We investigate black holes in asymptotically de Sitter space-times. We show that a trapped surface always appears inside the event horizon and the total area of the black holes does not decrease as in asymptotically flat space-times. We find, however, that there is an upper bound on the area of apparent horizons in a wide class of asymptotically de Sitter space-times, in contrast with asymptotically flat space-times. This implies that black holes with a large area cannot collide with each other in asymptotically de Sitter space-time, unless a naked singularity is formed. On the basis of these results we argue that the inflationary scenario works even for initially inhomogeneous universes.
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