Way Below is Approximating Relation

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

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

Proof
Let $x \in S$.

Define $\RR := \mathord \ll$.

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


 * $x^\ll = x^\RR$

where:


 * $x^\ll$ denotes the way below closure of $x$
 * $x^\RR$ denotes the $\RR$-segment of $x$

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

Thus by the axiom of approximation:
 * $x = \map \sup {x^\ll} = \map \sup {x^\RR}$

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