Dcpos built as graphs of functions

Let X and P be two dcpos, and let ψ be a map from X to P. When is the graph of ψ a dcpo? I will give you a funny sufficient condition, which involves the so-called d-topology, and Hausdorffness. I will briefly explain how this can be used to show that every Π⁰₂ subset of a continuous dcpo is domain-complete, namely, is homeomorphic to a G_δ subset of some other continuous dcpo. Read the full post.