Between a stone and a Hausdorff space

We consider the duality between General Relativity and the theory of Einstein algebras, in the extended setting where one permits non-Hausdorff manifolds. We show that the duality breaks down, and then go on to discuss a sense in which general relativity, formulated using non-Hausdorff manifolds, exhibits excess structure when compared to Einstein algebras. We discuss how these results bear on a class of algebraically-motivated deflationist views about spacetime ontology. We conclude with a conjecture concerning non-Hausdorff spacetimes with no bifurcate curves.