I know that the following two statements are correct.
- Every open set of $\mathbb{R}$ be written as countable union of disjoint open intervals ( including open rays and $\mathbb{R}$).
- Every open set of $\mathbb{R}$ be written as countable union of open bounded intervals.
Then is it true that every open set of $\mathbb{R}$ be written as countable union of disjoint open bounded intervals? I think following a similar argument as statement 1, we only need to show open rays and $\mathbb{R}$ satisfy this property.