Contractible space

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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.


[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:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article