Exercise 21, § 9

a) Let X be a Hausdorff space and let $\mathfrak{T}$ be the topology of X. Show that X is absolutely closed (Exercise 18) if and only if the semi-regular topology $\mathfrak{T}^*$ associated with $\mathfrak{T}$ (§ 8, Exercise 20) is the topology of a minimal space (Exercise 19).

b) If $\mathfrak{T}$ is completely Hausdorff (Exercise 20), then so is $\mathfrak{T}_*$ (use Exercise 20 of § 8). Deduce that an absolutely closed regular space is compact.