I'm pretty new to algebraic geometry. I have not clear what does it mean that the Zariski topology and the product topology over the affine space $\mathbb A^m × \mathbb A^n $ are different.
I'm convinced that if $X\subset \mathbb A^m$ and $Y\subset \mathbb A^n$ are affine variety, then $X×Y$ is an affine variety in $\mathbb A^m × \mathbb A^n$.
So actually, a closed set in the product topology is closed in Zariski topology as well.
I found in the notes that I'm reading an example of a set closed only in Zariski topology, however, I didn't understand it. It says "The subset $V(x-y)=\{(a,a):a∈K\}⊂\mathbb A^2$ is closed in the Zariski topology, but not in the product topology of $\mathbb A^1 × \mathbb A^1$".
However, shouldn't $\{(a,a):a∈K\}$ be closed in the product topology too?
Thanks in advance.