# Homeomorphism

(redirected from Self-homeomorphism)
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## homeomorphism

[¦hō·mē·ə¦mȯr‚fiz·əm]
(mathematics)
A continuous map between topological spaces which is one-to-one, onto, and its inverse function is continuous. Also known as bicontinuous function; topological mapping.

## Homeomorphism

one of the basic concepts of topology. Two figures (more precisely, two topological spaces) are said to be homeomorphic if there exists a one-to-one continuous mapping of any one onto the other, for which the inverse mapping is also continuous. In this case, the mapping itself is called a homeomorphism. For example, any circle is homeomorphic to any square; any two segments are homeomorphic, but a segment is not homeomorphic to a circle or a line. A line is homeomorphic to any interval (that is, a segment without end points). The concept of homeomorphism is the basis for defining the extremely important concept of a topological property. Specifically, a property of a figure F is said to be topological if it is found in all figures homeomorphic to F. Examples of topological properties are compactness and connectedness.

A. V. ARKHANGEL’SKII

References in periodicals archive ?
Based upon our calculations, the flat manifolds in Cases 3-5 have the property that they are not of Jiang-type but every self-homeomorphism has zero Nielsen number while N(f) = [absolute value of (L(f))] (see e.
In this section, we compute the Nielsen numbers of self-homeomorphisms of flat manifolds in the remaining 5 cases, 6-10.
The right stabilizer of f in Homeo [sub.+][0,1] contains a subgroup isomorphic to Homeo [sub.+][0,1], namely, a subgroup of self-homeomorphisms of the little interval [f.sup.-1] (x) preserving the endpoints and extended by the identity on the outside of [f.sup.-1] (x).

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