Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Chemistry A close similarity in the crystal forms of unlike compounds.
 n. Mathematics A continuous bijection between two figures whose inverse is also continuous.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. a continuous bijection from one topological space to another, with continuous inverse.
 n. a similarity in the crystal structure of unrelated compounds
from the GNU version of the Collaborative International Dictionary of English
 n. A near similarity of crystalline forms between unlike chemical compounds. See isomorphism.
from The Century Dictionary and Cyclopedia
 n. Similarity in crystalline form, but not necessarily in chemical composition.
 n. Same as isomorphism.
 n. Also homeomorphism.
Etymologies
Examples

A homeomorphism is, essentially, a onetoone correspondence (see any maths site for details).
On Thursday, the Legg report will be published along with...

One is that it is a homeomorphism invariant: if two spaces are homeomorphic, then they have the same fundamental group.

The topology on the countable product of the twopoint space '' D '' is induced by the metric The Cantor set may be embedded in the unit interval by the map which is a homeomorphism onto the subset of the unit interval obtained by iteratively deleting the middle third of each interval compact.

The Cantor set is [[homeomorphism  homeomorphic]] to a product of [[countable set  countably]] many copies of a twopoint space with the [[discrete metric]].

While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension it is possible to construct objects which are homeomorphic and not diffeomorphic  such objects are called "exotic
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