Continuous multi-utility representations of preorders

Let (X,t) be a topological space. Then a preorder ≾ on (X,t) has a continuous multi-utility representation if there exists a family of continuous and isotonic real-valued functions f on (X,≾,t) such that for all x∈X and all y∈X the inequalities x≾y mean that for all 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.