Exercise 3, § 2

Let $\mathfrak{F}_1, \mathfrak{F}_2$ be two distinct ultrafilters on an infinite set $X$. Show that the uniformities $\mathfrak{U}(\mathfrak{F}_1)$ and $\mathfrak{U}(\mathfrak{F}_2)$ ($§ 1$, Exercise 1) are not comparable, and that their least upper bound is the discrete uniformity. What is their greatest lower bound? Deduce that the set of Hausdorff uniformities on $X$ is equipotent to $\mathfrak{P}(\mathfrak{P}(X))$ [cf. Chapter I, § 8, Exercise 6 a)].