- Croatian: homeomorfizam
- Croatian: homeomorfizam
- 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.
DefinitionA function f between two topological spaces X and Y is called a homeomorphism if it has the following properties:
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.
- \mathbb^ and \mathbb^ are not homeomorphic for n\neq m
NotesThe third requirement, that f −1 be continuous, is essential. Consider for instance the function f : [0, 2π) → S1 defined by f(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism.
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.
- Every self-homeomorphism in S^1 can be extended to a self-homeomorphism of the whole disk D^2 (Alexander's Trick).
Informal discussionThe 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.
homeomorphism in Arabic: دالة هميومورفية
homeomorphism in Catalan: Homeomorfisme
homeomorphism in Czech: Homeomorfismus
homeomorphism in Danish: Homeomorfi
homeomorphism in German: Homöomorphismus
homeomorphism in Spanish: Homeomorfismo
homeomorphism in French: Homéomorphisme
homeomorphism in Hungarian: Homeomorfia
homeomorphism in Korean: 위상동형사상
homeomorphism in Italian: Omeomorfismo
homeomorphism in Hebrew: הומיאומורפיזם
homeomorphism in Georgian: ჰომეომორფიზმი
homeomorphism in Lithuanian: Homeomorfizmas
homeomorphism in Dutch: Homeomorfisme
homeomorphism in Japanese: 位相同型
homeomorphism in Polish: Homeomorfizm
homeomorphism in Portuguese: Homeomorfismo
homeomorphism in Russian: Гомеоморфизм
homeomorphism in Slovenian: Homeomorfizem
homeomorphism in Serbian: Хомеоморфизам
homeomorphism in Finnish: Homeomorfismi
homeomorphism in Swedish: Homeomorfi
homeomorphism in Tamil: இடவியல் உருமாற்றம்
homeomorphism in Turkish: Homeomorfizma
homeomorphism in Ukrainian: Гомеоморфізм
homeomorphism in Vietnamese: Phép đồng phôi
homeomorphism in Chinese: 同胚