Representing interval orders by a single real-valued function (Q1083995)

Let \(\succ\) be an interval order on a topological space X, and let \(x\succ^*y\) if and only if there exists x' with \(x\succcurlyeq x'\succ y\). Then \(\succ^*\) is a preference relation with transitive indifference, which, under suitable conditions, can be represented by a utility function v of the usual sort. In that case, maximisation of v (subject to contraints) leads to a most preferred point relative to \(\succ\). Following both the Debreu-Rader and the Arrow-Hahn approaches, we discuss the existence and continuity of such a function v.