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A countable chart cover detects manifold null sets

Statement

Let $M$ be a smooth manifold. If ${(U_{j}, φ_{j})}_{j \in N}$ is a countable smooth atlas with relatively compact domains, then a subset $E \subseteq M$ is null if and only if every $φ_{j} (E \cap U_{j})$ is null in $R^{\dim M}$ when $\dim M \geq 1$ , and every $φ_{j} (E \cap U_{j})$ is empty when $\dim M = 0$ .

Facts & Assumptions

Given: A countable smooth atlas ${(U_{j}, φ_{j})}_{j \in N}$ with relatively compact domains on a smooth manifold $M$ .

[F1]

On a $0$ -manifold, the only null subset is the empty set (Null subsets of a smooth manifold).

[L2]

Nullity is independent of the chosen smooth atlas (The null-set definition is independent of the smooth atlas).

Proof

technique · direct

1.1

If $\dim M = 0$ , [F1] says that $E$ is null exactly when $E = \emptyset$ . Because the chart domains cover $M$ , this is equivalent to every $E \cap U_{j}$ being empty, hence to every chart image being empty.

F1givencases

1.2

Assume $\dim M \geq 1$ . If $E$ is null, then every chart image $φ_{j} (E \cap U_{j})$ is null by definition.

givencases

2.1

Conversely, the given countable atlas is itself a smooth atlas, so [L2] says that being null with respect to this atlas is the same as being null with respect to any other. Therefore the displayed chartwise condition implies that $E$ is null.

L2step 1.1step 1.2

3.1

Hence this countable chart cover detects manifold null sets in every dimension.

step 1.1step 1.2step 2.1∎

Depends on

Used by

A null set has dense complement in a positive-dimensional manifold Proposition
An equidimensional C¹ map sends null sets to null sets Proposition
Countable unions and subsets of manifold null sets are null Proposition
The image of a lower-dimensional C¹ manifold is null Proposition
Morse-Sard for smooth manifolds Theorem

Dependency tree · two levels

9 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)