Let $X$ be a hereditarily Baire spaces i.e (every closed subspace of $X$ is baire space), and $Y$ be a subspace of $X$. If $Y= \bigcap_{i \in \mathbb{N}} (F_i \cup G_i)$, with each $F_i$ closed in $X$ and $G_i$ open in $X$, how to prove that $Y$ is baire spaces?
I know that every dense $G_\delta$ subspace of Baire spaces is Baire spaces, but I do not know how to prove that $Y$ (in my question) is Baire spaces.
I can only prove the claim for a perfect space $X$, that is when each closed subset $X$ is $G_\delta$ (which shows that possible counterexamples are hard to construct). Indeed, assume that for each $F_i$ there exists a family $\{G_{i,n}\}$ of open subsets of $X$ such that $F_i=\bigcap_{n\in\Bbb N} G_{i,n}$. Then
$$Y= \bigcap_{i \in \mathbb{N}} (F_i \cup G_i)= \bigcap_{i,n \in \mathbb{N}} G_{i,n}\cup G_i$$ is a $G_\delta$ subset of the space $X$, so it is a Baire space by the following proposition from “Baire spaces” by Haworth and McCoy (Warszawa, Panstwowe Wydawnictwo Naukowe, 1977).