The term "homeomorphic" is used in two different ways. Depending on context, it may mean
1. Possessing similarity of form, or
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.
An example of the term used in the "similarity of form" sense occurs in the definition of homeomorphic graphs.
