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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 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
isomorphisms in the
category of topological spaces—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. 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 in 1895.
Very roughly speaking, a topological space is a
geometric 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 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 th ...
and a
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 and a circle.
An often-repeated
mathematical joke 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 type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between two
topological spaces is a homeomorphism if it has the following properties:
*
is a
bijection (
one-to-one and
onto),
*
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 ...
is continuous (
is an
open mapping).
A homeomorphism is sometimes called a bicontinuous function. If such a function exists,
and
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 classes are called homeomorphism classes.
Examples
* The open
interval is homeomorphic to the
real numbers
for any
. (In this case, a bicontinuous forward mapping is given by
while other such mappings are given by scaled and translated versions of the or functions).
* The unit 2-
disc and the
unit square 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,
.
* 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 discre ...
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 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 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
is a
topological group
In mathematics, topological groups are logically 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 st ...
, its inversion map
is a homeomorphism. Also, for any
, the left translation
, the right translation
, and the inner automorphism
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 as a subspace of Euclidean R
''2'' but the real line is not compact.
*The one-dimensional intervals
, 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
In mathematics, topological groups are logically 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 st ...
.
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 ...
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
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, then the other is as well; if one of them is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, then the other is as well; if one of them is
Hausdorff, then the other is as well; their
homotopy 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 topolog ...
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 sets to open sets and
closed 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
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 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''—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''.
See also
*
*
* is an isomorphism between
uniform spaces
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
* is an isomorphism between
metric spaces
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
*
*
* (closely related to graph subdivision)
*
*
*
*
References
External links
*
{{Authority control
Theory of continuous functions
Functions and mappings