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

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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.

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A null set has dense complement in a positive-dimensional manifold

Statement

If $M$ is a positive-dimensional smooth manifold and $E \subseteq M$ is null, then $M ∖ E$ is dense in $M$ .

Facts & Assumptions

Given: A positive-dimensional smooth manifold $M$ and a null subset $E \subseteq M$ .

[L1]

A countable chart cover detects manifold nullity (A countable chart cover detects manifold null sets).

Proof

technique · direct

1.1

Suppose $M ∖ E$ were not dense. Then some nonempty open set $O \subseteq M$ would satisfy $O \subseteq E$ . Choose a chart $(U, φ)$ with $U \subseteq O$ and $φ (U) \subseteq R^{m}$ , where $m = \dim M \geq 1$ .

givenassume-contrachoose

2.1

Since $U \subseteq E$ , the chart image $φ (U)$ would be a null subset of $R^{m}$ by [L1]. But $φ (U)$ is a nonempty open subset of $R^{m}$ , so it contains a closed cube of positive side length and therefore cannot be null. Contradiction.

L1step 1.1contradiction

3.1

Hence every nonempty open subset of $M$ meets $M ∖ E$ , so $M ∖ E$ is dense.

discharge-contradiction: dense complementstep 2.1∎

Depends on

Used by

A smooth map from lower to higher dimension cannot be surjective Corollary
Regular values have null complement and are dense Corollary

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)