# Contractible space

From Encyclopedia of Mathematics

A topological space that is homotopy equivalent (see Homotopy type) to a one-point space; i.e., if there is a point and a homotopy from to the unique mapping . Such a mapping is called a contraction.

The cone over is contractible. For a pointed space , the requirement for contractibility is that there is a base-point-preserving homotopy from to the unique mapping .

A space is contractible if and only if it is a retract of the mapping cylinder of any constant mappping .

A set is starlike with respect to if for any the segment lies in . Convex subsets and starlike subsets in are contractible.

#### References

[a1] | C.T.J. Dodson, P.E. Parker, "A user's guide to algebraic topology" , Kluwer Acad. Publ. (1997) |

[a2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |

**How to Cite This Entry:**

Contractible space. M. Hazewinkel (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Contractible_space&oldid=16066

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098