Exercise 19, § 9

A Hausdorff space X is said to be minimal if its topology $\mathfrak{T}$ is such that every topology on X which is Hausdorff and coarser than $\mathfrak{T}$ necessarily coincides with $\mathfrak{T}$; then $\mathfrak{T}$ must be semi-regular (§ 8, Exercise 20). Show that a Hausdorff space X is minimal if and only if every filter base on X, which consists of open sets and has at most one cluster point, is convergent. It comes to the same thing to say that X is absolutely closed (Exercise 18) and that every filter base on X, which consists of open sets and has a single cluster point, converges to this point.