If we consider the open set $\mathbb{R}$, then for every $a \in \mathbb{R}$, you can find an open interval $(a-\epsilon, a+\epsilon)$.
I am probably over thinking this, but I am wondering: Why would it be an open interval if the boundary points are elements in $\mathbb{R}$? I know that $\mathbb{R}$ is both closed and open, but I don't see how the intervals would be open.