Open cover

Let $M$ be a smooth manifold .

An open cover of $M$ is a family of open sets $\{U_i\}_{i\in I}$ with $U_i\subset M$ such that

An open cover is called finite if $I$ is finite, and locally finite if every point of $M$ has a neighborhood meeting only finitely many $U_i$.

Open covers are the basic index sets for specifying local geometric data (charts, local frames, connection forms, etc.) together with overlap compatibilities, for instance the transition functions of a principal G-bundle or the local descriptions in an equivariant local trivialization .

Examples

1. Two-chart cover of the sphere. $S^2$ is covered by $U_N=S^2\setminus\{\text{south pole}\}$ and $U_S=S^2\setminus\{\text{north pole}\}$.
2. Cover by coordinate domains. Any smooth atlas on $M$ provides an open cover by chart domains.
3. Cover by a neighborhood basis refinement. If $\{V_\alpha\}$ is any family of open sets whose union is $M$, then for each $x\in M$ one may pick a smaller open neighborhood $U_x\subset V_{\alpha(x)}$; the resulting $\{U_x\}_{x\in M}$ is again an open cover, often tailored to local constructions.