CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

A nonempty space is contractible if and only if its identity map is nullhomotopic

Statement

For a nonempty topological space $X$ , the following are equivalent:

$X$ is contractible.
The identity map ${id}_{X}$ is nullhomotopic.

Facts & Assumptions

Given: A nonempty topological space $X$ .

[A1]

$X$ is contractible when every continuous map from $X$ to every topological space is nullhomotopic; a map is nullhomotopic when it is homotopic to a constant map (Nullhomotopic maps and contractible spaces).

[L1]

Postcomposition by a continuous map preserves homotopies (Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form, claim 2).

Proof

technique · direct

1.1

If $X$ is contractible, apply [A1] to the continuous map ${id}_{X} : X \to X$ to conclude that ${id}_{X}$ is nullhomotopic.

1.2

Conversely suppose ${id}_{X} ≃ c_{x_{0}}$ for some $x_{0} \in X$ , and let $f : X \to Y$ be any continuous map. Postcomposition by $f$ gives $f = f \circ {id}_{X} ≃ f \circ c_{x_{0}} = c_{f (x_{0})}$ by [L1]. Thus $f$ is nullhomotopic.

assume-hypL1A1

2.1

Since $Y$ and $f$ in step 1.2 were arbitrary, every continuous map out of $X$ is nullhomotopic, so $X$ is contractible by [A1]. Together with step 1.1 this proves the equivalence.

step 1.1step 1.2A1∎

Depends on

Used by

Contractible nonempty spaces have the homology of a point Corollary
Every nonempty contractible space is path-connected Corollary
Every nonempty convex subset of ℝⁿ is contractible Corollary
Every nonempty retract of a contractible space is contractible Corollary
FALSE: every retract is a deformation retract False statement
FALSE: homotopy-equivalent spaces must be homeomorphic False statement
A contractible space has trivial fundamental group Lemma

Dependency tree · two levels

6 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)