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**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.