Way Below is Approximating Relation

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice.

Then $\ll$ is an approximating relation on $S$.

Proof
Let $x \in S$.

Define $\mathcal R := \mathord\ll$.

By definitions of way below closure and $\mathcal R$-segment:


 * $x^\ll = x^{\mathcal R}$

where:


 * $x^\ll$ denotes the way below closure of $x$
 * $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$

By definition of continuous:
 * $L$ satisfies axiom of approximation.

Thus by axiom of approximation:
 * $x = \sup \left({x^\ll}\right) = \sup \left({ x^{\mathcal R} }\right)$

Hence $\ll$ is an approximating relation on $S$.