# Partition of unity

In mathematics, a partition of unity of a topological space X is a set of continuous functionsi} 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,

[itex]\sum_{i\in I} \rho_i(x) = 1 \mbox{ for all } x \in X.[itex]

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

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