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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.

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Hint: $G$ acts properly discontinously iff $\forall x\in X$ the orbit is locally finite.

let the orbit has a limit point $p\in X$ then by local compactness there would exist a compact set containing $p$ that does not satisfy the locally finite condition.

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