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Encyclopedia > Atlas (topology)

In topology, a branch of mathematics, 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). A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

More precisely, an atlas for a complicated space is constructed out of the following pieces of information:

• A list of spaces that are considered simple.
• For each point in the complicated space, a neighborhood of that point that is homeomorphic to a simple space. The homeomorphism is called a chart.
• We require the different charts to be compatible. At the minimum, we require that the composite of one chart with the inverse of another be a homeomorphism (known as a change of coordinates or a transition function), but we usually impose stronger requirements, such as smoothness.

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). This is a glossary of some terms used in the branch of mathematics known as topology. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... For other meanings of Atlas, see Atlas (disambiguation). ... Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ...

Different choices for simple spaces and compatibility conditions give different objects. For example, if we choose for our simple spaces Rn, we get topological manifolds. If we also require the coordinate changes to be diffeomorphisms, we get differentiable manifolds. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This article or section is in need of attention from an expert on the subject. ...

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). In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ...

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. In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ... In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ...

Results from FactBites:

 Atlas (topology) - encyclopedia article about Atlas (topology). (850 words) In topology Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. Topology is concerned with the study of the so-called topological properties of figures, that is to say, properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms). Each individual map in an atlas of the world gives a neighborhood of each point on the globe that is homeomorphic to the plane u + s v + t w where s and t range over all real numbers, and u, v and w are given vectors defining the plane.
 Atlas (topology) - Wikipedia, the free encyclopedia (412 words) 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. 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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