Jointly continuous utility functions on submetrizable kωkω-spaces

A Hausdorff topological space X   is a submetrizable kωkω-space if it is the inductive limit of an increasing sequence of metric compact subspaces of X  . These spaces have nice properties and they seem to be very interesting in the study of the utility representation problem. Every closed preorder on a submetrizable kωkω-space has a continuous utility representation and some theorems on the existence of jointly continuous utility functions have been recently proved.When the commodity space is locally compact second countable, Back proved the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. In this paper we generalize Back's Theorem to submetrizable kωkω-spaces with a family of not necessarily total preorders.The continuous utility representation theorems on submetrizable kωkω-spaces have some economic applications. In fact, an example of a submetrizable kωkω-space is the space of tempered distributions, which has been used to define a new state preference model in the infinite dimensional case.