could any one tell me how to prove: when $G$ acts properly discontinuously on riemann surface then the orbits are closed sets? $G$ is the group of all homeomorphism on $X$ say.
so we have $f:G\times X\rightarrow X$ be a proper discont. action so $\forall x\in X \exists V open$ containing $x$ $f(V)\cap V=\phi$ and orbit of $x$, $O_x=\{gx:g\in G\}$
we need to show $O_x$ is closed.