Let $M$ be a [topological manifold](/page/Topological%20Manifold) of dimension $n$, where topological manifolds are assumed to be second-countable. If $M\neq\varnothing$, then there exists an atlas $\{(U_j,\varphi_j)\}_{j\in J}$ for $M$ whose index set $J$ is finite or countably infinite. If $M=\varnothing$, then the empty collection is an atlas for $M$.