 Homeomorphism
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Homeomorphism A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic -- only a continuous mapping with a continuous inverse function.

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 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. The word homeomorphism comes from the Greek words (homoios) = similar or same and (morph?) = shape, 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 can't 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 $f^{-1}$ 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 A trefoil knot is homeomorphic to a torus, but not isotopic in R3. Continuous mappings are not always realizable as deformations.
• The open interval ${\textstyle (a,b)}$ is homeomorphic to the real numbers ${\textstyle \mathbf {R} }$ for any ${\textstyle a . (In this case, a bicontinuous forward mapping is given by ${\textstyle f(x)={\frac {1}{a-x}}+{\frac {1}{b-x}}}$ while other such mappings are given by scaled and translated versions of the tan or arg tanh functions).
• The unit 2-disc ${\textstyle D^{2}}$ 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, $(\rho ,\theta )\mapsto \left({\frac {\rho }{\max(|\cos \theta |,|\sin \theta |)}},\theta \right)$ .
• 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 $x\mapsto x^{-1}$ 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 $y\mapsto xyx^{-1}$ are homeomorphisms.

### Non-examples

• Rm and Rn are not homeomorphic for
• The Euclidean real line is not homeomorphic to the unit circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact.
• The one-dimensional intervals $[0,1]$ and $(0,1)$ are not homeomorphic because no continuous bijection could be made.