Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both Hausdorff and locally euclidean. I am worried about whether $E$ needs to be second countable.
If $E$ is not connected, then it does not need to be second countable. Take the standard covering map $\coprod_{i \in \mathbb{R}} M \to M$.
Question: If $E$ is connected, then why is it second countable?