CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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Excision fails without closure inside interior

Statement refuted

The condition $\overline{Z} \subseteq int (A)$ cannot be omitted from excision. Take $G = Z$ , $X = [0, 1]$ , and $A = Z = {0}$ .

Counterexample

Given: The indicated $X, A, Z$ .

Proof technique: direct.

1.1givenconstruct

The excised pair is $((0, 1], \emptyset)$ , whose $H_{0}$ is $Z$ ; the original pair has $H_{0} ([0, 1], {0}) = 0$ by its pair sequence.

2.1step 1.1algebra∎

Hence the inclusion cannot induce an isomorphism in degree zero. Here $\overline{Z} = {0}$ is not contained in the empty interior of $A$ , exposing the missing hypothesis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Theorem 2.20 (standard reference, not scraped)