Interior, closure and boundaries of $\mathbb{Q} \times \mathbb{Q}$ in $\mathbb{R}^2$

Question

Find the interior, closure and boundary of the set $\mathbb{Q} \times \mathbb{Q}$ when viewed as a subset of $\mathbb{R}^2$. Assume the usual Euclidean norm

I am stuck with this.

I think the interior is the empty set, because I can’t image the set having any open subsets.

I am not sure about the closure and boundary.

I also think the set is neither open or closed.

Answer 1

Interior is empty set as between any two rational there is a irrational. Now $\mathbb{Q}$ is dense in $\mathbb{R}$ therefore the closure of $\mathbb{Q}\times \mathbb{Q}$ in $\mathbb{R^2}$ is whole $\mathbb{R^2}$. Now as interior is empty set then boundary is also whole $\mathbb{R^2}$.

Answer 2

It is easy to prove it if you know that $\mathbb{R}\setminus\mathbb{Q} $ and $\mathbb{Q} $ are dense into $\mathbb{R}$ : the interior is empty, the closure is $\mathbb{R}^2$, it admits no boundaries (look at the sequence $(n,m)_{n,m\in \mathbb{Z}}$).