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  "body": "\n## Proof\n\n1. Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \\subseteq X$, and let $\\overline{S}$ be the [[closure|0DRKT1QJR7]] of $S$.\n2. By [[closure is the smallest closed superset|0SHC6Z304Z]], we have $S \\subseteq \\overline{S}$, and $\\overline{S}$ is [[closed|0G5AV9Q4RR]].\n3. Suppose that $S$ is closed. Then $S$ is a closed set with $S \\subseteq S$.\n4. By the same result, $\\overline{S} \\subseteq S$. With step 2, $\\overline{S} = S$.\n5. Conversely, suppose that $\\overline{S} = S$. By step 2, $\\overline{S}$ is closed, so $S$ is closed.\n\n## Idea\n\nThe closure always contains $S$ and is always closed. If $S$ is already closed, it is one of the closed sets the closure must fit inside. So the two sets agree.\n",
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