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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $x \\in X$ and let $r \\ge 0$.\nThen the [[closed ball|05DSV40EB9]] $\\bar{B}(x, r)$ is a [[closed set|0G5AV9Q4RR]] of $X$.\n\n## Remarks\n\nProof: let $y \\notin \\bar{B}(x, r)$ and let $\\varepsilon = d(x, y) - r \u003e 0$. If $d(y, z) \u003c \\varepsilon$, then $d(x, z) \\ge d(x, y) - d(y, z) \u003e r$ by the [[triangle inequality|01M6CW9VM6]]. Thus the complement of $\\bar{B}(x, r)$ is open.\nFor $r = 0$, this shows that each one-point set $\\{x\\}$ is closed in a metric space.\n\n## Explanations\n\n### L1 human\n\nA point outside the closed ball is at some positive distance beyond its edge.\nA small ball about that point stays outside, so the outside is open.\n",
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