Scott-closed set

Let (D, \sqsubseteq) be a partial order. A set X \subseteq D is Scott-closed just when

1. It is a lower set. In other words, y \sqsubseteq x \in X \Longrightarrow y \in X.
2. Its directed subsets are closed under suprema. If Y \subseteq X is directed and has a supremum \bigsqcup Y \in D, then \bigsqcup Y \in X.

Scott topology

The Scott-closed sets are the closed sets in the Scott topology.

Lattice structure

The Scott-closed sets form a complete lattice, which is also distributive.

The closure under binary union is proven (by Andreas Blass!) here.

References

• StackExchange: proof that Scott-closed sets are closed under union