Definition 20.1:

Let $Y$ be a topological space on which a group $G$ acts. We say that the action is properly discontinuous if each point $y \in Y$ has a neighborhood $U$ such that for any pair of distinct elements $g^{\prime}, g^{\prime\prime} \in G$,

$$g^{\prime}U\cap g^{\prime\prime}U = \emptyset.$$