I am studying topology on my own, and I am having trouble proving the following.
For a Hausdorff, connected, locally euclidean paracompact space $X$, there exists a countable basis for $X$.
I think if I possibly get any countable open cover of $X$ which consists of coordinate balls (that is, homeomorphic to open ball in euclidean space) since each coordinate balls are second countable. However, I cannot get a clue of how to find them. Am I doing right?