Topology MOC

Retraction

A retraction is a continuous map from a topological space to a subspace that does not move points in the subspace, i.e. for all . Alternatively, if is the natural inclusion of the subspace topology, then a retraction is a continuous left-inverse of , i.e. . topology A subspace for which such a retraction exists is called a retract of .

A special kind of retraction is a Deformation retraction, which has the additional property that .

Examples

• The boundary of a ball is not a retract
• Every one-point subset of is a retract, but only a deformation retract if is contractible.

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