Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological group with respect to the compact open topology $\tau_C$.
What can be said about the connectedness of $(H(X),\tau_C)$?
I would really like to get some references regarding this question and/or the more general one: for a connected space $X$, what topologies (eg. compact-open,pointwise, uniform) make $C(X)$ connected?
These questions arose from this specific one:
How can I prove that the group $H_+(S^1)$ (orientation preserving homeomorphisms of the circle) is connected?