P-adic Numbers is Hausdorff Topological Space/Proof 1
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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
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.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$