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In the
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
field of
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, a homeomorphism, topological isomorphism, or bicontinuous function is a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
between
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s that has a continuous
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
. Homeomorphisms are the
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
s in the
category of topological spacesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
—that is, they are the mappings that preserve all the
topological propertiesIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
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#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
in 1895. Very roughly speaking, a topological space is a
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ...

geometric
object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

square
and a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
are homeomorphic to each other, but a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
and a
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

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 In knot theory, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...

trefoil knot
and a circle. An often-repeated
mathematical joke A mathematical joke is a form of humor Humour ( Commonwealth English) or humor (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the E ...
is that topologists cannot 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
f : X \to Y between two
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s is a homeomorphism if it has the following properties: * f is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
(
one-to-one One-to-one or one to one may refer to: Mathematics and communication *One-to-one function, also called an injective function *One-to-one correspondence, also called a bijective function *One-to-one (communication), the act of an individual commun ...

one-to-one
and
onto In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

onto
), * f is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, * the
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
f^ is continuous (f is an
open mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on topological spaces. Its
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es are called homeomorphism classes.


Examples

* The open interval (a,b) is homeomorphic to the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s \mathbf for any a < b. (In this case, a bicontinuous forward mapping is given by f(x) = \frac + \frac while other such mappings are given by scaled and translated versions of the or functions). * The unit 2-
disc Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), ...
D^2 and the
unit square In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

polar coordinates
, (\rho, \theta) \mapsto \left( \frac, \theta\right). * The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of a
differentiable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

differentiable function
is homeomorphic to the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
of the function. * A differentiable parametrization of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
is a homeomorphism between the domain of the parametrization and the curve. * A
chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and ca ...
of a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
is a homeomorphism between an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of the manifold and an open subset of a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. * The
stereographic projection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

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 Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
). * If G is a
topological group 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a group ...
, its inversion map x \mapsto x^ 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^ are homeomorphisms.


Non-examples

* R''m'' and R''n'' are not homeomorphic for * The Euclidean
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is not homeomorphic to the unit circle as a subspace of R''2'', since the unit circle is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
as a subspace of Euclidean R''2'' but the real line is not compact. *The one-dimensional intervals ,1/math> and (0,1) are not homeomorphic because no continuous bijection could be made.


Notes

The third requirement, that f^ be continuous, is essential. Consider for instance the function f : ,2\pi)_\to_S^1_(the_unit_circle_in_\mathbb^2)_defined_byf(\phi)_=_(\cos\phi,\sin\phi)._This_function_is_bijective_and_continuous,_but_not_a_homeomorphism_(S^1_is_compact_ Compact_as_used_in_politics_may_refer_broadly_to_a_pact_ A_pact,_from_Latin_''pactum''_("something_agreed_upon"),_is_a_formal_agreement._In_international_relations_ International_relations_(IR),_international_affairs_(IA)_or_internationa_...
_but_[0,2\pi)_is_not)._The_function_f^_is_not_continuous_at_the_point_(1,0),_because_although_f^_maps_(1,0)_to_0,_any_
,2\pi)_\to_S^1_(the_unit_circle_in_\mathbb^2)_defined_byf(\phi)_=_(\cos\phi,\sin\phi)._This_function_is_bijective_and_continuous,_but_not_a_homeomorphism_(S^1_is_compact_ Compact_as_used_in_politics_may_refer_broadly_to_a_pact_ A_pact,_from_Latin_''pactum''_("something_agreed_upon"),_is_a_formal_agreement._In_international_relations_ International_relations_(IR),_international_affairs_(IA)_or_internationa_...
_but_[0,2\pi)_is_not)._The_function_f^_is_not_continuous_at_the_point_(1,0),_because_although_f^_maps_(1,0)_to_0,_any_Neighbourhood_(mathematics)">neighbourhood_ A_neighbourhood_(British_English,_Hiberno-English,_Hibernian_English,_Australian_English_and_Canadian_English)_or_neighborhood_(American_English;_American_and_British_English_spelling_differences,_see_spelling_differences)_is_a_geographicall_...
_of_this_point_also_includes_points_that_the_function_maps_close_to_2\pi,_but_the_points_it_maps_to_numbers_in_between_lie_outside_the_neighbourhood. Homeomorphisms_are_the_isomorphism_ In_mathematics_ Mathematics_(from_Greek:_)_includes_the_study_of_such_topics_as_numbers_(arithmetic_and_number_theory),_formulas_and_related_structures_(algebra),_shapes_and_spaces_in_which_they_are_contained_(geometry),_and_quantities_and_...

s_in_the_category_of_topological_spacesIn_mathematics Mathematics_(from_Ancient_Greek,_Greek:_)_includes_the_study_of_such_topics_as_quantity_(number_theory),_mathematical_structure,_structure_(algebra),_space_(geometry),_and_calculus,_change_(mathematical_analysis,_analysis)._It_ha_...
._As_such,_the_composition_of_two_homeomorphisms_is_again_a_homeomorphism,_and_the_set_of_all_self-homeomorphisms_X_\to_X_forms_a_group_(mathematics).html" "title="Neighbourhood_(mathematics).html" "title="unit_circle.html" ;"title=",2\pi) \to S^1 (the unit circle">,2\pi) \to S^1 (the unit circle in \mathbb^2) defined byf(\phi) = (\cos\phi,\sin\phi). This function is bijective and continuous, but not a homeomorphism (S^1 is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
but [0,2\pi) is not). The function f^ is not continuous at the point (1,0), because although f^ maps (1,0) to 0, any Neighbourhood (mathematics)">neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
of this point also includes points that the function maps close to 2\pi, but the points it maps to numbers in between lie outside the neighbourhood. Homeomorphisms are the
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
s in the
category of topological spacesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X \to X forms a group (mathematics)">group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, called the homeomorphism group of ''X'', often denoted \text(X). This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a
topological group 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, \text(X,Y), is a
torsor In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a gr ...
for the homeomorphism groups \text(X) and \text(Y), and, given a specific homeomorphism between X and Y, all three sets are identified.


Properties

* Two homeomorphic spaces share the same
topological propertiesIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
. For example, if one of them is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
, then the other is as well; if one of them is connected, then the other is as well; if one of them is , then the other is as well; their
homotopy In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...

homotopy
and
homology group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s will coincide. Note however that this does not extend to properties defined via a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
; 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and a closed mapping; that is, it maps
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s to open sets and
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s to closed sets. * Every self-homeomorphism in S^1 can be extended to a self-homeomorphism of the whole disk D^2 (
Alexander's trickAlexander's trick, also known as the Alexander trick, is a basic result in geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one ...
).


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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...

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 In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from Ancient Greek, Greek ὁμός ''homós'' "same, similar" and τόπος ''tópos'' " ...
. There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an between the on ''X'' and the homeomorphism from ''X'' to ''Y''.


See also

* * * is an isomorphism between
uniform spaces In the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathemati ...
* is an isomorphism between
metric spaces Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
* * * (closely related to graph subdivision) * * * *


References


External links

* {{Authority control Continuous mappings Functions and mappings