The way-below relation of function spaces over semantic domains

For partially ordered sets that are continuous in the sense of D.S. Scott, the way-below relation is crucial. It expresses the approximation of an ideal element by its finite parts. We present explicit characterizations of the way-below relation on spaces of continuous functions from topological spaces into continuous posets. Although it is well known in which cases these function spaces are continuous posets, such characterizations were lacking until now.