Upper Closure is Closure Operator

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $T^\succeq$ be the upper closure of $T$ for each $T \subseteq S$.

Then $\cdot^\succeq$ is a closure operator.

Inflationary
Let $T \subseteq S$.

Let $t \in T$.

Then since $T \subseteq S$, $t \in S$ by the definition of subset.

Since $\preceq$ is reflexive, $t \preceq t$.

Thus by the definition of upper closure, $t \in T^\succeq$.

Since this holds for all $t \in T$, $T \subseteq T^\succeq$.

Since this holds for all $T \subseteq S$:
 * $\cdot^\succeq$ is inflationary.

Order-Preserving
Let $T \subseteq U \subseteq S$.

Let $x \in T^\succeq$.

Then by the definition of upper closure: for some $t \in T$, $t \preceq x$.

By the definition of subset:
 * $t \in U$

Thus by the definition of upper closure:
 * $x \in U^\succeq$

Since this holds for all $x \in T^\succeq$:
 * $T^\succeq \subseteq U^\succeq$

Since this holds for all $T$ and $U$:
 * $\cdot^\succeq$ is order-preserving.

Idempotent
Let $T \subseteq S$.

By Upper Closure is Upper Section, $T^\succeq$ is an upper section.

Thus by Upper Section: Definition 3:
 * $\paren {T^\succeq}^\succeq = T^\succeq$

Since this holds for all $T$:
 * $\cdot^\succeq$ is idempotent.

Also see

 * Lower Closure is Closure Operator