Understanding Topology

Topology is a branch of mathematics that explores the properties of shapes and spaces that remain unchanged when objects are stretched, twisted, or bent—but not torn or glued. Imagine playing with a lump of clay: you can mold it into a ball or stretch it into a donut shape, and topology helps us understand what these shapes have in common.

Topological Equivalence

In topology, two objects are considered the same (or "homeomorphic") if you can turn one into the other through continuous deformation, like squishing or stretching, without tearing or cutting.

What Topology Ignores

Topology doesn't care about exact measurements like length, angle, or area. Instead, it focuses on more flexible properties, such as the number of holes in a shape.

Fields Within Topology

• Point-set topology: Studies basic concepts like open and closed sets.
• Algebraic topology: Uses algebraic tools to study topological spaces.
• Geometric topology: Focuses on the properties of manifolds and higher-dimensional spaces.