Is $\Bbb Q$ disconnected under general topology from $\Bbb R$? I just completed the proof of that fact that $\Bbb Q$ is disconnected. My question is how to find a separation of $\Bbb Q$.
Let $X$ be a topological space. We say $A$ and $B$ forms a separation of $X$ if $A\cap \bar{B}$ and $\bar{A}\cap B$ are empty and $A\cup B=X$.
Here $\Bbb Q$ denotes the set of rational numbers.
Can someone please help with hints.
Thanks.

$U= (-\infty, \sqrt{2}) \cap \mathbb{Q}$, is open in $\mathbb{Q}$ as the intersection of an open subset of $\mathbb{R}$ with $\mathbb{Q}$ (definition of subspace topology) and similarly $V=(\sqrt{2}, +\infty) \cap \mathbb{Q}$ is also open in $\mathbb{Q}$.
As $\sqrt{2} \notin \mathbb{Q}$ we have $\mathbb{Q} = U \cup V$ and $\overline{U} = U$ (closure in $\mathbb{Q}$) and $\overline{V}=V$ and clearly $U \cap V=\emptyset$, so $\{U,V\}$ forms a separation of $\mathbb{Q}$.
