Ultrametric space

Summary

An ultrametric space is a metric space 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.