
Originally Posted by
Plato
Because $\displaystyle \mathbb{Q} \times \mathbb{Q} \subseteq \mathbb{R}^2 $ we use the metric topology of $\displaystyle \mathbb{R}^2$.
Every point of $\displaystyle \mathbb{Q} \times \mathbb{Q}$ is a limit point of $\displaystyle \mathbb{R}^2$.
And every point of $\displaystyle \mathbb{R}^2 \backslash \left( {\mathbb{Q} \times \mathbb{Q}} \right)$ is a limit point of $\displaystyle \mathbb{Q} \times \mathbb{Q}$.
Can you answer your questions now?
BTW: The set $\displaystyle \mathbb{R}^2 \backslash \left( {\mathbb{Q} \times \mathbb{Q}} \right)$ is pathwise connected. It is actually easy to exhibit the path between two points.