We introduce an upper semicontinuity condition concerning a not necessarily total preorder on a topological space, namely strong upper semicontinuity, and in this way we extend to the nontotal case the famous Rader’s theorem,which guarantees the existence of an upper semicontinuous order-preserving function for an upper semicontinuous total preorder on a second countable topological space. We show that Rader’s theorem
is not generalizable if we adopt weaker upper semicontinuity conditions already
introduced in the literature. We characterize the existence of an upper semicontinuous
order-preserving function for all strongly upper semicontinuous preorders on a
metrizable topological space.
