Solution
= Solution
$X_1$ is the set of real algebraic numbers and is countable because each nonzero <polynomial> has finitely many roots. For $n\ge2$, $X_n$ is uncountable: the <polynomial> $x_1$ vanishes on $\{0\}\times\mathbb R^{n-1}$.
= Solution
$X_1$ is the set of real algebraic numbers and is countable because each nonzero <polynomial> has finitely many roots. For $n\ge2$, $X_n$ is uncountable: the <polynomial> $x_1$ vanishes on $\{0\}\times\mathbb R^{n-1}$.