Can any collection of open sets in $\mathbb{R}$ be covered by a countable subcollection? Let $A$ be a collection of open sets in $\mathbb{R}$. is there a countable subcollection $G_i$ of $A$ such that $$\cup_{G\in A} G=\cup_{i=1}^\infty G_i$$
I guess there must be such subcollection, but I don't know how to establish it.

Hint: What is the topology on $\mathbb R$ generated by the set of open intervals $(-\infty,b)$ and $(a, \infty)$ for $a,b \in \mathbb Q$?
