A fellow grad student asked me this, I have been playing for a while but have not come up with anything. Note that $\mathbb{C}$ is homeomorphic to $\mathbb{C} - \{0\}$ in the Zariski topology - just take any bijection and the closed sets (finite sets) will biject as well. Concocting a similar thing for the plane is harder though.

I think I can show that the rational plane and the rational plane minus the origin are homeomorphic by enumerating the irreducible curves and using a back and forth argument, but I have not written it all up formally to see if I am missing something yet.

I know the question isn't natural from the point of view of algebraic geometry, because one of the objects isn't even a variety. I think it is still interesting just to see how weird the zariski topology really is.

`$\mathbb{C}^2$`

(resp.`$\mathbb{C}^2\setminus\{(0,0)\}$`

) with a face connecting certain curves when they have a nonempty intersection — are these two structures isomorphic? It may not be simpler that way, but it puts the emphasis on curves, and also suggests looking at it from the model-theoretic point of view: the back-and-forth argument can be phrased by saying in terms of an Ehrenfeucht-Fraïssé game between these structures. (And I'm running out of space.) $\endgroup$4more comments