Cofibration

This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces

Definition

A subspace $A$ of a topological space $X$ is said 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 ${\tilde {F}}(x,0)=f(x)\ \forall \ x\in X$.