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
After a bit of research I found that A locally compact space that is Hausdorff (LCH) will be sigma-locally-compact. Also that a LCCB will be metrizable (with a complete metric) and separable thus Polish too. thanks
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
As for why an open subspace of a Polish space is Polish, again from the same link:
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: