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In the mathematical field of topology, a **homeomorphism** (from Greek * *ὅμοιος* (homoios)* 'similar, same', and * *μορφή* (morphē)* 'shape, form', named by Henri Poincaré^{[2]}^{[3]}), also called **topological isomorphism**, or **bicontinuous function**, is a bijective and 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.

Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous deformation 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. A homeomorphism that is a continuous deformation is a homotopy.

**^**Hubbard, John H.; West, Beverly H. (1995).*Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems*. Texts in Applied Mathematics. Vol. 18. Springer. p. 204. ISBN 978-0-387-94377-0.**^**Poincaré, H. (1895).*Analysis Situs*. Journal de l'Ecole polytechnique. Gauthier-Villars. OCLC 715734142. Archived from the original on 11 June 2016. Retrieved 29 April 2018.

Poincaré, Henri (2010).*Papers on Topology: Analysis Situs and Its Five Supplements*. Translated by Stillwell, John. American Mathematical Society. ISBN 978-0-8218-5234-7.**^**Gamelin, T. W.; Greene, R. E. (1999).*Introduction to Topology*(2nd ed.). Dover. p. 67. ISBN 978-0-486-40680-0.