Exercise 23, § 9

a) Let $X$ be a countable absolutely closed space. Show that the set of isolated points of $X$ is dense in $X$. [Use reductio ad absurdum, by arranging the points of $X$ in a sequence, and constructing by induction a countable open covering $(U_n)$ of $X$ such that no finite union of the $\overline{U}_n$ covers $X$].

*b) Give an example of a completely Hausdorff absolutely closed topological space which is not a Lindelöf space [in the product $[0, 1] \times [0, 1]$; consider the complements of the sets $A \times \{0\}$, where $A$ runs through the set of all subsets of $[0, 1]$, and apply the general method of $§ 8$, Exercise 20 c)]. This space satisfies condition $(D_{II})$ of $§ 1$, Exercise 1, but does not satisfy condition $(D_{III})$.

c) Let $X_0$ be a compact space with no isolated points, let $\mathfrak{M}$ be the set of complements of countable subsets of $X_0$ [so that the sets of $\mathfrak{M}$ are dense in $X_0$, cf. a)]. Let $\mathfrak{T}$ be the topology generated by the open sets of $X_0$ and the sets of $\mathfrak{M}$. Show that the space $X$ obtained by giving $X_0$ the topology $\mathfrak{T}$ is absolutely closed and completely Hausdorff and is a Lindelöf space which does not satisfy condition $(D_{II})$ of $§ 1$, Exercise 7; show also that no point of $X$ has a countable fundamental system of neighbourhoods and that no sequence in $X$, all of whose terms are distinct, has a cluster point. If every open subspace of $X_0$ is Lindelöf [cf. Exercise 16 c)], show that the same is true of $X$ and hence in particular that $X$ satisfies $(D_{III})$ [Exercise 16 a)]. This is the case when $X_0 = [0, 1]$. Show however that in this case there are countable closed subsets of $X$ which are not countable intersections of open sets of $X$ (cf. Chapter IX, $§ 5$, no. 3). \*