Does there exist an uncountable set $A \subset \mathbb{R}$, s.t for every $a \in A$ & every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$? 
Does there exist an uncountable set $A \subset \mathbb{R}$, such that for every $a \in A$ and every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$?

I am not sure what the answer is, but I am having trouble trying to construct such a set. Any hints?

Hint: The set of rational numbers is dense and countable.
