- The set of rational numbers $\mathbb{Q}$ is not a connected topological space
My Attempt. Let $\alpha \in \mathbb{R}$ be an irrational number. By definition, $\alpha \notin \mathbb{Q}$. Consider the sets: $$ \begin{array}{l}{S:=\mathbb{Q} \cap(-\infty, \alpha)} \\ {T:=\mathbb{Q} \cap(\alpha , \infty)}\end{array} $$
So since $S, T$ are open sets on $\mathbb{Q}$. Then
$$S \cup T=\mathbb{Q}$$ $$S \cap T=\varnothing$$ $$S, T \neq \varnothing$$
So $S,T$ are disjoint sets of $Q$. Hence, $\mathbb{Q}$ is not connected.
My question is: How can I show $S,T$ are open sets on $\mathbb{Q}$?