Can we find a dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of points, depending the cardinal of $U^{\prime}$ is finite or not?

Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of points, depending the cardinal of $U^{\prime}$ is finite or not?

Let's consider a Zariski dense subset $U$ of the affine n-space over an algebraically cloed field, say, $C$. Then we could define a map from the power set of $U$ to the set of closed subschemes of $A^n_C$ by sending a subset $U^{\prime}$ of $U$ to its Zariski Closure. Then my question is, what could we say about the image of this map? In case $U=$set of closed points of $A^n_C$, the map is surjective. However, I'm interested in more nontrivial examples, for example, can we find a dense subset $U$ of the affine plane over the complex number field, such that any subset of $U$ has its closure either the whole affine plane or finite number of points ?

Can we find a dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of points, depending the cardinal of $U^{\prime}$ is finite or not?