What's the point of the Zariski topology on $F^n$ when $F$ is not algebraically closed? $\DeclareMathOperator{\Spm}{Spm}$
$\DeclareMathOperator{\Spec}{Spec}$
Let $F$ be a field, and let $R = F[T_1, ... ,T_n]$.  On $\mathbb{A}^n_F := \Spm R$, the Zariski topology is given by defining the closed sets to be of the form 
$$V(I) = \{ \mathfrak m \in \mathbb{A}_F^n : I \subseteq \mathfrak m\}$$
for $I$ an ideal of $R$.
On $\mathbb{A}_F^n(F) = \textrm{Hom}_{\textrm{$F$-alg}}(R,F) = F^n$, the Zariski topology is given by defining the closed sets to be of the form
$$Z(I) = \{ p \in F^n : f(p) = 0 \textrm{ for all } f \in I \}$$
The image of the injection $\mathbb{A}_F^n(F) \rightarrow \mathbb{A}_F^n, \phi \mapsto \textrm{Ker } \phi$ consists of all maximal ideals $\mathfrak m$ of $R$ for which the residue field $R/\mathfrak m$ is equal to $F$.  Identifying $\mathbb{A}_F^n(F)$ with its image, we see that the Zariski topology on $\mathbb{A}_F^n(F)$ is just the induced topology from $\mathbb{A}_F^n$.  When $F$ is algebraically closed, this is a bijection.
I have seen introductory algebraic geometry notes motivate the Zariski topology for arbitrary fields by defining it first on $F^n$ rather than on $\mathbb{A}_F^n$ (or $\Spec R$).  Is there any practical reason to specifically define the Zariski topology on $F$-rational points?  Or is this just strictly for motivation?
 A: This is mostly for motivation. A lot of classical algebraic geometry happened in $F^n$ for various choices of $F$, and there's some nice and useful results one can state (and that were discovered) in this language. The advantages of this is that it's fairly concrete and many people starting to learn the subject have thought about $F^n$ before, probably in the guise of linear algebra. This can certainly be an easier introduction than hearing about generic points and all the "extra" points associated to $\operatorname{Spec} F[x_1,\cdots,x_n]$. And picturing what's going on is important in a subject with geometry in its name.
On the other hand, the theoretical aspects of this are somewhat limiting (for example, all the technology of generic points is a bit of a pain to deal with). As time passed and it became clear that thinking about schemes provided a nice way to do algebraic geometry, the Zariski topology on $F^n$ became less used. This isn't to say that it's useless to math - intuition, motivation, and the story behind what's going on are useful - but it's of less practical use in that people don't often prove theorems with it these days.
