Cofibration

Definition
A subspace $$A$$ of a topological space $$X$$ is said to be a cofibration, or to have the homotopy extension property if the following holds: given any map $$f_0: X \to Y$$ and a homotopy $$F:A \times I \to Y$$ such that $$F(a,0) = f(a) \ \forall \ a \in A$$, we have a homotopy $$\tilde{F}:X \times I \to Y$$ whose restriction to $$A$$ is $$F$$, and such that $$\tilde{F}(x,0) = f(x) \ \forall \ x \in X$$.

Stronger properties

 * CW pair
 * Simplicial pair

Weaker properties

 * Closed subset in a Hausdorff space: