Topologically free non-Hausdorff groupoids

TL;DR

This paper investigates the properties of isotropy in non-Hausdorff étale groupoids, explores their distinctions through examples, and links topological freeness to foundational set-theoretic principles.

Contribution

It introduces an alternate characterization of topological freeness for non-Hausdorff groupoids and establishes equivalences between the Baire Category Theorem and étale groupoid properties.

Findings

Distinct properties of isotropy in non-Hausdorff groupoids

New characterization of topological freeness without assuming Hausdorffness

Equivalence between Baire Category Theorem and étale groupoid theorems

Abstract

We study three conditions that control the behaviour of isotropy in \'etale 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 \'etale groupoid theorem, along with similar equivalences to other weakenings of the Axiom of Choice.