I am trying to prove that for any group $G$, if $EG$ is a contractible CW-complex on which $G$ acts freely, and for which the action of $G$ is cellular and free on the cells of $EG$, then the projection $EG \to EG/G$ define a fibration. Proving this is equivalent to find an open set $U$ of $EG$ such that $g(U)\cap U = \emptyset$ for all $g\neq e$ in $G$. Given a decomposition : $$EG = \bigcup_{n\geq0}X^{(n)}$$
I am trying to construct $U$ as $\bigcup_{n\geq0}U_n$, where $U_n$ is an open set of $X^{(n)}$ (formed by the n-cells), and $U_n \subset U_{n+1}$. Following this idea, it is possible to construct the $U_n$ by induction, and even a contractible basis of open neighborhoods for each point in $EG$. But I can only get the property $g(U)\cap U=\emptyset$ in the case of a finite group. Does this result still holds where $G$ is not finite ?