Can any open subset of $\overline{\mathbb{R}}$ be written as a countable union of disjoint open intervals.

Question

The context of this question is that I am trying to prove that $\{[-\infty,a]\}_{a\in\mathbb{R}}$ generates the Borel set of the extended reals. And I used this property of the real line to prove a similar argument but for the Reals.

Answer 1

Yes, that is true. Let $A$ be an open subset of $\overline{\Bbb R}$. Then:

• If $A\cap\Bbb R$ is bounded, then it is an open subset of $\Bbb R$ and, since $A$ is open, $A=A\cap\Bbb R$. And you know that then $A$ is a disjoint union of open intervals.
• Otherwise, $A\cap\Bbb R$ is unbounded. Write it as a disjoint union of open intervals. If $A\cap\Bbb R$ has no lower bound, then one of those intervals will be of the form $(-\infty,a)$; replace it with $[-\infty,a)$. Do a similar thing if $A\cap\Bbb R$ has no upper bound. And then you will have your disjoint union of open intervals.