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.
Properties
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.