Topological atlas ($\mathcal{A}$)

A topological atlas is a family $\mathcal{A}=\{(U_i,{\phi }_i)\}$ of coordinate charts whose domains form an open cover of a topological manifold. Each point must belong to at least one chart domain; no individual domain needs to cover the entire topological manifold. On overlaps, topological transition maps describe the same points in two coordinate systems.