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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\nThe closure of $S$ is the [[intersection|0839XNQXDK]] of all [[closed sets|0G5AV9Q4RR]] $F$ of $X$ with $S \\subseteq F$.\nIt is written $\\overline{S}$ or $\\operatorname{cl} S$.\n\n## Remarks\n\nThe family of these closed sets is not empty, because it contains $X$.\nA point $x$ is in $\\overline{S}$ if and only if each open set that contains $x$ meets $S$.\nIn $\\mathbb{R}$, the closure of $(0, 1)$ is $[0, 1]$, and the closure of $\\mathbb{Q}$ is $\\mathbb{R}$.\nThe closure and the [[interior|0C38S0KMJK]] are related by $X \\setminus \\overline{S} = \\operatorname{int}(X \\setminus S)$.\n\n## Explanations\n\n### L1 human\n\nThe closure of a set adds to the set all the points that you can approach from inside it.\nFor the open interval $(0, 1)$, you add the two end points $0$ and $1$.\n",
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