Exercise 8, § 1

Let $X$ be a completely regular space and let $\beta X$ be its Stone-Čech compactification.

a) A filter $\mathscr{F}$ on $X$ is said to be completely regular if $\mathscr{F}$ has a base $\mathscr{B}$ consisting of open sets such that for each set $A \in \mathscr{B}$ there exists a set $B \in \mathscr{B}$ contained in $A$, and a continuous mapping $f : X \to [0, 1]$ which is equal to 0 on $B$ and 1 on $\overline{C}A$. A completely regular filter is said to be maximal if there exists no strictly finer completely regular filter. Show that, if $\mathscr{F}$ is any completely regular filter on $E$, there is a maximal completely regular filter finer than $\mathscr{F}$ (use Zorn’s lemma).

b) A completely regular filter $\mathscr{F}$ is maximal if and only if, for each pair of open sets $A, B$ of $X$ such that $B \subset A$ and such that there is a continuous mapping of $X$ into $[0, 1]$ which is equal to 0 on $B$ and equal to 1 on $\overline{C}A$, we have either $A \in \mathscr{F}$, or else $A \notin \mathscr{F}$ there is a set of $\mathscr{F}$ which does not meet $B$ (if all the sets of $\mathscr{F}$ meet $B$, consider the filter generated by the sets of $\mathscr{F}$ and the sets $\bar{f}^{-1}([0, a[),$ where $a \in ]0, 1[$).

c) Show that every maximal completely regular filter $\mathscr{F}$ is a Cauchy filter on $X$ with respect to the uniformity induced by that of $\beta X$ [show that each continuous mapping $f : X \to [0, 1]$ has a limit with respect to $\mathscr{F}$, by arguing by reductio ad absurdum and using b)].

d) Two distinct maximal completely regular filters $\mathscr{F}, \mathscr{F}'$ cannot converge to the same point of $\beta X$ [observe that by b) there exists an open set $A \in \mathscr{F}$ and an open set $A' \in \mathscr{F}'$ such that $A \cap A' = \varnothing$; then argue by contradiction, using axiom $(O_{IV})$].

e) The trace on $X$ of the neighbourhood filter of any point $x_0 \in \beta X$ is a maximal completely regular filter $\mathscr{B}$ (argue by contradiction, using the definitions of a completely regular filter and of the neighbourhood filter of a point in $\beta X$.