I am reading the book Introduction to complex analytic geometry by Stanislaw Lojasiewicz. In Chapter II, we have that a subset $Z$ of a complex manifold $M$ is said to be a principal analytic subset of $M$ if
$Z =\{x \in M \ : \ f(x)=0\}$,
where $f$ is a holomorphic function on $M$. Since any domain $\Omega$ in $\mathbb{C}^n$ is a complex manifold (of dimension $n$), we can talk of principal analytic subset of $\Omega$. For a domain $\Omega$ in $\mathbb{C}^n (n>1)$ and a complex polynomial $p$ in $n$-variable, it follows that the subset $V=Z(p) \cap \Omega$ is a principle analytic subset of $\Omega$ given the zero set $Z(p)$ of polynomial $p$ has non-empty intersection with $\Omega$.
I have been able to prove that the closure of $V$ in $\mathbb{C}^n$, denoted by $\overline{V}$, must intersect the topological boundary $\partial \Omega$ of $\Omega$.
Does the same hold for the connected components of $V$? Let $V_i$ be a connected component of $V=Z(p) \cap \Omega$. Then the closure $\overline{V_i}$ of $V_i$ in $\mathbb{C}^n$ must have non-empty intersection with $\partial \Omega$.
Intuitively, it seems pretty clear. How to prove it mathematically?