Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set

Theorem
Let $T = \struct {S, \preceq, \tau}$ be a relational structure with Scott topology

where $\struct {S, \preceq}$ is an up-complete ordered set.

Let $A$ be a subset of $S$.

Then $A$ is closed $A$ is lower and closed under directed suprema.

Sufficient Condition
Assume that
 * $A$ is closed.

By definition of closed set:
 * $\relcomp S A \in \tau$

By definition of Scott topology:
 * $\relcomp S A$ is upper and inaccessible by directed suprema.

By Complement of Complement:
 * $\relcomp S {\relcomp S A} = A$

Thus by Complement of Upper Section is Lower Section:
 * $A$ is a lower section.

Thus by Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema:
 * $A$ is closed under directed suprema.

Necessary Condition
Assume that $A$ is lower and closed under directed suprema.

By Complement of Lower Section is Upper Section and Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema:
 * $\relcomp S A$ is upper and inaccessible by directed suprema.

By definition of Scott topology:
 * $\relcomp S A \in \tau$

Thus by definition:
 * $A$ is closed set.