On continuous multi-utility representations of semi-closed and closed preorders

•We discuss the problem of characterizing all the topological spaces such that every closed and respectively semi-closed preorder admits a continuous multi-utility representation.•We prove three different results concerning the existence of a continuous multi-utility representation for a preorder on a metrizable space.•A very general restrictive result is obtained by negating the existence of weakly inaccessible cardinal numbers.•We show that in a Hausdorff space a closed preorder admits a continuous multi-utility representation if and only if it is normal.

On the basis of the classical continuous multi-utility representation theorem of Levin on locally compact and σσ-compact Hausdorff spaces, we present necessary and sufficient conditions on a topological space (X,t)(X,t) under which every semi-closed and closed preorder respectively admits a continuous multi-utility representation. This discussion provides the fundaments of a mainly topological theory that systematically combines topological and order theoretic aspects of the continuous multi-utility representation problem.