Definition:Property (S)
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Definition
Let $L = \struct {S, \preceq}$ be an up-complete ordered set.
Let $X$ be a subset of $S$.
Then $X$ has property (S) if and only if
- for all directed subsets $D$ of $S$: $\sup D \in X \implies \exists y \in D:\forall x \in D: y \preceq x \implies x \in X$
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 3