Homeomorphism

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Related to Homeomorphisms: Homeomorphism group

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 ?
Sundaram studied generalized continuous functions and generalized homeomorphism.
Piotrowski (1979) further investigated semi homeomorphisms and Noiri & Ahmad (1985) introduced semi-weakly continuous functions and these functions were further characterized by Dorsett (1990).
Among the topics are Branner-Hubbard motions and attracting dynamics, examples of Feigenbaum Julia sets with small Hausdorff dimension, Sierpinski carpets and gaskets as Julia sets of rational maps, homeomorphisms of the Mandelbrot set, Arnold disks and the moduli of Herman rings of the complex standard family, and stretching rays and their accumulations following Pia Willumsen.
KELLOGG, On homeomorphisms for an elliptic equation in domains with corners,, Differential Integral Equations, 8 (1995), pp.
Nagaveni, Studies on generalizations on homeomorphisms in topological spaces, Ph.
Maps are defined up to orientation-preserving homeomorphisms.
Among the topics are stationary dynamical systems, the expansion of rational numbers in Mobius number systems, horospheres and Farey fractions, ergodic abelian actions with homogeneous spectrum, the geometric entropy of geodesic currents on free groups, statistics of matrix products in hyperbolic geometry, and the infinite sequence of fixed point free pseudo-Anosov homeomorphisms on a family of genus two surface.