nyuuzyou/wikis huggingface dataset:
P-adic Norm not Complete on Rational Numbers/Proof 3

== Theorem ==
== Proof ==
By Rational Numbers are Countably Infinite, the set of rational numbers is countably infinite.
By P-adic Numbers are Uncountable, the set of $p$-adic numbers $\Q_p$ is uncountably infinite.
Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
The $p$-adic numbers $\Q_p$ is the quotient ring $\CC \, \big / \NN$ by definition.
By Embedding Division Ring into Quotient Ring of Cauchy Sequences, the mapping $\phi: \Q \to \Q_p$ defined by:
:$\map \phi r = \sequence {r, r, r, \dotsc} + \NN$
where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$, is a distance-preserving monomorphism.
By Corollary to Surjection from Natural Numbers iff Countable then $\phi$ is not a surjection.
Hence:
:$\exists \sequence {x_n} \in \CC: \sequence {x_n} + \NN \not \in \map \phi \Q$
By Cauchy Sequence Converges Iff Equivalent to Constant Sequence then $\sequence {x_n}$ is not convergent in $\struct {\Q, \norm {\,\cdot\,}_p}$.
The result follows.
Category:P-adic Norm not Complete on Rational Numbers
P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\ds \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion.
Let $a$ be the $p$-adic number, that is left coset, in $\Q_p$ containing $\ds \sum_{i \mathop = m}^\infty d_i p^i$.
Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion:
:$l = \min \set {i: i \ge m \land d_i \ne 0}$
Then:
:$\norm a_p = p^{-l}$
== Proof ==
For all $n \ge m$, let:
:$\alpha_n = \ds \sum_{i \mathop = m}^n d_i p^i$
By assumption:
:$\sequence{\alpha_n}$ is a representative of $a$
By definition of the induced norm:
:$\norm a_p = \ds \lim_{n \mathop \to \infty} \norm {\alpha_n}_p$
From Eventually Constant Sequence Converges to Constant it is sufficient to show:
:$\forall n \ge l + 1 : \norm {\alpha_n}_p = p^{-l}$
Let $n \ge l + 1$.
Then:
The sum $\ds \sum_{i \mathop = l + 1}^n d_i p^i$ can be rewritten:
:$\ds \sum_{i \mathop = l + 1}^n d_i p^i = p^{l + 1} \sum_{i \mathop = l + 1}^n d_i p^{i - \paren {l + 1} }$
By definition of divisor:
:$\ds p^{l + 1} \divides \sum_{i \mathop = l + 1}^n d_i p^i$
Then:
By definition of a $p$-adic expansion:
:$0 < d_l < p$
Then:
:$p^l \divides d_l p^l$
:$p^{l + 1} \nmid d_l p^l$
By definition of $p$-adic norm:
:$\norm {d_l p^l}_p = p^{-l}$
Thus:
:$\ds \norm {\sum_{i \mathop = l + 1}^n d_i p^i}_p < p^{-l} = \norm {d_l p^l}_p$
Finally:
The result follows.
== Sources ==
* : $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Lemma $1.31$
Category:P-adic Number Theory
P-adic Norm of p-adic Number is Power of p

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p: x \ne 0$.
Then:
:$\exists v \in \Z: \norm x_p = p^{-v}$
== Lemma ==
== Proof 1 ==
== Proof 2 ==
Category:P-adic Number Theory
Category:P-adic Norm of p-adic Number is Power of p
P-adic Norm of p-adic Number is Power of p/Lemma

== Theorem ==
Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
Let $\sequence {x_n}$ be a Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$ such that $\sequence {x_n}$ does not converge to $0$.
Then:
:$\exists v \in \Z: \ds \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
== Proof ==
From P-adic Norm on Rational Numbers is Non-Archimedean Norm, $p$-adic norm is a non-Archimedean norm on the rationals $\Q$.
Since $\sequence {x_n}$ does not converge to $0$, from Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary:
:$\exists N \in \N: \forall n, m > N: \norm {x_n}_p = \norm {x_m}_p$
By definition of the $p$-adic norm on $\Q$:
:$\exists v \in \Z: \norm {x_{N + 1} }_p = p^{-v}$
Therefore:
:$\forall n > N: \norm {x_n}_p = \norm {x_{N + 1} }_p = p^{-v}$
Let $\sequence {y_n}$ be the subsequence of $\sequence {\norm {x_n}}$ defined by:
:$\forall n: y_n = \norm {x_{N + n} }_p$
Then $\sequence {y_n}$ is the constant sequence $\tuple {p^{-v}, p^{-v}, p^{-v}, \dotsc}$.
Therefore:
Category:P-adic Norm of p-adic Number is Power of p
P-adic Norm of p-adic Number is Power of p/Proof 1

