CounterexampleConstruction: Literature-sourcedVerification: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The closure of a CW cell need not be a closed ball

Statement refuted

Every closed cell in a CW complex is homeomorphic to a closed ball.

Counterexample

Given: Attach one $1$ -cell to one $0$ -cell by sending both points of $S^{0}$ to that vertex.

Proof technique: direct.

1.1

By Cell attachment by a characteristic map, the closed-cell image is $D^{1} / \partial D^{1} ≅ S^{1}$ .

given

2.1

Its open cell is still the embedded open interval by The interior of an attached cell embeds openly in its closure, but $S^{1}$ is not homeomorphic to $D^{1}$ ; this refutes the statement.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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)