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Homeomorphism


For homeomorphisms in graph theory, see Homeomorphism (graph theory).
Not to be confused with homomorphism.
"Topological equivalence" redirects here. For topological equivalence in dynamical systems, see Topological conjugacy.
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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