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Homeomorphism

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From Wikipedia, the free encyclopedia

Mapping which preserves all topological properties of a given space

For homeomorphisms in graph theory, seeHomeomorphism (graph theory).

Not to be confused withHomomorphism.

"Topological equivalence" redirects here. For the concept in dynamical systems, seeTopological conjugacy.

An often-repeatedmathematical joke is that topologists cannot tell the difference between acoffee mug and adonut,^[1] since a sufficiently pliabledonut could be reshaped to the form of acoffee mug by creating a dimple and progressively enlarging it, while preserving the donut hole in the mug's handle. This illustrates that a coffee mug and a donut (torus) are homeomorphic.

Inmathematics and more specifically intopology, ahomeomorphism (from Greek roots meaning "similar shape", named byHenri Poincaré),^[2]^[3] also calledtopological isomorphism, orbicontinuous function, is abijective andcontinuous function betweentopological spaces that has a continuousinverse function. Homeomorphisms are theisomorphisms in thecategory of topological spaces—that is, they are themappings that preserve all thetopological properties of a given space. Two spaces with a homeomorphism between them are calledhomeomorphic, and from a topological viewpoint they are the same.

Very roughly speaking, a topological space is ageometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, asquare and acircle are homeomorphic to each other, but asphere and atorus are not. However, this description can be misleading. Some continuous deformations do not result into homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as the homeomorphism between atrefoil knot and a circle.Homotopy andisotopy are precise definitions for the informal concept ofcontinuous deformation.

Definition

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Afunction $f:X\to Y$ between twotopological spaces is ahomeomorphism if it has the following properties:

$f {\displaystyle f}$ is abijection (one-to-one andonto),
$f {\displaystyle f}$ iscontinuous,
theinverse function $f^{-1}$ is continuous ( $f {\displaystyle f}$ is anopen mapping).

A homeomorphism is sometimes called abicontinuous function. If such a function exists, $X {\displaystyle X}$ and $Y {\displaystyle Y}$ arehomeomorphic. Aself-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic" is anequivalence relation on topological spaces. Itsequivalence classes are calledhomeomorphism classes.

The third requirement, that ${\textstyle f^{-1}}$ becontinuous, is essential. Consider for instance the function ${\textstyle f:[0,2\pi )\to S^{1}}$ (theunit circle in⁠ $\mathbb {R} ^{2}$ ⁠) defined by ${\textstyle f(\varphi )=(\cos \varphi ,\sin \varphi ).}$ This function is bijective and continuous, but not a homeomorphism ( ${\textstyle S^{1}}$ iscompact but ${\textstyle [0,2\pi )}$ is not). The function ${\textstyle f^{-1}}$ is not continuous at the point ${\textstyle (1,0),}$ because although ${\textstyle f^{-1}}$ maps ${\textstyle (1,0)}$ to ${\textstyle 0,}$ anyneighbourhood of this point also includes points that the function maps close to ${\textstyle 2\pi ,}$ but the points it maps to numbers in between lie outside the neighbourhood.^[4]

Homeomorphisms are theisomorphisms in thecategory of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ${\textstyle X\to X}$ forms agroup, called thehomeomorphism group ofX, often denoted ${\textstyle \operatorname {Homeo} (X).}$ This group can be given a topology, such as thecompact-open topology, which under certain assumptions makes it atopological group.^[5]

In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other.Homotopy andisotopy 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, ${\textstyle \operatorname {Homeo} (X,Y),}$ is atorsor for the homeomorphism groups ${\textstyle \operatorname {Homeo} (X)}$ and ${\textstyle \operatorname {Homeo} (Y),}$ and, given a specific homeomorphism between $X {\displaystyle X}$ and $Y, {\displaystyle Y,}$ all three sets are identified.^{[clarification needed]}

Examples

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A thickenedtrefoil knot is homeomorphic to a solid torus, but notisotopic in⁠ $\mathbb {R} ^{3}.$ ⁠ Continuous mappings are not always realizable as deformations.

