Exercise 15, § 1

a) Let $X$ be a locally compact space and let $\Phi$ be the set of all continuous mappings $g : X \to [0, 1]$ which are zero on the complement of some compact set (depending on $g$). Let $\mathscr{U}_1$ be the coarsest uniformity on $X$ for which all the mappings $g \in \Phi$ are uniformly continuous. Show that $\mathscr{U}_1$ is compatible with the topology of $X$, that the completion $\hat{X}$ of $X$ with respect to $\mathscr{U}_1$ is compact, and that $X \setminus \hat{X}$ consists of at most one point (show that if $X$ is not compact, the filter of complements of relatively compact subsets of $X$ is a Cauchy filter with respect to $\mathscr{U}_1$).

b) Show that $\mathscr{U}_1$ is the coarsest uniformity compatible with the topology of $X$.

c) Conversely, let $X$ be a completely regular space such that the set of uniformities compatible with the topology of $X$ has a coarsest element $\mathscr{U}_1$. Show that $X$ is locally compact. (Using Exercise 6, show that $X$ is precompact with respect to $\mathscr{U}_1$; then show that the complement of $X$ in its completion with respect to $\mathscr{U}_1$ cannot have more than one point.)

d) Suppose that $X$ is locally compact and $\sigma$-compact, and therefore the union of an increasing sequence $(\mathbf{U}_n)$ of relatively compact open sets such that $\overline{\mathbf{U}}_n \subset \mathbf{U}_{n+1}$ (Chapter I, § 9, no. 9, Proposition 15). Show that there exists a continuous real-valued function $f$ on $X$, such that $f(x) \leq n$ for $x \in \overline{\mathbf{U}}_n$ and $f(x) \geq n$ for $x \in \mathbf{C} \mathbf{U}_n$ [cf. Exercise 10 a)]. Let $\mathscr{U}_2$ be the coarsest uniformity on $X$ for which $f$ and all the functions $g \in \Phi$ are uniformly continuous. Show that $\mathscr{U}_2$ is compatible with the topology of $X$ and that there is an entourage $V$ of $\mathscr{U}_2$ such that $V(x)$ is relatively compact in $X$ for all $x \in X$ (cf. Chapter II, § 4, Exercise 9).