Given any field $p$, I want to know what the algebraic sets in $\mathbb{A}_p^2$.
So, I know I have $\emptyset,\mathbb{A}_p^2$ and all finite sets. The idea, is that if you think $p$ as in $\mathbb{R}$. You can see intuitively that the remaining algebraic sets are going to be intersections of 1-manifolds, but this idea maybe only works when $p$ is $\mathbb{R}$.
An other way (more general) to see it, is to say that the remaining algebraic sets will be the image of continuous ''curves'' $\gamma: I \to \mathbb{A}_p^2$.
But this is just ideas I have by looking at examples and following some intuition. I cannot seem to formalize this thought, furthermore I do not know if it is true, maybe I am missing some category of algebraic sets.
Any help would be appreciated, thanks.