{\displaystyle \mathbb {R} ^{3}.} — A thickenedtrefoil knot is homeomorphic to a solid torus, but notisotopic in⁠ $\mathbb {R} ^{3}.$ ⁠ Continuous mappings are not always realizable as deformations.

The openinterval ${\textstyle (a,b)}$ is homeomorphic to thereal numbers⁠ $\mathbb {R}$ ⁠ for any ${\textstyle a<b.}$ (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 thetan orarg tanh functions).
The unit 2-disc ${\textstyle D^{2}}$ and theunit square in⁠ $\mathbb {R} ^{2}$ ⁠ 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, inpolar coordinates, $(\rho ,\theta )\mapsto \left({\frac {\rho }{\max(|\cos \theta |,|\sin \theta |)}},\theta \right).$
Thegraph of adifferentiable function is homeomorphic to thedomain of the function.
A differentiableparametrization of acurve is a homeomorphism between the domain of the parametrization and the curve.
Achart of amanifold is a homeomorphism between anopen subset of the manifold and an open subset of aEuclidean space.
Thestereographic projection is a homeomorphism between the unit sphere in⁠ $\mathbb {R} ^{3}$ ⁠ with a single point removed and the set of all points in⁠ $\mathbb {R} ^{2}$ ⁠ (a 2-dimensionalplane).
If $G {\displaystyle G}$ is atopological 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.

Counter-examples

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⁠ $\mathbb {R} ^{m}$ ⁠ and⁠ $\mathbb {R} ^{n}$ ⁠ are not homeomorphic form ≠n.
The Euclideanreal line is not homeomorphic to the unit circle as a subspace of⁠ $\mathbb {R} ^{2}$ ⁠, since the unit circle iscompact as a subspace of Euclidean⁠ $\mathbb {R} ^{2}$ ⁠ but the real line is not compact.
The one-dimensional intervals $[0,1]$ and $(0,1)$ are not homeomorphic because one is compact while the other is not.

Properties

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Two homeomorphic spaces share the sametopological properties. For example, if one of them iscompact, then the other is as well; if one of them isconnected, then the other is as well; if one of them isHausdorff, then the other is as well; theirhomotopy andhomology groups will coincide. Note however that this does not extend to properties defined via ametric; there are metric spaces that are homeomorphic even though one of them iscomplete and the other is not.
A homeomorphism is simultaneously anopen mapping and aclosed mapping; that is, it mapsopen sets to open sets andclosed sets to closed sets.
Every self-homeomorphism in $S^{1}$ can be extended to a self-homeomorphism of the whole disk $D^{2}$ (Alexander's trick).

Informal discussion

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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 aline 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 ofhomotopy, which is actuallydefined as a continuous deformation, but from onefunction 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 spaceX correspond to which points onY—one just follows them asX 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) anisotopy between theidentity map onX and the homeomorphism fromX toY.

References

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^Hubbard, John H.; West, Beverly H. (1995).Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems. Texts in Applied Mathematics. Vol. 18. Springer. p. 204.ISBN 978-0-387-94377-0.
^Poincaré, H. (1895).Analysis Situs. Journal de l'Ecole polytechnique. Gauthier-Villars.OCLC 715734142. Archived fromthe original on 11 June 2016. Retrieved29 April 2018.
Poincaré, Henri (2010).Papers on Topology: Analysis Situs and Its Five Supplements. Translated by Stillwell, John. American Mathematical Society.ISBN 978-0-8218-5234-7.
^Gamelin, T. W.; Greene, R. E. (1999).Introduction to Topology (2nd ed.). Dover. p. 67.ISBN 978-0-486-40680-0.
^Väisälä, Jussi (1999).Topologia I. Limes RY. 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):910–912.doi:10.2307/30037630.JSTOR 30037630.Archived(PDF) from the original on 16 September 2016.

External links

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"Homeomorphism",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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