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?
Let$\gamma_n:S^n\rightarrow \mathbb{R}P^n$ be the universal cover. Take $A=\mathbb{R}P^n$ and let $B=E=S^n$, and both maps to $A$ to be the covering projection. Then $\gamma_n^*S^n=S^n\times_{\mathbb{R}P^n}S^n\cong S^n\times \mathbb{Z}_2$ is not connected.