Exercise 9, § 9

Let $X$ be an accessible space and $\mathscr{K}$ a set of quasi-compact closed subsets of $X$, which contains all finite subsets of $X$ and is such that the union of any two subsets of $\mathscr{K}$ belongs to $\mathscr{K}$. Let $X'$ be a set which is the sum of $X$ and a set $\{ \omega \}$ consisting of a single point. Show that there exists a quasi-compact accessible topology on $X'$ which induces the given topology on $X$, and which is such that the complements in $X'$ of the sets of $\mathscr{K}$ form a fundamental system of open neighbourhoods of $\omega$. Give examples of distinct topologies on $X'$ corresponding to distinct sets $\mathscr{K}$.