Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Then the set of $p$-adic integers $\Z_p$ is compact.

Proof

By definition the $p$-adic integers $\Z_p$ is the closed ball $\map {B^-_1} 0$.

From Open and Closed Balls in P-adic Numbers are Compact Subspaces, $\map {B^-_1} 0$ is compact.

$\blacksquare$

Sources

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