  The existence of initial singularities in expanding universes is proved
without assuming the timelike convergence condition. The assumptions made in
the proof are ones likely to hold both in open universes and in many closed
ones. (It is further argued that at least some of the expanding closed
universes that do not obey a key assumption of the theorem will have initial
singularities on other grounds.) The result is significant for two reasons:
(a)~previous closed-universe singularity theorems have assumed the timelike
convergence condition, and (b)~the timelike convergence condition is known to
be violated in inflationary spacetimes. An immediate consequence of this
theorem is that a recent result on initial singularities in open,
future-eternal, inflating spacetimes may now be extended to include many closed
universes. Also, as a fringe benefit, the time-reverse of the theorem may be
applied to gravitational collapse.
