TheoremStatement: 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.

Cover-small chains compute singular homology

Statement

The inclusion $C_{∙}^{U} (X; G) ↪ C_{∙} (X; G)$ induces an isomorphism on homology.

Facts & Assumptions

Given: A family $U$ whose interiors cover $X$ .

Proof

technique · direct

1.1givenconstruct

For an ordinary cycle $z$ , choose a depth with $S^{r} z$ small. Repeatedly applying $1 - S = \partial T + T \partial$ shows $z$ and $S^{r} z$ are homologous, proving surjectivity.

2.1step 1.1algebra∎

If a small cycle bounds an ordinary chain $b$ , choose a depth making $S^{r} b$ small. The same prism identity says the original cycle and its small subdivision differ by a boundary already in the small complex; together with $\partial S^{r} b = S^{r} \partial b$ this proves injectivity.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Proposition 2.21