TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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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.
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Excision for singular homology

Statement

If $Z \subseteq X$ and $\overline{Z} \subseteq {int}_{X} (A)$ , then inclusion $(X - Z, A - Z) \to (X, A)$ induces isomorphisms $H_{n} (X - Z, A - Z; G) ≅ H_{n} (X, A; G)$ for every $n$ .

Facts & Assumptions

Given: $\overline{Z} \subseteq {int}_{X} (A)$ .

Proof

technique · direct

1.1givenconstruct

Use the cover ${X ∖ \overline{Z}, int A}$ : its cover-small quotient complex for $(X, A)$ consists, modulo small chains in $A$ , of precisely the small chains avoiding $Z$ .

2.1step 1.1algebra∎

The two inclusions from their small complexes to the corresponding full relative complexes are chain homotopy equivalences. The quotient identification in step 1.1 therefore induces the stated isomorphism.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Theorem 2.20