DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Compact exhaustions of a manifold

Definition

A compact exhaustion of a manifold $M$ is a sequence $(K_{n})_{n \geq 1}$ of compact subsets such that $K_{n} \subseteq int (K_{n + 1})$ for every $n \geq 1$ and $M = ⋃_{n \geq 1} K_{n}$ .

Remarks

The interior condition is what makes the exhaustion useful for local constructions.

Depends on

Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

Used by

Every manifold has a compact exhaustion Lemma

Dependency tree · two levels

6 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)