Britannica Sources used in this answer

Why is every simple polygon homeomorphic to a circle?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, a homeomorphism is a continuous, one-to-one mapping between two objects that possesses a continuous inverse. A simple polygon is a closed curve composed of line segments that do not intersect themselves. Because it is a continuous, non-self-intersecting closed loop, a simple polygon qualifies as a simple closed curve, also known as a Jordan curve.123

In the context of topology, properties such as "straightness" or the number of sides are not preserved, meaning they are not topological invariants. Instead, a simple polygon and a circle share the same fundamental topological structure: both are connected sets that remain connected if one point is removed but become disconnected if two points are removed. Because they share these essential properties and can be continuously deformed into one another without tearing or gluing, any simple polygon is homeomorphic to a circle.1