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In Topology , an atlas describes how a complicated Space called a Manifold is glued together from simpler pieces. Each piece is given by a '''chart''' (also known as '''coordinate chart''' or '''local coordinate system'''). More precisely, an atlas for a complicated space is constructed out of the following pieces of information:
This definition of atlas is exactly analogous to the non-mathematical meaning of Atlas . Each individual map in an atlas of the world gives a neighborhood of each point on the globe that is homeomorphic to the Plane . While each individual map does not exactly line up with other maps that it overlaps with (because of the Earth's curvature), the overlap of two maps can still be compared (by using latitude and longitude lines, for example). Different choices for simple spaces and compatibility conditions give different objects. For example, if we choose for our simple spaces Rn, we get Topological Manifold s. If we also require the coordinate changes to be Diffeomorphism s, we get Differentiable Manifold s. We call two atlases compatible if the charts in the two atlases are all compatible (or equivalently if the union of the two atlases is an atlas). Usually, we want to consider two compatible atlases as giving rise to the same space. Formally, (as long as our concept of compatibility for charts has certain simple properties), we can define an Equivalence Relation on the set of all atlases, calling two the same if they are compatible. In fact, the union of all atlases compatible with a given atlas is itself an atlas, called a '''complete (or maximal) atlas'''. Thus every atlas is contained in a unique complete atlas (N.B. we don't need Zorn's lemma as is sometimes assumed). By definition, a smooth Differentiable Structure (or Differential Structure ) on a manifold ''M'' is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps. |
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