# 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

$\bot \ll \bot$

where $\ll$ is the way below relation.

By definition:

$\bot$ is compact.

Thus by definition of compact subset:

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

$\blacksquare$