Upper Closure is Smallest Containing Upper Section
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Theorem
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$.
Let $U = T^\succeq$ be the upper closure of $T$.
Then $U$ is the smallest upper section containing $T$ as a subset.
Proof
Follows from Upper Closure is Closure Operator and Set Closure is Smallest Closed Set/Closure Operator.
$\blacksquare$