I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there is a continuous function $f$ from $B$ to the closed unit interval $[0,1]$ such that $f(A) = \{0\}$ and $f(a) = 1$). Show that $E$ is also completely regular.
I can't see how to do this. My main stumbling block seems to be that if $A$ is a closed subset of $E$ and $a$ is a point in $E - A$ it is quite possible for $A$ to intersect $p^{-1}(\{p(a)\})$ so I can't see how any continuous function from $B$ to the closed unit interval could be composed with the covering map $p$ to show complete regularity of $E$.
Does anyone know if this result is definitely correct and how it can be proved? Thanks for your help.