Manifold chart

A chart is a homeomorphism from an open subset of a manifold to an open subset of Euclidean space.

A smooth atlas is a collection of manifold charts covering a topological manifold whose coordinate changes are smooth transition maps. It determines a smooth manifold structure by adjoining every compatible chart.

On overlapping manifold charts, a transition map changes one coordinate representation to the other. In a smooth atlas it and its inverse are smooth on the corresponding open coordinate domains.

A nonempty compact manifold cannot be homeomorphic to an open subset of Euclidean space, since no nonempty Euclidean open set is compact.