Local Basis of P-adic Number/Cosets

Theorem

Let $p$ be a prime number.

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

Let $a \in \Q_p$.

Let $\Z_p$ be the $p$-adic integers.

Then the set $\set {a + p^n \Z_p: n \in Z}$ is a local basis of $a$ consisting of clopen sets.

Proof

From Local Basis of P-adic Number the set $\set { \map {B_{p^{-n} } } a : n \in \Z}$ is a local basis of clopen sets.

From Open Balls of P-adic Number:

$\set {\map {B_{p^{-n} } } a : n \in \Z} = \set {a + p^{n + 1} \Z_p : n \in \Z} = \set {a + p^n \Z_p : n \in \Z}$

The result follows.

$\blacksquare$

Also see

• Closed Balls of P-adic Number
• Open Balls of P-adic Number

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$