CorollaryStatement: AI-adaptedProof: AI-adaptedverified 2026-09-26 (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.

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A closed Euclidean submanifold has a smooth neighborhood retraction

Statement

Assume countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), which follows from the Axiom of Choice (The Axiom of Choice). Every closed embedded smooth submanifold $S \subseteq R^{m}$ has an open neighbourhood $U$ and a smooth retraction $r : U \to S$ .

Facts & Assumptions

Given: Countable choice and a closed embedded smooth submanifold $S \subseteq R^{m}$ .

[L1]

The Euclidean tubular neighbourhood theorem gives a diffeomorphism $E : Ω_{δ} \to U$ from a variable-radius normal neighbourhood onto an open neighbourhood $U$ of $S$ (The Euclidean tubular neighbourhood theorem).

Proof

technique · direct

1.1L1givenconstruct

Countable choice supplies the hypothesis of [L1]; it is used only in obtaining its tubular diffeomorphism. Let $E : Ω_{δ} \to U$ be the tubular diffeomorphism from [L1]. The bundle projection $π : Ω_{δ} \to S, π (p, v) = p,$ is smooth.

2.1step 1.1algebra∎

Define $r : = π \circ E^{- 1} : U \to S$ . This map is smooth, and for $p \in S$ one has $E^{- 1} (p) = (p, 0)$ , so $r (p) = p$ . Hence $r$ is a smooth retraction.

Depends on

Used by

Dependency tree · two levels

10 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, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)