  We introduce a natural generalization of marginally outer trapped surfaces,
called immersed marginally outer trapped surfaces, and prove that three
dimensional asymptotically flat initial data sets either contain such surfaces
or are diffeomorphic to R^3. We establish a generalization of the Penrose
singularity theorem which shows that the presence of an immersed marginally
outer trapped surface generically implies the null geodesic incompleteness of
any spacetime that satisfies the null energy condition and which admits a
non-compact Cauchy surface. Taken together, these results can be viewed as an
initial data version of the Gannon-Lee singularity theorem. The first result is
a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that
every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to
R^3 contains an embedded stable minimal surface. We also obtain an initial data
version of the spacetime principle of topological censorship. Under physically
natural assumptions, a 3-dimensional asymptotically flat initial data set with
marginally outer trapped boundary and no immersed marginally outer trapped
surfaces in its interior is diffeomorphic to R^3 minus a finite number of open
balls. An extension to higher dimensions is also discussed.
