# Homeomorphism

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Related to Topological equivalence: topology

## homeomorphism

[¦hō·mē·ə¦mȯr‚fiz·əm]## 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