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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]], let $K \\subseteq X$ be [[compact|00R0X1WGPX]] and let $F \\subseteq X$ be [[closed|0G5AV9Q4RR]] with $F \\subseteq K$.\nThen $F$ is compact.\n\n## Remarks\n\nProof: let $(U_i)$ be open sets of $X$ that cover $F$. Add the open set $X \\setminus F$ to get an open cover of $K$. A finite subcover of $K$ gives, after you remove $X \\setminus F$, a finite subcover of $F$.\nMore generally, if $K$ is compact and $F$ is closed, then $K \\cap F$ is compact.\nFor example, the [[Cantor set|0BXNDZFQME]] is compact, because it is a closed subset of $[0, 1]$.\n\n## Explanations\n\n### L1 human\n\nA closed part of a compact set is again compact.\nThe open complement of the closed part covers everything outside it, so it does the job of all the extra open sets.\n",
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