Dense $G_\delta$ of open interval This is not a hard question, I just want to make sure whether it is possible or not. Let $I$ be an open interval of $\mathbb R$. The question is it always possible to find $G_\delta$ dense subsets of $I$? I know it true to find countable dense of $I$ as it consider as a second countable subspace of $\mathbb R$. But the question is possible to have it $G_\delta$ dense subset of $I$. Any help will appreciated.    

Take any countable subset $C$ of $I$ and then $I\setminus C$ is a dense $G_\delta$. Or use the complement of a homeomorphic copy of a Cantor set (the standard Cantor set lies in $[0,1]\subseteq(-1,1)$ and all open intervals are homeomorphic, so we always have them). 
