User:Leigh.Samphier/Todo

P-adic last links

 * : $\S 3.3$ 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

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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 $4.1$

Uses of Local Basis for Review

 * Leigh.Samphier/Sandbox/Definition:Weakly Locally Connected Space - Weekly locally connected at a point definitions are equivalent. Weekly locally connected at a point Is different to Locally connected at a point. Weekly locally connected space is Locally connected space.


 * Leigh.Samphier/Sandbox/Topology is Locally Compact iff Ordered Set of Topology is Continuous

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