CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 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.

A smooth map from lower to higher dimension cannot be surjective

Statement

Assume the Axiom of Countable Choice. If $F : M^{m} \to N^{n}$ is smooth and $m < n$ , then $F$ is not surjective onto any nonempty $N$ .

Facts & Assumptions

Given: Countable Choice and a smooth map $F : M^{m} \to N^{n}$ with $m < n$ and $N \neq \emptyset$ .

[L1]

The image of a lower-dimensional $C^{1}$ manifold is null (The image of a lower-dimensional $C^{1}$ manifold is null).

[L2]

In a positive-dimensional manifold, a null set has dense complement (A null set has dense complement in a positive-dimensional manifold).

Proof

technique · direct

1.1L1given

By [L1] under the stated Countable Choice, $F (M)$ is a null subset of $N$ . Since $m < n$ , the target dimension is positive.

2.1L2step 1.1contradiction

If $F$ were surjective, then $N = F (M)$ would be null in itself. But [L2] would then force its complement $\emptyset$ to be dense in $N$ , impossible because $N$ is nonempty.

3.1discharge-contradiction: surjectivity impossiblestep 2.1∎

Hence $F$ cannot be surjective.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)