Homeomorphism

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré),[2][3] 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.

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.

Homotopy and isotopy are precise definitions for the informal concept of continuous deformation.

A self-homeomorphism is a homeomorphism from a topological space onto itself.

Being "homeomorphic" is an equivalence relation on topological spaces.

This function is bijective and continuous, but not a homeomorphism (

but the points it maps to numbers in between lie outside the neighbourhood.

[4] Homeomorphisms are the isomorphisms in the category of topological spaces.

[5] In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other.

Homotopy and isotopy are equivalence relations that have been introduced for dealing with such situations.

Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them,

[clarification needed] The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance.

It is thus important to realize that it is the formal definition given above that counts.

In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only a finite number of points, including a single point.

This characterization of a homeomorphism often leads to a confusion with the concept of homotopy, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another.

In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y—one just follows them as X deforms.

In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto.

There is a name for the kind of deformation involved in visualizing a homeomorphism.

It is (except when cutting and regluing are required) an isotopy between the identity map on X and the homeomorphism from X to Y.