Exercise 24, § 9

a) Let $X$ be a Hausdorff space, and let $\Gamma$ be the set of filters on $X$ which have a base consisting of open sets and which have no cluster point in $X$. Show that the set $\Gamma$ is inductive if it is not empty (i.e., if $X$ is not absolutely closed).

b) Let $\Omega$ be the set of maximal filters in $\Gamma$, and let $X'$ be the set which is the sum of $X$ and $\Omega$. For each $x \in X$ let $\mathfrak{B}(x)$ denote the set of neighbourhoods of $x$ in $X$, and for each $\omega \in \Omega$ let $\mathfrak{B}(\omega)$ denote the set of subsets of $X'$ of the form $\{ \omega \} \cup M$, where $M$ belongs to the filter $\omega$. Then there is a topology on $X'$ in which $\mathfrak{B}(x')$ is a fundamental system of neighbourhoods of $x'$ for each $x' \in X'$. Show that, with this topology, $X'$ is absolutely closed and has the following universal property : given any open continuous mapping $f : X \to Y$ of $X$ into an absolutely closed space $Y$, there is a unique continuous mapping $g : X' \to Y$ which extends $f$ (consider the images under $f$ of the filters which belong to $\Omega$, and observe that in an absolutely closed space, a filter which is maximal in the set of filters which have a base consisting of open sets must be convergent). In particular, the topology of $X'$ is $X$-maximal ($§ 3$, Exercise 11). Deduce that there are absolutely closed spaces whose topology is quasi-maximal ($§ 2$, Exercise 6).