Hypertopologies, functional differential equations and jointly continuous utility functions

In the setting of Functional Differential Equations, the topology tau_B, a finer one than the Attouch - Wets, was introduced to obtain existence and continuous dependence results. The topology tau_B found many applications also in Mathematical Economics. This topology is natural in metric spaces where it has many good properties without additional conditions on the domain. New characterizations of boundedly Atsuji spaces are given by the coincidence of tau_B and the topology of uniform convergence on bounded sets. A result of particular importance, among the applications in Functional Differential Equations Theory, is the "homeomorphic property" of topology tau_B with respect to the compact-open topology that allows us to study hereditary ordinary differential equations by using the classical fixed point theorems. The known hypertopologies (Hausdorff, Attouch-Wets, Fell) have not the analogous homeomorphic property. In locally compact metric spaces the topology tau_B coincides with the topology introduced by K. Back to study the existence of the so-called jointly continuous utility functions. Back, using a result of Levin, proved the existence of a continuous map from the space of preorders, endowed with the Fell topology, to the space of utility functions (partial maps) endowed with the topology tau_B.