Exercise 6, § 8

a) Let $X$ be a Hausdorff space and $D$ a dense subset of $X$. Show that $\operatorname{card}(X) \leq 2^{2^{\operatorname{card}(D)}}$ (observe that every point of $X$ is the limit of a filter base on $D$). Show that this upper bound for $\operatorname{card}(X)$ cannot be improved ($§ 4$, Exercise 5 b)) and deduce that on an infinite set $A$ the set of ultrafilters and the set of filters are each equipotent to $\mathfrak{P}(\mathfrak{P}(A))$.

b) Let $K$ be the discrete space consisting of the two numbers $0$ and $1$. Let $A$ be an infinite set. Deduce from a) that if $\operatorname{card}(A) \geq 2^{2^{\operatorname{card}(N)}}$, then the product space $K^A$ satisfies the condition $(D_{IV})$ of Exercise 7 of $§ 1$, but satisfies neither $(D_{II})$ nor $(D_{III})$. ($§ 4$, Exercise 4).