Jointly continuous utility functions on k_{\omega}-spaces

Let (X; \tau) be a submetrizable k_omega-space, that is the inclusion inductive limit of a decreasing sequence (F_n)_n of second countable and locally compact subspaces of X. Of course the family (F_n)_n determines the topology of X. It is well known that X is a quotient space of a locally compact second countable space. These spaces seem to be very interesting in the study of the utility representation problem. In [3] the authors prove the existence of a continuous representation of a topo- logical space that is inclusion inductive limit of a countable chain of compact subspaces (X_n)_n and <= is a preorder on X such that every <=|Xn is closed and order-separable. In [4, 2] it is proved that each closed preorder on X has a continuous utility representation and some well known theorems due to Levin [7] on the existence of the jointly continuous utility functions are generalized on submetrizable k_omega-spaces. Back in [1], using a result of Levin, proves the existence of a continuous map from the space of preorders topologized by closed convergence and the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact- open topology. The commodity space is locally compact and second countable. Recently Back's Theorem is generalized to non-metrizable commodity spaces, precisely, to a regular space submetrizable by a boundedly compact metric in [6] or to X submetrizable k_omega-space in [5]. The continuous utility representation theorems on submetrizable k_omega-spaces can have some economic implications, in fact an example of submetrizable k_omega-space is the space of tempered distributions, which seems to be of interest in the study of market models in the Decision Theory. References [1] K.Back, Concepts of similarity for utility functions, J. Math. Econ. 15 (1986), 129-142. [2] G.Bosi, A.Caterino, R. Ceppitelli, Existence of continuous utility functions for arbi- trary binary relations: some sucient conditions, Tatra Mountains Math. Publ. 46 (2010), 15-27. [3] J.C.Candeal, E.Indurain, G.B.Mehta, Some utility theorems on inductive limits of preordered topological spaces, Bull. Austral. Mat. Soc. 52 (1995), 235-246, . [4] A.Caterino, R.Ceppitelli, F.Maccarino, Continuous utility functions on submetrizable hemicompact k-spaces, Applied General Topology 10 (2009), 187-195. [5] A.Caterino, R.Ceppitelli, Jointly continuous utility functions on k_omega - spaces, preprint. [6] A.Caterino, R.Ceppitelli, L. Holà, A generalization of Back's Theorem, preprint. [7] V. L. Levin, A continuous utility theorem for closed preorders on a sigma-compact metriz- able space, Soviet. Math. Dokl. 28 (1983), 715-718.