Hawking has shown that if black holes were to exist in a universe that expands forever, black holes would completely evaporate, violating unitarity. Unitarity thus requires that the universe exist for only a finite future proper time. I develop this argument, showing that unitarity also requires the boundaries of all future sets to be Cauchy surfaces, and so no event horizons can exist. Thus, the null generators of the surfaces of astrophysical black holes must leave the surface in both time directions, allowing non-spherical topologies for black hole surfaces. Unitarity thus also requires the effective cosmological constant to be zero eventually, otherwise the universe would expand forever.