---
tag: A6DPR620B5
kind: concept
name: Ultrametric space
msc: ["54E35", 12J25]
specializes: [0VKD1TP7F3]
sources:
  - url: https://en.wikipedia.org/wiki/Ultrametric_space
same_as:
  wikidata: Q1897429
  wikipedia: Ultrametric space
---

## Summary

An ultrametric space is a [[metric space|0VKD1TP7F3]] whose distance satisfies the strong triangle inequality $d(x, z) \le \max(d(x, y), d(y, z))$.
The main examples are the $p$-adic numbers and spaces of infinite sequences, where two sequences are close when they agree for a long time.
Ultrametric geometry is strange at first: every triangle is isosceles and every point of a ball is a centre of that ball.
Balls are both open and closed, so ultrametric spaces are [[zero-dimensional|JJ7D0CYECZ]].

## Explanations

### L1 human

Think of a family tree, and measure the distance between two people by how far back their nearest common ancestor lives.
To go from $x$ to $z$ you never need a longer jump than the longest of the two jumps through $y$.
On $\Q$, the $3$-adic distance makes $9$ and $0$ closer than $1$ and $0$, since $9$ is divisible by a higher power of $3$.