PropositionStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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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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Assuming Countable Choice, every smooth manifold admits a countable smooth atlas with relatively compact domains

Statement

Assume the Axiom of Countable Choice. Every smooth manifold admits a countable smooth atlas whose chart domains have compact closures.

Facts & Assumptions

Given: The Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and a smooth manifold $M$ .

[L1]

Under Countable Choice, the trivial open cover ${M}$ has a countable subordinate cover by relatively compact coordinate balls (Every open cover of a manifold has a countable relatively compact coordinate-ball subcover). Applying this supplier in step 1.1 is the use of Countable Choice here.

[F1]

A smooth atlas is a cover by pairwise smoothly compatible smooth charts (Smooth atlases).

Proof

technique · direct

1.1L1given

By [L1], there are countably many coordinate balls $B_{1}, B_{2}, \dots$ covering $M$ , each with compact closure.

2.1F1step 1.1

Each $B_{n}$ carries its inherited smooth chart, and these charts are pairwise compatible because they come from the smooth structure of $M$ . Thus they form a countable smooth atlas by [F1].

3.1step 2.1∎

This is the required countable smooth atlas.

Depends on

Used by

An equidimensional C¹ map sends null sets to null sets Proposition
The image of a lower-dimensional C¹ manifold is null Proposition
Cut locus of a point has riemannian volume zero Theorem
Morse-Sard for smooth manifolds Theorem

Dependency tree · two levels

13 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)