Definition:Compact Closure

Definition
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$\rightset{}$