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There are two possible definitions:

1. Possessing similarity of form,

2. Continuous, One-to-One, Onto, and having a continuous inverse.

The most common meaning is possessing intrinsic topological equivalence. Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Homeomorphism ignores the space in which surfaces are embedded, so the deformation can be completed in a higher dimensional space than the surface was originally embedded. Mirror Images are homeomorphic, as are Möbius Strip with an Even number of half-twists, and Möbius Strip with an Odd number of half-twists.

In Category Theory terms, homeomorphisms are Isomorphisms in the Category of Topological Spaces and continuous maps.

See also Homomorphic, Polish Space

© 1996-9 Eric W. Weisstein