# Topology on the group of autohomeomorphisms

I am wondering whether the group of all autohomeomorphisms of a compact metric space can be given a reasonable topological group structure? (Preferably, can it be turned into a locally compact group?)

I think that moral reason should be no, but I might be wrong. Here's how I see it: Boolean algebras, which are discrete in nature correspond to compact, Hausdorff zero-dimensional spaces. The group of automorphisms of a Boolean algebra is then the same as the group of automorphisms of the corresponding Stone space...

I'll also add that we don't need to restrict ourselves to the topological category because, as pointed out above, the automorphism group tends to be unmanageable. If we restrict ourselves to metric spaces for instance, and consider the group of bijective autoisometries, the groups tend to be much more well-behaved. For instance the circle $S^1$ has automorphism group (in $\mathbf{Met}$) isomorphic to $S^1\ltimes\mathbb{Z}_2$ or sometimes called $Dih(S^1)$, the circular dihedral group, with the topology of $S^1\sqcup S^1$ inherited in the obvious way.