DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-09-23 (gpt-6-sol)

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 $M$ .

Depends on

Used by

Algebraic open inclusions surject on fundamental groups Lemma
Metastable approximation of maps by embeddings Lemma
A countable chart cover detects manifold null sets Proposition
A null set has dense complement in a positive-dimensional manifold Proposition
The image of a lower-dimensional C¹ manifold is null Proposition
The null-set definition is independent of the smooth atlas Proposition
Cut locus of a point has riemannian volume zero Theorem
For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions Theorem
Morse-Sard for smooth manifolds Theorem
Strong Whitney approximation by transverse maps Theorem

Dependency tree · two levels

18 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, 2nd ed. (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)