Exhibit a countable collection of open sets $U_j$ such that each open set can be written as a union of some of the sets $U_j$.

Question

Exhibit a countable collection of open sets $U_j$ such that each open set $\mathcal{O} \subseteq \mathbb{R}$ can be written as a union of some of the sets $U_j$.

I'm having trouble with this one. If I'm reading this, any open set in $\mathbb{R}$ including disjoint sets and those involving irrational numbers would need to be written as a union of sets from $U_j$. Any ideas on this one?

Answer 1

Try the collection of open intervals $(a,b)$ for all $a<b$ in $\mathbb{Q}$.