Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold

The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.