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 ?
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same combinatorics, then any homeomorphism h : ([T.
t,u,v]))) for all f : Z x X [right arrow] Y is a neutrosophic homeomorphism.
2]) is called a SBT homeomorphism if it has the following properties:
every left and right translations are semi homeomorphism.
Definition 5 (Skorohod metric) Let [LAMBDA] be the collection of all homeomorphisms (10) [lambda]: [0,1] [right arrow] [0,1] with [lambda](0) = 0 and [lambda](1) = 1.
n-m] respectively, so H is a topological group homeomorphism.
A homeomorphism, also known as a topological isomorphism, is the most basic space of this kind, revealing a continuous inverse function.
The notion of homeomorphism, applied in synergetic theory to describe the qualitative behaviour of gradual processes which show sudden discontinuities in their development (Bernardez, 1995; Haken, 1977; Thom, 1972; Prigogine, 1983; Wildgen, 1994) may help further explain the structure of the category.
LA] will also require that the token relation be a homeomorphism between [concat.
is a Caratheodory function and [empty set]: R [right arrow] R is an increasing homeomorphism such that [empty set](0) = 0.
We also recall classical homeomorphism results between probability spaces and the basic model infinite dimensional manifolds.
Serres notes the homeomorphism that runs across the series: