# Where is the second countability of a Riemann surface used in finding an exhaustion of the surface

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

• Otherwise you wouldn't have a countable cover consisting of relatively compact open sets. – Daniel Fischer Dec 8 '17 at 20:55
• Does paracompactness then allow us to find a locally finite refinement of that cover? – user135520 Dec 8 '17 at 21:08
• Yes, but for a locally compact and $\sigma$-compact Hausdorff space, one can directly construct a locally finite open cover consisting of relatively compact sets, so that $\overline{U_n} \subset U_{n+1}$ for all $n$ without explicitly using paracompactness. – Daniel Fischer Dec 8 '17 at 21:13
• Incidentally Riemann surfaces need not be 2nd countable (just take an uncountable disjoint union of open disks); the correct statement is that all connected Riemann surfaces are 2nd countable. – Moishe Kohan Dec 9 '17 at 1:18