Bottom in Compact Subset

Theorem

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Then $\bot \in K\left({L}\right)$

where $\bot$ denotes the smallest element of $L$,

$K\left({L}\right)$ denotes the compact subset of $L$.

Proof

By Bottom is Way Below Any Element:

$\bot \ll \bot$

where $\ll$ is the way below relation.

By definition:

$\bot$ is compact.

Thus by definition of compact subset:

$\bot \in K\left({L}\right)$

$\blacksquare$

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