Can every nonempty open set be written as a countable union of bounded open intervals of the form $(a_k,b_k)$, where $a_k$ and $b_k$ are real numbers (not $\pm\infty$)?
If yes, can someone point me toward a proof?
If not, counterexample?
Note that this is not the same question as the property "every nonempty open set is the disjoint union of a countable collection of open intervals."
Best Answer
Hint: Let $A$ be open, and consider all intervals $(p,q)$ such that $p$ and $q$ are rational and $(p,q)\subset A$.