Retract

Symbol-free definition
A subspace of a topological space is said to be a retract if it satisfies the following equivalent conditions:


 * There is a continuous map on the whole topological space that maps everything to within the subspace, and that is the identity map when restricted to the subspace.
 * There is a continuous idempotent map whose image-cum-fixed-point space is precisely the given subspace.
 * The inclusion of the subspace in the whole space has a left inverse.

Such a map (satisfying any of the three equivalent conditions) is termed a retraction.

Facts
Clearly the whole space is a retract of itself (the identity map being a retraction) and every one-point subspace is also a retract (the constant map to that one point being the retraction).

Stronger properties

 * Homotopy retract
 * Weak deformation retract
 * Deformation retract

Weaker properties

 * Homotopically injective subspace
 * Homologically injective subspace
 * Weak retract
 * Neighbourhood retract

Facts

 * In a Hausdorff space, any retract is a closed subset
 * Many properties of topological spaces are preserved on taking retracts. Examples are properties like being simply connected, weakly contractible, contractible, or having the fixed-point property. For a full list of such properties, refer: Category:Retract-hereditary properties of topological spaces

Textbook references

 * , Page 223, Exercise 4 (definition introduced in exercise)
 * , Page 4 (formal definition)
 * , Page 28 (formal definition)