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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $S \\subseteq X$ and let $x \\in X$.\nThen $x$ is in the [[closure|0DRKT1QJR7]] $\\overline{S}$ if and only if for each real $\\varepsilon \u003e 0$ there is $s \\in S$ with $d(x, s) \u003c \\varepsilon$.\n\n## Remarks\n\nEquivalently, $x \\in \\overline{S}$ if and only if each [[open ball|0DC3MPR7C9]] $B(x, \\varepsilon)$ meets $S$.\nFor nonempty $S$, this means $d(x, S) = 0$, where $d(x, S)$ is the [[distance from the point to the set|07TXYYVYZR]].\nIn $\\mathbb{R}$, the point $0$ is in the closure of $\\{1/n : n \\ge 1\\}$, because $1/n \u003c \\varepsilon$ for large $n$.\n\n## Explanations\n\n### L1 human\n\nA point is in the closure of $S$ when there are points of $S$ as close to it as you want.\nThe point does not need to be in $S$ itself.\n",
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