In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to ${\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}}$ for all ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$. Sometimes the associated metric is also called a non-Archimedean metric or super-metric.

An ultrametric on a set M is a real-valued function

(where ${\displaystyle \mathbb {R} }$ denotes the real numbers), such that for all x, y, z ∈ M:

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 ultrapseudometric on M. An ultrapseudometric space is a pair (M, d) consisting of a set M and an ultrapseudometric d on M.

In the case when M is an Abelian group (written additively) and d is generated by a length function ${\displaystyle \|\cdot \|}$ (so that ${\displaystyle d(x,y)=\|x-y\|}$), the last property can be made stronger using the Krull sharpening to:

We want to prove that if ${\displaystyle \|x+y\|\leq \max \left\{\|x\|,\|y\|\right\}}$, then the equality occurs if ${\displaystyle \|x\|\neq \|y\|}$. Without loss of generality, let us assume that ${\displaystyle \|x\|>\|y\|.}$ This implies that ${\displaystyle \|x+y\|\leq \|x\|}$. But we can also compute ${\displaystyle \|x\|=\|(x+y)-y\|\leq \max \left\{\|x+y\|,\|y\|\right\}}$. Now, the value of ${\displaystyle \max \left\{\|x+y\|,\|y\|\right\}}$ cannot be ${\displaystyle \|y\|}$, for if that is the case, we have ${\displaystyle \|x\|\leq \|y\|}$ contrary to the initial assumption. Thus, ${\displaystyle \max \left\{\|x+y\|,\|y\|\right\}=\|x+y\|}$, and ${\displaystyle \|x\|\leq \|x+y\|}$. Using the initial inequality, we have ${\displaystyle \|x\|\leq \|x+y\|\leq \|x\|}$ and therefore ${\displaystyle \|x+y\|=\|x\|}$.

From the above definition, one can conclude several typical properties of ultrametrics. For example, for all ${\displaystyle x,y,z\in M}$, at least one of the three equalities ${\displaystyle d(x,y)=d(y,z)}$ or ${\displaystyle d(x,z)=d(y,z)}$ or ${\displaystyle d(x,y)=d(z,x)}$ holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set.