Three Points in Ultrametric Space have Two Equal Distances/Corollary

Theorem

Let $\struct {X, d}$ be an ultrametric space.

Let $x, y, z \in X$.

Then:

at least two of the distances $\map d {x, y}$, $\map d {x, z}$ and $\map d {y, z}$ are equal.

Proof

Either:

$\map d {x, z} = \map d {y, z}$

or:

$\map d {x, z} \ne \map d {y, z}$

By Three Points in Ultrametric Space have Two Equal Distances:

$\map d {x, z} = \map d {y, z}$ or $\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$

In either case two of the distances are equal.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.3$: Topology