##### Douglas Harper's Etymology Dictionary

**homeomorphism**

1854, from *homeomorphous* (1832), from homeo- + *morphous* (see metamorphosis); originally of crystals. *Homeomorphic* is from 1902.

##### Wiktionary

**homeomorphism**

n. 1 (context topology English) a continuous bijection from one topological space to another, with continuous inverse. 2 (context chemistry English) a similarity in the crystal structure of unrelated compounds

##### Wikipedia

**Homeomorphism**

In the mathematical field of topology, a **homeomorphism** or **topological isomorphism** or **bi continuous 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 and *μορφή* (*morphē*) = shape, form.

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. An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, 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 a cup's handle.

**Homeomorphism (graph theory)**

In graph theory, two graphs *G* and *G*ʹ are **homeomorphic** if there is a graph isomorphism from some **subdivision** of *G* to some **subdivision** of *G*ʹ. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.