Exercise 4, § 4

Let $X$ be the locally compact space defined in Chapter I, § 9, Exercise 11. Show that every neighbourhood of the diagonal $\Delta$ in $X \times X$ contains a set of the form $[x, b[ \times [x, b[$. Deduce that there is only one uniformity on $X$ compatible with the topology of $X$, and that the completion of $X$ for this uniformity can be identified with the interval $X' = [a, b]$ carrying the topology $\mathfrak{T}_v(X)$. (Observe that every ultrafilter on $X$ must be a Cauchy filter with respect to any uniformity compatible with the topology of $X$).