Exercise 12, § 4

a) Let R be a Boolean algebra, and identify R with a Boolean algebra of subsets of the set Ω of maximal prefilters on R (Chapter I, § 6, Exercise 20). For each finite partition $\omega = (A_i)$ of Ω, formed of sets belonging to R, let $C_{\overline{\omega}}$ denote the subset $\bigcup_i (A_i \times A_i)$ of $\Omega \times \Omega$. Show that the sets $C_{\overline{\omega}}$ form a fundamental system of entourages of a Hausdorff uniformity on Ω. The completion, $\hat{\Omega}$, of Ω with respect to this uniformity is a totally disconnected compact space. A subset of $\hat{\Omega}$ is both open and closed in $\hat{\Omega}$ if and only if it is of the form $\overline{A}$, where $A \in R$. Show that $A \to \overline{A}$ is a bijective mapping of $R$ onto the set of subsets of $\hat{\Omega}$ which are both open and closed, and that $CA = C\overline{A}$, $A \cup B = \overline{A} \cup \overline{B}$ and $A \cap B = \overline{A} \cap \overline{B}$. Deduce that the canonical mapping $\Omega \to \hat{\Omega}$ is bijective.

b) Let $X$ be a Hausdorff topological space in which the sets that are both open and closed form a base for the topology. Take $R$ to be the Boolean algebra formed by subsets of $X$ which are both open and closed; then the topology induced on $X$ by the topology of the completion $\hat{X}$ of $X$ with respect to the uniformity $U$ defined in a) is the given topology on $X$. Further, the maximal elements of the set of filters on $X$ having a base formed of open sets, and the maximal elements of the set of filters on $X$ having a base formed of closed sets are Cauchy filters with respect to the uniformity $U$. Deduce that there are continuous surjections $\varphi : \tilde{X} \to \hat{X}$, $\psi : X' \to \hat{X}$, where $\tilde{X}, X'$ are the spaces defined in Chapter I, § 9, Exercises 25 and 24. Show that if $X$ is the rational line $Q$, then $\psi$ is not bijective. If $X$ is extremally disconnected (Chapter I, § 11, Exercise 21), then $\psi$ is bijective, $\hat{X}$ is extremally disconnected and can be identified with the semi-regular space associated with $X'$ (Chapter I, § 8, Exercise 20); thus extremally disconnected spaces can be characterized as dense subspaces of extremally disconnected compact spaces. Hence give an example of an extremally disconnected compact space which has no isolated points [(cf. Chapter I, § 11, Exercise 21 f)].

In particular, if we take $X$ to be a discrete space, then $\hat{X}$ can be identified with the ultrafilter space of $X$ (Chapter I, § 9, Exercise 26).