# Homeomorphism

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.[1][2]

Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.

An often-repeated mathematical joke is that topologists cannot tell the difference between a coffee cup and a donut,[3] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle.