Continuous multi-utility representations of preorders

Let (X,t) be a topological space. Then a preorder 7e on (X,t) has a continuous multi-utility representation if there exists a family of continuous and isotonic real-valued functions f on (X, 7e,t) such that for all x 08X and all y 08X the inequalities x 7ey mean that for all the inequalities f(x) 64f(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\u2019s concept of a normally preordered space