File:Blue_Trefoil_Knot.png · Wikimedia Commons · See Wikimedia Commons
homeomorphism
Sign in to saveAlso known as homeomorphic, homeomorphic map, topological equivalence, bi continuous function
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
A homeomorphism is a special kind of mathematical mapping between shapes or spaces that preserves all their topological properties—the qualities that remain unchanged even when a shape is stretched or bent (but not torn). Mathematicians consider two spaces connected by a homeomorphism to be essentially the same from a topological perspective, which makes homeomorphisms useful for understanding which shapes are fundamentally equivalent despite their different appearances.
AI-generated from the Wikipedia summary — may contain errors.
Wikidata facts
- Image
- Homeo tasse.png
Show 2 more facts
- Commons category
- Homeomorphisms
- Commons gallery
- Homeomorphism
Sources (1)
via Wikidata · CC0
~8 min read
Article
8 sectionsContents
- Definition
- Examples
- Counter-examples
- Properties
- Informal discussion
- See also
- References
- External links
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as the homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous deformation.