If $X$ is locally compact Hausdorff, then the following are all equivalent:
- $X$ is second countable,
- $X$ is metrizable and $\sigma$-compact,
- $X$ is metrizable and separable,
- $X$ is Polish.
I want to show that a LCH space satisfying any of these equivalent conditions also admits a proper metric, a metric whose closed balls are compact. The source quoted for this is Topologie Generale by Bourbaki, but I'm having a hard time getting my hands on it, don't speak French, and to be honest have no desire to read Bourbaki.