Local Basis of P-adic Number

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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 open balls $\set {\map {B_{p^{-n} } } a : n \in Z}$ is a local basis of $a$ consisting of clopen sets.


Corollary 1

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.


Corollary 2

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

Let $\BB_a$ be the set of all open balls of $Q_p$ centered on $a$.

That is:

$\BB_a = \set{\map {B_\epsilon} a : \epsilon \in \R_{>0}}$

From Open Balls Centered on P-adic Number is Countable:

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

From Open Balls form Local Basis for Point of Metric Space, $\BB_a$ is a local basis of $a$.


From P-adic Numbers form Non-Archimedean Valued Field:

the $p$-adic numbers is a non-Archimedean division ring.

From Open Balls are Clopen In Non-Archimedean Division Ring:

the set $\BB_a = \set {\map {B_{p^{-n} } } a : n \in \Z}$ is a local basis of clopen sets.

$\blacksquare$