Exercise 11, § 1

Let $X$ be a completely regular space. Two closed sets $A, B$ in $X$ are said to be completely separated if there exists a continuous mapping $f$ of $X$ into $[0, 1]$ such that $f(x) = 0$ on $A$ and $f(x) = 1$ on $B$.

a) Let $\beta X$ be the Stone-Čech compactification of $X$. Show that two closed sets $A$ and $B$ in $X$ are completely separated if and only if their closures in $\beta X$ do not intersect [use Exercises 7 a) and 10 a)].

b) Let $h$ be a homeomorphism of $X$ onto a dense subset $X'$ of a compact space $K$, and let $\overline{h}$ be the continuous mapping $\beta X \to K$ which extends $h$. Suppose that, for each pair of completely separated closed subsets $A, B$ of $X$, the closures of $h(A)$ and $h(B)$ in $K$ do not intersect. Show that $\overline{h}$ is a homeomorphism of $\beta X$ onto $K$. (Show that $\overline{h}$ is injective: if $a, b$ are distinct points of $\beta X$ such that $\overline{h}(a) = \overline{h}(b)$, consider a continuous mapping $f : \beta X \to [0, 1]$ such that $f(a) = 0, f(b) = 1$, and consider the sets $X \cap \overline{f}^{-1}[0, 1/3]$ and $X \cap \overline{f}^{-1}[2/3, 1]$.)