Leigh.Samphier/Sandbox/Todo

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Common

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Matroids

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Definition:Parallel (Matroid)
Definition:Parallel (Matroid)

P-adic Numbers

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P-adic Number is Limit of Unique P-adic Expansion
P-adic Expansion Converges to P-adic Number


Next

Create theorem P-adic Numbers are Uncountable

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

Fix

Definition:Product Space (Topology)

Topological Product with Singleton

Criterion for Ring with Unity to be Topological Ring

Closure in Subspace

Missing Pages

Intersection is Largest Subset/Set of Sets

Equivalence of Definitions of Convergent Sequence in Topology

Strongly Locally Compact Hausdorff Space is Locally Compact & Definition:Strongly Locally Compact Space

Strongly Locally Compact Space may not be Locally Compact

Definition:Locally Metrizable Space

Basis of Countable Finite Complement Topology consists of Countably Infinite Sets

Equivalence of Definitions of Disconnected Set

Equivalence of Definitions of Quaternion Conjugate

Intersection of Compact Sets is Compact

Union of Compact Sets is Compact

P-Norm is Norm & P-Seminorm is Seminorm

P-Product Metrics are Topologically Equivalent

P-adic Valuation is Additive

Path Component is Closed

Quotient Topology is Topology

Definition:Quotient Topology/Definition 2

Triangle Inequality/Quaternions

Definition:Discontinuous (Topology)

Definition:Open Neighborhood (Metric Space)

Definition:Neighborhood of Real Point

Definition:Open Unit Ball

Definition:P-Sequence Norm

Definition:Point of Topological Space

Cleanup

Group Inverse Reverses Ordering in Ordered Group

Real and Imaginary Part Projections are Continuous

Definition:Euclidean Metric

Definition:Open Neighborhood/Real Analysis


Further Ideas

Special:WantedPages

Nagata-Smirnov Metrization Theorem, Stephen Willard - General Topology

Stone-Weierstrass Theorem, Stephen Willard - General Topology

Stone-Cech Compactification, Stephen Willard - General Topology

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

Definition:Frames & Locales

Gelfand-Naimark Theorem

Jordan Curve Theorem

Gelfand-Mazur Theorem