Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point. A contractible space is precisely one with the homotopy type of a point.

For example, any convex subset of Euclidean space is contractible. On the other hand, spheres of any finite dimension are not contractible.

Since a contractible space is homotopy equivalent to a point, all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible.

For a topological space X the following are all equivalent (here Y is an arbitrary topological space):

• X is contractible (i.e. the identity map is null-homotopic).
• X is homotopy equivalent to a one-point space.
• Any two maps f,g : Y → X are homotopic.
• Any map f : Y → X is null-homotopic.
• Any map f : X → Y is null-homotopic.

Any space which deformation retracts onto a point is clearly contractible. The converse, however, is false. There are examples of contractible spaces which do not deformation retract onto any point.

The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one.