PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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The null-set definition is independent of the smooth atlas

Statement

If $A$ and $B$ are smooth atlases on the same smooth manifold $M$ , then a subset $E \subseteq M$ is $A$ -null if and only if it is $B$ -null.

Facts & Assumptions

Given: Smooth atlases $A$ and $B$ on a smooth manifold $M$ , and a subset $E \subseteq M$ .

[F1]

A set is atlas-null when every chart image in that atlas is Euclidean null (Null subsets of a smooth manifold).

[L1]

Local diffeomorphisms preserve null sets locally ( $C^{1}$ local diffeomorphisms preserve null sets locally).

[L2]

Every open cover admits a countable cover by relatively compact coordinate balls subordinate to it (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover).

Proof

technique · direct

1.1

Assume $E$ is $A$ -null. Fix a chart $(V, ψ) \in B$ . The sets $V \cap U$ with $(U, φ) \in A$ cover $V$ , so [L2] gives a countable cover of $V$ by relatively compact coordinate balls $W_{j} \subseteq V \cap U_{j}$ .

F1L2givenchoose

2.1

On each $W_{j}$ , the transition map $ψ \circ φ_{j}^{- 1}$ is a local diffeomorphism between Euclidean chart domains. Since $E$ is $A$ -null, [F1] makes $φ_{j} (E \cap W_{j})$ null. Applying [L1] to the transition map shows that $ψ (E \cap W_{j})$ is null for every $j$ .

F1L1step 1.1

3.1

The set $ψ (E \cap V)$ is the countable union of the null sets $ψ (E \cap W_{j})$ , hence is null. Since $(V, ψ)$ was arbitrary, $E$ is $B$ -null by [F1]. The reverse implication is symmetric.

F1step 2.1algebra∎

Depends on

Used by

A countable chart cover detects manifold null sets Proposition

Cited to discharge well-definedness by Null subsets of a smooth manifold.

Dependency tree · two levels

11 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)