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  "body": "\n## Definition\n\nLet $X$ be a set.\nAn *ultrametric* on $X$ is a [[metric|0VKD1TP7F3]] $d \\colon X \\times X \\to [0, \\infty)$ that satisfies the strong triangle inequality\n$$d(x, z) \\le \\max(d(x, y), d(y, z)) \\quad \\text{for all } x, y, z \\in X.$$\nAn *ultrametric space* is a set with an ultrametric.\n\n## Remarks\n\nThe strong triangle inequality implies the usual one, since $\\max(a, b) \\le a + b$ for $a, b \\ge 0$.\n\nExample: fix a prime $p$ and let $v_p$ be the [[p-adic valuation|03HDF50VN4]] on $\\Q$.\nPut $d(x, y) = p^{-v_p(x - y)}$ for $x \\ne y$ and $d(x, x) = 0$.\nSince $v_p(a + b) \\ge \\min(v_p(a), v_p(b))$, applying it to $a = x - y$ and $b = y - z$ gives the strong triangle inequality.\n\nNon-example: the usual metric $d(x, y) = \\abs{x - y}$ on $\\R$.\nHere $d(0, 2) = 2$, while $\\max(d(0, 1), d(1, 2)) = 1$.\n\nIn an ultrametric space, if $d(x, y) \\ne d(y, z)$, then $d(x, z) = \\max(d(x, y), d(y, z))$.\nEvery open ball $B(a, r)$ equals $B(b, r)$ for each $b \\in B(a, r)$.\n",
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