Question regarding closure of sets defined by the vanishing of holomorphic functions

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?

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This is not true. Take $n=2$, $f(x,y)=x$, $g(x,y)=y(y-1),\; h(x,y)=x-y$.