Let $G$ be a finite group and $E$ a topological space. We say that $G$ acts properly discontinuous in $E$ if $\forall x\in E$ exists $ U \subset E$ neighborhood of $x$ such that $gU \cap U = \emptyset \Leftrightarrow g \neq id$.
Suppose that $G$ acts properly discontinuous in $E$ and let $B=E/G$.
I know there is $\rho : E \rightarrow B$ covering map.
How can I show that $E$ is Hausdorff and second-countable space if and only if $B$ is Hausdorff and second-countable?
