I'm working on Example 1B.7 of Hatcher's book Algebraic Topology, and I have some troubles to show that the action of $G$ in $EG$ is a covering space action.
First, it's easy to show that, the action of $G$ on $EG$, that's taking the simplex $[g_0,\cdots,g_n]$ linearly onto the simplex $[g g_0, \cdots, g g_n]$ for each $g\in G$ has no fixed points. This action is induced by a simplicial map that takes the vertex $g_i$ to $g g_i$, so, if $g_i = g g_i$ then $g$ must be the identity element $e$ of $G$. Because $EG$ is a CW complex, in particular, this is a Hausdorff space, so it is sufficient to show, by the exercise 1.3.23, that $G$ acts properly discontinuously on $EG$.
So let $p \in EG$. I want to construct a neighborhood $U$ such that $\{ g \in G | U \cap g(U) \neq \emptyset \}$ is finite, but I have trouble constructing this neighborhood explicitly. For example, take a simplex $[g_0,g_1,g_2]$ then, the action take this simplex to the another simplex $[g g_0, g g_1, g g_2]$. Because this action is free, a neighborhood $U$ of $p \in [g_0,g_1,g_2]$ is disjoint of $g U$ in $[g g_0, g g_1, g g_2]$. Basically, it's impossible that this action takes a neighborhood in some simplex and leaves it in the same simplex. But how can I formalize this?