Homotopy theory MOC

Contractible space

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

• is contractible iff it is homotopy equivalent to the single point space, i.e. .²
• is contractible iff the identity is null-homotopic.³

Proof of definition equivalence

Let . Then iff there exists such that (immediately ). Since all constant maps have the form , 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 then is a continuous path from to .

Examples

• A circle is not contractible, since Circle endomorphisms are homotopic iff they are of equal degree and .

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Footnotes

1. German zusammenziehbar ↩

2. 2020, Topology: A categorical approach, p. 35 ↩

3. 2010, @looseAlgebraischeTopologie2010, p. 37 (definition 2.1.7) ↩