Definition:Element is Way Below

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $x, y \in S$.

$x$ is way below $y$, denoted $x \ll y$,
 * for every directed subset $D$ of $S$
 * if $D$ admits a supremum and $y \preceq \sup D$
 * then there exists $d \in D$ such that $x \preceq d$

$x$ is compact $x \ll x$