Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema

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Theorem

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set.

Let $X$ be a closed under directed suprema subset of $S$.


Then $\complement_S\left({X}\right)$ is inaccessible by directed suprema.


Proof

Let $D$ be a directed subset of $S$ such that

$\sup D \in \complement_S\left({X}\right)$

By definition of relative complement:

$\sup D \notin X$

By definition of closed under directed suprema:

$D \nsubseteq X$

By Complement of Complement:

$D \nsubseteq \complement_S\left({\complement_S\left({X}\right)}\right)$

Thus by Empty Intersection iff Subset of Relative Complement:

$D \cap \complement_S\left({X}\right) \ne \varnothing$

$\blacksquare$


Also See

Sources