P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. , where is the directed join. When is the poset of truth values, i.e. Sierpinski space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpinski space is the classifying space for open sets.

A subset O of a partially ordered set P is called Scott-open if it is an upper set