Theorem 1

Let $\mathfrak{F}$ be a filter on a quasi-compact space $X$ and let $A$ be the set of cluster points of $\mathfrak{F}$. Then every neighbourhood of $A$ belongs to $\mathfrak{F}$.

Let $V$ be a neighbourhood of $A$ and suppose it possible that every set of $\mathfrak{F}$ meets $\complement V$. The intersections of the sets of $\mathfrak{F}$ with $\complement V$ then form a base of a filter $\mathfrak{G}$ on $X$; $X$ is quasi-compact, so that $\mathfrak{G}$ has at least one cluster point $y$, which does not belong to $A$, because the neighbourhood $V$ of $A$ does not meet some of the sets of $\mathfrak{G}$. But since $\mathfrak{G}$ is finer than $\mathfrak{F}$, $y$ is also a cluster point of $\mathfrak{F}$, which is contrary to hypothesis.