I have tried to prove that $\mathbb{Q}\subset \mathbb{R}$ equipped with the subspace topology is a disconnected space. I'd like to make sure my attempted proof is correct since topological properties of $\mathbb{Q}$ seem like nice things to be familiar with.
First, a base for the subspace topology on $\mathbb{Q}$ is $\{(a,b)\cap \mathbb{Q}\,|\,a, yes?
To separate the rationals into two disjoint open sets (a space is disconnected if it can be written as the union of two nonempty disjoint open sets), we need the end of one 'interval' to lie between two rational numbers, which intuitively seems to be problematic at first, since we can find rational numbers arbitrarily close to each other.
However, the rationals are countable. So label them as $a_1, a_2, ...\,$. Can we then take two arbitrary rationals, $a_i$ and $a_{i+1}$, and then take the relatively open sets $(-\infty,a_i)$ and $(a_{i+1},\infty)$, which are clearly disjoint, and whose union is clearly $\mathbb{Q}$ as a separation of $\mathbb{Q}$?