# Difference between revisions of "Homeomorphism"

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[[Image:Mug and Torus morph.gif|thumb|right|A continuous deformation between a coffee [[mug]] and a [[torus | [[Image:Mug and Torus morph.gif|thumb|right|A continuous deformation between a coffee [[mug]] and a donut ([[torus]]) illustrating that they are homeomorphic. But there need not be a [[Homotopy|continuous deformation]] for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.]] | ||

In the [[mathematics|mathematical]] field of [[topology]], a '''homeomorphism''' or '''topological isomorphism''' or ''' | In the [[mathematics|mathematical]] field of [[topology]], a '''homeomorphism''' or '''topological isomorphism''' or '''bi continuous 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 [[map (mathematics)|mappings]] that preserve all the [[topological property|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 language|Greek]] words ''[[wikt:ὅμοιος|ὅμοιος]]'' (''homoios'') = similar and ''[[wikt:μορφή|μορφή]]'' (''morphē'') = shape, form. | ||

Roughly speaking, a topological space is a [[geometry|geometric]] object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus | Roughly speaking, a topological space is a [[geometry|geometric]] object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus]] are not. An often-repeated [[mathematical joke]] is that topologists can't tell their coffee cup from their donut,<ref>{{cite book|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|first1=John H.|last1=Hubbard|first2=Beverly H.|last2=West|publisher=Springer|series=Texts in Applied Mathematics|volume=18|year=1995|isbn=978-0-387-94377-0|page=204|url=http://books.google.com/books?id=SHBj2oaSALoC&pg=PA204&dq=%22coffee+cup%22+topologist+joke#v=onepage&q=%22coffee%20cup%22%20topologist%20joke&f=false}}</ref> 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 a cup's handle. | ||

Topology is the study of those properties of objects that do not change when homeomorphisms are applied. | Topology is the study of those properties of objects that do not change when homeomorphisms are applied. | ||

==Definition== | ==Definition== | ||

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==Examples== | ==Examples== | ||

[[Image:Trefoil knot arb.png|thumb|right|240|A [[trefoil knot]] is homeomorphic to a circle. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.]] | [[Image:Trefoil knot arb.png|thumb|right|240|A [[trefoil knot]] is homeomorphic to a circle, but not [[Homotopy#Isotopy|isotopic]]. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.]] | ||

* The unit 2-[[ball (mathematics)|disc]] D<sup>2</sup> and the [[unit square]] in '''R'''<sup>2</sup> are homeomorphic. | * The unit 2-[[ball (mathematics)|disc]] D<sup>2</sup> and the [[unit square]] in '''R'''<sup>2</sup> are homeomorphic.{{clarification needed|date=December 2014}} | ||

* The open [[interval (mathematics)|interval]] (a, b) is homeomorphic to the [[real number]]s '''R''' for any a < b. | * The open [[interval (mathematics)|interval]] (a, b) is homeomorphic to the [[real number]]s '''R''' for any a < b. (In this case, a bicontinuous forward mapping is given by {{math|''f''(''x'') {{=}} 1/(''x'' − ''a'') + 1/(''x'' − ''b'')}} while another such mapping is given by a scaled and translated version of the {{math|tan}} function). | ||

* The [[product topology|product space]] [[Sphere|S<sup>1</sup>]] × S<sup>1</sup> and the two-[[dimension]]al [[torus]] are homeomorphic. | * The [[product topology|product space]] [[Sphere|S<sup>1</sup>]] × S<sup>1</sup> and the two-[[dimension]]al [[torus]] are homeomorphic.{{clarification needed|date=December 2014}} | ||

* Every [[uniform isomorphism]] and [[isometric isomorphism]] is a homeomorphism. | * Every [[uniform isomorphism]] and [[isometric isomorphism]] is a homeomorphism. | ||

* The [[ | * The [[stereographic projection]] is a homeomorphism between the unit sphere in '''R'''<sup>3</sup> with a single point removed and the set of all points in '''R'''<sup>2</sup> (a 2-dimensional [[plane (mathematics)|plane]]). | ||

* | * If ''A'' is a commutative ring with unity, and ''S'' is a multiplicative subset of ''A'', then the [[spectrum of a ring|spectrum]] Spec(''A''<sub>''S''</sub>) is homeomorphic to {{nowrap|1={''p'' ∈ Spec(''A'') : ''p'' ∩ ''S'' = ∅}.}} | ||

* '''R'''<sup>''m''</sup> and '''R'''<sup>''n''</sup> are not homeomorphic for {{nowrap|1=''m'' ≠ ''n''.}} | * '''R'''<sup>''m''</sup> and '''R'''<sup>''n''</sup> are not homeomorphic for {{nowrap|1=''m'' ≠ ''n''.}} | ||

* The Euclidean [[real line]] is not homeomorphic to the unit circle as a subspace of '''R'''<sup>''2''</sup> as the unit circle is compact as a subspace of Euclidean '''R'''<sup>''2''</sup> but the real line is not compact. | * The Euclidean [[real line]] is not homeomorphic to the unit circle as a subspace of '''R'''<sup>''2''</sup> as the unit circle is compact as a subspace of Euclidean '''R'''<sup>''2''</sup> but the real line is not compact. | ||

* If G is a [[topological group]], its inversion map <math>x \mapsto x^{-1}</math> is a homeomorphism. Also, for any <math>x \in G</math>, the left translation <math>y \mapsto xy</math>, the right translation <math>y \mapsto yx</math>, and the inner automorphism <math>y \mapsto xyx^{-1}</math> are homeomorphisms. | |||

