Let $X\subseteq \mathbb A_k ^n$ be an affine algebraic set and let $U:=X\setminus V_X(f)$ be a standard open set, $f \in k[X]$.
Say I find a polynomial $G \in k[x_1,...,x_n]$ and an open neighbourhood $W \subseteq U$ such that $\forall x \in W : G(x) \neq 0$. Is it possible to prove that $G$ does not vanish on $U$?
I am asking since it appears that the standard open sets are very "small", so maybe one could lift that property from $W$ to $U$.
Consider the complex line and $U=\mathbb{C}$, consider $P=X$ it does not vanishes on $W=\mathbb{C}-\{0\}$ but it vanishes on $U$.