PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

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

Relative homology for the empty and total subspace

Statement

For every space $X$ and every integer $n$ , $H_{n} (X, \emptyset; G) ≅ H_{n} (X; G)$ and $H_{n} (X, X; G) = 0$ , including $n = 0$ .

Facts & Assumptions

Given: A space $X$ and an integer $n$ .

Proof

technique · direct

1.1

Since every singular chain group of $\emptyset$ is zero, $C_{∙} (X, \emptyset; G) = C_{∙} (X; G)$ as complexes.

givenalgebra

2.1

Since $C_{∙} (X; G) / C_{∙} (X; G)$ is the zero complex, its cycles and boundaries are both zero in every degree. Taking homology proves both assertions.

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, §2.1 (standard reference, not scraped)