Definition:Closed under Directed Suprema
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Definition
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set.
Let $X$ be a subset of $S$.
Then $X$ is closed under directed suprema if and only if
- for all directed subsets $D$ of $S$: $D \subseteq X \implies \sup D \in X$
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 2