I want to know if $p : Y \to X$ is covering and $Y$ is Hausdorff then is $X$ hausdorff (I dont know if this is true).
My attempt is this : let $a \ne b \in X.$ There exist nbd $U_a$ of $a$ and $U_b$ of $b$ such that $p^{-1} U_a = \sqcup U_{a,i}$ and $p^{-1} U_b = \sqcup U_{b,j}$ where $U_{a,i}$ mapped homeomorphically onto $U_a$ and $U_{b,j}$ onto $U_b.$ If $U_a$ and $U_b$ are disjoint we are done. If not there exist an element $z$ in the intersection. So each element in $p^{-1}(z)$ belongs exactly to only one of the intersection of $U_{a,i}$ and $U_{b,j}.$ Now I am not able to proceed.