\begin{definition}[Definition:Inaccessible by Directed Suprema]
Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $X$ be a subset of $S$.
Then $X$ is '''inaccessible by directed suprema''' {{iff}}:
:for all directed subsets $D$ of $S$:
::$\sup D \in X \implies X \cap D \ne \O$
\end{definition}