# Leigh.Samphier/Sandbox/Todo

## Contents

## Common

**Bold**

## Matroids

### Last Link

- 1976: Dominic Welsh:
*Matroid Theory*Chapter $1.$ $\S 4.$ Loops and parallel elements

- 1976: Dominic Welsh:
*Matroid Theory*Chapter $1.$ $\S 5.$ Properties of independent sets and bases

## P-adic Numbers

### Last Links

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*: $\S 1.4$ The field of $p$-adic numbers $\Q_p$, Proposition $1.29$

### 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

- 2000: John M. Lee:
*Introduction to Topological Manifolds*: $\S 4$ 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

Definition:Quotient Topology/Definition 2

Triangle Inequality/Quaternions

Definition:Discontinuous (Topology)

Definition:Open Neighborhood (Metric Space)

Definition:Neighborhood of Real Point

Definition:Point of Topological Space

### Cleanup

Real and Imaginary Part Projections are Continuous

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