Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers
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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 $3.3.8$