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	Topics Atlas (topology) Topic Home In topologyFacts About Topology Topology is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation ; these are ... , a branch of mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , an atlas describes how a complicated spaceTopological space Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.... called a manifoldManifold Summary A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in... 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: A list of spaces that are considered simple. For each point in the complicated space, a neighborhood of that point that is homeomorphicHomeomorphism In the mathematical field of topology a homeomorphism or topological isomorphism is a special isomorphism between top... to a simple space. The homeomorphism is called a chart. Different charts being compatible is required. At the minimum, it is required that the composite of one chart with the inverse of another be a homeomorphism (known as a change of coordinates or a transition functionTransition function In mathematics, a transition function has several different meanings:... ), yet usually stronger requirements, such as smoothnessSmooth function In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders:... , are imposed. 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 planePlane (mathematics) In mathematics, a plane is a fundamental two-dimensional object.... . 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 one chooses for simple spaces Rⁿ, topological manifoldTopological manifold A manifold is a topological space which is locally homeomorphic to Euclidean space.... s are obtained. If one also requires the coordinate changes to be diffeomorphismDiffeomorphism In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds.... s, differentiable manifoldDifferentiable manifold Informally, a differentiable manifold is a kind of topological space that is similar enough to Euclidean space to allow one ... s are obtained. Two atlases (over the same underlying topological space) are compatible if the charts in the two atlases are all compatible (or equivalently if the union of the two atlases is an atlas). Formally, (as long as the concept of compatibility for charts satisfies certain simple properties), compatibility defines an equivalence relationEquivalence relation In mathematics, an equivalence relation, denoted by an infix "~", is a binary relation on a set X that is reflexive,... on the set of all atlases. Usually, we consider compatible atlases as giving rise to the same manifold (we don't care how the manifold was "glued together", only what is left after "taking away the glue"), and so each of the equivalence classes corresponds to one manifold. 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. By definition, a smooth differentiable structureDifferential structure Overview In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifol... (or differential structureDifferential structure In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifol... ) on a manifold M is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps. External links by Rowland, Todd