Way-below relation

Remarks

If $x \ll y$, then $x \le y$, by taking $D = \set{y}$. If $w \le x \ll y \le z$, then $w \ll z$. An element $c$ is compact if and only if $c \ll c$. Example: in $[0, 1]$ with the usual order, $x \ll y$ holds if and only if $x = 0$ or $x < y$; for $x < y \le \bigvee D$, some $d \in D$ exceeds $x$ by the definition of the supremum. Non-example: in $[0, 1]$, $1/2 \not\ll 1/2$, because $D = \set{1/2 - 1/n : n \ge 3}$ is directed with supremum $1/2$ and no member is at least $1/2$.