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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a homeomorphism, topological isomorphism, or bicontinuous function is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
and
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
between topological spaces 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 X ...
. Homeomorphisms are the
isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
s 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 Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. ** Proto-Greek language, the assumed last common ancestor ...
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 mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "T ...
in 1895. Very roughly speaking, a topological space is a
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
are homeomorphic to each other, but a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
and a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
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, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest k ...
and a circle. An often-repeated
mathematical joke A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathema ...
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 $f : X \to Y$ between two topological spaces is a homeomorphism if it has the following properties: * $f$ is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
( one-to-one and
onto In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
), * $f$ 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 X ...
$f^$ is continuous ($f$ is an open mapping). 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. Each equivalence relatio ...
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 a ...
es are called homeomorphism classes.

# 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 $D^2$ 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 coordinate ...
in $\mathbf^2$ 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 In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, $\left(\rho, \theta\right) \mapsto \left\left( \frac, \theta\right\right)$. * 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 disc ...
of a differentiable function 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 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 represent tab ...
of a manifold is a homeomorphism between an open subset 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. * The
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter throu ...
is a homeomorphism between the unit sphere in $\mathbf^3$ with a single point removed and the set of all points in $\mathbf^2$ (a 2-dimensional plane). * If $G$ is a topological 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 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 or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
as a subspace of Euclidean R''2'' but the real line is not compact. *The one-dimensional intervals

# Notes

The third requirement, that $f^$ be continuous, is essential. Consider for instance the function $f : [0,2\pi) \to S^1$ (the unit circle in $\mathbf^2$) defined by$f(\phi) = (\cos\phi,\sin\phi)$. 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 or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
but 
_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_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_them.__The_word_i_...
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_\to_X_forms_a_group_(mathematics).html" ;"title="Neighbourhood_(mathematics).html" "title=",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, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 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 them. The word i ...
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 $X \to X$ forms a group (mathematics)">group, 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. 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, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
. 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 ''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 non-e ...
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 properties. 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, 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, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
and
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
s 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 set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
s to open sets and
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
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, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, 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''.