== Theorem ==
== Proof ==
From Rational Numbers are Dense Subfield of P-adic Numbers $\Q$ is dense in $\Q_p$.
By the definition of a dense subset then $\map \cl \Q = \Q_p$.
By Closure of Subset of Metric Space by Convergent Sequence then:
:there exists a sequence $\sequence {x_n} \subseteq \Q$ that converges to $x$.
That is:
:$\ds \lim_{n \mathop \to \infty} x_n = x$
From Modulus of Limit:
:$\ds \lim_{n \mathop \to \infty} \norm{x_n}_p = \norm x_p$
By Convergent Sequence in Normed Division Ring is Cauchy Sequence, $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
From Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers, $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.
From Lemma:
:$\exists v \in \Z: \norm x_p = \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.1$
Category:P-adic Norm of p-adic Number is Power of p
P-adic Norm of p-adic Number is Power of p/Proof 2

== Theorem ==
== Proof ==
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:
:$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$
:$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Then $x$ is a left coset in $\CC \, \big / \NN$.
Let $\sequence{x_n}$ be any Cauchy sequence in $x$.
From Lemma:
:$\exists v \in \Z: \ds \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
By definition of the $p$-adic norm on the $p$-adic numbers:
:$\norm x_p = \ds \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$
== Sources ==
* : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
Category:P-adic Norm of p-adic Number is Power of p
P-adic Norm satisfies Non-Archimedean Norm Axioms

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \to \R_{\ge 0}$.
Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms:
== Proof ==
From P-adic Numbers form Non-Archimedean Valued Field:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a non-Archimedean norm
Category:P-adic Number Theory
P-adic Norms are Not Equivalent

== Theorem ==
Let $p_1$ and $p_2$ be prime numbers such that $p_1 \ne p_2$.
Let $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ be the $p$-adic norms on the rationals $\Q$.
Then $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ are not equivalent norms.
That is, the topology induced by $\norm {\,\cdot\,}_{p_1}$ does not equal the topology induced by $\norm {\,\cdot\,}_{p_2}$.
== Proof ==
Consider $p_1/p_2 \in \Q$. 
With $\norm {\,\cdot\,}_{p_1}$:
On the other hand, with $\norm {\,\cdot\,}_{p_2}$:
By open unit ball equivalence, $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ are not equivalent norms.
By Equivalence of Definitions of Equivalent Division Ring Norms and topological equivalence then the topology induced by $\norm {\,\cdot\,}_{p_1}$ does not equal the topology induced by $\norm {\,\cdot\,}_{p_2}$.
== Sources ==
* : $\S 3.1.2$ Absolute Values on $\Q$, Problem $68$
* : $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Remark $1.28$ 
Category:Norm Theory
Category:P-adic Number Theory
P-adic Number has Unique P-adic Expansion Representative

