PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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Every smooth manifold admits a countable smooth atlas with relatively compact domains

Statement

Every smooth manifold admits a countable smooth atlas whose chart domains have compact closures.

Facts & Assumptions

Given: A smooth manifold $M$ .

[L1]

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).

[F1]

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

Proof

technique · direct

1.1

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

L1given

2.1

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].

F1step 1.1

3.1

This is the required countable smooth atlas.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)