Exercise 7, § 4

Let $X$ be a non-compact, completely regular space. a) Show that the following properties are equivalent: α) The subalgebra of $C^\infty(X; \mathbf{R})$ consisting of functions of the form $c + f$, where $c$ is a constant and $f$ has compact support (Chapter IX, § 4, no. 3), is dense in $C^\infty(X; \mathbf{R})$. β) If $\beta X$ is the Stone-Čech compactification of $X$, then $\beta X - X$ consists of a single point. γ) There is only one uniformity on $X$, compatible with its topology, with respect to which $X$ is precompact. δ) If $A, B$ are two completely separated closed subsets of $X$ (Chapter IX, § 1, Exercise 11) then one of $A, B$ is compact. ζ) If $\mathscr{U}$ is any uniformity on $X$ compatible with its topology and if $V$ is any entourage of $\mathscr{U}$, there is a subset $L$ of $X$ whose complement is compact, such that $L \times L \subset V$. θ) There is only one uniformity on $X$ compatible with its topology. [Note that α) implies that $X$ is locally compact and therefore open in $\beta X$, and hence show that $\alpha) \Longrightarrow \beta)$; to prove that $\beta) \Longrightarrow \alpha)$ use the Weierstrass-Stone theorem. To show that $\beta) \Longrightarrow \gamma)$, use the fact that the uniformity induced on $X$ by that of $\beta X$ is the finest uniformity compatible with the topology of $X$, with respect to which $X$ is precompact. To show that $\gamma) \Longrightarrow \delta)$, use Chapter IX, § 1, Exercise 11. To show that $\delta) \Longrightarrow \zeta)$, show first that if $V$ is any entourage of $\mathscr{U}$, there exists $z \in X$ such that $V(z)$ is not compact; otherwise $X$ would be complete with respect to $\mathscr{U}$ and paracompact (Chapter II, § 4, Exercise 9), therefore normal; show then with the help of δ) that $X$ would be countably compact (Chapter IX, § 2, Exercise 14), and finally use Chapter

II, § 4, Exercise 6. Then apply this result to an entourage V defined by $f(x, x') \leq 1$, where $f$ is a continuous pseudometric on $X$, and deduce from $\delta$) that there is a subset $L$ of $X$, whose complement is compact, such that $L \times L \subset V$. Finally, to establish that $\zeta) \Longrightarrow \theta$), observe that every ultrafilter on $X$ either converges or is finer than the filter of complements of relatively compact subsets of $X$.

b) Show that a space $X$ which satisfies the equivalent conditions of a) is pseudo-compact (Chapter IX, § 1, Exercise 22). [Use Exercise 12 of Chapter IX, § 1, or else observe that if $X$ is not pseudo-compact, there is a sequence $(x_n)$ of points of $X$ which has no cluster point and a continuous mapping $f : X \to [0, 1]$ such that $f(x_{2n}) = 0$ and $f(x_{2n+1}) = 1$.]

c) Let $Y$ be a non-compact, completely regular space. Show that the subspace $X$ of $\beta Y$ which is the complement of a point of $\beta Y - Y$ satisfies the conditions of a).