Countable Basis for P-adic Numbers

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Theorem

Let $p$ be a prime number.

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

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.


Then:

$\mathcal B_p = \set{ \map {{B_{p^{-n}}}} q : q \in \Q, n \in \Z}$

is a countable basis for $\struct{\Q_p, \tau_p}$.


Corollary 1

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.


Then:

$\mathcal B_p = \set{ \map {{B^{\,-}_{p^{-n}}}} q : q \in \Q, n \in \Z}$

is a countable basis for $\struct{\Q_p, \tau_p}$.


Corollary 2

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


Then:

$\mathcal B_p = \set{q + p^n \Z_p : q \in \Q, n \in \Z}$

is a countable basis for $\struct{\Q_p, \tau_p}$.


Proof

From Sequence of Powers of Number less than One, $\sequence{p^{-n}}$ is a real null sequence.

From Null Sequence induces Local Basis in Metric Space, for all $a \in \Q_p$ the set $\set { \map {B_{p^{-n}}} a : n \in \Z}$ is a local basis of $a$.

From Union of Local Bases is Basis, the set:

$\mathcal B’ = \displaystyle \bigcup_{a \in \Q_p} \set { \map {B_{p^{-n}}} a : n \in \Z} = \set { \map {B_{p^{-n}}} a : a \in Q_p, n \in \Z}$

is a basis for $\tau_p$.


Let $a \in \Q_p$ and $n \in \Z$.

By definition of the $p$-adic numbers, $\Q$ is everywhere dense in $\Q_p$.

Hence:

$\exists q \in \Q: q \in \map {B_{p^{-n}}} a$

From Characterization of Open Ball in P-adic Numbers,

$\map {B_{p^{-n}}} q = \map {B_{p^{-n}}} a$

Hence:

$\map {B_{p^{-n}}} a \in \mathcal B_p$

Since $q$ and $n$ were arbitrary, then:

$\mathcal B’ \subseteq \mathcal B_p$

It follows that $\mathcal B_p$ is a basis for $\tau_p$.


It remains to show that $\mathcal B_p$ is countable.

For all $q \in \Q$, let $\map {\mathcal B_p} q = \set { \map {B_{p^{-n}}} q : n \in \Z}$.

Then:

$\mathcal B_p = \displaystyle \bigcup_{q \in \Q} \map {\mathcal B_p} q$

From Integers are Countably Infinite and Surjection from Countably Infinite Set iff Countable:

$\forall q \in \Q: \map {\mathcal B_p} q$ is a countable set.

From Rational Numbers are Countably Infinite:

$\mathcal B_p$ is a countable union of countable sets.

From Countable Union of Countable Sets is Countable:

$\mathcal B_p$ is countable.

$\blacksquare$

Sources