CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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 closed subset of a manifold is the zero set of a smooth nonnegative function

Statement

Every closed subset $A$ of a smooth manifold $M$ is the zero set of some smooth nonnegative function $g : M \to [0, \infty)$ .

Facts & Assumptions

Given: A closed subset $A$ of a smooth manifold $M$ .

[L1]

Every open cover of a manifold has a countable cover by relatively compact coordinate balls subordinate to it (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover).

[L2]

A countable cover by coordinate balls with compact closures has a countable locally finite shrinking $W_{k} ⋐ V_{k}$ (A countable coordinate-ball cover has a countable locally finite shrinking).

[L3]

For every compact set inside an open set there is a smooth manifold bump equal to $1$ near that compact set and supported in the open set (A manifold bump for a compact set inside an open set).

[L4]

A locally finite sum of smooth functions is smooth (A locally finite sum of smooth functions is smooth).

Proof

technique · direct

1.1

Apply [L1] to the one-set open cover ${M ∖ A}$ of the open manifold $M ∖ A$ to obtain a countable cover by coordinate balls with compact closures contained in $M ∖ A$ . Then apply [L2] to obtain a countable locally finite shrinking $W_{k} ⋐ V_{k}$ of that cover. For each $k$ , apply [L3] to $\overline{W_{k}} \subseteq V_{k}$ to obtain a smooth function $b_{k} : M \to [0, 1]$ that is positive on $W_{k}$ and supported in $M ∖ A$ .

L1L2L3givenchoose

2.1

The family $(b_{k})$ is locally finite, so $g : = \sum_{k \geq 1} 2^{- k} b_{k}$ is smooth and nonnegative by [L4]. One has $g = 0$ on $A$ because every $b_{k}$ vanishes there, and $g > 0$ on $M ∖ A$ because each point there lies in some $W_{k}$ .

L4step 1.1

3.1

Therefore $A = g^{- 1} (0)$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

14 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)