Let $\text{Conf}_n(X) = \{(z_1, \cdots, z_n)\ \in X^n: z_i \neq z_j \text{ if } i\neq j\}$ be the configuration space of a topological space $X$. We can define on it an action of the permutation group $S_n$ by $\forall \sigma \in S_n, \forall z = (z_1, \cdots, z_n) \in \text{Conf}_n(X)$:
$$\sigma(z) = (z_{\sigma(1)}, \cdots, z_{\sigma(n)})$$
My question is: this action is always properly discontinuous? if so, how can I prove it? I believe that I have proved this if $X$ is a metric space in a way thats makes me wonder if I need $X$ to be at least Hausdorff.
P.S.: what I mean by properly discontinuous is as defined by Munkres, i.e. an action that for every point $x \in X$, there exists an open neighbourhood $U$ of $x$, such that $g(U) \cap U = \emptyset$ for every $g \neq 1$ in $G$