A closed set $O$ is a subset of the real numbers such that $\exists x \in O, \forall \epsilon>0$ s.t. $(x-\epsilon,x+\epsilon)$ is not a subset of $O$.
Are the only two $x \in O$ s.t. $(x-\epsilon,x+\epsilon)$ is not a subset of $O$, the infimum/minimum or the supremum/maximum of that set $O$?
That's not what a closed set is. You just defined what it means for a set to be not open. A set is closed if the complement is open. But certainly neither is required. For example, take $r\in\mathbb{Q}$. Then for all $\epsilon>0$, the set $(r-\epsilon,r+\epsilon)$ is not a subset of $\mathbb{Q}.$ I think you see open and closed as opposites rather than describing a relationship between the set and its complement.