P-adic Numbers is Locally Compact 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 locally compact.

Proof

From Local Basis of P-adic Number:

$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of $a$.

From Open and Closed Balls in P-adic Numbers are Compact Subspaces;

$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of compact sets of $a$.

Hence $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is locally compact by definition.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Corollary $3.3.8$