Closure is Closed

Theorem

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

Let $\cl: S \to S$ be a closure operator.

Let $x \in S$.

Then $\map \cl x$ is a closed element of $S$ with respect to $\cl$.

When the ordering in question is the subset relation on a power set, the result can be expressed as follows:

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$.

Then $\map \cl T$ is a closed set with respect to $\cl$.

By the definition of closure operator, $\cl$ is idempotent.

Therefore:

$\map \cl {\map \cl x} = \map \cl x$

It follows by definition that $\map \cl x$ is a closed element.

$\blacksquare$