Partition of unity

Summary

A partition of unity is a family of nonnegative functions on a space that add up to $1$ at every point. Usually each function is required to live inside a chosen open set. It is the standard tool for gluing local constructions into global ones. On a smooth manifold one builds an object in each coordinate chart, multiplies it by the matching function, and adds the results. Smooth partitions of unity exist on every smooth manifold, which is why Riemannian metrics and many other structures exist on all of them.