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  "body": "\n## Definition\n\nA [[topological space|0CNQXVBHCX]] $X$ is compact when each [[open cover|0NVPNC12JY]] of $X$ has a finite subcover.\n\n## Remarks\n\nSome authors, mainly in the French tradition, also require a compact space to be Hausdorff, and they call the general notion quasi-compact.\nEach finite topological space is compact. An infinite set with the [[discrete topology|0S2TEHX4E1]] is not compact.\nFor metric spaces, compactness is the same as [[sequential compactness|0CARW59WD2]], by [[this theorem|0K8SGQSV7G]].\nThe [[continuous image of a compact space is compact|0K9AAJ8G51]].\n\n## Explanations\n\n### L1 human\n\nIn a compact space, each way to cover the space by open sets can be reduced to finitely many of them.\nThis finiteness lets you turn local facts, which hold near each point, into global facts with one bound for the whole space.\n",
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