CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Good pairs and quotient reduced homology

Statement

Suppose $A$ is a nonempty closed subspace of $X$ and is a deformation retract of an open neighborhood $V$ in $X$ ; this is the good-pair hypothesis used here. Then the quotient map gives $H_{n} (X, A; G) ≅ {\tilde{H}}_{n} (X / A; G)$ for every $n$ .

Facts & Assumptions

Given: A nonempty closed subspace $A \subseteq X$ and an open neighborhood $V$ that deformation retracts onto $A$ through a homotopy fixing $A$ .

Proof

technique · direct

1.1

Since $V$ deformation retracts onto $A$ , $H_{*} (V, A; G) = 0$ . The quotient homotopy gives the same conclusion for $(V / A, A / A)$ . The long exact sequences of the short exact chain-complex sequences $0 \to C_{*} (V, A; G) \to C_{*} (X, A; G) \to C_{*} (X, V; G) \to 0$ and its quotient analogue therefore give $H_{*} (X, A; G) ≅ H_{*} (X, V; G), H_{*} (X / A, A / A; G) ≅ H_{*} (X / A, V / A; G) .$

givenalgebra

2.1

Because $A$ is closed and lies in the open set $V$ , excision applies to $(X, V)$ and to $(X / A, V / A)$ after removing respectively $A$ and the quotient point. The resulting removed pairs are homeomorphic, so the quotient map induces $H_{*} (X, V; G) ≅ H_{*} (X / A, V / A; G)$ . Combining with step 1.1 gives $H_{*} (X, A; G) ≅ H_{*} (X / A, A / A; G)$ . Since $A \neq \emptyset$ , $A / A$ is one point, and the augmented singular complex identifies the latter groups with ${\tilde{H}}_{*} (X / A; G)$ , including degree zero.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Proposition 2.22 (standard reference, not scraped)