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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a continuous function from M to N is null-homotopic if it is homotopic to a constant function. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

For example, any star domain of a Euclidean space is contractible. On the other hand, spheres of any finite dimension are not contractible. A star domain which is not convex. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... For other uses, see sphere (disambiguation). ...

Since a contractible space is homotopy equivalent to a point, all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

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 deformation retract onto any point. In topology, a retraction, as the name suggests, retracts an entire space into a subspace. ...

The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one. In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point. ...

Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X. In topology, a retraction, as the name suggests, retracts an entire space into a subspace. ...