User:Leigh.Samphier/Todo

Common
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$\begin{cases} 1 & : i = j \\ 0 & : i \ne j \end{cases}$

Last Link

 * Chapter $1.$ $\S 4.$ Loops and parallel elements
 * Definition:Parallel (Matroid)


 * Chapter $1.$ $\S 5.$ Properties of independent sets and bases
 * All Bases of Matroid have same Cardinality

Last Links

 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
 * P-adic Number is Limit of Unique P-adic Expansion


 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$, Proposition $1.29$
 * Unique Integers Close to Rationals in Valuation Ring of P-adic Norm

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 $4.1$

Fix
Definition:Product Space (Topology)

Topological Product with Singleton

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
Real and Imaginary Part Projections are Continuous

Definition:Euclidean Metric

Definition:Open Neighborhood/Real Analysis

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