Contractible

A set in  which can be reduced to one of its points, say , by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to  through a path with the properties that

1. Each path runs entirely inside the set.

2. Nearby points move on "neighboring" paths.

Condition (1) implies that a disconnected set, i.e., a set consisting of separate parts, cannot be contractible.

Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points  and  lying on different sides of a point . The paths connecting  and  with  are either opposite each other or have different lengths. A similar argument shows that, in general, for all , the -sphere (i.e., the boundary of the -dimensional ball) is not contractible.

A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example, the "house with two rooms." In a case like this, it is not evident how to construct a transformation of the type described above. However, its existence is assured by the formal definition of contractibility of a set , namely that  is homotopic to one of its points . This means that there is a continuous map  such that  is the identity map and  is the constant map sending each point to . Thus,  describes a continuous path from  to  as  varies from 0 to 1, and (1) is fulfilled. Moreover, since the map  is also continuous with respect to the second component, the path starting at  varies continuously with respect to , as required by (2).

REFERENCES:

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.