Let (X, t) be a topological space. Then a preorder≾ 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,≾, t) such that for all x∈ X and all y∈ X the inequalities x≾ y mean that for all f∈ 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 …

Chipman contended, in stark contrast to the conventional view, that, utility is not a real
number but a vector, and that it is inherently lexicographic in nature. On the other hand, in
recent years continuous multi-utility representations of a preorder on a topological space,
which proved to be the best kind of continuous representation, have been deeply studied. In
this paper, we first state a general result, which guarantees, for every preordered topological
space, the existence of a lexicographic order-embedding of the Chipman type. Then, we …