Definition:Continuous Ordered Set

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Definition

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.


Sources