  The aim of this work is to present an initial data version of Hawking's
theorem on the topology of back hole spacetimes in the context of manifolds
with boundary. More precisely, we generalize the results of G. J. Galloway and
R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free
boundary stable marginally outer trapped surface (MOTS) $\Sigma$ in an initial
data set with boundary satisfying natural dominant energy conditions (DEC) is
of positive Yamabe type, i.e. $\Sigma$ admits a metric of positive scalar
curvature with minimal boundary, provided $\Sigma$ is outermost. To do so, we
prove that if $\Sigma$ is a compact free boundary stable MOTS which does not
admit a metric of positive scalar curvature with minimal boundary in an initial
data set satisfying the interior and the boundary DEC, then an outer
neighborhood of $\Sigma$ can be foliated by free boundary MOTS $\Sigma_t$,
assuming that $\Sigma$ is weakly outermost. Moreover, each $\Sigma_t$ has
vanishing outward null second fundamental form, is Ricci flat with totally
geodesic boundary, and the dominant energy conditions saturate on $\Sigma_t$.
