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

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

Common
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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.

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

Locally Connected iff Components of Open Subsets are Open

Equivalence of Definitions of Locally Path-Connected Space

Locally Path-Connected iff Path Components of Open Subsets are Open

Equivalence of Definitions of Locally Compact Hausdorff Space

Necessary and Sufficient Condition for Convergent Sequence (Metric Space)

Residue Field of P-adic Norm on Rationals/Lemma 1

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