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-timesymmetric version of a theorem ofMeeks-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.