My idea:
I first observe its component, for example $\dfrac{3}{2}$. Then, I consider in a metric space $\mathbb{Q}$. I know that $\left\{\dfrac{3}{2}\right\}$ is relatively closed to $\mathbb{Q}$. However, the closure of $\mathbb{Q}\setminus\left\{\dfrac{3}{2}\right\}$ is $\mathbb{Q}$. It would become $\left\{\dfrac{3}{2}\right\}\cap\mathbb{Q}$ instead of $\emptyset$.
I could not find out two closed set to disconnect the metric space $\mathbb{Q}$. Where am I wrong since intuitively $\mathbb{Q}$ is not connected in $\mathbb{R}$?
$\mathbb Q \cap (-\infty, \sqrt 2)$ and $\mathbb Q \cap ( \sqrt 2,\infty)$ are closed and disjoint subsets of $\mathbb Q$ whose union is $\mathbb Q$.