P-adic Numbers is Totally Disconnected 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 totally disconnected.

Proof

From P-adic Numbers form Non-Archimedean Valued Field:

$\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

From Non-Archimedean Division Ring is Totally Disconnected:

$\struct {\Q_p, \tau_p}$ is totally disconnected.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$