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.