Topologically free non-Hausdorff groupoids

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.

💡 Research Summary

The paper investigates three isotropy‑controlling properties of étale groupoids—effectiveness, topological principality, and topological freeness—and clarifies how they relate under various additional hypotheses. After recalling basic notions of groupoids, étale groupoids, and several forms of the Axiom of Choice (AC, CC, CC₍fin₎, DC, DMC), the authors define the three properties precisely. An étale groupoid G is effective if the interior of its isotropy subgroupoid coincides with the unit space, topologically principal if the set of units with trivial isotropy is dense in the unit space, and topologically free if the interior of the non‑trivial isotropy (i.e., Iso(G) \ G⁽⁰⁾) is empty. For each notion a “strongly” version is introduced, requiring the property to persist when restricting to any closed invariant subset.

The first main result (Theorem 3.4) shows, using only ZF, that effectiveness implies topological freeness, topological principality implies topological freeness, and that a closed unit space together with topological principality forces effectiveness. Thus, without any separability or Hausdorff assumptions, the three notions are linked by one‑way implications.

When the groupoid is locally compact, second‑countable, and the Countable Choice axiom (CC) is assumed, Theorem 3.6 proves that effectiveness, topological principality, and topological freeness become equivalent. This extends earlier results that required both Hausdorffness and second‑countability, demonstrating that the equivalence survives in non‑Hausdorff settings as long as CC holds.

To illustrate that the three conditions are genuinely distinct in the absence of these extra hypotheses, the authors present three concrete examples. Example (a) (the “two‑headed snake” groupoid) is non‑Hausdorff, topologically principal, but not effective. Example (b) is a transformation groupoid built from the Cantor set; it is effective yet fails both topological principality and freeness. Example (c) adjoins a formal arrow γₓ to each rational point x, yielding a groupoid that is topologically free but neither effective nor topologically principal. These constructions confirm the strictness of the implications in Theorem 3.4.

Section 4 provides a new characterisation of topological freeness that works without Hausdorffness. Proposition 4.1 states that for a locally compact étale groupoid G, the following are equivalent: (i) for every open subset U of the unit space and every compact subset K of non‑unit isotropy, there exists a non‑empty open V ⊂ U with V K V = ∅; (ii) G is topologically free. The proof proceeds by contraposition for (i)⇒(ii) and by an inductive covering argument for (ii)⇒(i), the latter requiring only the Hausdorffness of the unit space to obtain neighbourhoods separating points from compact isotropy. This result generalises earlier characterisations that needed both Hausdorffness and second‑countability.

The final section (Section 5) connects the groupoid framework with set‑theoretic choice principles. Working in ZF, the authors prove that the statement “every étale groupoid (not necessarily Hausdorff) is topologically free” is equivalent to the Baire Category Theorem. Moreover, they derive corollaries linking this statement to various weakenings of the Axiom of Choice: CC, CC₍fin₎, DC, and DMC. For instance, Corollary 5.5 shows that CC is equivalent to the assertion that every locally compact, second‑countable étale groupoid is topologically free; Corollary 5.6 relates CC₍fin₎ + DMC to the Dependent Choice axiom, and Corollary 5.7 ties the Baire Category Theorem directly to topological freeness. These equivalences illustrate how properties of étale groupoids can serve as a bridge between operator‑algebraic structures and foundational set‑theoretic axioms.

In summary, the paper makes three substantive contributions: (1) it clarifies the precise logical relationships among effectiveness, topological principality, and topological freeness for étale groupoids, especially highlighting the role of second‑countability and choice axioms; (2) it furnishes a Hausdorff‑free characterisation of topological freeness via compact‑isotropy avoidance; and (3) it reveals deep connections between groupoid‑theoretic freeness and classical set‑theoretic principles such as the Baire Category Theorem and various forms of the Axiom of Choice. These results enrich both the theory of non‑Hausdorff groupoids in operator algebras and the understanding of how algebraic‑topological structures reflect foundational logical assumptions.