In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property holds only approximately. The notion was introduced by Coram and Duvall in 1977. A manifold approximate fibration is a proper approximate fibration between manifolds. Some authors believe that manifold approximate fibrations are the "correct bundle theory for topological manifolds and singular spaces".
References
Hughes, C.B.; Taylor, L.R.; Williams, E.B. (July 1995). "Rigidity of fibrations over nonpositively curved manifolds". Topology. 34 (3): 565–574. doi:10.1016/0040-9383(94)00035-J.
Further reading nLab - approximate fibration