CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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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.
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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 $\operatorname{id}_X$ 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 $\operatorname{id}_X:X\to X$ to conclude that $\operatorname{id}_X$ is nullhomotopic.

1.2

Conversely suppose $\operatorname{id}_X\simeq c_{x_0}$ for some $x_0\in X$ , and let $f:X\to Y$ be any continuous map. Postcomposition by $f$ gives $f=f\circ\operatorname{id}_X\simeq f\circ c_{x_0}=c_{f(x_0)}$ 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

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 11 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)