Partition of unity

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In mathematics, a partition of unity of a topological space X is a set of continuous functions, $\{\rho_i\}_{i\in I}$ , from X to the unit interval [0,1] such that for every point, $x\in X$ ,

there is a neighbourhood of x where all but a finite number of the functions are identically zero, and
the sum of all the respective function values at x is identically 1, i.e., $\;\sum_{i\in I} \rho_i(x) = 1$ .

A partition of unity of a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphing purposes. The dashed line on top is the sum of the functions in the partition.

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

The existence of partitions of unity assumes two distinct forms:

Given any open cover {U_i}_i∈I of a space, there exists a partition {ρ_i}_i∈I indexed over the same set I such that supp ρ_i⊆U_i. Such a partition is said to be subordinate to the open cover {U_i}_i.
Given any open cover {U_i}_i∈I of a space, there exists a partition {ρ_j}_j∈J indexed over a possibly distinct index set J such that each ρ_j has compact support and for each j∈J, supp ρ_j⊆U_i for some i∈I.

Thus one chooses either to have the supports indexed by the open cover, or the supports compact. If the space is compact, then there exist partitions satisfying both requirements.

Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity. Depending on the category which the space belongs to, it may also be a sufficient condition. The construction uses mollifiers (bump functions), which exist in the continuous and smooth categories, but not the analytic category. Thus analytic partitions of unity do not exist.