Lower Closure is Dual to Upper Closure

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

Let $a, b \in S$.

The following are dual statements:


 * $b \in \mathop{\bar \downarrow} \left({a}\right)$, the weak lower closure of $a$
 * $b \in \mathop{\bar \uparrow} \left({a}\right)$, the weak upper closure of $a$

Proof
By definition of weak lower closure, $b \in \mathop{\bar \downarrow} \left({a}\right)$ iff:


 * $b \preceq a$

The dual of this statement is:


 * $a \preceq b$

by Dual Pairs (Order Theory).

By definition of weak upper closure, this means $b \in \mathop{\bar \uparrow} \left({a}\right)$.

The converse follows from Dual of Dual Statement (Order Theory).

Also see

 * Duality Principle (Order Theory)