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

Used by

A continuous map homotopic to a homotopy equivalence is itself a homotopy equivalence Corollary
A homotopy equivalence induces a bijection between path components Corollary
A weakly contractible CW complex is contractible Corollary
Whitehead theorem fails without CW type Counterexample
Fiber and fiber homotopy equivalence Definition
A one-point space and ℝ are homotopy equivalent but not homeomorphic Example
FALSE: homotopy-equivalent spaces must be homeomorphic False statement
The first Stiefel–Whitney class classifies orientability Proposition
De rham cohomology is smooth homotopy invariant Theorem
Having the same homotopy type is an equivalence relation on topological spaces Theorem
Homotopy equivalences induce isomorphisms on singular homology Theorem
The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse Theorem
Whitehead theorem Theorem

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

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)