Definition:Inaccessible by Directed Suprema

Definition

Let $L = \struct {S, \preceq}$ be an up-complete ordered set.

Let $X$ be a subset of $S$.

Then $X$ is inaccessible by directed suprema if and only if:

for all directed subsets $D$ of $S$:

$\sup D \in X \implies X \cap D \ne \O$

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 WAYBEL11:def 1