Exercise 10, § 9

Let $X$ be an accessible space and let $R$ be a set of closed and quasi-compact subsets of $X$, containing the set of finite subsets and such that the union of two sets of $R$ belongs to $R$. Let $X′$ be a sum set of $X$ and of a set $\{\omega\}$ reduced to a point. Now prove that there exists on $X′$ a quasi-compact and accessible topology, which induces the given topology on $X$, and is such that the complements in $X′$ of the sets of $R$ form a fundamental system of open neighbourhoods of $\omega$. Give examples of distinct topologies on $X′$, corresponding to distinct sets $R$.