text: Q:
  
  Quantity of partition sets intersecting a compact set
  
  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 $X$?
  
  A:
  
  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 $X.$
  
  