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Then $a$ has exactly one representative that is a $p$-adic expansion.
== Proof ==
=== Case 1 ===
Let $\norm a_p \le 1$.
From P-adic Integer has Unique P-adic Expansion Representative, $a$ has exactly one representative that is a $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
=== Case 2 ===
Let $\norm a_p > 1$.
From P-adic Number times P-adic Norm is P-adic Unit,
:$\exists m \in \Z: p^m a \in \Z_p^\times$
where $\Z_p^\times$ denotes the $p$-adic units.
By definition a $p$-adic unit is a $p$-adic integer.
From P-adic Integer has Unique P-adic Expansion Representative:
:$p^m a$ has exactly one representative that is a $p$-adic expansion of the form:
::$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
From P-adic Unit has Norm Equal to One:
:$\norm {p^m a}_p = 1 = p^0$
From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
:$d_0 \neq 0$
Let $\sequence{x_n}$ be a representative of $a \in \Q_p$.
From Rational Numbers are Dense Subfield of P-adic Numbers: 
:the constant sequence $\tuple {p^m, p^m, p^m, \dotsc}$ represents $p^m \in \Q_p$.
By definition of the product in a quotient ring:
:the Cauchy sequence $\sequence{p^m x_n}$ is a representative of $p^m a \in \Q_p$.
From Representatives of same P-adic Number iff Difference is Null Sequence:
:$\sequence {p^m x_n - \ds \sum_{i \mathop = 0}^n d_i p^i}$ is a null sequence
From Multiple Rule for Cauchy Sequences in Normed Division Ring:
:$\sequence {x_n - \ds p^{-m} \sum_{i \mathop = 0}^n d_i p^i}$ is a null sequence
From Representatives of same P-adic Number iff Difference is Null Sequence:
:$\sequence {\ds p^{-m} \sum_{i \mathop = 0}^\infty d_i p^i}$ is a representative of $a$.
That is:
:$\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is a representative of $a$
where:
:$\forall i \ge -m: e_i = d_{i + m}$
By definition of a $p$-adic expansion;
:$\forall i \ge 0: 0 \le d_i < p$
Then:
:$\forall i \ge -m: 0 \le e_i = d_{i + m} < p$
Since $d_0 \neq 0$, then $e_{-m} = d_0 \neq 0$.
By definition of a $p$-adic expansion;
:$\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is a $p$-adic expansion that represents $a$
From P-adic Expansion Representative of P-adic Number is Unique:
:$\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is the only $p$-adic expansion that represents $a$
== Also see ==
* P-adic Number is Limit of Unique P-adic Expansion
* P-adic Integer has Unique Coherent Sequence Representative
* P-adic Integer has Unique P-adic Expansion Representative
== Sources ==
* : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
Category:P-adic Number Theory
P-adic Number is Limit of Unique P-adic Expansion

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p$.
Then $x$ is the limit of a unique $p$-adic expansion.
== Proof ==
Let $\Z_p$ be the $p$-adic integers.
From P-adic Number times Integer Power of p is P-adic Integer:
:$\exists m \in \Z_{\ge 0} : x p^m \in \Z_p$
From P-adic Integer is Limit of Unique P-adic Expansion, there exists a unique $p$-adic expansion of the form:
:$\ds \sum_{n \mathop = 0}^\infty d_n p^n$
such that:
:$\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = 0}^n d_i p^i = x p^m$
From Multiple Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = 0}^n d_i p^{i - m} = x$
Re-indexing from $-m$:
:$\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = -m}^n d_i' p^i = x$
where:
:$\forall i \in \Z_{\ge -m} : d_i' = d_{i + m}$
Then the $p$-adic expansion $\ds \sum_{n \mathop = -m}^\infty d_n' p^n$ converges to $x$.
Let $\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = -m}^n e_i p^i = x$.
From Multiple Rule for Sequences in Normed Division Ring:
:$\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = -m}^n e_i p^{i + m} = x p^m$
Re-indexing from $0$:
:$\ds \lim_{n \mathop \to \infty} \sum_{i \mathop = 0}^n e_i' p^i = x p^m$
where:
:$\forall i \in \Z_{\ge -m} : e_i' = e_{i - m}$
From P-adic Integer is Limit of Unique P-adic Expansion:
:$\forall i \in \Z_{\ge 0} : e_i' = d_i$
Re-indexing from $-m$:
:$\forall i \in \Z_{\ge -m} : e_i = d_i'$
Hence $\ds \sum_{n \mathop = -m}^\infty d_n' p^n$ is the unique $p$-adic expansion that converges to $x$.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$: Corollary $3.3.12$
Category:P-adic Number Theory
P-adic Number is Power of p Times P-adic Unit

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p^\times$ be the $p$-adic units.
Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the $p$-adic valuaton on the $p$-adic numbers.
Let $a \in \Q_p$.
Then there exists $u \in \Z_p^\times$ such that:
:$a = p^{\map {\nu_p} a} \cdot u$
== Proof ==
From P-adic Number times P-adic Norm is P-adic Unit, there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\times$
where
:$p^n = \norm a_p$
We have:
Let $u = p^n a$.
Then:
Category:P-adic Number Theory
Category:P-adic Units
P-adic Number times Integer Power of p is P-adic Integer

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Then:
:$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$
== Proof ==
Let $a \in \Q_p$.
=== Case: $\norm a_p \le 1$ ===
Let $\norm a_p \le 1$.
By definition of the $p$-adic integers:
:$a \in \Z_p$.
Hence:
:$p^0 a \in \Z_p$.
=== Case: $\norm a_p > 1$ ===
Let $\norm a_p > 1$.
From P-adic Number times P-adic Norm is P-adic Unit, there exists $n \in \Z$ such that $p^n a$ is a $p$-adic unit.
Then:
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$
Category:P-adic Number Theory
P-adic Number times P-adic Norm is P-adic Unit

