In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρ_i} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1,

Partitions of unity are useful because they often allow one to extend local constructions to the whole space.

See also: paracompact space