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

On the basis of the classical continuous multi-utility representation theorem of Levin on locally compact and $\sigma$-compact Hausdorff spaces, we present necessary and sufficient conditions on a topological space $(X,t)$ 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.