Local Basis of P-adic Number/Cosets
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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
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$