# Contractible space

A topological space $X$ that is homotopy equivalent (see Homotopy type) to a one-point space; i.e., if there is a point $x \in X$ and a homotopy from $\operatorname{id}: X \rightarrow X$ to the unique mapping $p : X \rightarrow \{ x \}$. Such a mapping is called a contraction.

The cone over $X$ is contractible. For a pointed space $( X , * )$, the requirement for contractibility is that there is a base-point-preserving homotopy from $\operatorname{id}: ( X , * ) \rightarrow ( X , * )$ to the unique mapping $p : ( X , * ) \rightarrow ( * , * )$.

A space is contractible if and only if it is a retract of the mapping cylinder of any constant mappping $p : X \rightarrow \{ x \}$.

A set $X \subset {\bf R} ^ { n }$ is starlike with respect to $x _ { 0 } \in X$ if for any $x \in X$ the segment $[ x _ { 0 } , x ]$ lies in $x$. Convex subsets and starlike subsets in ${\bf R} ^ { n }$ 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Contractible_space&oldid=50240