# Lower Closure of Directed Subset is Ideal

## Theorem

Let $\mathscr S = \left({S, \preceq}\right)$ be an ordered set.

Let $D$ be a directed subset of $S$.

Then

$D^\preceq$ is an ideal in $\mathscr S$

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

## Proof

$D^\preceq$ is directed.
$D^\preceq$ is a lower set.

Thus by definition

$D^\preceq$ is an ideal in $\mathscr S$

$\blacksquare$