Zariski Topology and Euclidean Ball

Question

I don't understand why an open Euclidean ball is not open in the Zariski topology. I understand that there's a definition, but everything about the definition seems completely arbitrary.

I'm working through some topology right now, but I'm having difficulty understanding basic concepts or what's going on. My strength was never abstract algebra or anything of that kind. Anyways, I don't get the basic idea of a Zariski topology. What exactly is it supposed to be? What are we trying to do with it? I'm just confused and don't get anything. It seems like everything has just come out of nowhere for no reason.

I do not like algebra or other topics like it, but I realized I needed to learn some topology for my research. So I'm taking topology right now, but I'm really struggling to understand what's going on. The visual material that involves transformations or other things like that, I'm getting, but I do not understand all the other stuff. How the hell do I understand these basic concepts or ideas.

Answer 1

We suppose the ground field is $C$ the field of complex numbers. An subset for the Zariski topology is the complementary of a closed subset which has the form $V(I)$ where $I$ is an ideal of $C[X_1,...,X_n]$. $I$ is generated by a finite number of polynomial functions. Thus $V(I)$ can't contain an open subset of $C^n$ as the complementary of the open ball unless it is $C^n$.

Answer 2

A basis of the topology is the set open sets of the form $D(f)=\bigl\{\text{prime ideals }\mathfrak p \text{ which do not contain }f\bigr\}$, where $f$ is a polynomial in $\mathbf C[X_1,\dots,X_n]$. This topology in general is not Hausdorff.

Thus, roughly, we can say that in $\mathbf C^2$, a basic open set, is the complement of an algebraic (complex) curve. This clearly has very little to do with Euclidean open balls.