Topologically free non-Hausdorff groupoids

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.