Definition:Continuous Ordered Set

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

Then $\left({S, \preceq}\right)$ is continuous
 * (for all elements $x$ of $S$: the way below closure $x^\ll$ of $x$ is directed) and
 * $\left({S, \preceq}\right)$ is up-complete and satisfies axiom of approximation.