Let $(X, d)$ be a non-empty complete metric space with no isolated points, $G$ an open dense subset of $X$ and $E$ a countable dense subset of $X$.

(i) Stating clearly any standard theorem you use, prove that $G \backslash E$ is a dense subset of $X$.

(ii) If $G$ is only assumed to be uncountable and dense in $X$, does it still follow that $G \backslash E$ is dense in $X$ ? Justify your answer.