nLab
approximate fibration

The approximate homotopy lifting property is a weak version of the homotopy lifting property in the setup of metric spaces.

A proper map $p:E\to B$ between locally compact metric absolute neighborhood retracts (ANRs) satisfies the approximate homotopy lifting property for a space $X$ if for any open covering? $U$ of B, and any map $h:X\to E$ with a homotopy $H:X\times I\to B$ such that $p\circ h=H_0$, there exists a homotopy $G:X\times I\to E$ such that $G_0=h$ and the maps $p\circ G$ and $H$ are $U$-close?.

A proper map $p:E\to B$ between locally compact metric ANRs is an approximate fibration if $p$ has the approximate homotopy lifting property for all metric spaces.

It is straightfoward to generalize this notion to the level maps of inverse systems of locally compact metric ANRs.