Sigma-compact space

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In topology, a branch of mathematics, a σ-compact space is a topological space that is the union of countably many compact subsets, which also called countable at infinity. This sequence of compact sets is called an exhaustion by compact sets. Obviously, every compact space is σ-compact. Moreover, every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover). The reverse implications do not hold. For example, standard Euclidean space (Rⁿ) is σ-compact but not compact, and the lower limit topology on the real line is Lindelöf but not σ-compact or compact.

A space is said to be σ-locally compact if it is both σ-compact and locally compact.