I have the following construction from the proof of Prohorov's Theorem. Let $S$ be a metric space and choose compact sets $K_u$ in such a way that $K_1 \subset K_2 \subset \cdots$ and $P_n K_u > 1-u^{-1}$ for all $u$ and $n$. Then the set $\cup_u K_u$ is separable. I cannot see any special properties of the $K_n$'s except that they are compact. I cannot find any source that says a countable union of compact sets is separable. So how is this union separable?
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5Just choose a countable dense set for each $K_n$ (compact metric spaces are separable), and take the union of those. – David Mitra Aug 01 '19 at 01:31
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What is $P_nK_u$? – Paul Frost Aug 01 '19 at 16:45