A Note on Continuity and Continuous Representability of Interval Orders

We introduce the notion of weak continuity relative to an interval order on a topological space (X, τ). Then we show that a weakly continuous interval order on a second countable topological space (X, τ) is represented by a pair (u, v) of continuous real-valued functions (in the sense that, for all x, y ∈ X, x y if and only if u(x) ≤ v(y)). In this way we generalize the famous continuous utility representation theorem of Debreu according to which a total preorder on a second countable topological space (X, τ) admits a continuous utility represen-tation