  Let $X$ be a closed $3$-manifold, $\mathcal{M}_{R>0}$ the space of metrics on
$X$ with positive scalar curvature, and $\text{Diff}(X)$ the group of
diffeomorphisms of $X$. Marques proves the fundamental result that
$\mathcal{M}_{R>0}/ \text{Diff}(X)$ is path connected. Using this and the
theorem of Cerf in differential topology, Marques shows that the space of
asymptotically flat metrics with nonnegative scalar curvature on $\mathbb{R}^3$
is path connected. Based on Carlotto-Li's generalization of Marques' result to
the case of compact manifold with boundary, we show that the space of
asymptotically flat metrics with nonnegative scalar curvature and mean convex
boundary on $\mathbb{R}^3\backslash B^3$ is path connected. The differential
topology part of Marques' argument no longer yields the desired result for
$\mathbb{R}^3\setminus B^3$, but we bypass this issue by finding a more
elementary proof. We also include a path-connectedness result for the space of
black hole initial data sets, which can be thought of as a necessary condition
for the Final State Conjecture.
