Topologically free non-Hausdorff groupoids
Lisa Orloff Clark, Ryan Thompson, Ilija Tolich
TL;DR
This work clarifies and relates key isotropy conditions for étale groupoids, showing that in the customary second-countable Hausdorff setting the properties of being effective, topologically principal, and topologically free coincide, while highlighting their distinctions in general. It defends prioritizing topologically free as the robust notion, extending Anantharaman-Delaroche's characterization to non-Hausdorff contexts, and connecting these ideas to Baire category phenomena under ZF without AC. A central achievement is a non-Hausdorff analogue of the topological freeness criterion and a Baire-type theorem that yields several AC weakenings (CC, DC, DMC) as equivalent formulations in the groupoid setting. Collectively, the results illuminate when isotropy can be controlled in non-Hausdorff étale groupoids and offer practical guidance for analysis of associated groupoid C*-algebras and related algebraic structures.
Abstract
We study three conditions that control the behaviour of isotropy in étale groupoids, and their relationships under the additional assumptions of second-countability and Hausdorffness. We examine a number of examples that show these properties are distinct. Working under the assumption of the Zermelo-Fraenkel axioms, excluding choice, we then examine an alternate characterization of topological freeness, first introduced by Anantharaman-Delaroche, in the non-Hausdorff setting. Finally, we prove an equivalence between the Baire Category Theorem and an étale groupoid theorem, along with similar equivalences to other weakenings of the Axiom of Choice.