In topology, an atlas describes how a complicated space 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 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), 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).
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.
The choice of atlas for a space is not unique, but we can always choose a unique maximal atlas: an atlas of charts refines another one if it adds charts (in such a way that the overlap functions remain compatible). The existence of maximal atlases, that cannot further be refined, follows from Zorn's lemma.
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.
Charts need not be geometric projections, and the number of charts is a matter of some choice.
A coordinate map, a coordinatechart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.
Charts in an atlas may overlap and a single point of a manifold may be represented in several charts.
Given two coordinatecharts, one containing p (representing an observer) and another containing q (another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results.
The idea of coordinatecharts as 'local observers who can perform measurements in their vicinity' also makes good physical sense, as this is how one actually collects physical data - locally.
In general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the Riemann tensor.
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