  We study pocket universes which have zero cosmological constant and
non-trivial boundary topology. These arise from bubble collisions in eternal
inflation. Using a simplified dust model of collisions we find that boundaries
of any genus can occur. Using a radiation shell model we perform analytic
studies in the thin wall limit to show the existence of geometries with a
single toroidal boundary. We give plausibility arguments that higher genus
boundaries can also occur. In geometries with one boundary of any genus a
timelike observer can see the entire boundary. Geometries with multiple
disconnected boundaries can also occur. In the spherical case with two
boundaries the boundaries are separated by a horizon. Our results suggest that
the holographic dual description for eternal inflation, proposed by Freivogel,
Sekino, Susskind and Yeh, should include summation over the genus of the base
space of the dual conformal field theory. We point out peculiarities of this
genus expansion compared to the string perturbation series.
