Sigma-compact space ($X={\bigcup }_{n\ge 1}{K}_n,K_n\text{ compact}$)

A space is sigma-compact if it is a countable union of compact subsets. A locally compact, sigma-compact metrizable space is second countable: each compact metrizable subset has a countable dense set, their union is countable and dense, and a separable metric space has a countable basis.