Partition of unity

In mathematics, a partition of unity on a topological space ⁠

X

{\displaystyle X}

⁠ is a set ⁠

R

{\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, and

the sum of all the function values at ⁠

x

{\displaystyle x}

⁠ is 1, i.e.,

∑

ρ

∈

R

ρ

(

x

)

=

1.

{\textstyle \sum _{\rho \in R}\rho (x)=1.}

Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions.

Existence

The existence of partitions of unity assumes two distinct forms:

Given any open cover

{

U

i

}

i

∈

I

{\displaystyle \{U_{i}\}_{i\in I}}

of a space, there exists a partition of unity

{

ρ

i

}

i

∈

I

{\displaystyle \{\rho _{i}\}_{i\in I}}

indexed over the same set ⁠

I

{\displaystyle I}

⁠ such that supp

ρ

i

⊆

U

i

.

{\displaystyle \rho _{i}\subseteq U_{i}.}

Such a partition is said to be subordinate to the open cover

{

U

i

}

i

.

{\displaystyle \{U_{i}\}_{i}.}

Given any open cover

{

U

i

}

i

∈

I

{\displaystyle \{U_{i}\}_{i\in I}}

of a locally compact space, there exists a partition of unity

{

ρ

j

}

j

∈

J

{\displaystyle \{\rho _{j}\}_{j\in J}}

indexed over a possibly distinct index set ⁠

J

{\displaystyle J}

⁠ such that each ⁠

ρ

j

{\displaystyle \rho _{j}}

⁠ has compact support and for each ⁠

j

∈

J

{\displaystyle j\in J}

⁠ there is an ⁠

i

∈

I

{\displaystyle i\in I}

⁠ with supp

ρ

j

⊆

U

i

{\displaystyle \rho _{j}\subseteq U_{i}}

.

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

A finite open cover always has a continuous partition of unity subordinate to it, provided the space is locally compact and Hausdorff.

Paracompactness of the space is a necessary condition to guarantee the existence of a partition of unity subordinate to any open cover.