Exercise 25, § 2

Let $X$ be a metrizable space, let $\beta X$ be its Stone-Čech compactification ($§ 1$, Exercise 7) and let $d$ be a bounded metric compatible with the topology of $X$. For each $x \in X$, let $f_x(y)$ denote the real-valued function obtained by extending $y \to d(x, y)$ by continuity to $\beta X$.

a) Show that $f_x(z) + f_y(z) \geq d(x, y)$ whenever $x$ and $y$ are in $X$ and $z$ is in $\beta X$. For each $y \in \beta X$ other than $x \in \beta X$, show that $f_x(y) > 0$.

b) Show that if $X$ is complete with respect to the metric $d$, then $X$ is the intersection of a sequence of open sets in $\beta X$. [Consider the set $G_n$ of all $y \in \beta X$ such that $f_x(y) < 1/n$ for at least one point $x \in \varphi X$, and show by using $a$ that $X = \bigcap_n G_n$.]

c) Conversely, show that if $X$ is the intersection of a sequence of open sets $G_n$ in $\beta X$, there exists a metric $d'$ on $X$ which is compatible with the topology of $X$, and with respect to which $X$ is complete. [Considering only the case in which $X \neq \beta X$, note that $\beta X - X$ is the union of a family of compact sets $F_n = \beta X - G_n$; for each $n$, let $g_n(x) = \inf_{y \in F_n} f_x(y)$, and show by using $a$ that $g_n(x) > 0$ for all $x \in X$, and that $g_n$ is continuous on $X$. Finish the proof as in Exercise 16 b) of $§ 1$.]