Contractible space

A space is contractible when its identity map is homotopic to a constant map. Every map into a contractible space is null-homotopic.

A contraction is a homotopy from the identity map of

X

to a constant map. Its existence says that

X

is a contractible space. For a tree, choosing a root and moving each point along its unique finite path to that root gives a contraction, with continuity understood in the weak topology of a CW complex.

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In the field of topology, a **contractible space** is a type of topological space that is homotopically equivalent to a single point. This means that there exists a continuous deformation (a homotopy) that can transform the entire space into a point while keeping the structure of the space intact.