Exercise 26, § 6

Let E be an infinite set and let $\mathfrak{F}$ be a set of subsets of E such that for each $A \in \mathfrak{F}$ we have

Show that there exists a partition $(B_\iota)_{\iota \in I}$ of E such that

for all $\iota \in I$, and such that $A \cap B_\iota \neq \emptyset$ for all $A \in \mathfrak{F}$ and all $\iota \in I$.

(With the notation of Exercise 24 (a), consider first a surjective mapping $f$ of $\omega_\alpha$ into $\mathfrak{F}$ such that for each $\mathrm{A} \in \mathfrak{F}$ the set of all $\xi \in \omega_\alpha$ such that $f(\xi) = \mathrm{A}$ has cardinal equal to $\mathfrak{a}$. Then, by transfinite induction, define a bijection $g$ of $\omega_\alpha$ onto E such that $g(\xi) \in f(\xi)$ for every $\xi \in \omega_\alpha$.)