# Leigh.Samphier/Sandbox

RED

## Contents

## $p$-adic Numbers

### Continuing Svetlana Katok Book

Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique

Leigh.Samphier/Sandbox/Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion

### Every P-adic Number is Limit of P-adic Expansion

P-adic Number is Limit of Unique P-adic Expansion - Complete the uniqueness

Leigh.Samphier/Sandbox/Set of P-adic Units is Unit Sphere

Leigh.Samphier/Sandbox/Characterization of Rational P-adic Units

Leigh.Samphier/Sandbox/P-adic Expansion of P-adic Units

## Matroids

### Loops and Parallel Elements

To be completed.

### Properties of Independent Sets and Bases

Leigh.Samphier/Sandbox/Matroid satisfies Base Axiom

Leigh.Samphier/Sandbox/Alternative Axiomatization of Matroid

## Topology

### Fix Product Space Issues

Leigh.Samphier/Sandbox/Product Local Basis Induced from Factor Spaces Local Bases

Leigh.Samphier/Sandbox/Countable Product of First-Countable Spaces is First-Countable

Leigh.Samphier/Sandbox/Countable Product of Separable Spaces is Separable

Leigh.Samphier/Sandbox/Factor Spaces are T4 if Product Space is T4

Leigh.Samphier/Sandbox/Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact

Leigh.Samphier/Sandbox/Finite Product Space is Connected iff Factors are Connected

Leigh.Samphier/Sandbox/Finite Product Space is Connected iff Factors are Connected/General Case

Leigh.Samphier/Sandbox/Metric is Continuous

Leigh.Samphier/Sandbox/Paracompactness is Preserved under Projections

Leigh.Samphier/Sandbox/Points in Product Spaces are Near Open Sets

Leigh.Samphier/Sandbox/Product of Hausdorff Factor Spaces is Hausdorff

Leigh.Samphier/Sandbox/Product of Hausdorff Factor Spaces is Hausdorff/General Result

Leigh.Samphier/Sandbox/Product Space is T2 iff Factor Spaces are T2

Leigh.Samphier/Sandbox/Product Space is T3 1/2 iff Factor Spaces are T3 1/2

Leigh.Samphier/Sandbox/Product Space is T3 iff Factor Spaces are T3

Leigh.Samphier/Sandbox/Products of Products are Homeomorphic to Collapsed Products

Leigh.Samphier/Sandbox/Subspace of Product Space Homeomorphic to Factor Space

Leigh.Samphier/Sandbox/Topological Product with Singleton

Leigh.Samphier/Sandbox/Tychonoff's Theorem

Leigh.Samphier/Sandbox/Tychonoff's Theorem Without Choice

Leigh.Samphier/Sandbox/Product Space is Product in Category of Topological Spaces