Exercise 23, § 9

a) Let X be a separated space, A a dense subset of X. Suppose that every filter on A admits at least one adherent point in X. Now prove that X is absolutely closed (verify axiom (AF’) of Exercise 19 of I, p. 108). b) Let $X_0$ be a compact space in which there exists a dense and non-open set A for the topology $\mathfrak{T}_0$ of $X_0$. Let $\mathfrak{T}$ be the topology on $X_0$ generated by the open parts for $\mathfrak{T}_0$ and by A, so that $\mathfrak{T}^* = \mathfrak{T}_0$ (I, p. 103, Exercise 20). If X is the non-compact absolutely closed space obtained by endowing $X_0$ with $\mathfrak{T}$ (Exercise 22), now prove that every filter on A admits an adherent point in X.