Exercise 5, § 4

Let $K$ be the discrete space consisting of the two numbers $0, 1$; let $A$ be an infinite set and let $X$ denote the product space $K^{P(A)}$.

a) Let $J$ be any finite subset of $A$; suppose $J$ has $p$ elements. For each subset $H$ of $J$, let $M_H$ be the set of subsets $L$ of $A$ such that $L \cap J = H$; the $M_H$ form a partition $\tilde{\omega}_J$ of $P(A)$ into $2^p$ subsets. Let $F_J$ be the subset of $X$ consisting of those points $(x_L)_{L \subset A}$ such that $x_L = x_M$ whenever $L$ and $M$ belong to the same set of the partition $\tilde{\omega}_J$. $F_J$ is a finite set, with $2^{2^p}$ elements. If $F$ is the union of the $F_J$ as $J$ runs through all finite subsets of $A$, show that $F$ is equipotent to $A$.

b) Show that $F$ is dense in $X$. Taking $A = \mathbf{N}$, deduce that $X$ satisfies condition $(D_{II})$ of § 1, Exercise 7, but does not satisfy condition $(D_{III})$ (Exercise 4).