Topologically free non-Hausdorff groupoids
Operator Algebras
2026-02-04 v1 Logic
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.
Keywords
Cite
@article{arxiv.2602.02926,
title = {Topologically free non-Hausdorff groupoids},
author = {Lisa Orloff Clark and Ryan Thompson and Ilija Tolich},
journal= {arXiv preprint arXiv:2602.02926},
year = {2026}
}