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. 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.
- Any two maps f,g : Y → X are homotopic.
- Any map f : Y → X 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 strongly deformation retract onto any point.
The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one.
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.
[edit] 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. Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected.
[edit] Examples and counterexamples
- Any star domain of a Euclidean space is contractible.
- Spheres of any finite dimension are not contractible.
- The unit sphere in Hilbert space is contractible.
- The house with two rooms is standard example of a space which is contractible, but not intuitively so.
- The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected.
- All manifolds and CW complexes are locally contractible.
