PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

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CW skeleta are closed and cells form a disjoint partition

Statement

For a CW complex $X$ , every skeleton $X^{n}$ is closed, and $X^{n} ∖ X^{n - 1}$ is the disjoint union of the open $n$ -cells.

Facts & Assumptions

Given: A CW complex and its skeleta as in Skeleta, CW subcomplexes, and relative CW complexes.

Proof

technique · direct

1.1

Let $χ_{e} : D^{m} \to X$ be the characteristic map of an $m$ -cell. If $m \leq n$ , then $χ_{e}^{- 1} (X^{n}) = D^{m}$ . If $m > n$ , this inverse image lies in $S^{m - 1}$ and equals the inverse image there of the already closed lower skeleton $X^{n} \subseteq X^{m - 1}$ ; induction on $m$ makes it closed in $S^{m - 1}$ and hence in $D^{m}$ . The weak-topology clause now makes $X^{n}$ closed in $X$ .

given

2.1

The open-cell interiors are disjoint by the cellular partition, and The interior of an attached cell embeds openly in its closure identifies precisely the new interior points after $X^{n - 1}$ . This gives the asserted difference.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

4 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, Chapter 0 (standard reference, not scraped)