Bottom is Compact

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Theorem

Let $L$ be a bounded below ordered set.


Then $\bot$ is a compact element

where $\bot$ is the smallest element in $L$.


Proof

By Bottom is Way Below Any Element:

$\bot \ll \bot$

where $\ll$ denotes the way below relation.

Hence $\bot$ is a compact element.

$\blacksquare$


Sources