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:EBp:E\to B between locally compact metric absolute neighborhood retracts (ANRs) satisfies the approximate homotopy lifting property for a space XX if for any open covering UU of B, and any map h:XEh : X\to E with a homotopy H:X×IBH : X \times I \to B such that ph=H 0p\circ h = H_0, there exists a homotopy G:X×IEG : X\times I\to E such that G 0=hG_0 = h and the maps pGp\circ G and HH are UU-close?.

A proper map p:EBp : E\to B between locally compact metric ANRs is an approximate fibration if pp 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)