THE COSMIC CENSOR FORBIDS NAKED TOPOLOGY

For any asymptotically flat spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and s atisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume g lobal hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptio ns concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial infinity ('i(0)-avoidance' ), and that no future incomplete null geodesic is visible from future null infinity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Fur thermore, we prove lemmas that have as a consequence that if a future incomplete null geodesic were visible from infinity, then, given our i (0)-avoidance assumption, it would also be visible from points of spac etime that can communicate with infinity, and so would signify a true naked singularity.