Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$ and $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} (\mathbb{C}[t]/(t^n))$ to $Y$ such that its composition with the natural morphism from $\mathrm{Spec}(\mathbb{C})$ to $\mathrm{Spec}(\mathbb{C}[t]/(t^n))$ corresponds to the closed point $x$, we have that this morphism factors through $X$. Does this imply that there exists an open neighbourhood $U$ of $x$ in $Y$ such that $U$ is contained in $X$?