Homeomorphisms

Homeomorphisms are continuous functions with continuous inverses, establishing topological equivalence between spaces.

Definition

A function \(f: X \to Y\) is a homeomorphism if it is bijective, continuous, and has a continuous inverse \(f^{-1}\).

Examples

Stretching a circle into an ellipse is a homeomorphism; cutting it into a line segment is not.

Significance

Homeomorphisms classify spaces: a coffee cup and a donut are homeomorphic (one hole), but not a sphere.