A simple characterization of the existence of upper semicontinuous order-preserving functions

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