CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-07-31

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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.

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A continuous map homotopic to a homotopy equivalence is itself a homotopy equivalence

Statement

Let $f_{0}, f : X \to Y$ be continuous maps with $f ≃ f_{0}$ . If $f_{0}$ is a homotopy equivalence, then $f$ is a homotopy equivalence. Every homotopy inverse of $f_{0}$ is also a homotopy inverse of $f$ .

Facts & Assumptions

Given: Continuous maps $f_{0}, f : X \to Y$ , a homotopy $f ≃ f_{0}$ , and a homotopy inverse $g : Y \to X$ of $f_{0}$ .

[A1]

$g \circ f_{0} ≃ {id}_{X}$ and $f_{0} \circ g ≃ {id}_{Y}$ (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[L1]

Precomposition and postcomposition by continuous maps preserve homotopies (Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form).

[L2]

Homotopy is transitive (Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations).

Proof

technique · direct

1.1

Postcomposing $f ≃ f_{0}$ by $g$ gives $g \circ f ≃ g \circ f_{0}$ by [L1], and [A1] with transitivity gives $g \circ f ≃ {id}_{X}$ .

L1A1L2

1.2

Precomposing $f ≃ f_{0}$ by $g$ gives $f \circ g ≃ f_{0} \circ g$ by [L1], and [A1] with transitivity gives $f \circ g ≃ {id}_{Y}$ .

L1A1L2

2.1

Steps 1.1 and 1.2 show that $g$ is a homotopy inverse of $f$ , so $f$ is a homotopy equivalence.

step 1.1step 1.2A1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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