Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Homeomorphism
Summary
In the mathematical field of topology, a homeomorphism (, 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 —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 the homeomorphism is 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 are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle. A homeomorphism that is a continuous deformation
is a homotopy.
A function between two topological spaces is a homeomorphism if it has the following properties:
is a bijection (one-to-one and onto),
is continuous,
the inverse function is continuous ( is an open mapping).
A homeomorphism is sometimes called a bicontinuous function. If such a function exists, and are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes.
The third requirement, that be continuous, is essential. Consider for instance the function (the unit circle in \R^2) defined by This function is bijective and continuous, but not a homeomorphism ( is compact but is not). The function is not continuous at the point because although maps to any neighbourhood of this point also includes points that the function maps close to but the points it maps to numbers in between lie outside the neighbourhood.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.