Net-Compact Hausdorff Topologies and Continuous Multi-Utility Representations for Closed Preorders

In this paper, we deal with continuous multi-utility representations for closed preorders. We introduce the definition of a net-compact topology, which generalizes the concept of a sequentially compact topology. Indeed, a sequentially compact and first countable topological space is net-compact. First, we show that if every closed preorder on a net-compact Hausdorff topological space has a continuous multi-utility representation, then the topology is normal. Second, we prove that every closed preorder on a normal and net-compact Hausdorff topological space admits a continuous multi-utility representation.