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. 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: Y → X are homotopic.
- Any map f: Y → X 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).
Locally contractible spaces
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
- Any Euclidean space is contractible, as is any star domain on a Euclidean space.
- The Whitehead manifold is contractible.
- Spheres of any finite dimension are not contractible.
- The unit sphere in an infinite-dimensional Hilbert space is contractible.
- The house with two rooms is standard example of a space which is contractible, but not intuitively so.
- Dunce hat
- The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected.