Exercise 22, § 9

(a) Let $X$ be a separated space, and let $\mathfrak{T}$ be its topology. Show that for $X$ to be absolutely closed (Exercise 19), it is necessary and sufficient that the semi-regular topology $\mathfrak{T}^*$ associated with $\mathfrak{T}$ (I, p. 103, Exercise 20) be a topology of a minimal space (Exercise 20).

(b) If $\mathfrak{T}$ is completely separated (Exercise 21), the same is true of $\mathfrak{T}^*$ (use Exercise 20 of I, p. 103). Deduce that a regular absolutely closed space is compact.