\begin{definition}[Definition:Continuous Ordered Set]
Let $\left({S, \preceq}\right)$ be an ordered set.
Then $\left({S, \preceq}\right)$ is '''continuous''' {{iff}}
:(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.
\end{definition}