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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $x \\in X$ and let $r \\ge 0$ be a real number.\nThe closed ball with centre $x$ and radius $r$ is the set\n$$\\bar{B}(x, r) = \\{ y \\in X : d(x, y) \\le r \\}.$$\n\n## Remarks\n\nThe [[open ball|0DC3MPR7C9]] $B(x, r)$ is a subset of $\\bar{B}(x, r)$.\nIn $\\mathbb{R}$ with the usual metric, $\\bar{B}(x, r)$ is the closed interval $[x - r, x + r]$.\nThe closed ball is not always the closure of the open ball. In the [[discrete metric|0NA4FK7B8S]], $B(x, 1) = \\{x\\}$ is closed, but $\\bar{B}(x, 1) = X$.\nFor $r = 0$, the closed ball is $\\{x\\}$.\n\n## Explanations\n\n### L1 human\n\nThe closed ball contains the points at distance at most $r$, so it also contains its edge.\nIn the plane, it is the disc of radius $r$ together with its circle.\n",
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