The elaboration and reflection on utility theory can be traced back to early economic theory (Hervés-Beloso & del Valle-Inclán Cruces, 2019), but the topic is definitely multidisciplinary (e.g. psychology, cognitive sciences, chemistry, physics, etc.). For instance, the problem of utility representation has been addressed on connected separable topological spaces in Debreu, 1954, Debreu, 1964 and Eilenberg (1941). Interval orders received a clear status as objects of research through Fishburn’s contributions (Fishburn, 1970a, Fishburn, 1970b, Fishburn, 1973, Fishburn, 1985 and Monjardet, 1988). Numerical representations of preferences beyond utility functions have known a strong interest that remains actual and recent (Bosi, Campión et al., 2007, Bosi, Candeal et al., 2007, Bosi et al., 2015, Bridges, 1985, Bridges, 1986, Bridges and Mehta, 1995, Chateauneuf, 1987). Interestingly some partial strict orders already appeared in the early scientific works of Wiener (1914). These works dealt with binary relations over temporal events. This fact is acknowledged in Fishburn and Monjardet (1992). As one may have guessed interval orders are the natural candidates to model temporal events through the relation PIP⊂P where P stands for “before” and I for “simultaneous”. We foster the use of Wiener’s condition, namely IP∩PI⊂P. This is a special case of bi-weak orders where the precedence relation IP and the succession relation PI are precisely the strict weak order extensions realizing P. A further step now is to obtain, on connected separable topological spaces, numerical representations with continuous bi-utility functions for these special interval orders.

The structure of the paper is as follows, Section 2 reminds the usual definitions, then Section 3 introduces basic properties concerning interval orders and Wiener’s condition. In Section 4 we remind some representation theorems on connected separable topological spaces and we gather results dealing with bi-utility representations for interval orders. Section 5 provides simple characterization of strict weak orders based on Wiener’s condition. Section 6 offers worked out examples. Section 7 concludes our study.