The maximality of a family of pairwise disjoint meager open sets implies the denseness of its union Consider the following theorem from A. Kechris's Classical Descriptive Set Theory:

(8.29) Theorem. Let $X$ be a topological space and $A \subseteq X$. Put $$U(A) = \bigcup \{ U\text{ open} : U \Vdash A \}.$$ Then $U(A) \setminus A$ is meager, and if $A$ has the [Baire Property], $A \setminus U(A)$, and thus $A \mathop{\Delta} U(A)$, is meager, so $A =^* U(A)$.

(Here $U \Vdash A$ means that $A$ is comeager in $U$, or $U \setminus A$ is meager.)
In following the proof for the case $A = \varnothing$, we begin with the family $$\mathscr U=\{\mathcal U: \mathcal U\text{ is a pairwise disjoint subfamily of meager open subsets of }X \}.$$ Then let $ \{U_i \}_{i\in I} $ be a maximal element of $\mathscr U$. (Such a family $ \{U_i \}_{i\in I} $ can be chosen by Zorn's Lemma.) Letting $W=\bigcup_{i\in I} U_i$, the maximality of $\{ U_i \}_{i \in I}$ implies that $W$ is dense in $U(\varnothing)$.
My question is, why does the maximality of $\{ U_i \}_{i \in I}$ imply the denseness of the set $W$ in $U(\varnothing)$?

If $W$ is not dense in $U(\varnothing)$, then $U(\varnothing) \not\subseteq \overline{W}$, and so $U(\varnothing) \setminus \overline{W}$ is a nonempty open set. Picking any $x \in U(\varnothing) \setminus \overline{W}$ by definition of $U(\varnothing)$ there must be an open meager $U$ containing $x$. In particular, $U \cap (U(\varnothing) \setminus \overline{W})$ is a nonempty open set. Now this set is meager because it is a subset of the meager $U$, and is disjoint from each $U_i$, contradicting the maximality of $\{ U_i \}_{i \in I}$.
