General Topology - Boundary

Question

Consider the topology on $\mathbb{R}$, $\tau(\mathcal{E})$ where $\mathcal{E} = \{(-\infty, a] : a \in \mathbb{R}\}$. What is the boundary of $(0,1)$?

I know that $bd(0,1) = cl(0,1) \setminus int(0,1)$, but am not sure how to determine $cl(0,1)$ and $int(0,1)$.

Answer 1

Each basic open set (-∞, a] intersects both (0,1) and its complement. Therefore, every point on the real line is a boundary point of (0,1), and ∂(0,1) = R.