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p^\times$ be the $p$-adic units.
Let $a \in \Q_p$.
Then there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\times$
where
:$p^n = \norm a_p$
== Proof ==
From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.
Let $n = -v$.
Now:
== Sources ==
* : $\S 1.5$ Arithmetical operations in $\Q_p$: Proposition $1.37$
Category:P-adic Number Theory
Category:P-adic Units
P-adic Numbers are Generated Ring Extension of P-adic Integers

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Then:
:$Q_p = \Z_p \sqbrk {1 / p}$
where $\Z_p \sqbrk {1 / p}$ denotes the ring extension generated by $1 / p$.
== Proof ==
Let $a \in \Q_p$.
From P-adic Number times Integer Power of p is P-adic Integer, there exists $n \in \N$ such that $p^n a \in \Z_p$.
Since $n \in \N$ and $p^n a \in \Z_p$, let $\map f X \in \Z_p \sqbrk X$ be the polynomial:
:$\paren {p^n a} X^n$
Then:
:$\map f {1 / p} = \paren {p^n a} \paren {1 / p}^n = a$.
Hence:
:$a \in \Z_p \sqbrk {1 / p}$.
Since $a \in \Q_p$ was arbitrary it follows that $\Q_p \subseteq \Z_p \sqbrk {1 / p }$.
By definition of a generated ring extension, $\Z_p \sqbrk {1 / p } \subseteq \Q_p$.
The result follows.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.5$
Category:P-adic Number Theory
P-adic Numbers are Uncountable

== Theorem ==
Let $p$ be any prime number.
The set of $p$-adic numbers $\Q_p$ is an uncountable set.
== Proof ==
Let $P$ be the set of sequences on $\set{i : i \in \N : 0 \le i < p}$.
That is:
:$P = \set{\sequence{d_n} : d_n \in \N : 0 \le d_n < p}$
From Cantor's Diagonal Argument:
:$P$ is an uncountable set
Let $f: P \to \Q_p$ be the mapping from $P$ to $\Z_p$ defined by:
:$\forall \sequence{d_n} \in P : \map f {\sequence{d_n}} = \ds \sum_{n = 0}^\infty d_n p^n$
where $Z_p$ denotes the $p$-adic integers and $\ds \sum_{n = 0}^\infty d_n p^n$ denotes a $p$-adic expansion
From P-adic Integer has Unique P-adic Expansion Representative:
:$f$ is bijective
Hence:
:$\Z_p$ is an uncountable set
Recall that $\Z_p \subseteq \Q_p$.
From Sufficient Conditions for Uncountability:
:$\Q_p$ is an uncountable set
== Sources ==
* : $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Exercise $20$
Category:P-adic Number Theory
P-adic Numbers form Completion of Rational Numbers with P-adic Norm

== Theorem ==
Let $p$ be a prime number.
Let $\norm {\,\cdot\,}^\Q_p$ be the $p$-adic norm on the rationals $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Then $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$
== Proof ==
Let $\norm {\,\cdot\,}^\Q_p$ be the $p$-adic norm on the rationals $\Q$.
From $p$-adic Norm on Rational Numbers is Non-Archimedean Norm:
:$\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ is a valued field with non-Archimedean norm $\norm {\,\cdot\,}_p$ 
By :
:$\Q_p$ is the quotient ring $\CC \, \big / \NN$
where:
:$\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
and
:$\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
By :
:$\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ is defined by:
::$\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}^\Q_p$
:where $\eqclass{x_n}{}$ denotes the left coset of $\sequence{x_n}$ in $\Q_p$
From Completion of Normed Division Ring:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
Category:P-adic Number Theory
P-adic Numbers is Hausdorff Topological Space

== 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 $\norm {\,\cdot\,}_p$.
Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff.
== Proof 1 ==
== Proof 2 ==
Category:Topology of P-adic Numbers
Category:P-adic Numbers is Hausdorff Topological Space
P-adic Numbers is Hausdorff Topological Space/Proof 1

== Theorem ==
== Proof ==
Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$.
By definition of the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$, $\tau_p$ is the topology induced by the metric $d_p$.
From Metric Space is Hausdorff, it follows that $\struct{\Q_p, \tau_p}$ is Hausdorff.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$
Category:P-adic Numbers is Hausdorff Topological Space
P-adic Numbers is Hausdorff Topological Space/Proof 2

