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:''Topological equivalence redirects here; see also Topological Equivalence (dynamical Systems) .'' and a Donut illustrating that they are homeomorphic. But there does not need to be a continuous deformation for two spaces to be homeomorphic.]] In the Mathematical field of Topology , a homeomorphism or '''topological isomorphism''' (from the Greek words ''homoios'' = ''similar'' and ''μορφή (morphē)'' = shape = form (Latin deformation of morphe)) is a special Isomorphism between Topological Space s which respects Topological Properties . 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 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. DEFINITION A Function ''f'' between two Topological Space s ''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 between 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'''. EXAMPLES is homeomorphic to a Torus . While this may seem illogical, in four dimensions they can easily be deformed continuously.]]
NOTES The third requirement, that ''f'' −1 be continuous, is essential. Consider for instance the function ''f'' : Homeomorphisms are the Isomorphism s 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 . PROPERTIES
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 . 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''. SEE ALSO
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