\begin{definition}[Definition:Directed Subset]
Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is a '''directed subset''' of $S$ {{iff}}:
:$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$
\end{definition}