1) $d(x,y)\ge 0$ for all $x,y\in X$, and $d(x,y)= 0$ if and only if $x=y$
2) $d(x,y)=d(y,x)$ for all $x,y\in X$
3) $d(x,y)\le d(x,z)+d(y,z)$ for all $x,y,z\in X$
This gives us a metric space $(X,d)$. A metric space is called ultrametric if it's metric is non-Archimedian, meaning that for property 3) we have the following:
$d(x,y) \le {\max}\{d(x,z),d(y,z)\}$
It's easy to check that when this holds, property 3) always holds.
Geometry
The topology of an ultrametric space is rather interesting, below are some geometric properties for an ultrametric space $(X,d)$.
All Triangles are Isosceles
Let $x,y,z$ be the three vertices of a triangle in the space. Then one can show that we always have two of the distances $d(x,y), d(y,z), d(x,z)$ to be equal.
Balls
As usual, we define an open ball of radius $r>0$ centered at a point $x\in X$ as $B_r(x) = \{y\in X | d(y,x)<r\}$ and a closed ball of radius $r$ centered at $x$ as $\overline{B_r(x)} = \{y\in X | d(y,x)\le r\}$.
The balls in an ultrametric space have the following properties:
1) They are clopen (both open and closed).
2) Every point of the ball is the center.
3) No two balls can overlap, and one has to be contained in another.
An Example - p-adic numbers
Definition Let $p$ be a prime. Given a non-zero rational $x=\frac{m}{n}$, where $m,n\in\mathbb{Z}$,we can write it as follows,
\begin{equation*}
x = p^{v(x)}\frac{a'}{b'}
\end{equation*}
such that $p$ doesn't divide $a'$ and $p$ doesn't divide $b'$. By unique prime factorization, we know $v(x)\in\mathbb{Z}$ is unique. Basically, we want to extract from $b$ and from $c$ as high a power of the prime number $p$ as possible.
The p-adic absolute value is defined as follows,
\begin{equation*}
|x|_p = p^{-v(x)}
\end{equation*}
and we define $|0|_p=0$.
This absolute value on $\mathbb{Q}$ measures how divisible a rational is by the prime number $p$, and since $0$ can be viewed as infinitely divisible by $p$, it is reasonable to define it as $0$.
Properties The p-adic absolute value can be easily checked to have the following properties,
$|a|_p = 0 \Leftrightarrow a=0$
$|ab|_p=|a|_p|b|_p$
$|a+b|_p\le {\max}\{|a|_p,|b|_p\}\le |a|_p + |b|_p$
As seen above, $|\ |_p$ is a non-Archimedian absolute value, and actually, it is the only non-trivial absolute value on $\mathbb{Q}$ where $p$ is allowed to be $\infty$ besides prime numbers.
Definition Let $p$ be a prime. Given a non-zero rational $x=\frac{m}{n}$, where $m,n\in\mathbb{Z}$,we can write it as follows,
\begin{equation*}
x = p^{v(x)}\frac{a'}{b'}
\end{equation*}
such that $p$ doesn't divide $a'$ and $p$ doesn't divide $b'$. By unique prime factorization, we know $v(x)\in\mathbb{Z}$ is unique. Basically, we want to extract from $b$ and from $c$ as high a power of the prime number $p$ as possible.
The p-adic absolute value is defined as follows,
\begin{equation*}
|x|_p = p^{-v(x)}
\end{equation*}
and we define $|0|_p=0$.
This absolute value on $\mathbb{Q}$ measures how divisible a rational is by the prime number $p$, and since $0$ can be viewed as infinitely divisible by $p$, it is reasonable to define it as $0$.
Properties The p-adic absolute value can be easily checked to have the following properties,
$|a|_p = 0 \Leftrightarrow a=0$
$|ab|_p=|a|_p|b|_p$
$|a+b|_p\le {\max}\{|a|_p,|b|_p\}\le |a|_p + |b|_p$
As seen above, $|\ |_p$ is a non-Archimedian absolute value, and actually, it is the only non-trivial absolute value on $\mathbb{Q}$ where $p$ is allowed to be $\infty$ besides prime numbers.
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