About residual sets and dense sets

Question

Suppose $E\subset X$ is some subset of a (complete) metric space $X$. To each $p\in E$ we associate a small ball $B(p,\epsilon_p)$ centered at $p$, and we want to consider the union $$ \hat E:=\bigcup_{p\in E} B(p,\epsilon_p) $$

Question: When $E$ is only a dense subset, then $\hat E$ may not be the total space. A counter example is that $E=\mathbb Q$ and $X=\mathbb R$. If we further require $E$ is residual, can we conclude $\hat E=X$? If not, any counter example?

Answer 1

E.g. if $E$ is the set of irrationals in the reals (which is residual), we can take $r_p =|p|$ for all irrational $p$ and then $0 \notin \hat{E}$ for that those radii.

So there is no general guarantee that $\hat{E} = X$; we can always avoid specific points of the complement if we so desire.