Bottom in Compact Subset
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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