The interior of $\mathbb{R} \times \mathbb{Q}$

A question says, find the closure and interior of the sets $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb{Q}$. The answers say $\mathbb{R}^2$ and $\emptyset$ respectively for both. Why isn't the interior of $\mathbb{R} \times \mathbb{Q}$, $\mathbb{R} \times \emptyset$ because the interior of $\mathbb{R}$ is $\mathbb{R}$? Does $\mathbb{R} \times \emptyset$ even make sense as a set?

Edit: For clarity I mean in $\mathbb{R}^2$ with the usual topology. I was told one should always assume the usual topology when it is not specified.

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2022-07-25 20:46:22
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