and a donut (torus
) illustrating that they are homeomorphic. But there need not be a continuous deformation
for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function.
In the mathematics|mathematical
field of topology
, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function
between topological space
s that has a continuous inverse function
. Homeomorphisms are the isomorphism
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
'' (''homoios'') = similar or same and ''μορφή
'' (''morphē'') = shape, form, introduced to mathematics by Henri Poincaré
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
and a circle
are homeomorphic to each other, but a sphere
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.
between two topological space
s is a homeomorphism if it has the following properties:
is a bijection
* the inverse function
is continuous (
is an open mapping
A homeomorphism is sometimes called a bicontinuous function. If such a function exists,
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 class
es are called homeomorphism classes.
._Continuous_mappings_are_not_always_realizable_as_deformations..html" style="text-decoration: none;"class="mw-redirect" title="Homotopy#Isotopy">isotopic in R3
. Continuous mappings are not always realizable as deformations.">Homotopy#Isotopy">isotopic in R3
. Continuous mappings are not always realizable as deformations.
* The open interval
is homeomorphic to the real number
. (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
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
of a differentiable function
is homeomorphic to the domain
of the function.
* A differentiable parametrization
of a curve
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
* The 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
is a topological group
, its inversion map
is a homeomorphism. Also, for any
, the left translation
, the right translation
, and the inner automorphism
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
)_defined_byf(\phi)_=_(\cos\phi,\sin\phi)._This_function_is_bijective_and_continuous,_but_not_a_homeomorphism_(S^1_is_[[Compact_space.html" style="text-decoration: none;"class="mw-redirect" title="unit_circle.html" style="text-decoration: none;"class="mw-redirect" title=",2\pi) \to S^1 (the ,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_space">compact_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).html" style="text-decoration: none;"class="mw-redirect" title="unit circle">,2\pi) \to S^1 (the [[unit circle in ) defined by. This function is bijective and continuous, but not a homeomorphism ( is [[Compact space">compact but is not). The function is not continuous at the point , because although maps to , any [[Neighbourhood (mathematics)">neighbourhood of this point also includes points that the function maps close to but the points it maps to numbers in between lie outside the neighbourhood.
Homeomorphisms are the [[isomorphisms in the [[category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms forms a group, called the homeomorphism group of ''X'', often denoted . 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.
Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, is a torsor for the homeomorphism groups and , and, given a specific homeomorphism between and , all three sets are identified.
* Two homeomorphic spaces share the same topological properties. For example, if one of them is compact, 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 and homology groups 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 can be extended to a self-homeomorphism of the whole disk (Alexander's trick).
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''.
* is an isomorphism between uniform spaces
* is an isomorphism between metric spaces
* (closely related to graph subdivision)
Category:Functions and mappings