Retract

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

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.

Definition with symbols

Fill this in later

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).

Relation with other properties

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

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 223, Exercise 4 (definition introduced in exercise)
• An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 4 (formal definition)
• Algebraic Topology by Edwin H. Spanier^{More info}, Page 28 (formal definition)