From Bredon, we say that the $G$-action on a space $X$ is properly discontinuous if
"Each point $x \in X$ has a neighborhood $U$ such that $g(U) \cap U \neq \emptyset$" implies "$g = e$, an identity element of $G$".
But, if we take $U=X$, then $g(U)=U=X$ so any point in $X$ would have such neighborhood. I think that there is some problem in my understanding or the definition in the textbook is misleading.
Please help me, thanks.