View source for Homeomorphism

{{Short description|Mapping which preserves all topological properties of a given space}}
{{hatnote group|{{distinguish|Homomorphism}}{{For|homeomorphisms in graph theory|Homeomorphism (graph theory)}}}}
{{redirect|Topological equivalence|the concept in dynamical systems|Topological conjugacy}}

In [[mathematics]] and more specifically in [[topology]], a '''homeomorphism''' ([[Neoclassical compound|from Greek roots]] meaning "similar shape", named by [[Henri Poincaré]]),<ref>{{cite book |url=http://serge.mehl.free.fr/anx/ana_situs.html |title=Analysis Situs |author-link= Henri Poincaré|last=Poincaré |first=H. |date=1895 |publisher=Gauthier-Villars |series=Journal de l'Ecole polytechnique |oclc=715734142  |access-date=29 April 2018|url-status=dead|archive-url=https://web.archive.org/web/20160611022329/http://serge.mehl.free.fr/anx/ana_situs.html|archive-date=11 June 2016}}<br/>{{cite book |last=Poincaré |first=Henri |year=2010 |title=Papers on Topology: Analysis Situs and Its Five Supplements |translator-first=John |translator-last=Stillwell |publisher=American Mathematical Society |isbn=978-0-8218-5234-7}}</ref><ref>{{cite book |last1=Gamelin |first1=T. W. |author-link=Theodore Gamelin |last2=Greene |first2=R. E. |author-link2=Robert Everist Greene |year=1999 |title=Introduction to Topology |publisher=Dover |isbn=978-0-486-40680-0 |edition=2nd |page=67 |url=https://books.google.com/books?id=thAHAGyV2MQC&pg=PA67 }}</ref> also called '''topological isomorphism''', or '''bicontinuous function''', is a [[bijective]] and [[Continuous function#Continuous functions between topological spaces|continuous function]] between [[topological space]]s that has a continuous [[inverse function]]. Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]&mdash;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.

Very roughly speaking, a topological space is a [[geometry|geometric]] object, and a homeomorphism results from a continuous deformation 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. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from  continuous deformations, such as the homeomorphism between a [[trefoil knot]] and a circle. [[Homotopy]] and [[homotopy#Isotopy|isotopy]] are precise definitions for the informal concept of ''continuous deformation''.

==Definition==
A [[function (mathematics)|function]] <math>f : X \to Y</math> between two [[topological space]]s is a '''homeomorphism''' if it has the following properties:

* <math>f</math> is a [[bijection]] ([[injective function|one-to-one]] and [[onto]]),
* <math>f</math> is [[Continuity (topology)|continuous]],
* the [[inverse function]] <math>f^{-1}</math> is continuous (<math>f</math> is an [[open mapping]]).

A homeomorphism is sometimes called a ''bicontinuous'' function. If such a function exists, <math>X</math> and <math>Y</math> 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 class]]es are called '''homeomorphism classes'''.

The third requirement, that <math display="inline">f^{-1}</math> be [[Continuous function|continuous]], is essential. Consider for instance the function <math display="inline">f : [0,2\pi) \to S^1</math> (the [[unit circle]] in {{tmath|\R^2}}) defined by<math display="inline">f(\varphi) = (\cos\varphi,\sin\varphi).</math> This function is bijective and continuous, but not a homeomorphism (<math display="inline">S^1</math> is [[Compact space|compact]] but <math display="inline">[0,2\pi)</math> is not). The function <math display="inline">f^{-1}</math> is not continuous at the point <math display="inline">(1,0),</math> because although <math display="inline">f^{-1}</math> maps <math display="inline">(1,0)</math> to <math display="inline">0,</math> any [[Neighbourhood (mathematics)|neighbourhood]] of this point also includes points that the function maps close to <math display="inline">2\pi,</math> but the points it maps to numbers in between lie outside the neighbourhood.<ref>{{cite book |last=Väisälä |first=Jussi |title=Topologia I |publisher=Limes RY |date=1999 |page=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 <math display="inline">X \to X</math> forms a [[group (mathematics)|group]], called the '''[[homeomorphism group]]''' of ''X'', often denoted <math display="inline">\operatorname{Homeo}(X).</math> This group can be given a topology, such as the [[compact-open topology]], which under certain assumptions makes it a [[topological group]].<ref>{{cite journal|last1=Dijkstra|first1=Jan J.|title=On Homeomorphism Groups and the Compact-Open Topology|journal=The American Mathematical Monthly|date=1 December 2005|volume=112|issue=10|pages=910–912|doi=10.2307/30037630|jstor=30037630 |url=http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf|url-status=live|archive-url=https://web.archive.org/web/20160916112245/http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf|archive-date=16 September 2016}}</ref>

In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other. [[Homotopy]] and  [[Homotopy#Isotopy|isotopy]] are equivalence relations that have been introduced for dealing with such situations.

Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, <math display="inline">\operatorname{Homeo}(X,Y),</math> is a [[torsor]] for the homeomorphism groups <math display="inline">\operatorname{Homeo}(X)</math> and <math display="inline">\operatorname{Homeo}(Y),</math> and, given a specific homeomorphism between <math>X</math> and <math>Y,</math> all three sets are identified.{{clarify|reason=Which?|date=July 2023}}

==Examples==
[[Image:Blue Trefoil Knot.png|thumb|upright=1.15|A thickened [[trefoil knot]] is homeomorphic to a solid torus, but not [[Homotopy#Isotopy|isotopic]] in  {{tmath|\R^3.}} Continuous mappings are not always realizable as deformations.]]
* The open [[interval (mathematics)|interval]] <math display="inline">(a,b)</math> is homeomorphic to the [[real number]]s {{tmath|\R}} for any <math display="inline">a < b.</math> (In this case, a bicontinuous forward mapping is given by <math display="inline">f(x) = \frac{1}{a-x} + \frac{1}{b-x} </math> while other such mappings are given by scaled and translated versions of the {{math|tan}} or {{math|arg tanh}} functions).
* The closed unit [[disk (mathematics)|disk]] <math display="inline">D^2</math> centered at the origin and the square <math display="inline">[-1,1]^2</math> in {{tmath|\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]], <math display=block>(\rho, \theta) \mapsto \left( \frac{\rho}{ \max(|\cos \theta|, |\sin \theta|)}, \theta\right).</math>
* The [[graph of a function|graph]] of a [[differentiable function]] is homeomorphic to the [[domain of a function|domain]] of the function.
* A differentiable [[parametric equation|parametrization]] of a [[curve]] is a homeomorphism between the domain of the parametrization and the curve.
* A [[chart (topology)|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 {{tmath|\R^3}} with a single point removed and the set of all points in {{tmath|\R^2}} (a 2-dimensional [[plane (mathematics)|plane]]). 
* If <math>G</math> 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.

===Counter-examples===
* {{tmath|\R^m}} and {{tmath|\R^n}} are not homeomorphic for {{math|1=''m'' &ne; ''n''.}} This is a non-trivial result, usually proved via the [[domain invariance theorem]].
* The Euclidean [[real line]] is not homeomorphic to the unit circle as a subspace of {{tmath|\R^2}}, since the unit circle is [[Compact space|compact]] as a subspace of Euclidean {{tmath|\R^2}} but the real line is not compact.
*The one-dimensional intervals <math>[0,1]</math> and <math>(0,1)</math> are not homeomorphic because one is compact while the other is not.

==Properties==
* Two homeomorphic spaces share the same [[topological property|topological properties]]. For example, if one of them is [[compact space|compact]], then the other is as well; if one of them is [[connectedness|connected]], then the other is as well; if one of them is [[Hausdorff space|Hausdorff]], then the other is as well; their [[homotopy]] and [[homology group]]s will coincide. Note however that this does not extend to properties defined via a [[metric space|metric]]; there are metric spaces that are homeomorphic even though one of them is [[completeness (topology)|complete]] and the other is not.
* A homeomorphism is simultaneously an [[open mapping]] and a [[closed mapping]]; that is, it maps [[open set]]s to open sets and [[closed set]]s to closed sets.
* Every self-homeomorphism in <math>S^1</math> can be extended to a self-homeomorphism of the whole disk <math>D^2</math> ([[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&mdash;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''&mdash;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 [[homotopy|isotopy]] between the [[identity function|identity map]] on ''X'' and the homeomorphism from ''X'' to ''Y''.

== See also ==

* {{annotated link|Local homeomorphism}}
* {{annotated link|Diffeomorphism}}
* {{annotated link|Uniform isomorphism}} is an isomorphism between [[uniform spaces]]
* {{annotated link|Isometric isomorphism}} is an isomorphism between [[metric spaces]]
* {{annotated link|Homeomorphism group}}
* {{annotated link|Dehn twist}}
* {{annotated link|Homeomorphism (graph theory)}} (closely related to graph subdivision)
* {{annotated link|Homotopy#Isotopy}}
* {{annotated link|Mapping class group}}
* {{annotated link|Poincaré conjecture}}
* {{annotated link|Universal homeomorphism}}

==References==
{{reflist}}

==External links==
*{{springer|title=Homeomorphism|id=p/h047600}}

{{Topology}}

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[[Category:Theory of continuous functions]]
[[Category:Functions and mappings]]
[[Category:Homeomorphisms| ]]