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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Note that if there exists a homeomorphism h : ([P.sup.2], [C.sup.1]) [right arrow] ([P.sup.2], [C.sup.2]), h(E) = E always holds.
The mapping [[tau].sub.C] is a homeomorphism. The pair (V, [[tau].sub.C]) defines a complex chart for the regular part in a neighborhood of the singular point [x.sub.1] = [chi]([m.sub.1]).
Then, it is easy to check that h is homeomorphism and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Let (X, [tau]) and (Y, [tau]) be two topological spaces, A([not equal to] [empty set]) [subset not equal to] X, and f: (X, [tau]) [right arrow] (Y, [tau]) a homeomorphism. If (X, [tau]) is [A]-[theta]-paracompact then (Y, [theta]) is [f(A)][theta]-paracompact.
Thus, we define the homeomorphism [PSI] that corresponds to H
there exist a homeomorphism : [X.sub.A.sub.(n)] [right arrow] [X.sub.A] such that [Mathematical Expression Omitted].
Let H be the homeomorphism and lattice isomorphism from ([I.sub.P],[T.sub.P]) onto the dense subspace of (C(Y),k) defined in the previous section.
It is clear that by Remark 3.1, {[A.sub.1]} x [S.sub.2] is g[ALEPH] homeomorphism to [S.sub.2] and hence by Remark 3.2, {[A.sub.1]} x [S.sub.2] is g[ALEPH] compact.
This study showed that normalization of probabilistic quality measures with a homographic homeomorphism is more powerful than the normalization homeomorphism refines initiated by Andre Totohasina.
In particular, they are not conjugate to the uniform cover case via a homeomorphism. We show that, with at most countable exceptions, every irrational rotation number can be realized when the degree is two.
Topology ideally aims to find the homeomorphism type of a topological space.