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,
\sum_{i\in I} \rho_i(x) = 1 \mbox{ for all } x \in X.
Partitions of unity are useful because they often allow one to extend local constructions to the whole space. See also: paracompact space

 

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