Strong deformation retract

Symbol-free definition
A subspace of a topological space is termed a strong deformation retract (sometimes simply a deformation retract) if there is a homotopy between the identity map on the whole space, and a retraction onto the subspace, such that the map at every intermediate stage, restricts to identity on the subspace. Such a homotopy is termed a strong deformation retraction.

Definition with symbols
A subspace $$A$$ of a topological space $$X$$ is termed a strong deformation retract (sometimes simply deformation retract) of $$X$$ if there is a homotopy $$F: X \times I \to X$$ such that:


 * $$f(x,0) = x \ \forall \ x \in X$$
 * $$f(a,t) = a \ \forall \ a \in A, t \in I$$
 * $$f(x,1) \in A \ \forall \ x \in X$$

The second condition is what distinguishes deformation retracts from the weaker notion of homotopy retract.

Weaker properties

 * Homotopy retract
 * Retract
 * Neighbourhood retract

Metaproperties
If $$A$$ is a strong deformation retract of $$B$$ and $$B$$ is a strong deformation retract of $$C$$ then $$A$$ is a deformation retract of $$C$$.

If $$A_i$$ is a deformation retract of $$B_i$$ for $$i=1,2$$ then $$A_1 \times A_2$$ is a deformation retract of $$B_1 \times B_2$$.