I got the following statements:
Again, I'm super unsure on how to prove something is wrong. I think 4 is definitely right, but I'm unsure about each of the other statements. We have the Definition, that a Subset of $k^n$ is called Zarisiki-closed if it is the set of solutions to a set of Polynomials.
The topology $\mathbb{A}^1_k$ is the cofinite topology. In our case, $\overline{\mathbb{Q}}^1 = \mathbb{A}^1_{\overline{\mathbb{Q}}}$ so the closed sets are precisely the entire space and finite sets. Therefore, $(4)$ is closed and the others are not closed.