Deformation retraction

Definition
Let $$X$$ be a topological space and $$A$$ a subspace. A deformation retraction from $$X$$ to $$A$$ is a homomotopy $$F: X \times I \to X$$ such that:


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

If a deformation retraction exists from $$X$$ to $$A$$, we say that $$A$$ is a deformation retract of $$X$$.

Stronger properties

 * Sudden deformation retraction
 * Semi-sudden deformation retraction

Weaker properties

 * Weak deformation retraction