homeomorphism

In the mathematical field of

topology

, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the

isomorphism

s 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. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895.

Very roughly speaking, a topological space is a

geometric

object, and the homeomorphism is a continuous stretching and bending 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.

An often-repeated mathematical joke is that topologists cannot tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle.

Definition

A function $f : X \to Y$ between two topological spaces is a homeomorphism if it has the following properties:

* $f$ is a

bijection

(

one-to-one

and

onto

),
* $f$ is continuous,
* the inverse function $f^$ is continuous ($f$ is an open mapping).

A homeomorphism is sometimes called a bicontinuous function. If such a function exists, $X$ and $Y$ 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.

Examples

* The open interval $(a,b)$ is homeomorphic to the real numbers $\mathbf$ for any $a < b$. (In this case, a bicontinuous forward mapping is given by $f(x) = \frac + \frac$ while other such mappings are given by scaled and translated versions of the or functions).
* The unit 2-disc $D^2$ and the unit square in R² are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in

polar coordinates

, $(\rho, \theta) \mapsto \left( \frac, \theta\right)$.
* The

graph

of a

differentiable function

is homeomorphic to the domain of the function.
* A differentiable parametrization of a

curve

is a homeomorphism between the domain of the parametrization and the curve.
* A chart of a

manifold

is a homeomorphism between an open subset of the manifold and an open subset of a Euclidean space.
* The

stereographic projection

is a homeomorphism between the unit sphere in R³ with a single point removed and the set of all points in R² (a 2-dimensional plane).
* If $G$ is a topological group, its inversion map $x \mapsto x^$ is a homeomorphism. Also, for any $x \in G$, the left translation $y \mapsto xy$, the right translation $y \mapsto yx$, and the inner automorphism $y \mapsto xyx^$ are homeomorphisms.

Non-examples

* R^''m'' and R^''n'' are not homeomorphic for
* The Euclidean real line is not homeomorphic to the unit circle as a subspace of R^''2'', since the unit circle is compact as a subspace of Euclidean R^''2'' but the real line is not compact.
*The one-dimensional intervals ,1/math> and $(0,1)$ are not homeomorphic because no continuous bijection could be made.

Notes

The third requirement, that $f^$ be continuous, is essential. Consider for instance the function f : ,2\pi)_\to_S^1_(the_unit_circle_in_\mathbb^2)_defined_byf(\phi)_=_(\cos\phi,\sin\phi)._This_function_is_bijective_and_continuous,_but_not_a_homeomorphism_(S^1_is_compact_
Compact_as_used_in_politics_may_refer_broadly_to_a_pact_

A_pact,_from_Latin_''pactum''_("something_agreed_upon"),_is_a_formal_agreement._In_international_relations_

International_relations_(IR),_international_affairs_(IA)_or_internationa_...