Continuous multi-utility representations of preorders

Let (X, t) be a topological space. Then a preorder less than or similar to on (X, t) has a continuous multi-utility representation if there exists a family F of continuous and isotonic real-valued functions f on (X, less than or similar to, t) such that for all x is an element of X and all y is an element of X the inequalities x less than or similar to y mean that for all f is an element of F the inequalities f(x) <= f (y) hold. We discuss the existence of a continuous multi-utility representation by using suitable concepts of continuity of a preorder. In addition, we clarify in detail the relation between the concept of a continuous multi-utility representation and Nachbin's concept of a normally preordered space. (C) 2012 Elsevier B.V. All rights reserved.