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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $A$ be a nonempty subset of $X$ and let $x \\in X$.\nThe distance from $x$ to $A$ is the real number\n$$d(x, A) = \\inf \\{ d(x, a) : a \\in A \\}.$$\n\n## Remarks\n\nThe [[infimum|04KFTA34C5]] exists, because the set is nonempty and bounded below by $0$.\nIf $x \\in A$, then $d(x, A) = 0$. The converse is false: in $\\mathbb{R}$, $d(0, (0, 1)) = 0$ but $0 \\notin (0, 1)$.\nIn general $d(x, A) = 0$ if and only if $x$ is in the [[closure|0DRKT1QJR7]] of $A$.\nFor all $x, y \\in X$, $|d(x, A) - d(y, A)| \\le d(x, y)$. Thus $x \\mapsto d(x, A)$ is [[Lipschitz|0TMZ21WKCN]] with constant $1$.\n\n## Explanations\n\n### L1 human\n\nThe distance from a point to a set is the shortest distance from the point to the set, when a shortest distance exists.\nFor example, the distance from $3$ to the interval $[0, 1]$ is $2$.\n",
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