CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Contractible nonempty spaces have the homology of a point

Statement

If $X$ is a nonempty contractible topological space, then for every $n \geq 0$ and every abelian group $G$ , $H_{n}^{s i n g} (X; G) ≅ H_{n}^{s i n g} (*; G),$ where $*$ denotes a one-point space.

Facts & Assumptions

Given: A nonempty contractible topological space $X$ , an abelian group $G$ , and an integer $n \geq 0$ .

[L1]

For a nonempty space, contractibility is equivalent to the identity map being nullhomotopic (A nonempty space is contractible if and only if its identity map is nullhomotopic).

[L2]

A map that is homotopic to a constant map is nullhomotopic, and a space is contractible exactly when every map out of it is nullhomotopic (Nullhomotopic maps and contractible spaces).

[L3]

Homotopy equivalences induce isomorphisms on singular homology (Homotopy equivalences induce isomorphisms on singular homology).

Proof

technique · direct

1.1

By [L1], the identity map on $X$ is homotopic to the constant map at some point $x_{0} \in X$ . Let $p : X \to *$ be the unique map and let $i : * \to X$ send the point of $*$ to $x_{0}$ . Then $i \circ p$ is the constant map at $x_{0}$ , so $i \circ p ≃ {id}_{X}$ , while $p \circ i = {id}_{*}$ . Hence $X$ is homotopy equivalent to a point.

L1L2given

2.1

Apply [L3] to the map $p : X \to *$ . It induces an isomorphism on every singular homology group, which is exactly the displayed conclusion.

L3step 1.1∎

Depends on

Used by

The homology of an interval from contractibility Example

Dependency tree · two levels

9 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 (standard reference, not scraped)