Exercise 22, § 9

a) Let $X$ be a Hausdorff space and let $A$ be a dense subset of $X$. Show that if every filter on $A$ has at least one cluster point in $X$, then $X$ is absolutely closed [verify axiom (AF') of Exercise 18].

b) Let $X_0$ be a compact space which has a non-open dense subset $A$, and let $\mathfrak{T}_0$ be the topology of $X_0$. Let $\mathfrak{T}$ be the topology on $X_0$ generated by $A$ and the subsets of $X_0$ which are open in $\mathfrak{T}_0$, so that $\mathfrak{T}^* = \mathfrak{T}_0$ ($§ 8$, Exercise 20). If $X$ denotes the non-compact absolutely closed space obtained by endowing the set $X_0$ with the topology $\mathfrak{T}$ (Exercise 21) show that every filter on $A$ has a cluster point in $X$.