TheoremStatement: 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.

Long exact sequence of a pair

Statement

For $A \subseteq X$ there is an exact sequence $\dots \to H_{n} (A; G) \to H_{n} (X; G) \to H_{n} (X, A; G) \overset{δ}{\to} H_{n - 1} (A; G) \to H_{n - 1} (X; G) \to \dots .$

Facts & Assumptions

Given: A subspace $A \subseteq X$ .

Proof

technique · direct

1.1

Inclusion and quotient form a degreewise short exact sequence $0 \to C_{∙} (A; G) \to C_{∙} (X; G) \to C_{∙} (X, A; G) \to 0$ .

givenconstruct

2.1

The long-exact-sequence theorem for a short exact sequence of complexes applied to step 1.1 gives precisely the displayed sequence, with the third homology group equal to relative homology by definition.

step 1.1algebra∎

Depends on

Used by

Good pairs and quotient reduced homology Corollary
Excision fails without closure inside interior Counterexample
Relative connecting homomorphism on cycles Definition
Relative homology of a disk and its boundary Example
Relative homology of an interval and its endpoints Example
Naturality of the pair long exact sequence Theorem

Dependency tree · two levels

10 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 2.16 (standard reference, not scraped)