I'm reading the first chapter (pg 11) Hubbard's Teichmüller Theory where we are considering the following construction for a Reimann surface $X$
Choose a locally finite cover of $X$ by relatively compact open sets $U_{n}$ and smooth partition of unity $\varphi_{n}$ subordinate to this cover. We then have that $$g(x) := \sum^{\infty}_{n=0}n\varphi_{n}(x)$$ is a proper function.
It was shown earlier that all connected Riemann surfaces are second countable and the author points out that second countability was used in the above construction.
I was wondering if someone could point out explicitly how secound countablity is used here