LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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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.

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Homology of an infinite CW complex is the colimit of skeletal homology

Statement

Assume the Axiom of Choice. For a CW complex $X$ , the natural map ${\underset{\to}{l i m}}_{n} H_{k} (X^{n}; G) \to H_{k} (X; G)$ is an isomorphism for every $k$ .

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and a singular cycle or a singular bounding chain in $X$ .

Proof

technique · direct

1.1given

A singular chain has finitely many simplices, so its image is compact; The image of a compact space lies in a finite CW subcomplex places it in one finite subcomplex and hence in some skeleton. Thus every homology class is in the image.

2.1step 1.1∎

If a class from $H_{k} (X^{n})$ dies in $H_{k} (X)$ , a finite singular bounding chain has compact image and lies in some $X^{m}$ . It already kills the class there, proving injectivity of the colimit map.

Depends on

Used by

Cellular homology of an infinite-dimensional projective space Example

Dependency tree · two levels

13 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, Appendix A (standard reference, not scraped)