Exercise 2, § 3

a) Let $X$ be an infinite set and let $\mathfrak{F}$ be a non-trivial ultrafilter on $X$. Show that if $X$ carries the uniformity $\mathfrak{U}(\mathfrak{F})$ ($§ 1$, Exercise 1) then the complement of $X$ in its completion $\hat{X}$ consists of a single point, and that the topological space $\hat{X}$ can be identified with the space associated with the ultrafilter $\mathfrak{F}$ (Chapter I, § 6, no. 5, Example). b) Deduce from a) an example of a uniformity on $\mathbf{R}$ which is finer than the additive uniformity and with respect to which $\mathbf{R}$ is not complete.