Let $X \subset \mathbb{P}_{\mathbb{C}}^n$ a complex smooth projective variety, $x \in X$ a closed point and $S \subset X$ a constructible subset containing $x$.
Recall, on $X$ we can consider two topologies:
- Zariski topology from variety structure
- classical topology; we can use analytic implicit function theorem
In more modern fashion the later topology arise as natural topology of $X^{an}$ where there is an analytification functor $X \to X^{an}$ sending complex varieties to complex analytic spaces, a generalization of complex manifolds; since we assume $X$ to be smooth, in our case $X^{an}$ is a complex manifold. For core details on construction I recommend to take a look in Mumford's Algebraic Geometry I, Complex Projective Varieties [p 10].
In what follows now we not differ between $X$ and $X^{an}$ but consider instead $X$ as uderlying set with two different topologies living on and going to study relationship between both. We assumed $S$ to be constructible, that is $S= \bigcup_{i=1}^n O_i \cap C_i$ where $O_i$ are Zariski open and $O_i$ are Zariski closed in $X$.
Now assume that $S$ contains an open with respect classical analytical topology neighbourhood $U_{cl}$ of $x$.
Question: How to see that $S$ contains also a Zariski open neighbourhood $U_{Za}$ of $x$? The problem is closely related to that one.