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. 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).
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.
Read more about Contractible Space: Locally Contractible Spaces, Examples and Counterexamples
Famous quotes containing the word space:
“In the United States there is more space where nobody is is than where anybody is.”
—Gertrude Stein (18741946)