P-adic Numbers is Hausdorff Topological Space

Theorem
Let $p$ be a prime number.

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.

Proof 1
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.

Proof 2
Let $x, y \in \Q_p$ such that $x \ne y$.

By :
 * $\norm {x - y}_p > 0$

Let:
 * $r := \dfrac {\norm {x - y}_p} 2$

Let $\map {B_r} x$ and $\map {B_r} y$ be the $r$-open balls of $x$ and $y$ respectively.

Then:
 * $x \in \map {B_r} x$
 * $y \in \map {B_r} y$
 * $\map {B_r} x \cap \map {B_r} y = \O$