  We analyze the horizon structure of families of space times obtained by
evolving initial data sets containing apparent horizons with several connected
components. We show that under certain smallness conditions the outermost
apparent horizons will also have several connected components. We further show
that, again under a smallness condition, the maximal globally hyperbolic
development of the many black hole initial data constructed by Chrusciel and
Delay, or of hyperboloidal data of Isenberg, Mazzeo and Pollack, will have an
event horizon, the intersection of which with the initial data hypersurface is
not connected. This justifies the "many black hole" character of those
space-times.
