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

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Smooth partitions of unity subordinate to an open cover

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources