CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $i:A\hookrightarrow X$ , and a retraction $r:X\to A$ .

[A1]

Retraction means $r\circ i=\operatorname{id}_A$ (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 $\operatorname{id}_T$ 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], $\operatorname{id}_X\simeq c_{x_0}$ for some $x_0\in X$ . Precompose this homotopy by $i$ and postcompose by $r$ . By [L2], $r\circ\operatorname{id}_X\circ i\simeq r\circ c_{x_0}\circ i$ .

L1L2

2.1

The left side of step 1.1 is $r\circ i=\operatorname{id}_A$ by [A1], and the right side is the constant map on $A$ with value $r(x_0)$ . Thus $\operatorname{id}_A$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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