  An extension of Penrose's singularity theorem is proved for spacetimes where
black holes are allowed to form from non-singular initial data. With standard
assumptions about the spacetime, and assuming the existence of a trapped
surface which lies outside of black hole horizons and is not completely
surrounded by horizons, we show that the spacetime region outside (or on) the
horizons must contain singularities. If the trapped surface is surrounded by
horizons, we show that the horizons divide spacetime into causally disconnected
pieces. Unlike the original Penrose's theorem, our theorems provide some
information about the location of singularities. We illustrate how they can be
used to rule out some cosmological scenarios.
