Autohomeomorphism groups

The compact-open topology is the unique minimum group topology on the group of autohomeomorphisms of any one dimensional metric manifold. Every compact metric zero-dimensional topological group is topologically isomorphic to the autohomeomorphism group, with the compact-open topology, of a compact, connected, locally connected, metric space. Hence there are two compact, connected, locally connected, metric spaces with algebraically, but not topologically, isomorphic autohomeomorphism groups, with compact-open topology.