Equivalence of Definitions of Kuratowski Closure Operator

Theorem
The two definitions of Kuratowski closure operator are equivalent.

Proof
That Definition 2 implies Definition 1 follows immediately from the definition of closure operator.

Let $X$ be a set and let $\operatorname{cl}$ be a Kuratowski closure operator on $X$ by Definition 1.

By the definition of closure operator, all we need to prove is that $\operatorname{cl}$ is increasing.

Let $A \subseteq B \subseteq X$.

Then by Definition 1 and Union with Superset is Superset, $\operatorname{cl} (B) = \operatorname{cl} \left({A \cup B}\right) = \operatorname{cl}(A) \cup \operatorname{cl}(B)$.

By Subset of Union, $\operatorname{cl}(A) \subseteq \operatorname{cl}(A) \cup \operatorname{cl} (B) = \operatorname{cl}(B)$.