User:Leigh.Samphier/Todo

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 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.4$
 * P-adic Integers is Metric Completion of Integers


 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$, Proposition $3.2.12$
 * Definition:P-adic Number

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Leigh.Samphier/Sandbox/P-adic Valuation Extends to P-adic Numbers

Common
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with zero $0_R$ and unity $1_R$.

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $p$ be a prime number.

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

Topology

 * : $\S 4$ Connectedness and Compactness, Proposition $4.1$

Missing Pages
Equivalence of Definitions of Weakly Locally Connected at Point

Cleanup
Necessary and Sufficient Condition for Convergent Sequence (Metric Space)

Group Inverse Reverses Ordering in Ordered Group

Real and Imaginary Part Projections are Continuous

Definition:Euclidean Metric

Definition:Open Neighborhood/Real Analysis

Add Definition:Normed Division Algebra to Norm

 * Definition:Normed Division Algebra

Further Ideas
Definition:Stone Space Stone's Representation Theorem for Boolean Algebras

Definition:Frames & Locales

Nagata-Smirnov Metrization Theorem

Stone-Weierstrass Theorem

Gelfand-Naimark Theorem

Jordan Curve Theorem

Gelfand-Mazur Theorem