Ok, so the rules are the following:
- You can erase parts of the curve; which means you ignore, or take away some amount (not necessarily finite) of points of the graph*
- You can only use hypersurfaces, meaning it has to be zeroes of ONE polynomial (Not necessarily irreducible though) of two varibles with real coefficients
- $0$ is not a valid polynomial, since you could erase all the points you don't want on your curve
- For the same reason as in (3), $\mathbb R^2$ is not a valid curve
*With graph I mean the set of points of the hypersurface
Note that if you have a hypersurface $V(F)$ and $V(G)$ you can combine their graphs simply because $V(F)\cup V(G)=V(FG)$, in this case, the erasing must be done once you paste all these parts together, don't know if it makes a difference, but if your strict with the definition of erasing, it must be that way.
Basically, the question can be reframed as
Which are all the sets of points $X\subsetneq\mathbb R^2$ such that for every hypersurface $V\subsetneq\mathbb A^2_\mathbb R$, $X\not\subset V$?
But I wanted to explain what was my point on asking this was.
I know how to construct any set of finitely many points, any curve that's just made out of straight lines and in general any figure that's just made out of polynomial sections; but I don't know if you can construct stuff like the curve $y=\sin x $, it is clear that you can't without the erasing trick, but I don't know if this erasing trick fixes all of these problems
Note: Even if a hypersurface has to be irreducible, every time I say hypersurface in this question I'm just trying to say the zeroes of a single polynomial, doesn't have to be irreducible necessarily