Partition of unity

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 0, and
• the sum of all the function values at x is 1, i.e., $\;\sum_{i\in I} \rho_i(x) = 1$.

Sometimes, the requirement is not as strict: the sum of all the function values at a particular point is only required to be positive rather than a fixed number for all points in the space

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:

1. 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.
2. 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 subordinate to any open cover. 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 manifold categories, but not the analytic category. Thus analytic partitions of unity do not exist. See analytic continuation.

Applications

A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over a manifold: One first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that the definition is independent of the chosen partition of unity.

A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.

See also

• gluing axiom
• fine sheaf

References

• Tu, Loring W. (2011), An introduction to manifolds, Universitext (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4419-7400-6, ISBN 978-1-4419-7399-3 , see chapter 13

External links

• General information on partition of unity at [Mathworld]
• Applications of a partition of unity at [Planet Math]