| Contractible |
Website Links For Space |
Information AboutContractible |
| CATEGORIES ABOUT CONTRACTIBLE SPACE | |
| topology | |
| homotopy theory | |
| properties of topological spaces | |
|
For example, any Star Domain of a Euclidean Space is contractible. On the other hand, Spheres of any finite dimension are not contractible. Since a contractible space is homotopy equivalent to a point, all the Homotopy Group s 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):
Any space which Deformation Retract s 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. 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''. |
|
|