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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *countably compact* when every [[countable|01TQB88H4C]] [[open cover|049PXYHW1T]] of $X$ has a finite subcover.\n\n## Remarks\n\nThe following are equivalent for any topological space $X$:\n\n1. $X$ is countably compact;\n2. every sequence in $X$ has a cluster point, that is, a point $x$ such that each neighbourhood of $x$ contains $x_n$ for infinitely many $n$;\n3. every countable family of closed sets with the finite intersection property has nonempty intersection.\n\nEvery [[compact|0TVJZYB6SH]] space and every [[sequentially compact|0FRMJH3CY7]] space is countably compact.\nEvery countably compact space is [[limit point compact|E8XKSQY76M]], and the converse holds for [[T1 spaces|0NW6CH38YR]].\nA countably compact space that is also [[Lindelöf|SW0SN1NEJE]] is compact.\n\nExample: $[0, 1]$ is countably compact, since it is compact by the Heine–Borel theorem.\nExample: the space $[0, \\omega_1)$ of countable ordinals with the order topology is countably compact but not compact.\n\nNon-example: $\\R$ is not countably compact.\nThe countable open cover by the intervals $(-n, n)$ for $n \\ge 1$ has no finite subcover.\n",
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