Definition:Continuous Ordered Set

Definition

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.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_3:def 6