Three Points in Ultrametric Space have Two Equal Distances

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Theorem

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

Let $x, y, z \in X$ with $\map d {x, z} \ne \map d {y, z}$.


Then:

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


Corollary 1

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.


Corollary 2

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $x, y \in R$ and $\norm x \ne \norm y$.

Then:

$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y}$


Corollary 3

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $x, y \in R$ and $\norm x \lt \norm y$.

Then:

$\norm {x + y} = \norm {x - y} = \norm {y - x} = \norm y$


Corollary 4

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $x, y \in R$.

Then:

  • $\norm {x + y} \lt \norm y \implies \norm x = \norm y$
  • $\norm {x - y} \lt \norm y \implies \norm x = \norm y$
  • $\norm {y - x} \lt \norm y \implies \norm x = \norm y$


Corollary 5

Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.

Let $a, b \in \Z_{\ne 0}$ be coprime:

$a \perp b$

Then:

$\norm a = 1$ or $\norm b = 1$


Proof

Without loss of generality, let $\map d {x, z} > \map d {y, z}$.


Then:

\(\ds \map d {x, y}\) \(\le\) \(\ds \max \set {\map d {x, z}, \map d {y, z} }\) Definition of Non-Archimedean Metric
\(\ds \) \(=\) \(\ds \map d {x, z}\) since $\map d {x, z} > \map d {y, z}$


On the other hand:

\(\ds \map d {x, z}\) \(\le\) \(\ds \max \set {\map d {x, y}, \map d {y, z} }\) Definition of Non-Archimedean Metric
\(\ds \) \(=\) \(\ds \map d {x, y}\) since $\map d {x, z} > \map d {y, z}$


Putting the two inequalities together then:

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

$\blacksquare$


Sources

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