I saw a proof of the following theorem.
Every open subset $\mathcal{O}$ of $\mathbb{R}$ can be written uniquely as a countable union of disjoint open intervals.
The proof was convincing, but can anyone help me writing out explicitly such a representation of the interval $(0,1)$? Or maybe $(0,1)$ itself is already such a representation? Can I write it as countable union of "smaller" disjoint open intervals?