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?

[If these questions should be asked on SE instead, please let me know and I'll move them.]

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?