LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passverified 2026-09-09 (Codex)

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.

Smooth extension from a closed neighbourhood

Statement

Assume countable choice ${A C}_{ω}$ . Let $C$ be a closed subset of a smooth manifold $M$ , let $U \subseteq M$ be open with $C \subseteq U$ , and let $f : U \to R$ be smooth. Then there exists a smooth function $F : M \to R$ such that $F = f$ on an open neighbourhood of $C$ and $supp (F) \subseteq U$ .

Separately, without any choice assumption, the same conclusion holds whenever a smooth cutoff $χ : M \to [0, 1]$ equal to one on an open neighbourhood of $C$ and satisfying $supp (χ) \subseteq U$ is supplied. Here supports are closures in $M$ of nonzero loci. Countable choice is used only to obtain the cutoff, not to extend a supplied one.

Facts & Assumptions

Given: A closed set $C \subseteq M$ , an open set $U \subseteq M$ containing $C$ , and a smooth function $f : U \to R$ ; for the existence clause assume countable choice, and for the conditional clause take a supplied cutoff as stated.

[L1]

Under countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), there is a smooth cutoff $χ : M \to [0, 1]$ equal to $1$ on a neighbourhood of $C$ and supported in $U$ (A smooth Urysohn lemma for a closed set in an open set). That theorem uses countable choice for countable coordinate witnesses, finite annular covers, and bumps.

[L2]

Smooth maps paste over an open cover (Smooth maps paste over an open cover).

[L3]

The one-variable product rule applies to each coordinate partial derivative (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ). Induction on derivative words expresses every iterated derivative of a product as a finite sum of products of derivatives of the factors, all continuous. Thus products of smooth real-valued functions are smooth in charts ( $C^{k}$ maps and multi-index derivative notation in Euclidean space). Constant zero is smooth because its positive-order derivatives vanish.

Proof

technique · direct

1.1L1givenconstruct

For the existence clause obtain $χ$ from [L1], spending countable choice only through that supplier. For the conditional clause use the supplied $χ$ instead, without invoking [L1]. Let $V$ be its supplied open neighbourhood of $C$ on which $χ = 1$ , and replace $V$ by $V \cap U$ . Put $S = supp (χ)$ and $W = M ∖ S$ . Then $S$ is closed, $W$ is open, and $U \cup W = M$ because $S \subseteq U$ .

2.1L2L3step 1.1

On $U$ take the smooth function $(χ ∣_{U}) f$ , smooth by [L3]; on $W$ take the smooth constant zero function. On $U \cap W$ these agree because $χ = 0$ outside $S$ . By [L2] they determine a smooth function $F : M \to R$ . This construction uses a two-member open cover, not a choice of a neighbourhood at each boundary point.

3.1step 1.1step 2.1∎

On $V$ one has $F = f$ because $χ = 1$ . Moreover $F$ vanishes on $W$ , so its nonzero locus lies in the closed set $S$ . Taking closure in $M$ gives $supp (F) \subseteq S \subseteq U$ . The construction includes $C = \emptyset$ , $U = \emptyset$ , and $M = \emptyset$ with the empty neighbourhood or empty local formula as appropriate. No choice is used after the cutoff has been supplied.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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)