## Isotopy, Ambient Isotopy, Homotopy, Retracts, and Deformation Retracts

Isotopic Embeddings on Topological Spaces

Isotopic and Non-Isotopic Embeddings on the Bounded Cone

Ambient Isotopic Embeddings on Topological Spaces

Homotopic Mappings Relative to a Subset of a Topological Space

Homotopically Equivalent Topological Spaces

Homeomorphic Topological Spaces are Homotopically Equivalent

Retract Subspaces of a Topological Space

Deformation Retract Subspaces of a Topological Space

## Fundamental Groups of a Topological Space

Products of Paths Relative to {0, 1} in a Topological Space

Associativity of Products of Paths in a Topological Space

Constant Paths in a Topological Space

Inverse Paths in a Topological Spaces

The Fundamental Group of a Topological Space at a Point

Fundamental Groups under Homeomorphisms on Topological Spaces

Simply Connected Topological Spaces

The Fundamental Groups of a Path Connected Topological Space are Isomorphic

The Fundamental Groups of Discrete and Indiscrete Topological Spaces

The Induced Mapping from the Fundamental Groups of Two Topological Spaces

The Fundamental Group of a Topological Product

The Fundamental Group of the Cylinder and the Torus

An Example of a Retract that is NOT a Deformation Retract

## Brouwer's Fixed Point Theorem

Brouwer's Fixed Point Theorem

Solutions to Systems of Equations by Brouwer's Fixed Point Theorem

The Jordan Curve Theorem

## Group Presentations

Words Over a Set

Group Presentations

A Group Presentation for Zn

A Group Presentation for S3

Tietze Transformations

Tietze Transformations Example 1

Tietze Transformations Example 2

The Free Product of Groups

## The Seifert-van Kampen Theorem

The Seifert-van Kampen Theorem

The Seifert-van Kampen Theorem Example 1

The Seifert-van Kampen Theorem Example 2

The Seifert-van Kampen Theorem Example 3