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

1.1

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.

givenconstruct

2.1

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.

step 1.1algebra∎

Depends on

Used by

No uniform subdivision depth for all singular simplices Counterexample
Cover-small chains for the two-arc cover of a circle Example
Cover-small chains compute singular homology Theorem
The cover-small inclusion is a chain homotopy equivalence Theorem

Dependency tree · two levels

22 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

Allen Hatcher, Algebraic Topology, Proposition 2.21 (standard reference, not scraped)