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.[1][2] 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. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.

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.
  • X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)
  • Any two maps f,g: YX are homotopic.
  • Any map f: YX is null-homotopic.

The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).

Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X.

Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0.

Locally contractible spaces[edit]

A topological space is locally contractible if every point has a local base of contractible neighborhoods. Contractible spaces are not necessarily locally contractible nor vice-versa. For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected.

Examples and counterexamples[edit]