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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\nThen $S$ is [[closed|0G5AV9Q4RR]] in $X$ if and only if $\\overline{S} = S$.\n\n## Remarks\n\nProof: if $S$ is closed, then $\\overline{S} \\subseteq S$ by [[closure is the smallest closed superset|0SHC6Z304Z]], and $S \\subseteq \\overline{S}$ always holds. If $\\overline{S} = S$, then $S$ is closed because the [[closure is closed|02TZDKG50Q]].\nThe dual statement is: $S$ is open if and only if $\\operatorname{int} S = S$.\n\n## Explanations\n\n### L1 human\n\nA closed set already contains all the points that you can approach from inside it.\nSo taking the closure adds nothing.\n",
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