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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


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


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

Weaker properties



Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 223, Exercise 4 (definition introduced in exercise)
  • An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. RotmanMore info, Page 4 (formal definition)
  • Algebraic Topology by Edwin H. SpanierMore info, Page 28 (formal definition)