Summary of Topology on P-adic Numbers

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 $\struct{\Q_p, \tau_p}$ is:

$(1): \quad$ Hausdorff

$(2): \quad$ second-countable

$(3): \quad$ totally disconnected

$(4): \quad$ locally compact

Proof

Follows from:

P-adic Numbers is Hausdorff Topological Space

P-adic Numbers is Second Countable Topological Space

P-adic Numbers is Totally Disconnected Topological Space

P-adic Numbers is Locally Compact Topological Space

$\blacksquare$

Sources

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