# 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.

Revised on December 24, 2009 02:25:27 by Toby Bartels (151.213.42.84)