LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 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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A compact subspace of a CW complex meets only finitely many cells

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Every compact subspace of a CW complex meets only finitely many open cells.

Facts & Assumptions

Given: A compact $K \subseteq X$ , where $X$ is a CW complex.

[A1]

The Axiom of Choice supplies a choice function on the nonempty subsets of the set of cells meeting $K$ , and a point in each chosen nonempty cell intersection (The Axiom of Choice).

Proof

technique · contradiction

1.1A1givenassume-contra

If $K$ met infinitely many cells, use [A1] recursively on the nonempty set of cells remaining after finitely many earlier choices, obtaining distinct met cells and points $x_{i} \in K$ in them. Put $S = {x_{i} : i \geq 1}$ . Induction over the skeleta shows that $S \cap X^{n}$ is closed in $X^{n}$ : on the boundary of each characteristic $n$ -disk this follows from the induction hypothesis, and the disk interior contains at most one $x_{i}$ . The weak topology then closes the induction.

2.1step 1.1discharge-contradiction∎

The same argument applies to every subset of $S$ , so $S$ is a closed discrete subspace of $K$ . An infinite discrete space is not compact, contradicting compactness of the closed subspace $S \subseteq K$ .

Depends on

Used by

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

Dependency tree · two levels

7 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, Proposition A.1 (standard reference, not scraped)