HMIC 2013 Problem 4

A subset $U \subset \mathbb{R}$ is open if for any $x \in U$, there exist real numbers $a, b$ such that $x \in(a, b) \subset U$. Suppose $S \subset \mathbb{R}$ has the property that any open set intersecting $(0,1)$ also intersects $S$. Let $T$ be a countable collection of open sets containing $S$. Prove that the intersection of all of the sets of $T$ is not a countable subset of $\mathbb{R}$. (A set $\Gamma$ is countable if there exists a bijective function $f: \Gamma \rightarrow \mathbb{Z}$.)