There are two possible definitions:
1. Possessing similarity of form,
2. Continuous, one-to-one, in surjection, 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.
