# Definition:Continuous Ordered Set

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