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 $S$.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_4:Lm12