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  "body": "\n## Summary\n\nA countably compact space is a topological space in which every countable open cover has a finite subcover.\nIt is a weakening of [[compactness|0TVJZYB6SH]] that is equivalent to the statement that every sequence has a cluster point.\nCompact spaces and [[sequentially compact|0FRMJH3CY7]] spaces are countably compact.\nFor metric spaces, countable compactness, compactness and sequential compactness all coincide.\n\n## Explanations\n\n### L1 human\n\nCompactness asks that every open cover can be cut down to finitely many sets.\nCountable compactness only asks this for covers with countably many sets.\nAn equivalent picture: every sequence in the space keeps coming back near some point.\nFor example, the interval $[0, 1]$ is countably compact, but $(0, 1]$ is not: the cover by the sets $(1/n, 1]$ has no finite subcover.\n",
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