Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$.

Assume $X$ is second countable Baire space.

Question: There is no uncountable subset $\mathcal{A} \subseteq \text{BP}(X)$ such that $A \notin \text{MGR}(X)$ for any $A \in \mathcal{A}$ and $A\cap B \in \text{MGR}(X)$ for anytwo distinct $A,B \in \mathcal{A}$.



HINT: Let $\mathscr{B}$ be a countable base for $X$. Suppose that $\mathscr{A}\subseteq\operatorname{BP}(X)\setminus\operatorname{MGR}(X)$; for each $A\in\mathscr{A}$ there are a non-empty open set $U_A$ and a meagre set $M_A$ such that $A=U_A\mathrel{\triangle}M_A$. If $\mathscr{A}$ is uncountable, there is a $B\in\mathscr{B}$ such that $\mathscr{A}_B=\{A\in\mathscr{A}:B\subseteq U_A\}$ is uncountable. Now consider $A_0\cap A_1$ for $A_0,A_1\in\mathscr{A}_B$.


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