Local Basis of P-adic Number/Closed Balls
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Then the set of closed balls $\set {\map {B^-_{p^{-n} } } a: n \in Z}$ 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 Ball in P-adic Numbers is Closed Ball:
- $\set {\map {B_{p^{-n} } } a: n \in \Z} = \set {\map {B^-_{p^{-\paren{n + 1} } } } a: n \in \Z} = \set {\map {B^-_{p^{-n} } } a : n \in \Z}$
The result follows.
$\blacksquare$