Contractible space

A topological space $𝑋$ is contractible¹ iff it can be continuously contracted to a single point. This yields the following two equivalent definitions #m/def/homotopy

$𝑋$ is contractible iff it is homotopy equivalent to the single point space, i.e. $𝑋 ≃ *$ .²
$𝑋$ is contractible iff the identity $i d 𝑋$ is null-homotopic.³

Proof of definition equivalence

Let $𝑇 : 𝑋 \to *$ . Then $𝑋 ≃ *$ iff there exists $𝑐 : * \to 𝑋$ such that $𝑐 𝑇 ≃ i d 𝑋$ (immediately $𝑇 𝑐 = i d *$ ). Since all constant maps have the form $𝑐 𝑇$ , the definitions are equivalent.

Contraction to a point may be generalised to retraction to a subspace.

Properties

Every contractible space is path-connected, since if $ℎ : i d 𝑋 ≃ 𝐶 𝑇$ then $ℎ (𝑥, -)$ is a continuous path from $𝑥$ to $𝐶 *$ .

Examples

A circle is not contractible, since Circle endomorphisms are homotopic iff they are of equal degree and $d e g 𝑐 𝑇 = 0 \neq 1 = d e g i d$ .

#state/tidy | #lang/en | #SemBr

German zusammenziehbar ↩
2020, Topology: A categorical approach, p. 35 ↩
2010, @looseAlgebraischeTopologie2010, p. 37 (definition 2.1.7) ↩

Contractible space

Properties

Examples

Footnotes