Transitive Closure is Closure Operator

Definition
Let $S$ be a set.

Let $\mathscr A$ be the set of all endorelations on $S$.

Then the transitive closure operator is a closure operator on $\mathscr A$.

Proof
Let $\QQ$ be the set of transitive relations on $S$.

By Intersection of Transitive Relations is Transitive, the intersection of any subset of $\QQ$ is in $\QQ$.

Recall the definition of transitive closure as the intersection of transitive supersets:


 * The transitive closure of a relation $\RR$ on $S$ is the intersection of elements of $\QQ$ that contain $S$.

From Closure Operator from Closed Sets we conclude that transitive closure is a closure operator.