Exercise 12, § 1

Let $X$ be a completely regular space and let $\beta X$ be its Stone-Čech compactification.

a) Let $f$ be a finite continuous real-valued function on $\beta X$, such that $f(x) > 0$ for all $x \in X$, and such that the set $A = \overline{f}(0) \subset \beta X - X$ is not empty. Then there is a sequence $(a_n)$ of points of $X$ such that the sequence of numbers $\lambda_n = f(a_n) > 0$ is strictly decreasing and such that $\lim_{n \to \infty} \lambda_n = 0$. For each integer $n > 0$, let $I_n$ be an open interval with centre $\lambda_n$ in $\mathbf{R}^*$, such that the closed intervals $\overline{I}_n$ are mutually disjoint, and let $M_n = \overline{f}(I_n) \cap X$. For each subset $H$ of $N$, let $M_H$ be the union of the $M_n$ for which $n \in H$; for each filter $\mathfrak{F}$ on $N$ which is finer than the Fréchet filter, let $\mathfrak{F}'$ denote the filter on $X$ which has the $M_H (H \in \mathfrak{F})$ as a base. Show that $\mathfrak{F}'$ is a completely regular filter (Exercise 8), and that if $\mathfrak{F}_1, \mathfrak{F}_2$ are two distinct ultrafilters on $N$, each of which is finer than the Fréchet filter, then there is no filter on $X$ which is finer than each of $\mathfrak{F}_1', \mathfrak{F}_2'$. Deduce that $\operatorname{Card}(A) \geq 2^{2^{\operatorname{Card}(N)}}$ [use Exercise 8 d) and Chapter I, § 4, Exercise 5].

b) Suppose that $X$ is infinite and discrete. Show that the uniformity induced on $X$ by that of $\beta X$ is the uniformity of finite partitions (Chapter II, § 2, no. 2 and § 4, Exercise 12) [use Exercise 11 b)]. Hence show that $\operatorname{Card}(\beta X) = 2^{2^{\operatorname{Card}(N)}}$ (cf. Chapter I, § 4, Exercise 5). Show that there is a continuous real-valued function $f \geq 0$ on $X$ such that $\overline{f}(0)$ is not empty and is contained in $\beta X - X$.

c) Under the same hypotheses as in b), show that if $A$ is an infinite closed subset of $\beta X$, then $\operatorname{Card}(A) \geq 2^{2^{\operatorname{Card}(N)}}$. [Show first that there exists an infinite sequence $(a_n)$ of points of $A$, and for each $n$ a neighbourhood $V_n$ of $a_n$ in $\beta X$, the $V_n$ being mutually disjoint; deduce that every bounded real-valued function $f$ defined on the set $D$ of the $a_n$ can be extended to a continuous function on $\beta X$; for this, consider a real-valued function $g$ defined on $X$ and equal to $f(a_n)$ at each point of $X \cap V_n$. Hence conclude that $\overline{D} \subset A$ is homeomorphic to the Stone-Čech compactification of $D$ (cf. § 4, Exercise 17).]