Let $G$ be a finite group acting on some Hausdorff space $X$. I have some feeling that the action of $G$ is always properly discontinuous, i.e. for all $x\in X$, there is an open neighbourhood $U$ of $x$ such that $g\cdot U\cap U=\emptyset$ for all $g\neq e_G\in G$. Is this true, or under what additional conditions might this be true?
Try to prove the following observation: If $g⋅x ≠ x$, then there is $U_g$ a neighborhood of $x$ such that $g ⋅ U_g ∩ U_g = ∅$. Moreover, for every smaller neighborhood $V$ of $x$, the same holds.