I came across the following problem on open sets:
Let $\mathcal{C}$ be a set of nonempty open sets in $\mathbb{R}$ such that for $U,V \in \mathcal{C}$, either $U=V$ or $U \cap V = \emptyset$. Prove that either $\mathcal{C}$ is finite or the sets in $\mathcal{C}$ can be listed in a sequence $(U_n)$.
Now we can enumerate the rational numbers using Farey sequences. So choose a rational number from each $U \in \mathcal{C}$. Form a neighborhood about each rational number from the open sets and see if they (the neighborhoods) intersect or not.
So it suffices that if we choose a rational number from each open set, by countability of the rationals $\mathcal{C}$ is countable?