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Contractible space

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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 : YX are homotopic.
  • Any map f : YX is null-homotopic.
  • Any map f : XY 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.

Last updated: 05-14-2005 14:02:05
Last updated: 05-13-2005 07:56:04