Contractible space

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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.


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098