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Manifold

In [mathematics] , more specifically in [differential geometry] and [topology] , a ** manifold ** is a [mathematical space] that on a small enough scale resembles the [Euclidean space] of a specific dimension, called the [dimension] of the manifold.

Thus a [line] and a [circle] are one-dimensional manifolds, a [plane] and [sphere] (the surface of a [ball] ) are two-dimensional manifolds, and [so forth] . More formally, every point of an * n * -dimensional manifold has a [neighborhood] [homeomorphic] to the * n * -dimensional space ** R ** n * . *

Although manifolds resemble Euclidean spaces near each point ("locally"), the global structure of a manifold may be more complicated. For example, any point on the usual two-dimensional surface of a [sphere] is surrounded by a circular region that can be flattened to a circular region of the plane, as in a geographical map. However, the sphere differs from the plane "in the large": in the language of topology, they are not homeomorphic. The structure of a manifold is encoded by a collection of * charts * that form an [atlas] , in analogy with an atlas consisting of charts of the surface of the Earth.

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