==Notes== | ==Notes== | ||

The third requirement, that ''f''<sup> −1</sup> be continuous, is essential. Consider for instance the function ''f'': <nowiki>[0, 2π)</nowiki> → S<sup>1</sup> defined by ''f''(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism (S<sup>1</sup> is compact but <nowiki>[0, 2π)</nowiki> is not). | The third requirement, that ''f''<sup> −1</sup> be continuous, is essential. Consider for instance the function ''f'': <nowiki>[0, 2π)</nowiki> → S<sup>1</sup> (the [[unit circle]] in <math>\mathbb{R}^2</math>) defined by ''f''(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism (S<sup>1</sup> is compact but <nowiki>[0, 2π)</nowiki> is not). The function ''f''<sup> −1</sup> is not continuous at the point (1, 0), because although ''f''<sup> −1</sup> maps (1, 0) to 0, any neighbourhood of this point also includes points that the function maps close to 1, but the points it maps to numbers in between lie outside the neighbourhood.<ref>Väisälä, Jussi: ''Topologia I'', Limes RY 1999, p. 63. ISBN 951-745-184-9.</ref> | ||

Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ''X'' → ''X'' forms a [[group (mathematics)|group]], called the '''[[homeomorphism group]]''' of ''X'', often denoted Homeo(''X''); this group can be given a topology, such as the [[compact-open topology]], making it a [[topological group]]. | Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ''X'' → ''X'' forms a [[group (mathematics)|group]], called the '''[[homeomorphism group]]''' of ''X'', often denoted Homeo(''X''); this group can be given a topology, such as the [[compact-open topology]], making it a [[topological group]]. | ||

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

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, Homeo(''X,'' ''Y''), is a [[torsor]] for the homeomorphism groups Homeo(''X'') and Homeo(''Y''), and given a specific homeomorphism between ''X'' and ''Y'', all three sets are identified. | Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, Homeo(''X,'' ''Y''), is a [[torsor]] for the homeomorphism groups Homeo(''X'') and Homeo(''Y''), and given a specific homeomorphism between ''X'' and ''Y'', all three sets are identified. | ||

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*[[Dehn twist]] | *[[Dehn twist]] | ||

*[[Homeomorphism (graph theory)]] (closely related to graph subdivision) | *[[Homeomorphism (graph theory)]] (closely related to graph subdivision) | ||

*[[Isotopy]] | *[[Homotopy#Isotopy]] | ||

*[[Mapping class group]] | *[[Mapping class group]] | ||

*[[Poincaré conjecture]] | *[[Poincaré conjecture]] | ||

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[[Category:Homeomorphisms| ]] | [[Category:Homeomorphisms| ]] | ||

[[Category:Functions and mappings]] | [[Category:Functions and mappings]] | ||

## Latest revision as of 07:25, 28 December 2014

Template:Distinguish {{#invoke:Hatnote|hatnote}} {{#invoke:Hatnote|hatnote}}Template:Main other

In the mathematical field of topology, a **homeomorphism** or **topological isomorphism** or **bi continuous 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 and *μορφή* (*morphē*) = shape, form.

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. An often-repeated mathematical joke is that topologists can't tell their coffee cup from their donut,^{[1]} 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 a cup's handle.

Topology is the study of those properties of objects that do not change when homeomorphisms are applied.

## Definition

A function *f*: *X* → *Y* between two topological spaces (*X*, *T _{X}*) and (

*Y*,

*T*) is called a

_{Y}**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 function with these three properties is sometimes called **bicontinuous**. If such a function exists, we say *X* and *Y* are **homeomorphic**. A **self-homeomorphism** is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called **homeomorphism classes**.

## Examples

- The unit 2-disc D
^{2}and the unit square in**R**^{2}are homeomorphic.Template:Clarification needed - The open interval (a, b) is homeomorphic to the real numbers
**R**for any a < b. (In this case, a bicontinuous forward mapping is given by*f*(*x*) = 1/(*x*−*a*) + 1/(*x*−*b*) while another such mapping is given by a scaled and translated version of the tan function). - The product space S
^{1}× S^{1}and the two-dimensional torus are homeomorphic.Template:Clarification needed - Every uniform isomorphism and isometric isomorphism is a homeomorphism.
- 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
*A*is a commutative ring with unity, and*S*is a multiplicative subset of*A*, then the spectrum Spec(*A*_{S}) is homeomorphic to {*p*∈ Spec(*A*) :*p*∩*S*= ∅}. **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}as the unit circle is compact as a subspace of Euclidean**R**^{2}but the real line is not compact. - If G 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.

## Notes

The third requirement, that *f*^{ −1} be continuous, is essential. Consider for instance the function *f*: [0, 2π) → S^{1} (the unit circle in ) defined by *f*(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism (S^{1} is compact but [0, 2π) is not). The function *f*^{ −1} is not continuous at the point (1, 0), because although *f*^{ −1} maps (1, 0) to 0, any neighbourhood of this point also includes points that the function maps close to 1, but the points it maps to numbers in between lie outside the neighbourhood.^{[2]}

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 *X* → *X* forms a group, called the **homeomorphism group** of *X*, often denoted Homeo(*X*); this group can be given a topology, such as the compact-open topology, making it a topological group.

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, Homeo(*X,* *Y*), is a torsor for the homeomorphism groups Homeo(*X*) and Homeo(*Y*), and given a specific homeomorphism between *X* and *Y*, all three sets are identified.

## Properties

- 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 & 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).

## Informal discussion

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.

This characterization of a homeomorphism often leads to 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*.

## See also

- Local homeomorphism
- Diffeomorphism
- Uniform isomorphism is an isomorphism between uniform spaces
- Isometric isomorphism is an isomorphism between metric spaces
- Dehn twist
- Homeomorphism (graph theory) (closely related to graph subdivision)
- Homotopy#Isotopy
- Mapping class group
- Poincaré conjecture

## References

## External links

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}