Lower Closure of Element is Closed under Directed Suprema

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

Let $x \in S$.

Then $x^\preceq$ is closed under directed suprema,

where $x^\preceq$ denotes the lower closure of $x$.


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

$D \subseteq x^\preceq$

By Lower Closure of Element is Ideal:

$x^\preceq$ is directed.

By definition of up-complete:

$D$ and $x^\preceq$ admit suprema.

By Supremum of Subset:

$\sup D \preceq \map \sup {x^\preceq}$

By Supremum of Lower Closure of Element:

$\sup D \preceq x$

Thus by definition of lower closure of element:

$\sup D \in x^\preceq$