Exercise 20, § 8

a) An open subset $U$ of a topological space $X$ is said to be regular if $U = \alpha(u)$ (§ 1, Exercise 3); equivalently, $U$ is regular if $U$ is the interior of a closed set. $X$ is said to be semi-regular (and its topology $\mathfrak{T}$ is said to be semi-regular) if the regular open subsets of $X$ form a base of $\mathfrak{T}$. Show that if $\mathfrak{T}$ satisfies axiom $(O_{III})$ then $\mathfrak{T}$ is semi-regular.

b) Prove that the intersection of two regular open sets is a regular open set. Deduce that the regular open sets with respect to $\mathfrak{T}$ form a base of a topology $\mathfrak{T}^*$ on $X$, which is coarser than $\mathfrak{T}$ and semi-regular; $\mathfrak{T}^*$ is said to be the semi-regular topology associated with $\mathfrak{T}$. Show that for each subset $U$ of $X$ which is open in $\mathfrak{T}$, the closure of $U$ with respect to $\mathfrak{T}^*$ is the same as its closure in $\mathfrak{T}$, and that if $X$ is accessible for $\mathfrak{T}$ the isolated points of $X$ with respect to $\mathfrak{T}^*$ are the same as the isolated points of $X$ with respect to $\mathfrak{T}$. Show that $\mathfrak{T}^*$ is Hausdorff if and only if $\mathfrak{T}$ is [use Exercise 3 d) of § 1], and finally that if $Y$ is a regular space then the continuous mappings of $X$ into $Y$ are the same in both topologies $\mathfrak{T}$ and $\mathfrak{T}^*$.

c) Let $X$ be a semi-regular space and let $\mathfrak{T}_0$ be its topology. Show that a topology $\mathfrak{T}$ on $X$ is such that $\mathfrak{T}^* = \mathfrak{T}_0$ if and only if there is a set $\mathfrak{M}$ of dense subsets of $X$ (in the topology $\mathfrak{T}_0$) such that every finite intersection of sets of $\mathfrak{M}$ belongs to $\mathfrak{M}$, and such that the topology $\mathfrak{T}$ is generated by the union of $\mathfrak{M}$ and the set of subsets of $X$ which are open in the topology $\mathfrak{T}_0$. [Consider the dense open subsets (in the topology $\mathfrak{T}$) and notice that every open set in $\mathfrak{T}$ is the intersection of a dense open set (in the topology $\mathfrak{T}$) and an open set (in the topology $\mathfrak{T}_0$).] Hence construct examples of non-regular Hausdorff topologies which are finer than a regular topology (and even finer than a topology of a compact metrizable space).