A set is called meager is it can written as a countable union of nowhere dense sets. A set $S$ is said to have the Baire property if for some open set $O$, the symmetric difference $S\Delta O$ is meager.
Problem: Assuming that there is a set of real numbers which does not have the Baire property construct a set $X$ which is not meager and such that for any non-empty open set $O$, $O\setminus X$ is not meager.
My feeble attempt: Suppose $S\subseteq\Bbb R$ does not have the Baire property. Then for all nonempty open sets $O$, $S\Delta O$ is not meager. Taking $\Bbb R$ as the open set $S\Delta\Bbb R=\Bbb R\setminus S$, which cannot be meager. I thought I could take the set $X$ as $\Bbb R\setminus S$. Now for any non-empty open set $O$, $O\setminus X=O\cap S$. I have no idea how to show $O\cap S$ is not meager. In fact I think my choice for $X$ is wrong.
Please suggest how to construct such $X$.