realization framework that applies even in non-Hausdorff and point-free contexts, using algebraic methods.

Contribution

It introduces a new algebraic proof method for topological realization theorems, generalizes results to groupoid actions, and characterizes potentially open relations in the Borel hierarchy.

Findings

Unified realization theorem for Polish and groupoid actions

Characterization of potentially open Borel relations

Proofs valid in non-Hausdorff and point-free settings

Abstract

We extend the Becker--Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker--Kechris theorems, as well as Sami's and Hjorth's sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms; and the equivalence of "potentially open" versus "orbitwise open" Borel sets. We also characterize "potentially open" $n$-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions, and prove a result subsuming Lupini's Becker--Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups, and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids, and more generally in the point-free context, for open localic groupoids.