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 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.
