DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-31

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.

Homotopy equivalences, homotopy inverses and spaces of the same homotopy type

Definition

Let $X, Y$ be topological spaces. A continuous map $f : X \to Y$ is a homotopy equivalence if there is a continuous map $g : Y \to X$ such that

$g \circ f ≃ {id}_{X} and f \circ g ≃ {id}_{Y}$

in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints. Such a $g$ is a homotopy inverse of $f$ .

The spaces $X$ and $Y$ have the same homotopy type, or are homotopy equivalent, written $X ≃ Y$ , when a homotopy equivalence $X \to Y$ exists.

The equations required of an ordinary inverse have been weakened to homotopies. Neither composite need equal the corresponding identity map, and a homotopy equivalence need not be bijective.

Depends on

Dependency tree · 2 levels

Sources

A. Hatcher, Algebraic Topology, Section 0