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 * ὅμοιος * (*homoios*) = similar or same and * μορφή * (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.^{ [1] }^{ [2] }

- Definition
- Examples
- Non-examples
- Notes
- Properties
- Informal discussion
- See also
- References
- External links

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,^{ [3] } 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.

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 open interval is homeomorphic to the real numbers for any . (In this case, a bicontinuous forward mapping is given by while other such mappings are given by scaled and translated versions of the tan or arg tanh functions).
- The unit 2-disc and the unit square in
**R**^{2}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, . - 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
**R**^{3}with a single point removed and the set of all points in**R**^{2}(a 2-dimensional plane). - If is a topological group, its inversion map is a homeomorphism. Also, for any , the left translation , the right translation , and the inner automorphism are homeomorphisms.

**R**^{m}and**R**^{n}are not homeomorphic for*m*≠*n*.- 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 and are not homeomorphic because no continuous bijection could be made.
^{ [4] }

The third requirement, that be continuous, is essential. Consider for instance the function (the unit circle in ) 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.^{ [5] }

Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms forms a group, called the ** homeomorphism group ** of *X*, often denoted . This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a topological group.^{ [6] }

For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group.

Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, is a torsor for the homeomorphism groups and , and, given a specific homeomorphism between and , all three sets are identified.

- Two homeomorphic spaces share the same topological properties. For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homotopy and homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces that are homeomorphic even though one of them is complete and the other is not.
- A homeomorphism is simultaneously an open mapping and a closed mapping; that is, it maps open sets to open sets and closed sets to closed sets.
- Every self-homeomorphism in can be extended to a self-homeomorphism of the whole disk (Alexander's trick).

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. Homotopy does lead to a relation on spaces: homotopy equivalence.

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*.

- Local homeomorphism
- Diffeomorphism – Isomorphism of smooth manifolds; a smooth bijection with a smooth inverse
- Uniform isomorphism – Uniformly continuous homeomorphism is an isomorphism between uniform spaces
- Isometric isomorphism is an isomorphism between metric spaces
- Homeomorphism group
- Dehn twist
- Homeomorphism (graph theory) (closely related to graph subdivision)
- Homotopy#Isotopy
- Mapping class group – Group of isotopy classes of a topological automorphism group
- Poincaré conjecture – Theorem in geometric topology
- Universal homeomorphism

In mathematics, a **diffeomorphism** is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

In the mathematical field of algebraic topology, the **fundamental group** of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, specifically algebraic topology, a **covering map** is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood **evenly covered** by . In this case, is called a **covering space** and the **base space** of the covering projection. The definition implies that every covering map is a local homeomorphism.

In topology, a branch of mathematics, a **fibration** is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

In mathematical analysis, a **space-filling curve** is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called *Peano curves*, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In geometric topology, a branch of mathematics, a **Dehn twist** is a certain type of self-homeomorphism of a surface.

In topology, especially algebraic topology, the **cone****of a topological space** is the quotient space:

In mathematics, in the subfield of geometric topology, the **mapping class group** is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

In mathematics, particularly topology, the **homeomorphism group** of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups.

In mathematics, specifically algebraic topology, the **mapping cylinder** of a continuous function between topological spaces and is the quotient

In mathematics, in particular homotopy theory, a continuous mapping

In mathematics, the **Teichmüller space** of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from to itself.

In mathematics, the **homotopy category** is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In algebraic topology, an area of mathematics, a **homeotopy group** of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

In mathematics, and more precisely in topology, the **mapping class group** of a surface, sometimes called the **modular group** or **Teichmüller modular group**, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

- ↑ "Analysis Situs selon Poincaré (1895)".
*serge.mehl.free.fr*. Archived from the original on 11 June 2016. Retrieved 29 April 2018. - ↑ Gamelin, T. W.; Greene, R. E. (1999).
*Introduction to Topology*. Courier. p. 67. - ↑ Hubbard, John H.; West, Beverly H. (1995).
*Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems*. Texts in Applied Mathematics.**18**. Springer. p. 204. ISBN 978-0-387-94377-0. - ↑ "Continuous bijection from (0,1) to [0,1]".
*Mathematics Stack Exchange*. 2011-06-01. Retrieved 2019-04-02. - ↑ Väisälä, Jussi:
*Topologia I*, Limes RY 1999, p. 63. ISBN 951-745-184-9. - ↑ Dijkstra, Jan J. (1 December 2005). "On Homeomorphism Groups and the Compact-Open Topology" (PDF).
*The American Mathematical Monthly*.**112**(10): 910. doi:10.2307/30037630. Archived (PDF) from the original on 16 September 2016.

- "Homeomorphism",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - "Homeomorphism".
*PlanetMath*.

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