Definition:Continuous Ordered Set

Let $\struct {S, \preceq}$ be an ordered set.
Then $\struct {S, \preceq}$ is continuous if and only if
(for all elements $x$ of $S$: the way below closure $x^\ll$ of $x$ is directed) and:
$\struct {S, \preceq}$ is up-complete and satisfies the axiom of approximation.