Ultrametric

An ultrametric is a Metric which satisfies the following strengthened version of the Triangle Inequality,

$$d(x,z) \leq \max (d(x,y),d(y,z))$$

for all $x,y,z$. At least two of $d(x,y)$, $d(y,z)$, and $d(x,z)$ are the same.

Let $X$ be a Set, and let $X^{\Bbb{N}}$ (where N is the Set of Natural Numbers) denote the collection of sequences of elements of $X$ (i.e., all the possible sequences $x_1$, $x_2$, $x_3$, ...). For sequences $a=(a_1,a_2,\dots)$, $b=(b_1,b_2,\dots)$, let $n$ be the number of initial places where the sequences agree, i.e., $a_1=b_1$, $a_2=b_2$, ..., $a_n=b_n$, but $a_{n+1} \not= b_{n+1}$. Take $n=0$ if $a_1 \not= b_1$. Then defining $d(a,b) = 2^{-n}$ gives an ultrametric.

The p-adic Number metric is another example of an ultrametric.

See also Metric, p-adic Number