Global analysis, analysis on manifolds Definition

Partition of unity

Builds on Topological space Continuous function Open cover Locally finite family Smooth manifold

Definition

Let $X$ be a topological space. For a function $\rho \colon X \to \R$ , its support $\operatorname{supp} \rho$ is the closure of $\set{x \in X : \rho(x) \neq 0}$ . A partition of unity on $X$ is a family $(\rho_i)_{i \in I}$ of continuous functions $\rho_i \colon X \to [0, 1]$ such that:

the family of supports $(\operatorname{supp} \rho_i)_{i \in I}$ is locally finite: each point of $X$ has a neighbourhood that meets only finitely many of them;
$\sum_{i \in I} \rho_i(x) = 1$ for every $x \in X$ .

Let $(U_i)_{i \in I}$ be an open cover of $X$ with the same index set. The partition of unity is subordinate to $(U_i)_{i \in I}$ if $\operatorname{supp} \rho_i \subset U_i$ for every $i$ . If $X$ is a smooth manifold and every $\rho_i$ is smooth, it is a smooth partition of unity.

Remarks

Example on $\R$ with the cover $U_1 = (-\infty, 1)$ and $U_2 = (-1, \infty)$ . Let $\rho_1(x) = 1$ for $x \le -1/2$ , $\rho_1(x) = 1/2 - x$ for $-1/2 \le x \le 1/2$ , and $\rho_1(x) = 0$ for $x \ge 1/2$ , and let $\rho_2 = 1 - \rho_1$ . Both are continuous with values in $[0, 1]$ , the sum is $1$ , and $\operatorname{supp} \rho_1 = (-\infty, 1/2] \subset U_1$ and $\operatorname{supp} \rho_2 = [-1/2, \infty) \subset U_2$ . Replacing the linear piece by a smooth monotone transition gives a smooth partition of unity.

Non-example: the constant functions $\rho_1 = \rho_2 = 1/4$ on $\R$ are continuous with locally finite supports, but their sum is $1/2$ , so clause 2 fails.

Clause 1 makes the sum in clause 2 a finite sum near each point, so it is continuous, or smooth in the smooth case. Every paracompact Hausdorff space has a partition of unity subordinate to any open cover. Every smooth manifold has a smooth partition of unity subordinate to any open cover; the proof uses bump functions and second countability.

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Tag	`KK8KZQNDVY`
URI	`m2://KK8KZQNDVY`
Kind	definition
Writing	W0 stub
MSC	58A05, 54D20
Defines	Partition of unity
Uses	Topological space, Continuous function, Open cover, Locally finite family, Smooth manifold
Changed	2 November 2026

Partition of unity

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