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. Such a 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.

Krantz, S. G. "The Concept of Homeomorphism." §6.4.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 86, 1999.

CITE THIS AS:

Weisstein, Eric W. "Homeomorphic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Homeomorphic.html