  We show that the homeomorphism group of a surface without boundary does not
admit a Hausdorff group topology strictly coarser than the compact-open
topology. In combination with known automatic continuity results, this implies
that the compact-open topology is the unique Hausdorff separable group topology
on the group if the surface is closed or the complement in a closed surface of
either a finite set or the union of a finite set and a Cantor set.
