A short proof of Rubin’s theorem

In a remarkable theorem, M. Rubin proved that if a group G acts in a locally dense way on a locally compact Hausdorff space X without isolated points, then the space X and the action of G on X are unique up to G-equivariant homeomorphism. Here we give a short, self-contained proof of Rubin’s theorem, using equivalence classes of ultrafilters on a poset to reconstruct the points of the space X.