A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions. A homeomorphism which also preserves distances is called an isometry. Affine transformations are another type of common geometric homeomorphism.
The similarity in meaning and form of the words "homomorphism" and "homeomorphism" is unfortunate and a common source of confusion.
See also
Affine Transformation, Homeomorphic, Homeomorphic Graphs, Homeomorphic Type, Homeomorphism Group, Homomorphism, Isometry, Module Homomorphism, Structure Homomorphism, Topologically Conjugate Explore this topic in the MathWorld classroom
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References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 101, 1967.Krantz, S. G. "The Concept of Homeomorphism." §6.4.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 86, 1999.Ore, Ø. Graphs and Their Uses. New York: Random House, 1963.
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Cite this as:
Weisstein, Eric W. "Homeomorphism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Homeomorphism.html