Upper Semicontinuous Representations of Semiorders as Interval Orders

We characterize the upper semicontinuous representability of a semiorder $\prec$ as an interval order (namely, by a pair $(u,v)$ of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the representing functions is a one-way utility for the characteristic weak order $\prec^0$ associated to the semiorder. Such a description generalizes the {\em upper semicontinuous threshold representation}. To this aim, we introduce a suitable upper semicontinuity condition concerning a semiorder, namely {\em strict upper semicontinuity}. We further characterize the mere existence of an upper semicontinuous one-way utility for this characteristic weak order, with a view to the identification of maximal elements on compact metric spaces.