  We extend the proof of automatic continuity for homeomorphism groups of
manifolds to non-compact manifolds and manifolds with marked points and their
mapping class groups. Specifically, we show that, for any manifold $M$
homeomorphic to the interior of a compact manifold, and a set $X \subset M$
homeomorphic to the union of a Cantor set and finite set, the relative
homeomorphism group $\mathrm{Homeo}(M, X)$ and the mapping class group
$\mathrm{Homeo}(M, X)/\mathrm{Homeo}_0(M,X)$ have the property that any
homomorphism from such a group to any separable topological group is
necessarily continuous.
