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Concept# Ultrametric space

Summary

In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to 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.
Formal definition
An ultrametric on a set M is a real-valued function
:d\colon M \times M \rightarrow \mathbb{R}
(where ℝ denote the real numbers), such that for all x, y, z ∈ M:
# d(x, y) ≥ 0;

# d(x, y) = d(y, x) (symmetry);

# d(x, x) = 0;

# if d(x, y) = 0 then x = y;

# d(x, z) ≤ max {d(x, y), d(y, z) } (strong triangle inequality or ultrametric inequality).

An ultrametric space is a pair (M, d) consisting of a set M together with an ultrametric d on M, which is called the space's associated distance function (also called a metric).
If d satisfies all of the conditions except possibly condition 4 then d is called an ult

Official source

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