# Leigh.Samphier/Sandbox

## Common

RED

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

## $p$-adic Numbers

• Definitions related to P-adic Number Theory can be found here.

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.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $\Z_p$ be the $p$-adic integers for some prime $p$.

### Continuing Svetlana Katok Book

Equivalence of Definitions of P-adic Integer
P-adic Unit has Norm Equal to One

### Continuing Fernando Q. Gouvea Book

P-adic Unit has Norm Equal to One
Leigh.Samphier/Sandbox/Characterization of Rational P-adic Unit Lemma 3.3.4 (i) link between Lemma 3.3.4 and Lemma 3.3.4 (ii)
Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence

## Matroids

• Definitions related to Matroid Theory can be found here.
• Results about Matroid Theory can be found here.

Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.

Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.

### Properties of Independent Sets and Bases

All Bases of Matroid have same Cardinality

References:

## Stone Spaces

• Definitions related to Stone Spaces can be found here.
• Results about Stone Spaces can be found here.
Join Semilattice Ordered Subset Not Always Subsemilattice

## 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