## English

### Alternative spellings

- homoeomorphic qualifier Commonwealth

- Having a homeomorphism
- 1974, Wesley E. Terry, "Any infinite-dimensional Fréchet space homeomorphic with its countable product is topologically a Hilbert space", Trans. Amer. Math. Soc. 196, pages 93-104.
- 2007, Andrjez Nowik, "A Vitali set can be homeomorphic to its complement", Acta Math. Hungar. 115:1-2, pages 145-154.
- 2007, Tim D. Austin,, "A pair of non-homeomorphic product measures on the Cantor set", Math. Proc. Cambridge Philos. Soc. 142:1, pages 103-110.

distinguish homomorphism

- Topological equivalence redirects here; see also topological equivalence (dynamical systems).

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 donut are not. An often-repeated
joke is that topologists can't tell the coffee cup from which they
are drinking from the donut they are eating, since a sufficiently
pliable donut could be reshaped to the form of a coffee cup by
creating a dimple and progressively enlarging it, while shrinking
the hole into a handle.

Intuitively, a homeomorphism maps points in the
first object that are "close together" to points in the second
object that are close together, and points in the first object that
are not close together to points in the second object that are not
close together. Topology is the study of those properties of
objects that do not change when homeomorphisms are applied.

- f is a bijection (1-1 and onto),
- f is continuous,
- the inverse function f −1 is continuous (f is an open mapping).

If such a function exists, we say X and Y are
homeomorphic. A self-homeomorphism is a homeomorphism of a
topological space and itself. The homeomorphisms form an equivalence
relation on the class
of all topological spaces. The resulting equivalence
classes are called homeomorphism classes.

- The unit 2-disc D2 and the unit square in R2 are homeomorphic.

- The open interval (−1, 1) is homeomorphic to the real numbers R.

- The product space S1 × S1 and the two-dimensional torus are homeomorphic.

- Every uniform isomorphism and isometric isomorphism is a homeomorphism.

- \mathbb^ and \mathbb^ are not homeomorphic for n\neq m

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 X → X forms a group,
called the homeomorphism group of X, often denoted Homeo(X).

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.

- 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 homology groups will coincide. Note however that this does not extend to properties defined via a metric; there are metric spaces which 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).

This characterization of a homeomorphism often
leads to 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.

- Local homeomorphism
- Diffeomorphism
- Uniform isomorphism is an isomorphism between uniform spaces
- Isometric isomorphism is an isomorphism between metric spaces
- Dehn twist
- Homeomorphism (graph theory) (closely related to graph subdivision)
- Isotopy
- Mapping class group

homeomorphic in Arabic: دالة هميومورفية

homeomorphic in Catalan: Homeomorfisme

homeomorphic in Czech: Homeomorfismus

homeomorphic in Danish: Homeomorfi

homeomorphic in German: Homöomorphismus

homeomorphic in Spanish: Homeomorfismo

homeomorphic in French: Homéomorphisme

homeomorphic in Hungarian: Homeomorfia

homeomorphic in Korean: 위상동형사상

homeomorphic in Italian: Omeomorfismo

homeomorphic in Hebrew: הומיאומורפיזם

homeomorphic in Georgian: ჰომეომორფიზმი

homeomorphic in Lithuanian: Homeomorfizmas

homeomorphic in Dutch: Homeomorfisme

homeomorphic in Japanese: 位相同型

homeomorphic in Polish: Homeomorfizm

homeomorphic in Portuguese: Homeomorfismo

homeomorphic in Russian: Гомеоморфизм

homeomorphic in Slovenian: Homeomorfizem

homeomorphic in Serbian: Хомеоморфизам

homeomorphic in Finnish: Homeomorfismi

homeomorphic in Swedish: Homeomorfi

homeomorphic in Tamil: இடவியல் உருமாற்றம்

homeomorphic in Turkish: Homeomorfizma

homeomorphic in Ukrainian: Гомеоморфізм

homeomorphic in Vietnamese: Phép đồng phôi

homeomorphic in Chinese: 同胚

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