Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$


where $f, g$ and $h$ are holomorphic functions. 
Let us also assume that $Y$ is non empty. In particular 
this would avoid something like $h(x)$ being a ``factor'' of 
$g(x)$.  

Then is it always 
true that 


$$ \bar{X} \subset X \cup \bar{Y} $$


where $\bar{X}$ and $\bar{Y}$ denote the respective closures taken inside $\mathbb{C}^n$?


If this is not true, then is there any reasonable condition on $f$, $g$ and $h$ to ensure that this is true? 



Question regarding closure of sets defined by the vanishing of holomorphic functions
This is not true. Take $n=2$, $f(x,y)=x$, $ g(x,y)=y(y-1),\; h(x,y)=x-y$.


