Ultrametric

In mathematics, an ultrametric space is a special kind of metric space in which the triangle inequality is replaced with ${\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}}$. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications.

Formally, an ultrametric space is a set of points ${\displaystyle M}$ with an associated distance function (also called a metric)

(where ${\displaystyle \mathbb {R} }$ is the set of real numbers), such that for all ${\displaystyle x,y,z\in M}$, one has:

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