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  "body": "\n## Summary\n\nAn ultrametric space is a [[metric space|0VKD1TP7F3]] whose distance satisfies the strong triangle inequality $d(x, z) \\le \\max(d(x, y), d(y, z))$.\nThe main examples are the $p$-adic numbers and spaces of infinite sequences, where two sequences are close when they agree for a long time.\nUltrametric geometry is strange at first: every triangle is isosceles and every point of a ball is a centre of that ball.\nBalls are both open and closed, so ultrametric spaces are [[zero-dimensional|JJ7D0CYECZ]].\n\n## Explanations\n\n### L1 human\n\nThink of a family tree, and measure the distance between two people by how far back their nearest common ancestor lives.\nTo go from $x$ to $z$ you never need a longer jump than the longest of the two jumps through $y$.\nOn $\\Q$, the $3$-adic distance makes $9$ and $0$ closer than $1$ and $0$, since $9$ is divisible by a higher power of $3$.\n",
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