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  "body": "\n## Proof\n\n1. Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\n2. Let $\\mathcal{F}$ be the family of all [[closed sets|0G5AV9Q4RR]] $F$ of $X$ with $S \\subseteq F$. By the definition of the [[closure|0DRKT1QJR7]], $\\overline{S}$ is the [[intersection|0839XNQXDK]] of the members of $\\mathcal{F}$.\n3. Let $s \\in S$. Each $F \\in \\mathcal{F}$ contains $S$, so $s \\in F$ for each $F \\in \\mathcal{F}$. Thus $s \\in \\overline{S}$, and $S \\subseteq \\overline{S}$.\n4. By [[closure is closed|02TZDKG50Q]], the set $\\overline{S}$ is closed.\n5. Let $F$ be a closed set of $X$ with $S \\subseteq F$. Then $F \\in \\mathcal{F}$.\n6. The intersection of a family is contained in each member of the family. So $\\overline{S} \\subseteq F$.\n\n## Idea\n\nThe closure is the intersection of all closed sets that contain $S$. Every one of them contains $S$, so the intersection does too. And the intersection sits inside each of them, so it is the smallest.\n",
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