Let $A$ is a subset of $\mathbb{R}$ and the cardinality of $A$ is $2^\omega$. The question is this: Does the closure of $A$ in $\mathbb{R}$ have a nonempty interior in $\mathbb{R}$?
Added: Thank you for helps. In fact, I can't understand the proof of the lemma 2.11 in this paper. In line 7 of the proof, I don't know why $cl_R(A_{N,M})$ has a non-empty interior in $R$?