Introduction

Topology is the branch of mathematics that studies the properties of shapes and spaces that stay the same under continuous stretching, bending, or twisting, as long as nothing is torn or glued together. Because of this, it is often called “rubber-sheet geometry,” where the focus is not on exact lengths or angles but on deeper features like connectedness, boundaries, and the number of holes. A classic example is that a circle and a square are considered the same topologically because one can be smoothly deformed into the other, while a figure-eight is different because its basic structure is not the same.

Contents

General Topology
Algebraic Topology

Low-Dimensional Topology ^2fc8e9

Knot Theory
Morse Theory
Topological Quantum Field Theory