# Let $X$ be a topological space, and let $G\subseteq Homeo(X)$ be a group of homeomorphisms of $X$ where $G$ acts properly discontinuously

Let $$X$$ be a topological space, and let $$G\subseteq \textrm{Homeo}(X)$$ be a group of homeomorphisms of $$X$$.

a) Prove that if $$G$$ is finite, then $$G$$ acts properly discontinuously on $$X$$.

b) Suppose $$X$$ is compact and that $$G$$ acts properly discontinuously on $$X$$. Prove that $$G$$ is finite.

Any help is appreciated. I have included the definition below:

Definition: A group $$G\subseteq Homeo(X)$$ acts properly continuously on $$X$$ if for any compact set $$C\subseteq X$$, $$|\{g\in G \mid C\cap gC \neq\emptyset\}| < \infty$$

• (a) is obvious : any subset of $G$ is finite, in this case. – J. Darné Nov 4 '18 at 19:40

Item $$a)$$ follows immediately from the fact that if $$G$$ is finite, then so is each set $$\{g\in G:C\cap gC\neq\emptyset\}$$.
Item $$b)$$ follows by applying the fact that, since $$X$$ is compact, the set $$\{g\in G:X\cap gX\neq\emptyset\}$$ is finite. But then it is clear that $$G=\{g\in G:X\cap gX\neq\emptyset\}$$, so $$G$$ is finite.