# Strict Lower Closure is Dual to Strict Upper Closure

## Theorem

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

Let $a, b \in S$.

The following are dual statements:

$b \in a^\prec$, the strict lower closure of $a$
$b \in a^\succ$, the strict upper closure of $a$

## Proof

By definition of strict lower closure:

$b \in a^\prec$
$b$ strictly precedes $a$

The dual of this statement is:

$b$ strictly succeeds $a$

By definition of strict upper closure, this means:

$b \in a^\succ$

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

$\blacksquare$