Definition:Compact Closure

Definition
Let $L = \left({S, \preceq}\right)$ be an ordered set.

Let $x \in S$.

Then compact closure of $x$, denoted $x^{\mathrm{compact}}$, is defined by
 * $x^{\mathrm{compact}} := \left\{ {y \in S: y \preceq x \land y}\right.$ is compact$\left.{}\right\}$