Let (D,⊑) be a partial order. A set X⊆D is Scott-closed just when
- It is a lower set. In other words, y⊑x∈X⟹y∈X.
- Its directed subsets are closed under suprema. If Y⊆X is directed and has a supremum ⨆Y∈D, then ⨆Y∈X.
The Scott-closed sets are the closed sets in the Scott topology.
The Scott-closed sets form a complete lattice, which is also distributive.
The closure under binary union is proven (by Andreas Blass!) here.