In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically
anti-de-Sitter spacetimes obeying natural causality conditions exhibit
topological censorship. We use this fact in this paper to derive in arbitrary
dimension relations between the topology of the timelike boundary-at-infinity,
\scri, and that of the spacetime interior to this boundary. We prove as a
simple corollary of topological censorship that any asymptotically anti-de
Sitter spacetime with a disconnected boundary-at-infinity necessarily contains
black hole horizons which screen the boundary components from each other. This
corollary may be viewed as a Lorentzian analog of the Witten and Yau result
hep-th/9910245, but is independent of the scalar curvature of \scri.
Furthermore, the topology of V', the Cauchy surface (as defined for
asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions
exterior to event horizons, is constrained by that of \scri. In this paper,
we prove a generalization of the homology results in gr-qc/9902061 in arbitrary
dimension, that H_{n-1}(V;Z)=Z^k where V is the closure of V' and k is the
number of boundaries Σi​ interior to Σ0​. As a consequence, V
does not contain any wormholes or other compact, non-simply connected
topological structures. Finally, for the case of n=2, we show that these
constraints and the onto homomorphism of the fundamental groups from which they
follow are sufficient to limit the topology of interior of V to either B^2 or
I×S1.Comment: Revtex, 20 page