The Cosmic Censor Forbids Naked Topology

For any asymptotically at spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume global hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptions concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial in nity (\i-avoidance"), and that no future incomplete null geodesic is visible from future null in nity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Furthermore, we prove lemmas which have as a consequence that if a future incomplete null geodesic were visible from in nity, then given our i-avoidance assumption, it would also be visible from points of spacetime that can communicate with in nity, and so would signify a true naked singularity.