DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Smooth partitions of unity subordinate to an open cover

Definition

Let $M$ be a smooth manifold and let $(U_{i})_{i \in I}$ be an open cover of $M$ . A family of smooth functions $(ϕ_{i})_{i \in I}$ with $ϕ_{i} : M \to [0, 1]$ is a smooth partition of unity subordinate to $(U_{i})_{i \in I}$ when:

the family $(supp (ϕ_{i}))_{i \in I}$ is locally finite;
$supp (ϕ_{i}) \subseteq U_{i}$ for every $i \in I$ ; and
$\sum_{i} ϕ_{i} (p) = 1$ for every $p \in M$ .

Remarks

Because the support family is locally finite, the pointwise sum in (3) is locally a finite sum.

Depends on

Used by

A continuous partition of unity need not be smooth False statement
A smooth Urysohn lemma for a closed set in an open set Theorem
Smooth locally defined functions can be glued by a partition of unity Theorem
Smooth partitions of unity exist on manifolds Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)