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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

Used by

No uniform subdivision depth for all singular simplices Counterexample
Relative cup product for an excisive triad Definition
Cover-small chains for the two-arc cover of a circle Example
The de Rham map commutes with Mayer–Vietoris connectors Lemma
The period pairing is well defined and computed by integration Lemma
Cover-small chains compute singular homology Theorem
Smooth singular mayer vietoris sequence Theorem
The cover-small inclusion is a chain homotopy equivalence Theorem
The symplectic period formula for integrals of wedge products Theorem

Dependency tree · two levels

23 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)