- Homeomorphism
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**"Topological equivalence**redirects here; see alsotopological equivalence (dynamical systems) ." In the mathematical field of

donut illustrating that they are homeomorphic. But there does not need to be a continuous deformation for two spaces to be homeomorphic.topology , a**homeomorphism**or**topological isomorphism**(from the Greek words "ὅμοιος (homoios)" = similar and "μορφή (morphē)" = shape = form (Latin deformation of morphe)) is a bicontinuous function between twotopological space s. Homeomorphisms are theisomorphism s in thecategory of topological spaces — that is, they are the mappings which 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.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 asphere 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.

**Definition**A function "f" between two

topological space s "X" and "Y" is called a**homeomorphism**if it has the following properties:* "f" is a

bijection (1-1 andonto ),

* "f" is continuous,

* theinverse function "f"^{ −1}is continuous (f is anopen mapping ).A function with these three properties is sometimes called

**bicontinuous**. 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 anequivalence relation on the class of all topological spaces. The resulting equivalence classes are called**homeomorphism classes**.**Examples*** The unit 2-disc D

^{2}and theunit square in**R**^{2}are homeomorphic.* The open interval (−1, 1) is homeomorphic to the

real number s**R**.* The product space S

^{1}× S^{1}and the two-dimension altorus are homeomorphic.* Every

uniform isomorphism andisometric isomorphism is a homeomorphism.* Any

2-sphere with a single point removed is homeomorphic to the set of all points in**R**^{2}(a 2-dimensional plane).* Let $A$ be a commutative ring with unity and let $S$ be a mutiplicative subset of $A$. Then Spec $(A\_S)$ is homeomorphic to $\{\; p\; in\; extrm\{Spec\; \}\; A\; :\; p\; cap\; S\; =\; emptyset\; \}$

*$mathbb\{R\}^\{n\}$ and $mathbb\{R\}^\{m\}$ are not homeomorphic for $n\; eq\; m$

* An example of a continuous bijection that is not a homeomorphism is the map that takes the half-open interval $[0,1)$ and wraps it around the circle. In this case the inverse - although it exists - fails to be continuous. The primage of certain sets which are actual open in the relative topology of the half-open inteval are not open in the more natural topology of the circle (they are half-open intevals).

**Notes**The third requirement, that "f"

^{ −1}be continuous, is essential. Consider for instance the function "f" :[0, 2π) → S^{1}defined by "f"(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism.Homeomorphisms are the

isomorphism s in thecategory 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 themapping class group .**Properties*** 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 group s 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 aclosed mapping , that is it mapsopen set s to open sets andclosed set 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 to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.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".**ee also***

Local homeomorphism

*Diffeomorphism

*Uniform isomorphism is an isomorphism betweenuniform spaces

*Isometric isomorphism is an isomorphism betweenmetric spaces

*Dehn twist

*Homeomorphism (graph theory) (closely related to graph subdivision)

*Isotopy

*Mapping class group **External links***planetmath reference|id=912|title=Homeomorphism

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