Let $k$ be an algebraically closed field and denote $\mathbb{A}^n$ as the $n$-tuple of elements in $k$ where we put the Zariski topology. Say we have $\mathbb{A}^n \times \mathbb{A}^m$ where we put the product topology. I was wondering does this correspond to $\mathbb{A}^{m+n}$ with Zariski topology? I was wondering how I can see this? (or that this is not the case) Thank you.
The reason why I ask is I am reading a set of notes and in it, they prove that $\mathbb{A}^n$ is not complete by the following argument: Consider the projection $\mathbb{A}^n \times \mathbb{A}^1 \rightarrow \mathbb{A}^1$. Consider a closed set subset $X \subseteq \mathbb{A}^n \times \mathbb{A}^1 = \mathbb{A}^{n+1}$ defined to be $X = \{(a_1, ..., a_n, b) : a_1..a_nb = 1 \}$. The image is $\mathbb{A}^1 \backslash \{0\}$ which is open. Thus $\mathbb{A}^n$ is not complete.
And I was wondering about $\mathbb{A}^n \times \mathbb{A}^1 = \mathbb{A}^{n+1}$ part in this argument.
The underlying topological space of a product scheme is almost never the same as the product of the underlying topological spaces of the schemes involved in the product. For instance, consider the product $\Bbb A^n\times \Bbb A^n$ for $n>0$ and suppose we're taking the product topology. The diagonal copy of $\Bbb A^n$ should be a perfectly good closed subvariety, but if it were, then $\Bbb A^n$ would be Hausdorff, which is absurd.
So what we see is that we can't put the product topology on products in the category of schemes - we need something else. Remembering that affine schemes are contravariantly equivalent to commutative rings via $\operatorname{Spec}$ and global sections functors, we see that a product in affine schemes should be the same as a coproduct in commutative rings. We know what that is - a coproduct in commutative rings is a tensor product. And sure enough, things behave nicely in this scenario - we see that $$\Bbb A^n\times \Bbb A^1 \cong \operatorname{Spec} \Bbb Z[x_1,\cdots,x_n]\times \operatorname{Spec} \Bbb Z[y] \cong \operatorname{Spec} \Bbb Z[x_1,\cdots,x_n]\otimes\Bbb Z[y] \cong \operatorname{Spec} \Bbb Z[x_1,\cdots,x_n,y] \cong \Bbb A^{n+1}.$$