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?



Connectedness properties of groups of homeomorphisms
Regarding the last part of your question:


Let $f\_{0}$ and $f\_{1}$ be two orientation preserving homeomorphism in $H\_{+}(S^{1})$. Lift $f\_{0}$ and $f\_{1}$ to homeomorphisms $F\_{0}$ and $F\_{1}$ on $\mathbb{R}$, as in <http://en.wikipedia.org/wiki/Rotation_number>.


Then consider the quotient of the homotopy $F\_{t}=tF\_{0}+(1-t)F\_{1}$ as a path of homeomorphism on $S^{1}$.


More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G\_{t}=P \circ F\_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G\_{t}(x+1)=G\_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f\_{0}$ to $f\_{1}$. 


