CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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.

Every open subset of a manifold is the cozero set of a smooth function

Statement

Assume the Axiom of Countable Choice. Every open subset $U$ of a smooth manifold $M$ is the cozero set of a smooth function on $M$ .

Facts & Assumptions

Given: Countable Choice and an open set $U \subseteq M$ .

[L1]

Under Countable Choice, every closed subset of a manifold is the zero set of a smooth nonnegative function (Every closed subset of a manifold is the zero set of a smooth nonnegative function).

Proof

technique · direct

1.1given

The complement $A : = M ∖ U$ is closed.

2.1L1step 1.1

By [L1], there is a smooth nonnegative function $g$ with $A = g^{- 1} (0)$ , so $U = {x : g (x) \neq 0}$ .

3.1step 2.1∎

Hence $U$ is a smooth cozero set.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)