Partition of unity
In mathematics, a partition of unity on a topological space X {\displaystyle X} is a set R {\displaystyle R} of continuous functions from X {\displaystyle X} to the unit interval [0,1] such that for every point x ∈ X {\displaystyle x\in X} : there is a neighbourhood of x {\displaystyle x} where all but a finite number of the functions of R {\displaystyle R} are zero, and the sum of all the function values at x {\displaystyle x} is 1, i.e., ∑ ρ ∈ R ρ ( x ) = 1. {\textstyle \sum _{\rho \in R}\rho (x)=1.} Partitions of unity are useful because they often allow one to extend local constructions to the whole space.