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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *σ-compact* when there are [[compact|0R6EWFKRCF]] subsets $K_1, K_2, K_3, \\ldots$ of $X$ with\n$$X = \\bigcup_{n \\ge 1} K_n.$$\n\n## Remarks\n\nThe family may be finite or [[countable|01TQB88H4C]]; a finite family can be padded by repeating a set.\nEvery σ-compact space is [[Lindelöf|SW0SN1NEJE]]: an open cover has a finite subcover of each $K_n$, and the union of these is a countable subcover.\nA closed subspace and a continuous image of a σ-compact space are σ-compact.\n\nExample: $\\R^n$ is σ-compact.\nIt 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.\nExample: $\\Q$ is σ-compact, as the countable union of its one-point subsets.\n\nNon-example: an uncountable set $X$ with the discrete topology is not σ-compact.\nIts compact subsets are finite, and a countable union of finite sets is countable.\n\nNotation: σ-compact, also written sigma-compact.\n",
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