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, $\operatorname{cl} (B) = \operatorname{cl} (A \cup (B \setminus A)) = \operatorname{cl}(A) \cup \operatorname{cl}(B \setminus A)$.

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