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