Compact Subset is Join Subsemilattice

Theorem

Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $\map K L$ be a compact subset of $L$.

Then $\map K L$ is join subsemilattice:

$\forall x, y \in \map K L: x \vee y \in \map K L$

Proof

Let $x, y \in \map K L$.

By definition of compact subset:

$x$ and $y$ are compact.

By definition of compact:

$x \ll x$ and $y \ll y$

By Way Below is Congruent for Join:

$x \vee y \ll x \vee y$

By definition:

$x \vee y$ is compact.

Thus by definition compact subset:

$x \vee y \in \map K L$

$\blacksquare$

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_8:3