Exact definition of a regular open set

Question

Let $X$ be a space.

(1) A subset $A$ of $X$ is regular open iff $A={\rm int}\ \bar{A}$.

(2) An open subset $A$ of $X$ is regular open iff $A={\rm int}\ \bar{A}$.

(1) implies that any subset $A$ of $X$ is regular closed as long as $A={\rm int}\ \bar{A}$, whereas (2) presupposes that $A$ be open in the first place. I saw both definitions around. Which one is correct? (I understand that a regular open set is open.)

Answer 1

(1) implies that $A$ is open since $A$ is the interior of some other set, so the definitions are equivalent.