Exercise 2, § 6

Let $X$ be a metrizable space, $d$ a bounded distance compatible with the topology of $X$, $\beta X$ the Stone-Čech compactification of $X$ (IX, p. 9). For every $x \in X$, denote by $f_x$ the numerical function obtained by extending by continuity the function $y \mapsto d(x, y)$ to $\beta X$. a) Now prove that for $x \in X, y \in X, z \in \beta X$, one has $f_x(z) + f_y(z) \geq d(x, y)$. For every point $y \in \beta X$ distinct from $x \in X$, now prove that $f_x(y) > 0$. b) Now prove that if $X$ is intersection of a sequence of open sets $G_n$ in $\beta X$, there exists on $X$ a distance compatible with the topology and for which $X$ is complete. (Remark that $\beta X - X$ is the union of the family of compact sets $F_n = \beta X - G_n$; for every $n$, let $g_n(x) = \inf_{y \in F_n} f_x(y)$; now prove, using $a$), that one has $g_n(x) > 0$ in $X$ and that $g_n$ is continuous in $X$. Finally use the method of IX, p. 90, exerc. 25.)