\begin{definition}[Definition:Compact Closure]
Let $L = \struct{S, \preceq}$ be an ordered set.
Let $x \in S$.
Then '''compact closure''' of $x$, denoted $x^{\mathrm{compact}}$, is defined by
:$x^{\mathrm{compact}} := \leftset {y \in S: y \preceq x \land y}$ is compact$\rightset{}$
\end{definition}