DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)

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.

Null subsets of a smooth manifold

Definition

Let $M$ be a smooth $m$ -manifold, and let $A$ be a smooth atlas on $M$ . A subset $A \subseteq M$ is $A$ -null when, for every chart $(U, φ) \in A$ , the set

$φ (A \cap U) \subseteq R^{m}$

is null in the Euclidean sense of Measure zero and content zero in $R^{m}$ by countable and finite cube covers when $m \geq 1$ , and is empty when $m = 0$ . Equivalently, on a $0$ -manifold the only null subset is the empty set.

Under the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the next proposition shows that this condition is independent of the chosen smooth atlas. Under that premise one may therefore speak simply of a null subset of $M$ .

Null subsets of a smooth manifold

Definition

Depends on

Dependency tree · 2 levels

Sources