Set of zeros of polynomial in several complex variable.

Question

Question: the question is already asked here(link→If $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ are non constant polynomial.)

$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant polynomials in complex variables with $k\ge 1$.

I need know if $$\{(z,w):p(z,w)=0\}$$

which of these are true or false:

$1$. bounded with empty interior

$2$. unbounded with empty interior

$3$. bounded with non empty interior

$4$. unbounded with non empty interior

they said that answer is (b) But, Wikipedia says, zero sets of complex analytic functions in more than one variable are never discrete (link→https://en.m.wikipedia.org/wiki/Analytic_function?wprov=sfla1 other reference Why are there no discrete zero sets of a polynomial in two complex variables?)

So how could zero set of $p(z,w)$ in $\mathbb{C×C}$ have empty interior if it is never discrete? Please help me.

Answer 1

I don't see a contradiction. For instance, in $\mathbb{R}^2$ the set $\{(x,0)\,|\,x\in\mathbb{R}\}$ has empty interior and it is not discrete.