Let $(X, \tau)$ be Cantor space, the collection of binary sequences equipped with the product of discrete topologies.
Does there exist a collection $(U_i)_{i \in I}$ of subsets of $X$ such that $I$ is an uncountable set, $U_i$ is second category for all $i \in I$, and $U_i \cap U_j = \emptyset$ for all $i \neq j \in I$?
Yes, such a family can easily be constructed by transfinite induction. There are only $\mathfrak{c}$ countable unions of closed nowhere dense subsets of $X$; let $(C_\beta)_{\beta<\mathfrak{c}}$ be an enumeration of them. Also, let $f:\mathfrak{c}\to\mathfrak{c}\times\mathfrak{c}$ be a bijection. Now construct a family $(U_\gamma)_{\gamma<\mathfrak{c}}$ by an induction of length $\mathfrak{c}$, where in the $\alpha$th step you choose a new element of $X\setminus C_\beta$ to be in $U_\gamma$, where $f(\alpha)=(\beta,\gamma)$. This is always possible since $X\setminus C_\beta$ has cardinality $\mathfrak{c}$, and you have only chosen $|\alpha|<\mathfrak{c}$ elements so far. In the end, you get a family $(U_\gamma)$ such that each $U_\gamma$ is not contained in any $C_\beta$, and thus has second category.