Let $A$ be a path connected, locally path connected space, so as $B \rightarrow A$ a continuous map from $B$ path connected, locally path connected, and let $E \rightarrow A$ be a connected covering. I can study the pullback of $E \times_A B$ over $B$ as a covering space.
Question: Is it connected?
I suppose that in general, assuming $B,A,E$ connected, it is not true that $B\times_A E$ is a connected covering, but I don't have a counter example. But in this case, can we hope for connectedness of the pullback?