Let $X$ be a compact metric space.


Let $\{X\_\alpha:\alpha\lt \mathfrak c\}$ be a partition of $X$ into $\mathfrak c=|\mathbb R|$ dense first category $F\_\sigma$-subsets of $X$.


Let $A$ be a non-empty closed subset of $X$ such that $A\cap X\_\alpha$ is first category in $A$ for each $\alpha<\mathfrak c$. 


It is clear that uncountably many of the $X\_\alpha$ must intersect $A$. But what can be said about $$\bigcup \{X\_\alpha:X\_\alpha\cap A\neq\varnothing\}?$$ 


Is this set equal to $X$? Does it at least contain a dense $G\_\delta$-subset of $X$?



Quantity of partition sets intersecting a compact set
**Counterexample.**


Let $\{\alpha:\alpha\lt\mathfrak c\}=I\cup J$ where $I\cap J=\emptyset,\ |I|=|J|=\mathfrak c.$


Let $A=\{t\_\alpha:\alpha\in J\}$ be the Cantor ternary set; $t\_\alpha\ne t\_\beta$ for $\alpha\ne\beta$.


Let $S$ be a dense $G\_\delta$-subset of $[0,1]$ which has Lebesgue measure zero and is disjoint from $A;$ then $T=[0,1]\setminus S$ is a first category subset of $[0,1].$


For every interval $(a,b)\subseteq[0,1]$ we have $|S\cap(a,b)|=|(T\setminus A)\cap(a,b)|=\mathfrak c.$


Let $\{S\_\alpha:\alpha\in I\}$ be a partition of $S$ into countable dense sets.


Let $\{T\_\alpha:\alpha\in J\}$ be a partition of $T$ into countable dense sets such that $T\_\alpha\cap A=\{t\_\alpha\}.$


For $\alpha\lt\mathfrak c$ define


$$X\_\alpha=\begin{cases}
S\_\alpha\ \text{ if }\ \alpha\in I,\\
T\_\alpha\ \text{ if }\ \alpha\in J.
\end{cases}$$


$X=[0,1]$ is a compact metric space, and $\{X\_\alpha:\alpha\lt\mathfrak c\}$ is a partition of $X$ into $\mathfrak c$ countable dense subsets.


$A$ is a nonempty closed subset of $X$ such that $X\_\alpha\cap A=\emptyset$ for $\alpha\in I$ and $X\_\alpha\cap A=\{t\_\alpha\}$ for $\alpha\in J,$ so that $X\_\alpha\cap A$ is first category in $A$ for each $\alpha\lt\mathfrak c.$


Finally, $\bigcup\{X\_\alpha:X\_\alpha\cap A\ne\emptyset\}=\bigcup\{X\_\alpha:\alpha\in J\}=T$ is first category in $X.$


