Background
I recently encountered this textbook exercise (from Ideals, Varieties, and Algorithms by Cox) in my abstract algebra course that I found rather interesting:
Let $R=\{(x,y)\in\mathbb{R}^2 \ | \ y>0\}$ be the upper half plane. Prove that $R$ is not an affine variety.
As an assignment, I am to come up with some exercises of my own, and I wanted to see if I can generalize this question at all, which brings us to my question.
Question
My question might seem strange, and it might not even have an answer, but what I'm wondering is this: can we extend this result (that the upper half of $\mathbb{R}^2$ is not an affine variety) further?
For example, would it be true also that
$R=\{(x,y,z)\in\mathbb{R}^3 \ | \ z>0\}$
is not an affine variety? Or more generally that
$R=\{(x_1,\dots,x_n)\in\mathbb{R}^n \ | \ x_n>0\}$
isn't an affine variety?
As always, thank you all for taking the time to help me out!