Exercise 26, § 9

a) Let X be an accessible space, and let $\Phi$ be the set of filters on X possessing a basis formed of closed sets. Now prove that the set $\Phi$ is inductive. Let $X''$ be the set of maximal filters in $\Phi$.

b) For every open set 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 sets in X, one has $(U \cap V)^* = U^* \cap V^*$ and $(U \cup V)^* = U^* \cup V^*$ (observe that if $\mathfrak{F} \in X''$ is such that $U \notin \mathfrak{F}$, then $X - U \in \mathfrak{F}$). Now prove that the $U^*$ form a basis of a topology on $X''$, for which $X''$ is quasi-compact (verify axiom (C'') by means of the preceding observation) and accessible (if $\mathfrak{F}, \mathfrak{G}$ are two distinct elements of $X''$, show that there exist in X two closed sets A, B having no point in common such that $A \in \mathfrak{F}, B \in \mathfrak{G}$).

c) For every $x \in X$, let $\varphi(x)$ be the ultrafilter on X having as basis the unique set $\{x\}$. Now prove that $\varphi$ is a homeomorphism of X onto an everywhere dense subset of $X''$. If X is quasi-compact (and accessible) $\varphi(X) = X''$.

d) Now prove that for every continuous mapping $f$ of X into a compact space Y, there exists one and only one continuous mapping $g : X'' \to Y$ such that $f = g \circ \varphi$.

e) Let $\tilde{X}$ be the associated universal separated space of $X''$ (I, p. 104, exerc.