I've begun studying manifolds, and an exercise asked to prove the existence of subordinate partitions of unity. One of the steps is this question:
Let $M$ be a paracompact, Hausdorff, smooth manifold. Show that every open cover of $M$ admits a countable, locally finite refinement by open sets.
Of course, the "locally finite" part comes directly from the definition of paracompactness. However, I'm stuck in the "countable" part. It's never implied that the manifold is second-countable, which almost all my search results use as hypothesis, so I either don't need it or I have to prove.
If this isn't related to manifolds, but it's a direct result of Hausdorfness, paracompactness or both, I'd like a proof as well.