CorollaryStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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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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Outside a null set every translation makes a chosen value a transverse zero

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $f : M \to R^{n}$ be smooth and let $q \in R^{n}$ . Then outside a null subset of $a \in R^{n}$ , the map $p \mapsto f (p) + a$ has $q$ as a regular value.

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth map $f : M \to R^{n}$ and a point $q \in R^{n}$ .

[L1]

Under ${A C}_{ω}$ , for the point submanifold ${q}$ , generic translations are transverse to it outside a null set (Generic translations of a Euclidean-valued map are transverse).

[L2]

Transversality to a point is the regular-value condition (Transversality to a point is the regular-value condition).

Proof

technique · direct

1.1L1given

Apply [L1] with $Z = {q}$ . Then outside a null subset of $a \in R^{n}$ , the translated map $f_{a} (p) = f (p) + a$ is transverse to ${q}$ .

2.1L2step 1.1

By [L2], $f_{a} ⋔ {q}$ is equivalent to $q$ being a regular value of $f_{a}$ .

3.1step 2.1∎

Therefore outside a null set of translations, the chosen value becomes a transverse zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)