Today I will add one closed graph type theorem to the list made by Terence Tao in his post Closed graph theorem in various categories.
I first recall a few basic notions of domain theory. If is a poset or partially ordered set, a subset of is directed if it is nonempty and for all there is some such that and . The poset is a dcpo (a directed-complete poset) if every directed subset has a supremum, written . A subset of is d-open if for every directed subset of . The collection of d-open subsets forms a topology called the d-topology.
Then a map is called d-continuous if it is continuous when and are equipped with their respective d-topologies. It happens that a map is order-preserving and d-continuous if and only if for every directed subset of (which is a characterization of Scott continuity). For a proof of this latter assertion and for more on the d-topology of a poset I refer the reader to the paper D-completions and the d-topology by Klaus Keimel and Jimmie Lawson.
Theorem (Closed graph theorem (dcpo theory)). Let be dcpo’s. Then an order-preserving map is d-continuous if and only if the graph is d-closed (in the poset equipped with the product topology).
Before giving a proof to this theorem, it is important to recall that the d-topology on the poset does not coincide in general with the product of the d-topologies on and .
Proof. Suppose that is d-closed, and let be a directed subset of . We define , which is a subset of , and a directed subset of the poset since is order-preserving. It is easily seen that has a supremum in equal to ; since is d-closed, this supremum is in , hence .
Question. Do we have a closed graph theorem in (continuous?) dcpo’s equipped with the Lawson topology?