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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and more specifically in
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a homeomorphism ( from Greek roots meaning "similar shape", named by
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
), also called topological isomorphism, or bicontinuous function, is a
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
and
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
between
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s that has a continuous
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
. Homeomorphisms are the
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
—that is, they are the mappings that preserve all the
topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
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
geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
are homeomorphic to each other, but a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
and a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
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 In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology ...
and a circle.
Homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
and isotopy are precise definitions for the informal concept of ''continuous deformation''.


Definition

A function f : X \to Y between two
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s is a homeomorphism if it has the following properties: * f is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
( one-to-one and
onto In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
), * f is continuous, * the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
f^ is continuous (f is an
open mapping In mathematics, more specifically in topology, an open map is a function (mathematics), function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the Image (mathem ...
). 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, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
on topological spaces. Its
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es are called homeomorphism classes. The third requirement, that f^ be continuous, is essential. Consider for instance the function f : ,2\pi) \to S^1 (the unit circle in ) defined byf(\varphi) = (\cos\varphi,\sin\varphi). 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 or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
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, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...
. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X \to X forms a group, called the homeomorphism group">group (mathematics)">group, called the homeomorphism group of ''X'', often denoted \operatorname(X). This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
. In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other.
Homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
and 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, \operatorname(X,Y), is a
torsor In mathematics, a principal homogeneous space, or torsor, for a 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 group ''G'' is a no ...
for the homeomorphism groups \operatorname(X) and \operatorname(Y), and, given a specific homeomorphism between X and Y, all three sets are identified.


Examples

* The open interval (a,b) is homeomorphic to the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s 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 D^2 and the
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
in 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, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, (\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 ...
of a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
is homeomorphic to the domain 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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is a homeomorphism between the domain of the parametrization and the curve. * A
chart A chart (sometimes known as a graph) is a graphics, graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can repres ...
of a
manifold 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 N ...
is a homeomorphism between an
open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
of the manifold and an open subset of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. * The
stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
is a homeomorphism between the unit sphere in with a single point removed and the set of all points in (a 2-dimensional plane). * If G is a
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
, 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.


Counter-examples

* and are not homeomorphic for * The Euclidean
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
is not homeomorphic to the unit circle as a subspace of , since the unit circle is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
as a subspace of Euclidean but the real line is not compact. *The one-dimensional intervals ,1/math> and (0,1) are not homeomorphic because one is compact while the other is not.


Properties

* Two homeomorphic spaces share the same
topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
. For example, if one of them is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, 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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
and
homology group In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
s will coincide. Note however that this does not extend to properties defined via a
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
; 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, more specifically in topology, an open map is a function (mathematics), function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the Image (mathem ...
and a closed mapping; that is, it maps
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s to open sets and
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
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 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 In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
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, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
, 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, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
. 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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on ''X'' and the homeomorphism from ''X'' to ''Y''.


See also

* * * is an isomorphism between
uniform spaces In the mathematical field of topology, a uniform space is a set with additional structure that is used to define '' uniform properties'', such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces a ...
* is an isomorphism between metric spaces * * * (closely related to graph subdivision) * * * *


References


External links

* {{Authority control Theory of continuous functions Functions and mappings