# 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 It might be outdated or ideologically biased.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

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.

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