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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

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What is a LCCB? – William Jul 8 '12 at 23:55
Probably Locally Compact with a Countable Base – ncmathsadist Jul 8 '12 at 23:56
yes, sorry just added it. – solojazz Jul 8 '12 at 23:57
Did you mean to ask if a locally compact second-countable metrizable space is neccessarily Polish? – tomasz Jul 9 '12 at 2:36
By locally compact with countable basis, do you mean sigma-locally-compact ? if yes, the space would have lindelöf property, and if it is endowed with a metric, it would be separable. – saposcat Jul 9 '12 at 3:23

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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

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