\begin{definition}[Definition:Property (S)]
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set.
Let $X$ be a subset of $S$.
Then $X$ has '''property (S)''' {{iff}}
:for all directed subsets $D$ of $S$: $\sup D \in X \implies \exists y \in D:\forall x \in D: y \preceq x \implies x \in X$
\end{definition}