1. ## The zariski topology

Hi guys,

I'm having trouble trying to prove this:

Show that the Zariski Topology on $\displaystyle \mathbb{A}^2$ is not the product topology on $\displaystyle \mathbb{A}^1 \times \mathbb{A}^1$.

I think my main problem is that I have no idea how you would apply the product topology on $\displaystyle \mathbb{A}^1$. Stupidly, the university have scheduled Algebraic Geometry and Topology such that they run concurrently, which means we've just been introduced to the concept of a Topological space. A push in the right direction would be much appreciated, and a brief explanation of how I could apply the concept of a product topology on affine algebraic varieties would be really good.

Let us pick the diagonal $\displaystyle \mathbb{V}(x-y)$, given by the set $\displaystyle \mathbb{V}_{\triangle} = \{ (x,y) : x-y=0 \}$. Following the definition of the Zariski topology on $\displaystyle \mathbb{A}^2$, this is closed. We just have to show that $\displaystyle \mathbb{V}_{\triangle}$ is not closed on the product topology $\displaystyle \mathbb{A}^1 \times \mathbb{A}^1$.