Openness of a subset in complex 2-plane

Question

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is Zariski open in $C$. Is $U$ Zariski open in $\mathbb{C}^{2}$? Or at least, is there a Zariski neighborhood $V$ of $0$ with $V\subset U$? Thanks!