If Compact Between then Way Below

Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Let $x, k, y \in S$ such that:

$x \preceq k$ and $k \preceq y$ and $k \in \map K L$

where $\map K L$ denotes the compact subset of $L$.

Then $x \ll y$

where $\ll$ denotes the way below relation.

Proof

By definition of compact subset:

$k$ is compact.

By definition of compact:

$k \ll k$

Thus by Preceding and Way Below implies Way Below:

$x \ll y$

$\blacksquare$

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_8:1