In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a chart. A chart for a topological space M (also called a coordinate chart, coordinate patch, coordinate map, or local frame) is a homeomorphism from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair . Formal definition of atlas An atlas for a topological space is an indexed family of charts on which covers (that is, ). If the codomain of each chart is the n-dimensional Euclidean space, then is said to be an n-dimensional manifold. The plural of atlas is atlases, although some authors use atlantes. An atlas on an -dimensional manifold is called an adequate atlas if the image of each chart is either or , is a locally finite open cover of , and , where is the open ball of radius 1 centered at the origin and is the closed half space. Every second-countable manifold admits an adequate atlas. Moreover, if is an open covering of the second-countable manifold then there is an adequate atlas on such that is a refinement of . Transition maps A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.) To be more precise, suppose that and are two charts for a manifold M such that is non-empty. The transition map is the map defined by Note that since and are both homeomorphisms, the transition map is also a homeomorphism. More structure One often desires more structure on a manifold than simply the topological structure. For example, if one would like