Let $M$ be a compact [topological manifold](/page/Topological%20Manifold) of dimension $n$, let $I$ be an index set, and for each $i\in I$ let $U_i\subset M$ be open, let $V_i\subset\mathbb R^n$ be open, and let $\varphi_i:U_i\to V_i$ be a homeomorphism. Suppose $\{(U_i,\varphi_i)\}_{i\in I}$ is an atlas for $M$, so that
Then there exists a finite subset $J\subset I$ such that $\{(U_j,\varphi_j)\}_{j\in J}$ is a finite subcollection of the atlas whose chart domains cover $M$; equivalently,