LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-09-23 (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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every open cover of a manifold has a countable relatively compact coordinate-ball subcover

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every open cover of a smooth manifold has a countable cover by coordinate balls with compact closures, each closure contained in one member of the original cover.

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth manifold $M$ , and an open cover $U$ of $M$ .

[L1]

Coordinate balls form a basis of the underlying topological manifold (Coordinate balls form a basis of a topological manifold).

[L2]

Under ${A C}_{ω}$ , second-countable spaces are Lindelof (Assuming countable choice, every second countable space is Lindelöf).

[A1]

By the library convention in Smooth manifolds and their smooth charts, every smooth manifold is second countable.

Proof

technique · direct

1.1L1givenconstruct

Form the set $B$ of all coordinate balls $B$ for which $\overline{B}$ is compact and $\overline{B} \subseteq U$ for at least one $U \in U$ . No member is chosen for each point. This family covers $M$ : for a given $p$ , one cover member $U$ contains it, and [L1] supplies one such ball within $U$ .

2.1A1L2step 1.1given

The family $B$ is an open cover, so [A1] and [L2] give a finite or countable subcover $(B_{n})$ . By its definition, each $B_{n}$ has a nonempty set of witness members $U \in U$ with $\overline{B_{n}} \subseteq U$ . Use ${A C}_{ω}$ once to choose a witness $U_{n}$ for each listed ball (finite choice suffices in the finite case). Thus $\overline{B_{n}}$ is compact and contained in $U_{n}$ .

3.1step 2.1∎

Thus the original cover has a countable subordinate cover by relatively compact coordinate balls.

Depends on

Used by

Dependency tree · two levels

15 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)