\begin{definition}[Definition:Closed under Directed Suprema]
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set.
Let $X$ be a subset of $S$.
Then $X$ is '''closed under directed suprema''' {{iff}}
:for all directed subsets $D$ of $S$: $D \subseteq X \implies \sup D \in X$
\end{definition}