Exercise 7, § 1

Let $X$ be a completely regular space. If $\mathscr{U}_0$ denotes the universal uniformity on $X$ (Exercise 5), then $\mathscr{U}_0^*$ (Exercise 6) is the coarsest uniformity on $X$ for which all continuous mappings of $X$ into $[0, 1]$ are uniformly continuous. Let $\beta X$ denote the compact space obtained by completing $X$ with respect to the uniformity $U_0^*$. $\beta X$ is called the Stone-Čech compactification of $X$.

a) Show that every continuous mapping of $X$ into a compact space $K$ can be extended to a continuous mapping of $\beta X$ into $K$ (cf. Set Theory, Chapter IV, § 3).

b) Let $f$ be a homeomorphism of $X$ onto a dense subset $X'$ of a compact space $K$, and let $\bar{f}$ be the continuous mapping $\beta X \to K$ which extends $f$. Show that $\bar{f}(\beta X - X) = K - X'$. [Suppose that there is a point $a \in \beta X - X$ such that $\bar{f}(a) = f(b)$, where $b \in X$; consider a continuous mapping $g : \beta X \to [0, 1]$ such that $g(a) = 1$ and $g(b) = 0$, and let $P$ denote the set of all $x \in \beta X$ such that $g(x) \geq 1/2$. There is a continuous mapping $h : K \to [0, 1]$ such that $h(f(b)) = 0$ and $h(f(x)) = 1$ for all $x \in P$; show that the relation $h(f(a)) = 0$ contradicts the fact that $a$ is in the closure of $P$ in $\beta X$.]