Homeomorphism

:''This word should not be confused with homomorphism.'' ---- In the mathematics field of topology a homeomorphism or topological isomorphism (from the Greek language words ''homeos'' = identical and ''morphe'' = shape) is a special isomorphism between topological spaces which respects topological property. Two spaces with a homeomorphism between them are called homeomorphic. 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. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is 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_(mathematics) ''f'' between two topological spaces ''X'' and ''Y'' is called homeomorphism if it has the following properties * ''f'' is a bijection, * ''f'' is Continuity (topology), * the inverse function ''f''⁻¹ is continuous. If such a function exists we say ''X'' and ''Y'' are homeomorphic. The homeomorphisms form an equivalence relation on the class (set theory) of all topological spaces. The resulting equivalence class are called homeomorphism classes. == Examples == * The unit 2-ball (mathematics) D² and the unit square in R² are homeomorphic. * The open interval (mathematics) (-1, 1) is homeomorphic to the real numbers R. * The product topology S¹ × S¹ and the two-dimensional torus are homeomorphic. * Every uniform isomorphism and isometric isomorphism is a homeomorphism. == Notes == The 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 (mathematics), called the homeomorphism group of ''X'', often denoted Homeo(''X''). == Properties == * two homeomorphic spaces share the same topological properties. For example, if one of them is compact space, then the other is as well; if one of them is connectedness, then the other is as well; if one of them is Hausdorff space, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric space; there are metric spaces which are homeomorphic even though one of them is completeness (topology) and the other is not. * a homeomorphism is an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets. == 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 homotopy between the identity map on ''X'' and the homeomorphism from ''X'' to ''Y''. == See also == * local homeomorphism * homotopy * diffeomorphism * uniform isomorphism is an isomorphism between uniform spaces * isometric isomorphism is an isomorphism between metric spaces General topology Topology

Homeomorphism

Isn't the correct spelling ''homoeomorphism'' rather than homeomorphism? --fil Not in everyday use. User:Charles Matthews 15:20, 9 Feb 2004 (UTC) : Ok I don't use this term ''every day'', and I also thought homeomorphic is the correct spelling; but my spell checker (ispell-emacs) suggests ''homoeomorphic'' instead of ''homeomorphic''. Maybe one should point out the alternative spelling. --fil On the whole, I'd stick with the way humans spell it. User:Charles Matthews 16:28, 9 Feb 2004 (UTC)

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Words begining with Homeomorphism:

Homeomorphism
Homeomorphism
Homeomorphisms
Homeomorphism_(graph_theory)
Homeomorphism_group