Given a set $S\subseteq \mathbb{ R}$ with the property that for every $ x\in S$ there is an open interval $I$ containing $x$
such that $I\cap S$ is countable. Prove that S is countable.

I am trying to write $S$ as a countable union of countable sets by using the given property of intersection, but not getting the suitable idea. Please suggest me. Thanks a lot.