In this paper, we go further on the problem of the continuous numerical representability
of interval orders defined on topological spaces. A new condition of compatibility between
the given topology and the indifference associated to the main trace of an interval order
is introduced. Provided that this condition is fulfilled, a semiorder has a continuous
interval order representation through a pair of continuous real-valued functions. Other
necessary and sufficient conditions for the continuous representability of interval orders
are also discussed, and, in particular, a characterization is achieved for the particular
case of interval orders defined on a topological space of finite support