Can the closure of isolated point set be uncountable

Question

Let $S\subset [0,1]$ be a countable set, $\bar{S}$ is the closure of $S$. Can $\bar{S}$ be uncountable? I think it can not be, but I can't find a proof.

Edit: I just realized that my previous question is trivial. But what I really mean is that suppose $S$ is countable and with isolated points can $\bar{S}$ be uncountable?

Answer 1

Closure of the rationals $\mathbb{Q} \cap [0,1]$ is $\mathbb{R} \cap [0,1]$, which is uncountable.