P-adic Numbers is Hausdorff Topological Space

Theorem

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff.

Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$.

By definition of the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$, $\tau_p$ is the topology induced by the metric $d_p$.

From Metric Space is Hausdorff, it follows that $\struct{\Q_p, \tau_p}$ is Hausdorff.

$\blacksquare$

Let $x, y \in \Q_p$ such that $x \ne y$.

$r := \norm {x - y}_p > 0$

Then, for all $z\in\Q_p$ we have:

$\ds \max\set {\norm {x - z}_p, \norm {z - y}_p}$

$\ge$

$\ds \norm {(x-z) + (z-y)} _p$

$\ds $

$=$

$\ds \norm {x - y} _p$

$\ds $

$=$

$\ds r$

Therefore, the $r$-open balls $\map {B_r} x$ and $\map {B_r} y$ are disjoint.

$\blacksquare$