ar X iv : g r - qc / 9 60 90 07 v 1 1 S ep 1 99 6 The Cosmic Censor Forbids Naked Topology

For any asymptotically flat spacetime with a suitable causal structure obeying Penrose's cosmic censorship conjecture (weak form) and satisfying conditions guaranteeing focus-ing 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 infinity (" i 0-avoidance "), and that no 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. Furthermore, we prove lem-mas which have as a consequence that if an incomplete null geodesic were visible from infinity, then given our i 0-avoidance assumption, it would also be visible from points of spacetime that can communicate with infinity, and so would signify a true naked singularity.