I am trying the show the following:
(10) Show that polynomials $F\in k[x_1,\dotsc,x_n]$ are continuous maps $f\colon\mathbb{A}^n\to\mathbb{A}^1$ (for the Zariski topology on $\mathbb{A}^n$ and on $\mathbb{A}^1$).
$\textbf{My Attempt:}$
In this case, to show that this map is continuous we need to show that the preimage of a closed set is closed. We know $f$ is a function that maps the zeroes of a certain affine set to the zeroes of another affine set. I'm not sure how to proceed further. Would there be a simple topological argument hidden in here?
let $Z$ be a closed set. Then $f(v) \in Z$ if and only if $f(z)$ belongs to the zero locus of some collection of polynomials $h_1,\dots h_n$. But then $0=h_1\circ f=\dots =h_n\circ f$. But these are all polynomials, so $f^{-1}(Z)$ must be closed.
I've written it this way, since it generalizes, but really the point is that we can argue for a single point, since if the preimage $f^{-1}(a)$ is closed, then the preimage of a finite set of points is a finite union of closed sets, which is closed.
Now note that
$f(v)=a$ if and only if $f(v)-a=0$, which is a specialization of $h_1 \circ f$ when $h_1(x)=x-a$.