Exercise 1, § 1

Let $X$ be an infinite set and let $\mathfrak{F}$ be an ultrafilter on $X$ such that the sets of $\mathfrak{F}$ have empty intersection. For each $A \in \mathfrak{F}$, let $V_A$ denote the subset $\Delta \cup (A \times A)$ of $X \times X$. Show that, as $A$ runs through $\mathfrak{F}$, the sets $V_A$ form a fundamental system of entourages for a uniformity $\mathfrak{U}(\mathfrak{F})$ on $X$, and that the topology induced by this uniformity is discrete.