TITLE: Is the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?
QUESTION [5 upvotes]: I know the closure of $\mathbb Q$ is $\mathbb R$, but does this imply that the closure of $\mathbb Q \times \mathbb Q$ equal to $\mathbb R \times \mathbb R$?

REPLY [6 votes]: In fact , $$\overline{A\times B}=\bar {A}\times \bar{B} $$for any two subsets $A,B\in \mathbb R$.
Here, $$\overline{\mathbb Q\times \mathbb Q}=\bar {\mathbb Q}\times \bar{ \mathbb Q}=\mathbb R\times  {\mathbb R}$$