Preordered topological spaces, utility and jointly utility functions

Classical theorems of Eilenberg and Debreu state that every continuous total order, defined respectively on a connected separable space and on a second countable space, has a continuous representation. The notions of network and netweight provide useful tools in the theory of representation. In [8], by using these concepts, it is proved the following result that includes the above mentioned results. Let X be a topological preordered space, where the preorder is continuous and total. If X is a union of a countable family of spaces X_i, where every X_i is connected separable or has countable network, then the preorder has a continuous representation. In [2] the existence of a continuous utility function for a weakly continuous binary relation is characterized. In particular, it is proved that every weakly continuous binary relation on a topological space with countable network has a continuous utility representation. The utility representation problem was solved by Levin (1983) for non linear closed preorders defined in locally compact second countable spaces. A generalization of Levin's Theorem to submetrizable k_w-spaces is presented in [7] and, by using a different technique, it is also proved in [2]. The k_w-spaces have nice properties. In [10] the author proves that every k_w-space, equipped with a closed preorder, is normally preordered and that every second countable normally pre-ordered space has a countable utility (multi-utility) representation. More generally, in [3] it is proved that every normally preordered space with countable netweight has a countable utility representation. Therefore, every submetrizable k_w-space has a countable utility representation too. If X is a topological space and P is a set of closed preorders on X, the problem of the existence of jointly continuous utility functions is to find topological conditions on P and X in order that there exists a continuous function u from P x X to R such that u(<,.) is a utility function for every preorder < belonging to P. In [9] Levin proved some results on the existence of jointly continuous utility functions, for example by assuming that P and X are locally compact and second countable. In [7] Levin's results have been generalized to some on-metrizable cases, for instance if P and X are submetrizable hemicompact spaces and P x X is a k-space. Back in [1] revisited Levin's Theorems by using partial maps and hypertopologies. He considered the set P of all linear closed preorders defined on closed subsets of a locally compact second countable space X endowed with the topology of closed convergence and the set U of all continuous real partial maps defined on closed subsets of X with the t_c topology, a generalization of the compact-open topology. He showed the existence of a continuous map n from P to U such that n(<) is a utility function for every n belonging to P. Under the hypotheses that X is regular and submetrizable by a boundedly compact metric, in [6] we have associated to the space of preorders P a suitable space of preorders satisfying the hypotheses of Back's Theorem. So, by using Back's result, we proved the existence of a continuous map n from P to U such that n(<) is an isotone function for every < belonging to P, where the topology on U is just the t_c topology, while the topology in P , defined by mean of a suitable convergence structure, is reminiscent of the topology of closed convergence. Moreover, n(<) is a utility function if and only if the smallest d-closed preorder S(<) containing the closed preorder < satisfies the following property: if a < b then it is not true that bS(<)a. Another generalization of Back's theorem to submetrizable k_w-spaces is presented in [5]. References [1] K. Back, Concepts of similarity for utility functions, Journ. of Math. Econ., 15 (1986), 129-142. [2] G. Bosi, A. Caterino, R. Ceppitelli, Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions, Tatra Mt. Math. Publ., 47 (2010) 1-13. [3] G. Bosi, A. Caterino, R. Ceppitelli, Multi-utility representation and k_w-spaces, preprint 2012. [4] J. C. Candeal, E. Indurain, G. B. Mehta, Some utility theorems on inductive limits of preordered topological spaces, Bull. Austral. Math. Soc., 52 (1995), 235-246. [5] A. Caterino, R. Ceppitelli, Jointly continuous utility functions on k_w-spaces, preprint 2012. [6] A. Caterino, R. Ceppitelli, L. Hola, A generalization of Back's Theorem, preprint 2011. [7] A. Caterino, R. Ceppitelli, F. Maccarino, Continuous utility functions on submetrizable hemicompact k-spaces, Applied General Topology, 10 (2009), 187-195. [8] A. Caterino, R. Ceppitelli, G. B. Mehta, Preference Orders and Continuous Representations, Math. Slovaca 61 (2011), No. 1, 93-106. [9] V. L. Levin, A continuous utility theorem for closed preorders on a -compact metrizable space, Soviet Math. Dokl. 28 (1983), 715-718. [10] E. Minguzzi, Normally preordered spaces and utilities, Order (2011), DOI: 10.1007/s11083-011-9230-4, arXiv:1106.4457v2.