Fibration

This article defines a property that can be evaluated for a map between topological spaces. Note that the map is not assumed to be continuous

Definition

A continuous map $p:E\to B$ of topological spaces is termed a fibration or is said to have the homotopy lifting property if, given any map $F:X\times I\to B$ and a map ${\tilde {f}}:X\to E$ such that $p({\tilde {f}}(x))=f(x,0)$, there exists a map ${\tilde {F}}:X\times I\to E$ satisfying:

• $p\circ {\tilde {F}}=F$
• $F(x,0)={\tilde {f}}(x)$

This is dual to the notion of a cofibration.