CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

Every nonempty retract of a contractible space is contractible

Statement

Let $X$ be contractible and let a nonempty subspace $A \subseteq X$ be a retract of $X$ . Then $A$ is contractible.

Facts & Assumptions

Given: A contractible space $X$ , a nonempty retract $A \subseteq X$ , its inclusion $i : A ↪ X$ , and a retraction $r : X \to A$ .

[A1]

Retraction means $r \circ i = {id}_{A}$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[L1]

For every nonempty space $T$ , the space $T$ is contractible if and only if ${id}_{T}$ is homotopic to a constant map (A nonempty space is contractible if and only if its identity map is nullhomotopic, Nullhomotopic maps and contractible spaces).

[L2]

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

Proof

technique · direct

1.1

By [L1], ${id}_{X} ≃ c_{x_{0}}$ for some $x_{0} \in X$ . Precompose this homotopy by $i$ and postcompose by $r$ . By [L2], $r \circ {id}_{X} \circ i ≃ r \circ c_{x_{0}} \circ i$ .

L1L2

2.1

The left side of step 1.1 is $r \circ i = {id}_{A}$ by [A1], and the right side is the constant map on $A$ with value $r (x_{0})$ . Thus ${id}_{A}$ is nullhomotopic.

step 1.1A1

3.1

Since $A$ is nonempty, step 2.1 and [L1] applied to $A$ show that $A$ is contractible.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)