What does homeomorphism mean?
A homeomorphism in mathematics, especially topology, is a one-to-one correspondence between two topological spaces that is continuous in both directions. In other words, it pairs every point in one space with exactly one point in the other and preserves the spaces' topological structure. Because it does not need to preserve distances, angles, or size, a square and a circle can be related by a homeomorphism even though one has corners. The term is used to prove that spaces are topologically equivalent, classify shapes, and construct maps between mathematical objects. It is formal and specialized vocabulary, but it captures an important distinction: two figures may look different geometrically while having the same underlying topological form. A homeomorphism therefore expresses equivalence through a continuous, reversible correspondence rather than visual similarity.
nounA continuous one-to-one correspondence between two topological spaces that has a continuous inverse. Such a mapping preserves topological properties such as connectedness and compactness.
- A continuous bijection between two topological spaces whose inverse is also continuous, thereby establishing their topological equivalence.
"The theorem establishes a homeomorphism between the two spaces, showing that they have the same topological structure."
"The theorem proves that every continuous bijection with a continuous inverse is a homeomorphism."
"In algebraic geometry, the researcher used a homeomorphism to transfer the argument from one compact Riemann surface to another."
The regular plural adds -s to the singular form.
"The classification of these spaces depends on the relationships between their homeomorphisms."
A homeomorphism shows that a square and a circle are the same shape topologically, even though one has corners.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does homeomorphism come from?
The word is built from the Greek roots hómoios, meaning "similar" or "like," and morphḗ, meaning "form." Its components express the idea of preserving the same form or structure under a transformation. English usage dates from the 19th century, and the term became especially important in the development of topology and the formal study of continuous transformations.
Related word forms
How homeomorphism is actually used
This is specialized, formal mathematical vocabulary, used chiefly in topology and related fields. British speakers commonly use /ˌhəʊmiəˈmɔːfɪzəm/, while the transcription given is the usual American pronunciation.
Easily confused with homeomorphism
A homeomorphism is an isomorphism of topological spaces that is continuous and has a continuous inverse, whereas isomorphism is also used more broadly for structure-preserving bijections in algebra and other fields.
A homeomorphism concerns topology and preserves topological structure, while a homomorphism usually preserves algebraic structure, such as operations in a group or ring.