# Definition:Directed Subset

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## Definition

Let $\struct {S, \precsim}$ be a preordered set.

Let $H$ be a non-empty subset of $S$.

Then $H$ is a directed subset of $S$ if and only if:

$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$