Reflexive Closure is Closure Operator

Definition
Let $S$ be a set.

Let $R$ be the set of all endorelations on $S$.

Then the reflexive closure operator is a closure operator on $R$.

Reflexive Closure is Idempotent
Let $\mathcal Q$ be the set of reflexive relations on $S$.

By Intersection of Reflexive Relations is Reflexive, the intersection of any subset of $\mathcal Q$ is in $Q$.

By the definition of reflexive closure as the intersection of reflexive supersets:
 * The reflexive closure of a relation $\mathcal R$ on $S$ is the intersection of elements of $\mathcal Q$ that contain $S$.

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