Sigma-compact space

Remarks

The family may be finite or countable; a finite family can be padded by repeating a set. Every σ-compact space is Lindelöf: an open cover has a finite subcover of each $K_n$, and the union of these is a countable subcover. A closed subspace and a continuous image of a σ-compact space are σ-compact.

Example: $\R^n$ is σ-compact. It is the union of the closed balls $\set{x : \norm{x} \le m}$ for $m \ge 1$, which are compact by the Heine–Borel theorem. Example: $\Q$ is σ-compact, as the countable union of its one-point subsets.

Non-example: an uncountable set $X$ with the discrete topology is not σ-compact. Its compact subsets are finite, and a countable union of finite sets is countable.

Notation: σ-compact, also written sigma-compact.