LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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.

Finite chains eventually become cover-small

Statement

If the interiors of $U$ cover $X$ , then for every finite singular chain $c$ there is $r \geq 0$ with $S^{r} c \in C_{∙}^{U} (X; G)$ .

Facts & Assumptions

Given: A finite chain $c$ and a cover whose interiors cover $X$ .

Proof

technique · direct

1.1givenconstruct

The image of each simplex of $c$ is compact, so its pulled-back cover has a Lebesgue number; mesh decay supplies a subdivision depth making every subsimplex image lie in one cover member.

2.1step 1.1algebra∎

There are finitely many original simplices, so the maximum of their finitely many depths works for all of them. Thus $S^{r} c$ is cover-small; no depth is claimed for all singular simplices at once.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Proposition 2.21