== Theorem ==
== Proof ==
Let $x, y \in \Q_p$ such that $x \ne y$.
By :
:$r := \norm {x - y}_p > 0$
Then, for all $z\in\Q_p$ we have:
Therefore, the $r$-open balls $\map {B_r} x$ and $\map {B_r} y$ are disjoint.
Category:P-adic Numbers is Hausdorff Topological Space
P-adic Numbers is Locally Compact Topological Space

== 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 $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is locally compact.
== Proof ==
From Local Basis of P-adic Number:
:$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of $a$.
From Open and Closed Balls in P-adic Numbers are Compact Subspaces;
:$\forall a \in \Q_p$: the set $\set {\map {B_{p^{-n} } } a: n \in Z}$ is a local basis of compact sets of $a$.
Hence $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is locally compact by definition.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$: Corollary $3.3.8$
Category:Topology of P-adic Numbers
Category:Locally Compact Spaces
P-adic Numbers is Second Countable Topological Space

== 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 $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is second-countable.
== Proof ==
From Countable Basis for P-adic Numbers, the topological space $\struct {\Q_p, \tau_p}$ has a countable basis.
By definition, $\struct {\Q_p, \tau_p}$ is second-countable.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$
Category:Topology of P-adic Numbers
P-adic Numbers is Totally Disconnected Topological Space

== 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 $\norm {\,\cdot\,}_p$.
Then the topological space $\struct {\Q_p, \tau_p}$ is totally disconnected.
== Proof ==
From P-adic Numbers form Non-Archimedean Valued Field:
:$\norm {\,\cdot\,}_p$ is a non-Archimedean norm.
From Non-Archimedean Division Ring is Totally Disconnected:
:$\struct {\Q_p, \tau_p}$ is totally disconnected.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$
Category:Topology of P-adic Numbers
P-adic Open Ball is Instance of Open Ball of a Norm

== 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 $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $B \subseteq \Q_p$.
Then: 
:$B$ is an open ball in $p$-adic numbers with radius $\epsilon$ and centre $a$
:
:$B$ is an open ball of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$ .
That is, the definition of an open ball in $p$-adic numbers is a specific instance of the general definition of an open ball in a normed division ring.
== Proof ==
From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a non-Archimedean valued field.
The definition of an open ball in $p$-adic numbers is identical to the definition of an open ball of a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$.
Category:Open Balls 
Category:P-adic Number Theory
P-adic Sphere is Instance of Sphere of a Norm

== 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 $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $S \subseteq \Q_p$.
Then: 
:$S$ is a sphere in $p$-adic numbers with radius $\epsilon$ and centre $a$
:
:$S$ is a sphere of the normed division ring $\struct {\Q_p, \norm {\,\cdot\,}_p}$ with radius $\epsilon$ and centre $a$ 
That is, the definition of a closed ball in $p$-adic numbers is a specific instance of the general definition of a sphere in a normed division ring.
== Proof ==
From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a non-Archimedean valued field.
The definition of a sphere in $p$-adic numbers is identical to the definition of a sphere in a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$.
Category:P-adic Number Theory
P-adic Unit has Norm Equal to One

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ denote the $p$-adic integers.
Let $x \in \Q_p$.
Then x is a $p$-adic unit $\norm x_p = 1$
== Proof ==
=== Necessary Condition ===
Let $x$ be a $p$-adic unit.
Then:
:$x \in \Z_p$
:$x^{-1} \in \Z_p$
By definition of the $p$-adic integers:
:$\norm x_p \le 1$
:$\norm {x^{-1} }_p \le 1$
From Norm of Inverse in Division Ring:
:$\norm x_p \ge 1$
It follows that:
:$\norm x_p = 1$
=== Sufficient Condition ===
Let $\norm x_p = 1$.
From Norm of Inverse in Division Ring:
:$\norm {x^{-1} }_p = 1$
By definition of the $p$-adic integers:
:$x \in \Z_p$
:$x^{-1} \in \Z_p$
It follows that $x$ is an invertible element in $\Z_p$.
That is, x is a $p$-adic unit.
== Sources ==
* : $\S 3.3$ Exploring $\Q_p$
* : $\S 1.5$ Arithmetical operations in $\Q_p$: Exercise $26$
Category:P-adic Units