I want to investigate about if the set of points of the form $(x,1)$ with $x\neq 0$ (horizontal line without a point) is closed or not in $\mathbb{A}_k^n$ with the zariski topology.
My intuition say me yes. I think that the smaller algebraic set containg this set of points is $V(y-1)$, which consist of the entire horizontal line. However, I do not have idea about how I can avance...
You have two constraints: one which is closed and one which is open.
First, you constrain $y = 1$, which is a zariski closed condition, since this is the same as saying $y - 1 = 0$. Then, you constrain $x \neq 0$, which is a zariski open condition.
As such, your set will be an open subset of a closed subset in $\mathbb{A}^2$. These are typically not open or closed in $\mathbb{A}^2$.