In any Baire space, is there a $G_\delta$ set of first category not nowhere dense?

Question

In any Baire space, is there a $G_\delta$ set of first category not nowhere dense?

The case for $F_\sigma$ set is rather easy to see, but for $G_\delta$ it seems not really trivial to me. Specifically, I am interested in proving or disproving the statement that,

Given a complete metric space X, for any $G_\delta$ set it is of first category iff it is nowhere dense.

Answer 1

In any Baire space, a $G_\delta$ set of first category is nowhere dense.

Let $X$ be a Baire space and let $S$ be a $G_\delta$ set of first category in $X$; say $S=\bigcap_{n=1}^\infty S_n=\bigcup_{n=1}^\infty A_n$ where each $S_n$ is open and each $A_n$ is nowhere dense. Assume for a contradiction that there is a nonempty open set $U$ such that $S\cap U$ is dense in $U$.

Then $R_n=(X\setminus\overline U)\cup(S_n\setminus\overline{A_n})$ is a dense open subset of $X$. Since $X$ is a Baire space, $\bigcap_{n=1}^\infty R_n$ is dense in $X$. But $U\cap\bigcap_{n=1}^\infty R_n\subseteq\bigcap_{n=1}^\infty(S_n\setminus A_n)=\emptyset$, a contradiction.