I'm puzzled by something that might be complete silly. Halmos writes in his "Naive Set Theory":
If $\mathcal C$ is a collection of subsets of a set $E$ (that is, $\mathcal C$ is a subcollection of $\mathcal P(E)$), then write
$\mathfrak D = \{X \in \mathcal P(E):X^\prime \in \mathcal C\}$
It is customary to denote the union and intersection of $\mathfrak D$ by the symbols:
$\bigcup_{X \in \mathcal C}X' \;\;\text{ and } \;\;\bigcap_{X \in \mathcal C}X'$
He is defining the union and intersection of the complements of the sets $X \in \mathcal C$, which are subsets of the set $E$. $E$ is the "everything" set, which contains all sets considered in the current section.
I'm a little confused by why he is considering the set $\mathfrak D$. Why not just consdier the set of complements? $\mathfrak D$ is apparently, as I understand, the set of all subsets of $E$ such that the complement of $X$, namely $X'$ is in the collection of subsets of $E$, $\mathcal C$. By definition of the powerset of $E$,
$\mathcal C \subset \mathcal P(E)$
so $\mathfrak D$ is the set of subsets of $E$, $X$ such that $X'$ is also a subset of $E$.
I still can't see how $\mathfrak D$ enters in the picture here.