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 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 $F(a,0)=f(a)\mathrm{\forall }a\in A$, we have a homotopy $\tilde{F}:X\times I\to Y$ whose restriction to $A$ is $F$, and such that $\tilde{F}(x,0)=f(x)\mathrm{\forall }x\in X$.

Relation with other properties

Stronger properties

• CW pair
• Simplicial pair

Weaker properties

• Closed subset in a Hausdorff space: For full proof, refer: Cofibration implies closed subset in Hausdorff space

Metaproperties

Transitivity

This property of subspaces of topological spaces is transitive. In other words, if $A$ satisfies the property as a subspace of $B$ and $B$ satisfies the property as a subspace of $C$ then $A$ satisfies the property as a subspace of $C$