Homotopy retract

Symbol-free definition
A subspace of a topological space is termed a homotopy retract if the identity map from the whole space to itself is homotopic to the retraction onto that subspace.

Definition with symbols
A subspace $$A$$ of a topological space $$z$$ is termed a homotopy retract of $$X$$ if there exists a map $$F: X \times I \to X$$ such that:


 * $$F(x,0) = x \ \forall \ x \in X$$
 * $$F(x,1) \in A \ \forall \ x \in X$$

Note that unlike in the stronger notion of deformation retract, we do not require that at intermediate times, $$F$$ should restrict to the identity on $$A$$.

Stronger properties

 * Deformation retract

Weaker properties

 * Retract