TITLE: Are Polish space and LCCB space related?
QUESTION [8 upvotes]: It seems these spaces are the most useful ones for doing probabilities. Are LCCB (locally compact with countable basis) somewhat more general spaces that when endowed with a metric become Polish? I think I once knew the answer to this question. Thanks

REPLY [5 votes]: Theorem. Every locally-compact second-countable Hausdorff space is a Polish space.
I thought I'd quote a sketch of proof of above fact from somewhere else.
The following sketch is from https://golem.ph.utexas.edu/category/2008/08/polish_spaces.html

if X is second countable locally compact Hausdorff, then the one-point
  compactification X+ is metrizable and compact, hence complete (under
  any metric), and of course second countable, hence X+ is Polish. An
  open subspace of a Polish space is Polish, hence X is Polish.
Bourbaki (which I found as a Google Book result) provided invaluable
  assistance.

As for why an open subspace of a Polish space is Polish, again from the same link:

I had no idea that an open set of a Polish space was Polish — it
  seemed pretty tough removing that extra point and finding a complete
  metric on what’s left.
I now see how easy this is: we start with a metric on the one-point
  compactification X+, remove the point at infinity, and ‘stretch’ the
  metric near the removed point to get a complete metric on X.

As for why the one-point compactification is metrizable, the selected answer from the following thread explains how to build a local countable basis at infinity:
If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable