Exercise 25, § 9

a) Let $X$ be an accessible space, and let $\Phi$ be the set of filters on $X$ which have a base consisting of closed sets. Show that the set $\Phi$ is inductive, and let $X''$ be the set of maximal filters in $\Phi$.

b) For each open subset $U$ of $X$, let $U^*$ be the set of filters $\mathfrak{F} \in X''$ such that $U \in \mathfrak{F}$. If $U, V$ are two open subsets of $X$, then

(remark that if $\mathfrak{F} \in X''$ is such that $U \notin \mathfrak{F}$, then $X = U \in \mathfrak{F}$). Show that the sets $U^*$ form a base of a topology on $X''$ and that in this topology $X''$ is quasi-compact [verify axiom (C''''), using the remark above] and accessible [if $\mathfrak{F}, \mathfrak{G}$ are two distinct elements of $X''$, show that there exist two disjoint closed subsets $A$ and $B$ in $X$ such that $A \in \mathfrak{F}$ and $B \in \mathfrak{G}$].

c) For each $x \in X$ let $\varphi(x)$ be the ultrafilter on $X$ which has the single set $\{ x \}$ as a base. Show that $\varphi$ is a homeomorphism of $X$ onto a dense subset of $X''$ and that if $X$ is quasi-compact (and accessible), then $\varphi(X) = X''$.

d) Show that for each continuous mapping $f$ of $X$ into a compact space $Y$ there is a unique continuous mapping $g : X'' \to Y$ such that $f = g \circ \varphi$.

e) Let $\tilde{X}$ be the universal Hausdorff space associated with $X''$ ($§ 8$, Exercise 27); show that the canonical mapping $\psi : X'' \to \tilde{X}$ is surjective and that $\tilde{X}$ is compact. Thus every continuous mapping of $X$ into a compact space $Y$ can be written uniquely in the form $f = h \circ \psi \circ \varphi$, where $h : \tilde{X} \to Y$ is continuous.