Quasi-fibration

In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom.

Roughly speaking, it is a continuous map p: E → B having the same behaviour as a fibration regarding the (relative) homotopy groups of E, B and p−1(x).

A continuous surjective map of topological spaces p: E → B is called a quasifibration if it induces isomorphisms for all x ∈ B, y ∈ p−1(x) and i ≥ 0.

To see this, recall that Fb is the fibre of q under b where q: Ep → B is the usual path fibration construction.

To see that the latter statement holds, one only needs to bear in mind that continuous images of compact sets in B already lie in some Bn.