A continuous map r: X → A from a topological space X onto a subspace A ⊆ X such that r restricted to A is the identity map on A (r|_A = id_A).

Principle

A retraction is a projection-like map that leaves the designated subspace fixed while collapsing or projecting the remainder of the ambient space onto it; algebraically it provides a right inverse to the inclusion A ↪ X.

Demonstration

If X = A × [0,1] then the projection r(a,t)=a is a retraction onto A × {0}. In Euclidean space a linear orthogonal projection from R^n onto a linear subspace L is a retraction of R^n onto L when one chooses a complementary linear subspace.

Misapplication

Treating every inclusion A ↪ X as admitting a retraction (for example assuming the closed unit disk retracts continuously onto its boundary sphere) — such a map need not exist and in many classical cases it provably does not.

Consequence

If r: X → A is a retraction and i: A ↪ X is the inclusion then r ◦ i = id_A, so the induced maps on fundamental groups and homology satisfy r_* ◦ i_* = id; in particular i_* is injective and r_* is surjective.

Reversal

The inverse idea is an extension (or section) problem: given a map f: A → Y, an extension is a map F: X → Y with F|_A = f; unlike a retraction, an extension need not split an inclusion and typically is a weaker condition.

Boundary

Requires a specified subspace A of X and continuity of the map; excludes homotopy-retractions that only exist up to homotopy unless explicitly stated as deformation retractions or strong deformation retractions.

Semantic Tension

Retraction versus deformation retraction: a retraction is a single continuous map with r|_A = id, while a deformation retraction is a homotopy through maps from X to X that both fixes A and ends at a retraction; the latter is strictly stronger.

Synthesis

A retraction is the concrete, pointwise splitting of an inclusion A ↪ X by a continuous map r: X → A; it fixes A pointwise, yields algebraic splittings on induced invariants, and must be distinguished from homotopical notions that only hold up to deformation.