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Partitions of unity
In mathematics, a partition of unity on a topological spaceX{\displaystyle X} is a setR{\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, andthe sum of all the function values at x{\displaystyle x} is 1, i.e., ∑ρ∈Rρ(x)=1.{\textstyle \sum _{ ho \in R} ho (x)=1.}Partitions of unity are useful because they often allow one to extend local constructions to the whole space.
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