An often-repeated mathematical joke is that topologists cannot tell the difference between a coffee mug and a donut,[1] since a sufficiently pliable donut could be reshaped to the form of a coffee mug by creating a dimple and progressively enlarging it, while preserving the donut hole in the mug's handle. This illustrates that a coffee mug and a donut (torus) are homeomorphic.

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.

  1. ^ 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.
  2. ^ 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.
  3. ^ Gamelin, T. W.; Greene, R. E. (1999). Introduction to Topology (2nd ed.). Dover. p. 67. ISBN 978-0-486-40680-0.

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