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

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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

1.1

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.

givenconstruct

2.1

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.

step 1.1algebra∎

Depends on

Used by

Mayer–Vietoris sequence in singular homology Theorem
Naturality of singular Mayer–Vietoris Theorem
The cover-small inclusion is a chain homotopy equivalence Theorem

Dependency tree · two levels

8 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)